semigroupoids-5.2: src/Data/Functor/Bind/Class.hs

{-# LANGUAGE CPP #-}

#ifndef MIN_VERSION_semigroups
#define MIN_VERSION_semigroups(x,y,z) 1
#endif

#if __GLASGOW_HASKELL__ >= 702 && __GLASGOW_HASKELL <= 706 && defined(MIN_VERSION_comonad) && !(MIN_VERSION_comonad(3,0,3))
{-# LANGUAGE Trustworthy #-}
#endif

{-# OPTIONS_HADDOCK not-home #-}

#if __GLASGOW_HASKELL__ >= 708 && __GLASGOW_HASKELL__ < 710
{-# OPTIONS_GHC -fno-warn-amp #-}
#endif

{-# OPTIONS_GHC -fno-warn-deprecations #-}

-----------------------------------------------------------------------------
-- |
-- Copyright   :  (C) 2011-2015 Edward Kmett
-- License     :  BSD-style (see the file LICENSE)
--
-- Maintainer  :  Edward Kmett <ekmett@gmail.com>
-- Stability   :  provisional
-- Portability :  portable
--
-- This module is used to resolve the cyclic we get from defining these
-- classes here rather than in a package upstream. Otherwise we'd get
-- orphaned heads for many instances on the types in @transformers@ and @bifunctors@.
----------------------------------------------------------------------------
module Data.Functor.Bind.Class (
  -- * Applyable functors
    Apply(..)
  -- * Wrappers
  , WrappedApplicative(..)
  , MaybeApply(..)
  -- * Bindable functors
  , Bind(..)
  , apDefault
  , returning
  -- * Biappliable bifunctors
  , Biapply(..)
  ) where

import Data.Semigroup
import Control.Applicative
import Control.Applicative.Backwards
import Control.Applicative.Lift
import Control.Arrow
import Control.Category
import Control.Monad (ap)
import Control.Monad.Trans.Cont
import Control.Monad.Trans.Error
import Control.Monad.Trans.Except
import Control.Monad.Trans.Identity
import Control.Monad.Trans.Maybe
import Control.Monad.Trans.Reader
import Control.Monad.Trans.List
import qualified Control.Monad.Trans.RWS.Lazy as Lazy
import qualified Control.Monad.Trans.State.Lazy as Lazy
import qualified Control.Monad.Trans.Writer.Lazy as Lazy
import qualified Control.Monad.Trans.RWS.Strict as Strict
import qualified Control.Monad.Trans.State.Strict as Strict
import qualified Control.Monad.Trans.Writer.Strict as Strict
import Data.Biapplicative
import Data.Bifunctor.Biff
import Data.Bifunctor.Clown
import Data.Bifunctor.Flip
import Data.Bifunctor.Joker
import Data.Bifunctor.Join
import Data.Bifunctor.Product as Bifunctor
import Data.Bifunctor.Tannen
import Data.Bifunctor.Wrapped
import Data.Functor.Compose
import Data.Functor.Constant
import Data.Functor.Identity
import Data.Functor.Product as Functor
import Data.Functor.Reverse
import Data.Functor.Extend
import Data.List.NonEmpty
import Data.Orphans ()
import Prelude hiding (id, (.))

#ifdef MIN_VERSION_containers
import qualified Data.IntMap as IntMap
import Data.IntMap (IntMap)
import qualified Data.Map as Map
import Data.Map (Map)
import Data.Sequence (Seq)
import Data.Tree (Tree)
#endif

#ifdef MIN_VERSION_tagged
import Data.Tagged
#endif

#if defined(MIN_VERSION_tagged) || MIN_VERSION_base(4,7,0)
import Data.Proxy
#endif


#ifdef MIN_VERSION_comonad
import Control.Comonad
import Control.Comonad.Trans.Env
import Control.Comonad.Trans.Store
import Control.Comonad.Trans.Traced
#else
($>) :: Functor f => f a -> b -> f b
($>) = flip (<$)
#endif

infixl 1 >>-
infixl 4 <.>, <., .>

-- | A strong lax semi-monoidal endofunctor.
-- This is equivalent to an 'Applicative' without 'pure'.
--
-- Laws:
--
-- @
-- ('.') '<$>' u '<.>' v '<.>' w = u '<.>' (v '<.>' w)
-- x '<.>' (f '<$>' y) = ('.' f) '<$>' x '<.>' y
-- f '<$>' (x '<.>' y) = (f '.') '<$>' x '<.>' y
-- @
--
-- The laws imply that `.>` and `<.` really ignore their
-- left and right results, respectively, and really
-- return their right and left results, respectively.
-- Specifically,
--
-- @
-- (mf '<$>' m) '.>' (nf '<$>' n) = nf '<$>' (m '.>' n)
-- (mf '<$>' m) '<.' (nf '<$>' n) = mf '<$>' (m '<.' n)
-- @
class Functor f => Apply f where
  (<.>) :: f (a -> b) -> f a -> f b

