Cofree.hs

{-# LANGUAGE DeriveDataTypeable #-}
{-# LANGUAGE DeriveGeneric #-}
{-# LANGUAGE Rank2Types #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE UndecidableInstances #-}
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE Safe #-}
{-# LANGUAGE StandaloneDeriving #-}

-----------------------------------------------------------------------------
-- |
-- Module      :  Control.Comonad.Cofree
-- Copyright   :  (C) 2008-2013 Edward Kmett
-- License     :  BSD-style (see the file LICENSE)
--
-- Maintainer  :  Edward Kmett <ekmett@gmail.com>
-- Stability   :  provisional
-- Portability :  MPTCs, fundeps
--
-- Cofree comonads
--
----------------------------------------------------------------------------
module Control.Comonad.Cofree
  ( Cofree(..)
  , ComonadCofree(..)
  , section
  , coiter
  , coiterW
  , unfold
  , unfoldM
  , hoistCofree
  -- * Lenses into cofree comonads
  , _extract
  , _unwrap
  , telescoped
  , telescoped_
  , shoots
  , leaves
  ) where

import Control.Applicative
import Control.Comonad
import Control.Comonad.Trans.Class
import Control.Comonad.Cofree.Class
import Control.Comonad.Env.Class
import Control.Comonad.Store.Class as Class
import Control.Comonad.Traced.Class
import Control.Comonad.Hoist.Class
import Control.Category
import Control.Monad(ap, (>=>), liftM)
import Control.Monad.Zip
import Data.Functor.Bind
import Data.Functor.Classes
import Data.Functor.Extend
import Data.Functor.WithIndex
import Data.Data
import Data.Distributive
import Data.Foldable
import Data.Foldable.WithIndex
import Data.Semigroup
import Data.Traversable
import Data.Traversable.WithIndex
import Data.Semigroup.Foldable
import Data.Semigroup.Traversable
import GHC.Generics hiding (Infix, Prefix)
import Prelude hiding (id,(.))


infixr 5 :<

-- | The 'Cofree' 'Comonad' of a functor @f@.
--
-- /Formally/
--
-- A 'Comonad' @v@ is a cofree 'Comonad' for @f@ if every comonad homomorphism
-- from another comonad @w@ to @v@ is equivalent to a natural transformation
-- from @w@ to @f@.
--
-- A 'cofree' functor is right adjoint to a forgetful functor.
--
-- Cofree is a functor from the category of functors to the category of comonads
-- that is right adjoint to the forgetful functor from the category of comonads
-- to the category of functors that forgets how to 'extract' and
-- 'duplicate', leaving you with only a 'Functor'.
--
-- In practice, cofree comonads are quite useful for annotating syntax trees,
-- or talking about streams.
--
-- A number of common comonads arise directly as cofree comonads.
--
-- For instance,
--
-- * @'Cofree' 'Maybe'@ forms the comonad for a non-empty list.
--
-- * @'Cofree' ('Const' b)@ is a product.
--
-- * @'Cofree' 'Identity'@ forms an infinite stream.
--
-- * @'Cofree' ((->) b)'@ describes a Moore machine with states labeled with values of type a, and transitions on edges of type b.
--
-- Furthermore, if the functor @f@ forms a monoid (for example, by
-- being an instance of 'Alternative'), the resulting 'Comonad' is
-- also a 'Monad'. See
-- <http://www.cs.appstate.edu/~johannp/jfp06-revised.pdf Monadic Augment and Generalised Shortcut Fusion> by Neil Ghani et al., Section 4.3
-- for more details.
--
-- In particular, if @f a ≡ [a]@, the
-- resulting data structure is a <https://en.wikipedia.org/wiki/Rose_tree Rose tree>.
-- For a practical application, check
-- <https://web.archive.org/web/20161208002902/http://www.cs.le.ac.uk/people/ak155/Papers/CALCO-07/GK07.pdf Higher Dimensional Trees, Algebraically> by Neil Ghani et al.
data Cofree f a = a :< f (Cofree f a)
  deriving (Generic, Generic1)

deriving instance (Typeable f, Data (f (Cofree f a)), Data a) => Data (Cofree f a)