  -- | @ a '.>' b = 'const' 'id' '<$>' a '<.>' b @
  (.>) :: f a -> f b -> f b
  a .> b = const id <$> a <.> b

  -- | @ a '<.' b = 'const' '<$>' a '<.>' b @
  (<.) :: f a -> f b -> f a
  a <. b = const <$> a <.> b

#ifdef MIN_VERSION_tagged
instance Apply (Tagged a) where
  (<.>) = (<*>)
  (<.) = (<*)
  (.>) = (*>)
#endif

#if defined(MIN_VERSION_tagged) || MIN_VERSION_base(4,7,0)
instance Apply Proxy where
  (<.>) = (<*>)
  (<.) = (<*)
  (.>) = (*>)
#endif

instance Apply f => Apply (Backwards f) where
  Backwards f <.> Backwards a = Backwards (flip id <$> a <.> f)

instance (Apply f, Apply g) => Apply (Compose f g) where
  Compose f <.> Compose x = Compose ((<.>) <$> f <.> x)

instance Semigroup f => Apply (Constant f) where
  Constant a <.> Constant b = Constant (a <> b)
  Constant a <.  Constant b = Constant (a <> b)
  Constant a  .> Constant b = Constant (a <> b)

instance Apply f => Apply (Lift f) where
  Pure f  <.> Pure x  = Pure (f x)
  Pure f  <.> Other y = Other (f <$> y)
  Other f <.> Pure x  = Other (($ x) <$> f)
  Other f <.> Other y = Other (f <.> y)

instance (Apply f, Apply g) => Apply (Functor.Product f g) where
  Functor.Pair f g <.> Functor.Pair x y = Functor.Pair (f <.> x) (g <.> y)

instance Apply f => Apply (Reverse f) where
  Reverse a <.> Reverse b = Reverse (a <.> b)

instance Semigroup m => Apply ((,)m) where
  (m, f) <.> (n, a) = (m <> n, f a)
  (m, a) <.  (n, _) = (m <> n, a)
  (m, _)  .> (n, b) = (m <> n, b)

instance Apply NonEmpty where
  (<.>) = ap

instance Apply (Either a) where
  Left a  <.> _       = Left a
  Right _ <.> Left a  = Left a
  Right f <.> Right b = Right (f b)

  Left a  <.  _       = Left a
  Right _ <.  Left a  = Left a
  Right a <.  Right _ = Right a

  Left a   .> _       = Left a
  Right _  .> Left a  = Left a
  Right _  .> Right b = Right b

instance Semigroup m => Apply (Const m) where
  Const m <.> Const n = Const (m <> n)
  Const m <.  Const n = Const (m <> n)
  Const m  .> Const n = Const (m <> n)

instance Apply ((->)m) where
  (<.>) = (<*>)
  (<. ) = (<* )
  ( .>) = ( *>)

instance Apply ZipList where
  (<.>) = (<*>)
  (<. ) = (<* )
  ( .>) = ( *>)

instance Apply [] where
  (<.>) = (<*>)
  (<. ) = (<* )
  ( .>) = ( *>)

instance Apply IO where
  (<.>) = (<*>)
  (<. ) = (<* )
  ( .>) = ( *>)

instance Apply Maybe where
  (<.>) = (<*>)
  (<. ) = (<* )
  ( .>) = ( *>)

instance Apply Option where
  (<.>) = (<*>)
  (<. ) = (<* )
  ( .>) = ( *>)

instance Apply Identity where
  (<.>) = (<*>)
  (<. ) = (<* )
  ( .>) = ( *>)

instance Apply w => Apply (IdentityT w) where
  IdentityT wa <.> IdentityT wb = IdentityT (wa <.> wb)

instance Monad m => Apply (WrappedMonad m) where
  (<.>) = (<*>)
  (<. ) = (<* )
  ( .>) = ( *>)

instance Arrow a => Apply (WrappedArrow a b) where
  (<.>) = (<*>)
  (<. ) = (<* )
  ( .>) = ( *>)

#ifdef MIN_VERSION_containers
-- | A Map is not 'Applicative', but it is an instance of 'Apply'
instance Ord k => Apply (Map k) where
  (<.>) = Map.intersectionWith id
  (<. ) = Map.intersectionWith const
  ( .>) = Map.intersectionWith (const id)

-- | An IntMap is not 'Applicative', but it is an instance of 'Apply'
instance Apply IntMap where
  (<.>) = IntMap.intersectionWith id
  (<. ) = IntMap.intersectionWith const
  ( .>) = IntMap.intersectionWith (const id)

instance Apply Seq where
  (<.>) = ap

instance Apply Tree where
  (<.>) = (<*>)
  (<. ) = (<* )
  ( .>) = ( *>)
#endif

-- MaybeT is _not_ the same as Compose f Maybe
instance (Functor m, Monad m) => Apply (MaybeT m) where
  (<.>) = apDefault

-- ErrorT e is _not_ the same as Compose f (Either e)
instance (Functor m, Monad m) => Apply (ErrorT e m) where
  (<.>) = apDefault

instance (Functor m, Monad m) => Apply (ExceptT e m) where
  (<.>) = apDefault

instance Apply m => Apply (ReaderT e m) where
  ReaderT f <.> ReaderT a = ReaderT $ \e -> f e <.> a e

instance Apply m => Apply (ListT m) where
  ListT f <.> ListT a = ListT $ (<.>) <$> f <.> a

-- unfortunately, WriterT has its wrapped product in the wrong order to just use (<.>) instead of flap
instance (Apply m, Semigroup w) => Apply (Strict.WriterT w m) where
  Strict.WriterT f <.> Strict.WriterT a = Strict.WriterT $ flap <$> f <.> a where
    flap (x,m) (y,n) = (x y, m <> n)

instance (Apply m, Semigroup w) => Apply (Lazy.WriterT w m) where
  Lazy.WriterT f <.> Lazy.WriterT a = Lazy.WriterT $ flap <$> f <.> a where
    flap ~(x,m) ~(y,n) = (x y, m <> n)

instance Bind m => Apply (Strict.StateT s m) where
  (<.>) = apDefault

instance Bind m => Apply (Lazy.StateT s m) where
  (<.>) = apDefault

instance (Bind m, Semigroup w) => Apply (Strict.RWST r w s m) where
  (<.>) = apDefault

instance (Bind m, Semigroup w) => Apply (Lazy.RWST r w s m) where
  (<.>) = apDefault

instance Apply (ContT r m) where
  ContT f <.> ContT v = ContT $ \k -> f $ \g -> v (k . g)

#ifdef MIN_VERSION_comonad
instance (Semigroup e, Apply w) => Apply (EnvT e w) where
  EnvT ef wf <.> EnvT ea wa = EnvT (ef <> ea) (wf <.> wa)

instance (Apply w, Semigroup s) => Apply (StoreT s w) where
  StoreT ff m <.> StoreT fa n = StoreT ((<*>) <$> ff <.> fa) (m <> n)

instance Apply w => Apply (TracedT m w) where
  TracedT wf <.> TracedT wa = TracedT (ap <$> wf <.> wa)
#endif

-- | Wrap an 'Applicative' to be used as a member of 'Apply'
newtype WrappedApplicative f a = WrapApplicative { unwrapApplicative :: f a }

instance Functor f => Functor (WrappedApplicative f) where
  fmap f (WrapApplicative a) = WrapApplicative (f <$> a)

instance Applicative f => Apply (WrappedApplicative f) where
  WrapApplicative f <.> WrapApplicative a = WrapApplicative (f <*> a)
  WrapApplicative a <.  WrapApplicative b = WrapApplicative (a <*  b)
  WrapApplicative a  .> WrapApplicative b = WrapApplicative (a  *> b)

instance Applicative f => Applicative (WrappedApplicative f) where
  pure = WrapApplicative . pure
  WrapApplicative f <*> WrapApplicative a = WrapApplicative (f <*> a)
  WrapApplicative a <*  WrapApplicative b = WrapApplicative (a <*  b)
  WrapApplicative a  *> WrapApplicative b = WrapApplicative (a  *> b)

instance Alternative f => Alternative (WrappedApplicative f) where
  empty = WrapApplicative empty
  WrapApplicative a <|> WrapApplicative b = WrapApplicative (a <|> b)