-- | Use coiteration to generate a cofree comonad from a seed.
--
-- @'coiter' f = 'unfold' ('id' 'Control.Arrow.&&&' f)@
coiter :: Functor f => (a -> f a) -> a -> Cofree f a
coiter psi a = a :< (coiter psi <$> psi a)

-- | Like coiter for comonadic values.
coiterW :: (Comonad w, Functor f) => (w a -> f (w a)) -> w a -> Cofree f a
coiterW psi a = extract a :< (coiterW psi <$> psi a)

-- | Unfold a cofree comonad from a seed.
unfold :: Functor f => (b -> (a, f b)) -> b -> Cofree f a
unfold f c = case f c of
  (x, d) -> x :< fmap (unfold f) d

-- | Unfold a cofree comonad from a seed, monadically.
unfoldM :: (Traversable f, Monad m) => (b -> m (a, f b)) -> b -> m (Cofree f a)
unfoldM f = f >=> \ (x, t) -> (x :<) `liftM` Data.Traversable.mapM (unfoldM f) t

hoistCofree :: Functor f => (forall x . f x -> g x) -> Cofree f a -> Cofree g a
hoistCofree f (x :< y) = x :< f (hoistCofree f <$> y)

instance Functor f => ComonadCofree f (Cofree f) where
  unwrap (_ :< as) = as
  {-# INLINE unwrap #-}

instance Distributive f => Distributive (Cofree f) where
  distribute w = fmap extract w :< fmap distribute (collect unwrap w)

instance Functor f => Functor (Cofree f) where
  fmap f (a :< as) = f a :< fmap (fmap f) as
  b <$ (_ :< as) = b :< fmap (b <$) as

instance Functor f => Extend (Cofree f) where
  extended = extend
  {-# INLINE extended #-}
  duplicated = duplicate
  {-# INLINE duplicated #-}

instance Functor f => Comonad (Cofree f) where
  extend f w = f w :< fmap (extend f) (unwrap w)
  duplicate w = w :< fmap duplicate (unwrap w)
  extract (a :< _) = a
  {-# INLINE extract #-}

-- | This is not a true 'Comonad' transformer, but this instance is convenient.
instance ComonadTrans Cofree where
  lower (_ :< as) = fmap extract as
  {-# INLINE lower #-}

instance Alternative f => Monad (Cofree f) where
  return = pure
  {-# INLINE return #-}
  (a :< m) >>= k = case k a of
                     b :< n -> b :< (n <|> fmap (>>= k) m)

instance (Alternative f, MonadZip f) => MonadZip (Cofree f) where
  mzip (a :< as) (b :< bs) = (a, b) :< fmap (uncurry mzip) (mzip as bs)

-- |
--
-- @'lower' . 'section' = 'id'@
section :: Comonad f => f a -> Cofree f a
section as = extract as :< extend section as

instance Apply f => Apply (Cofree f) where
  (f :< fs) <.> (a :< as) = f a :< ((<.>) <$> fs <.> as)
  {-# INLINE (<.>) #-}
  (f :< fs) <.  (_ :< as) = f :< ((<. ) <$> fs <.> as)
  {-# INLINE (<.) #-}
  (_ :< fs)  .> (a :< as) = a :< (( .>) <$> fs <.> as)
  {-# INLINE (.>) #-}

instance ComonadApply f => ComonadApply (Cofree f) where
  (f :< fs) <@> (a :< as) = f a :< ((<@>) <$> fs <@> as)
  {-# INLINE (<@>) #-}
  (f :< fs) <@  (_ :< as) = f :< ((<@ ) <$> fs <@> as)
  {-# INLINE (<@) #-}
  (_ :< fs)  @> (a :< as) = a :< (( @>) <$> fs <@> as)
  {-# INLINE (@>) #-}