-- | Transform a Apply into an Applicative by adding a unit.
newtype MaybeApply f a = MaybeApply { runMaybeApply :: Either (f a) a }

instance Functor f => Functor (MaybeApply f) where
  fmap f (MaybeApply (Right a)) = MaybeApply (Right (f     a ))
  fmap f (MaybeApply (Left fa)) = MaybeApply (Left  (f <$> fa))

instance Apply f => Apply (MaybeApply f) where
  MaybeApply (Right f) <.> MaybeApply (Right a) = MaybeApply (Right (f        a ))
  MaybeApply (Right f) <.> MaybeApply (Left fa) = MaybeApply (Left  (f    <$> fa))
  MaybeApply (Left ff) <.> MaybeApply (Right a) = MaybeApply (Left  (($a) <$> ff))
  MaybeApply (Left ff) <.> MaybeApply (Left fa) = MaybeApply (Left  (ff   <.> fa))

  MaybeApply a         <. MaybeApply (Right _) = MaybeApply a
  MaybeApply (Right a) <. MaybeApply (Left fb) = MaybeApply (Left (a  <$ fb))
  MaybeApply (Left fa) <. MaybeApply (Left fb) = MaybeApply (Left (fa <. fb))

  MaybeApply (Right _) .> MaybeApply b = MaybeApply b
  MaybeApply (Left fa) .> MaybeApply (Right b) = MaybeApply (Left (fa $> b ))
  MaybeApply (Left fa) .> MaybeApply (Left fb) = MaybeApply (Left (fa .> fb))

instance Apply f => Applicative (MaybeApply f) where
  pure a = MaybeApply (Right a)
  (<*>) = (<.>)
  (<* ) = (<. )
  ( *>) = ( .>)

instance Extend f => Extend (MaybeApply f) where
  duplicated w@(MaybeApply Right{}) = MaybeApply (Right w)
  duplicated (MaybeApply (Left fa)) = MaybeApply (Left (extended (MaybeApply . Left) fa))

#ifdef MIN_VERSION_comonad
instance Comonad f => Comonad (MaybeApply f) where
  duplicate w@(MaybeApply Right{}) = MaybeApply (Right w)
  duplicate (MaybeApply (Left fa)) = MaybeApply (Left (extend (MaybeApply . Left) fa))
  extract (MaybeApply (Left fa)) = extract fa
  extract (MaybeApply (Right a)) = a

instance Apply (Cokleisli w a) where
  Cokleisli f <.> Cokleisli a = Cokleisli (\w -> (f w) (a w))
#endif

-- | A 'Monad' sans 'return'.
--
-- Minimal definition: Either 'join' or '>>-'
--
-- If defining both, then the following laws (the default definitions) must hold:
--
-- > join = (>>- id)
-- > m >>- f = join (fmap f m)
--
-- Laws:
--
-- > induced definition of <.>: f <.> x = f >>- (<$> x)
--
-- Finally, there are two associativity conditions:
--
-- > associativity of (>>-):    (m >>- f) >>- g == m >>- (\x -> f x >>- g)
-- > associativity of join:     join . join = join . fmap join
--
-- These can both be seen as special cases of the constraint that
--
-- > associativity of (->-): (f ->- g) ->- h = f ->- (g ->- h)
--

class Apply m => Bind m where
  (>>-) :: m a -> (a -> m b) -> m b
  m >>- f = join (fmap f m)

  join :: m (m a) -> m a
  join = (>>- id)

#if __GLASGOW_HASKELL__ >= 708
  {-# MINIMAL (>>-) | join #-}
#endif

returning :: Functor f => f a -> (a -> b) -> f b
returning = flip fmap

apDefault :: Bind f => f (a -> b) -> f a -> f b
apDefault f x = f >>- \f' -> f' <$> x

instance Semigroup m => Bind ((,)m) where
  ~(m, a) >>- f = let (n, b) = f a in (m <> n, b)

#ifdef MIN_VERSION_tagged
instance Bind (Tagged a) where
  Tagged a >>- f = f a
  join (Tagged a) = a
#endif

#if defined(MIN_VERSION_tagged) || MIN_VERSION_base(4,7,0)
instance Bind Proxy where
  _ >>- _ = Proxy
  join _ = Proxy
#endif

instance Bind (Either a) where
  Left a  >>- _ = Left a
  Right a >>- f = f a

instance (Bind f, Bind g) => Bind (Functor.Product f g) where
  Functor.Pair m n >>- f = Functor.Pair (m >>- fstP . f) (n >>- sndP . f) where
    fstP (Functor.Pair a _) = a
    sndP (Functor.Pair _ b) = b

instance Bind ((->)m) where
  f >>- g = \e -> g (f e) e

instance Bind [] where
  (>>-) = (>>=)

instance Bind NonEmpty where
  (>>-) = (>>=)

instance Bind IO where
  (>>-) = (>>=)

instance Bind Maybe where
  (>>-) = (>>=)

instance Bind Option where
  (>>-) = (>>=)

instance Bind Identity where
  (>>-) = (>>=)

instance Bind m => Bind (IdentityT m) where
  IdentityT m >>- f = IdentityT (m >>- runIdentityT . f)

instance Monad m => Bind (WrappedMonad m) where
  WrapMonad m >>- f = WrapMonad $ m >>= unwrapMonad . f

instance (Functor m, Monad m) => Bind (MaybeT m) where
  (>>-) = (>>=) -- distributive law requires Monad to inject @Nothing@

instance (Apply m, Monad m) => Bind (ListT m) where
  (>>-) = (>>=) -- distributive law requires Monad to inject @[]@

instance (Functor m, Monad m) => Bind (ErrorT e m) where
  m >>- k = ErrorT $ do
    a <- runErrorT m
    case a of
      Left l -> return (Left l)
      Right r -> runErrorT (k r)

instance (Functor m, Monad m) => Bind (ExceptT e m) where
  m >>- k = ExceptT $ do
    a <- runExceptT m
    case a of
      Left l -> return (Left l)
      Right r -> runExceptT (k r)

instance Bind m => Bind (ReaderT e m) where
  ReaderT m >>- f = ReaderT $ \e -> m e >>- \x -> runReaderT (f x) e

instance (Bind m, Semigroup w) => Bind (Lazy.WriterT w m) where
  m >>- k = Lazy.WriterT $
    Lazy.runWriterT m >>- \ ~(a, w) ->
    Lazy.runWriterT (k a) `returning` \ ~(b, w') ->
      (b, w <> w')

instance (Bind m, Semigroup w) => Bind (Strict.WriterT w m) where
  m >>- k = Strict.WriterT $
    Strict.runWriterT m >>- \ (a, w) ->
    Strict.runWriterT (k a) `returning` \ (b, w') ->
      (b, w <> w')

instance Bind m => Bind (Lazy.StateT s m) where
  m >>- k = Lazy.StateT $ \s ->
    Lazy.runStateT m s >>- \ ~(a, s') ->
    Lazy.runStateT (k a) s'

instance Bind m => Bind (Strict.StateT s m) where
  m >>- k = Strict.StateT $ \s ->
    Strict.runStateT m s >>- \ ~(a, s') ->
    Strict.runStateT (k a) s'

instance (Bind m, Semigroup w) => Bind (Lazy.RWST r w s m) where
  m >>- k = Lazy.RWST $ \r s ->
    Lazy.runRWST m r s >>- \ ~(a, s', w) ->
    Lazy.runRWST (k a) r s' `returning` \ ~(b, s'', w') ->
      (b, s'', w <> w')

instance (Bind m, Semigroup w) => Bind (Strict.RWST r w s m) where
  m >>- k = Strict.RWST $ \r s ->
    Strict.runRWST m r s >>- \ (a, s', w) ->
    Strict.runRWST (k a) r s' `returning` \ (b, s'', w') ->
      (b, s'', w <> w')

instance Bind (ContT r m) where
  m >>- k = ContT $ \c -> runContT m $ \a -> runContT (k a) c

{-
instance ArrowApply a => Bind (WrappedArrow a b) where
  (>>-) = (>>=)
-}

#ifdef MIN_VERSION_containers
-- | A 'Map' is not a 'Monad', but it is an instance of 'Bind'
instance Ord k => Bind (Map k) where
  m >>- f = Map.mapMaybeWithKey (\k -> Map.lookup k . f) m

-- | An 'IntMap' is not a 'Monad', but it is an instance of 'Bind'
instance Bind IntMap where
  m >>- f = IntMap.mapMaybeWithKey (\k -> IntMap.lookup k . f) m

instance Bind Seq where
  (>>-) = (>>=)

instance Bind Tree where
  (>>-) = (>>=)
#endif

infixl 4 <<.>>, <<., .>>

class Bifunctor p => Biapply p where
  (<<.>>) :: p (a -> b) (c -> d) -> p a c -> p b d

  -- |
  -- @
  -- a '.>' b ≡ 'const' 'id' '<$>' a '<.>' b
  -- @
  (.>>) :: p a b -> p c d -> p c d
  a .>> b = bimap (const id) (const id) <<$>> a <<.>> b
  {-# INLINE (.>>) #-}

  -- |
  -- @
  -- a '<.' b ≡ 'const' '<$>' a '<.>' b
  -- @
  (<<.) :: p a b -> p c d -> p a b
  a <<. b = bimap const const <<$>> a <<.>> b
  {-# INLINE (<<.) #-}

instance Biapply (,) where
  (f, g) <<.>> (a, b) = (f a, g b)
  {-# INLINE (<<.>>) #-}

#if MIN_VERSION_semigroups(0,16,2)
instance Biapply Arg where
  Arg f g <<.>> Arg a b = Arg (f a) (g b)
  {-# INLINE (<<.>>) #-}
#endif

instance Semigroup x => Biapply ((,,) x) where
  (x, f, g) <<.>> (x', a, b) = (x <> x', f a, g b)
  {-# INLINE (<<.>>) #-}

instance (Semigroup x, Semigroup y) => Biapply ((,,,) x y) where
  (x, y, f, g) <<.>> (x', y', a, b) = (x <> x', y <> y', f a, g b)
  {-# INLINE (<<.>>) #-}

instance (Semigroup x, Semigroup y, Semigroup z) => Biapply ((,,,,) x y z) where
  (x, y, z, f, g) <<.>> (x', y', z', a, b) = (x <> x', y <> y', z <> z', f a, g b)
  {-# INLINE (<<.>>) #-}

instance Biapply Const where
  Const f <<.>> Const x = Const (f x)
  {-# INLINE (<<.>>) #-}

#ifdef MIN_VERSION_tagged
instance Biapply Tagged where
  Tagged f <<.>> Tagged x = Tagged (f x)
  {-# INLINE (<<.>>) #-}
#endif

instance (Biapply p, Apply f, Apply g) => Biapply (Biff p f g) where
  Biff fg <<.>> Biff xy = Biff (bimap (<.>) (<.>) fg <<.>> xy)
  {-# INLINE (<<.>>) #-}

instance Apply f => Biapply (Clown f) where
  Clown fg <<.>> Clown xy = Clown (fg <.> xy)
  {-# INLINE (<<.>>) #-}

instance Biapply p => Biapply (Flip p) where
  Flip fg <<.>> Flip xy = Flip (fg <<.>> xy)
  {-# INLINE (<<.>>) #-}

instance Apply g => Biapply (Joker g) where
  Joker fg <<.>> Joker xy = Joker (fg <.> xy)
  {-# INLINE (<<.>>) #-}

-- orphan mess
instance Biapply p => Apply (Join p) where
  Join f <.> Join a = Join (f <<.>> a)
  {-# INLINE (<.>) #-}
  Join a .> Join b = Join (a .>> b)
  {-# INLINE (.>) #-}
  Join a <. Join b = Join (a <<. b)
  {-# INLINE (<.) #-}

instance (Biapply p, Biapply q) => Biapply (Bifunctor.Product p q) where
  Bifunctor.Pair w x <<.>> Bifunctor.Pair y z = Bifunctor.Pair (w <<.>> y) (x <<.>> z)
  {-# INLINE (<<.>>) #-}

instance (Apply f, Biapply p) => Biapply (Tannen f p) where
  Tannen fg <<.>> Tannen xy = Tannen ((<<.>>) <$> fg <.> xy)
  {-# INLINE (<<.>>) #-}

instance Biapply p => Biapply (WrappedBifunctor p) where
  WrapBifunctor fg <<.>> WrapBifunctor xy = WrapBifunctor (fg <<.>> xy)
  {-# INLINE (<<.>>) #-}