instance Alternative f => Applicative (Cofree f) where
  pure x = x :< empty
  {-# INLINE pure #-}
  (<*>) = ap
  {-# INLINE (<*>) #-}

instance (Show1 f) => Show1 (Cofree f) where
  liftShowsPrec sp sl = go
    where
      goList = liftShowList sp sl
      go d (a :< as) = showParen (d > 5) $
        sp 6 a . showString " :< " . liftShowsPrec go goList 5 as

instance (Show1 f, Show a) => Show (Cofree f a) where
  showsPrec = showsPrec1

instance (Read1 f) => Read1 (Cofree f) where
  liftReadsPrec rp rl = go
    where
      goList = liftReadList rp rl
      go d r = readParen (d > 5)
        (\r' -> [(u :< v, w) |
                (u, s) <- rp 6 r',
                (":<", t) <- lex s,
                (v, w) <- liftReadsPrec go goList 5 t]) r

instance (Read1 f, Read a) => Read (Cofree f a) where
  readsPrec = readsPrec1

instance (Eq1 f, Eq a) => Eq (Cofree f a) where
  (==) = eq1

instance (Eq1 f) => Eq1 (Cofree f) where
  liftEq eq = go
    where
      go (a :< as) (b :< bs) = eq a b && liftEq go as bs

instance (Ord1 f, Ord a) => Ord (Cofree f a) where
  compare = compare1

instance (Ord1 f) => Ord1 (Cofree f) where
  liftCompare cmp = go
    where
      go (a :< as) (b :< bs) = cmp a b `mappend` liftCompare go as bs

instance Foldable f => Foldable (Cofree f) where
  foldMap f = go where
    go (a :< as) = f a `mappend` foldMap go as
  {-# INLINE foldMap #-}
  length = go 0 where
    go s (_ :< as) = foldl' go (s + 1) as

instance Foldable1 f => Foldable1 (Cofree f) where
  foldMap1 f = go where
    go (a :< as) = f a <> foldMap1 go as
  {-# INLINE foldMap1 #-}

instance Traversable f => Traversable (Cofree f) where
  traverse f = go where
    go (a :< as) = (:<) <$> f a <*> traverse go as
  {-# INLINE traverse #-}

instance Traversable1 f => Traversable1 (Cofree f) where
  traverse1 f = go where
    go (a :< as) = (:<) <$> f a <.> traverse1 go as
  {-# INLINE traverse1 #-}

instance FunctorWithIndex i f => FunctorWithIndex [i] (Cofree f) where
  imap f (a :< as) = f [] a :< imap (\i -> imap (f . (:) i)) as
  {-# INLINE imap #-}

instance FoldableWithIndex i f => FoldableWithIndex [i] (Cofree f) where
  ifoldMap f (a :< as) = f [] a `mappend` ifoldMap (\i -> ifoldMap (f . (:) i)) as
  {-# INLINE ifoldMap #-}

instance TraversableWithIndex i f => TraversableWithIndex [i] (Cofree f) where
  itraverse f (a :< as) = (:<) <$> f [] a <*> itraverse (\i -> itraverse (f . (:) i)) as
  {-# INLINE itraverse #-}

instance ComonadHoist Cofree where
  cohoist = hoistCofree

instance ComonadEnv e w => ComonadEnv e (Cofree w) where
  ask = ask . lower
  {-# INLINE ask #-}

instance ComonadStore s w => ComonadStore s (Cofree w) where
  pos (_ :< as) = Class.pos as
  {-# INLINE pos #-}
  peek s (_ :< as) = extract (Class.peek s as)
  {-# INLINE peek #-}

instance ComonadTraced m w => ComonadTraced m (Cofree w) where
  trace m = trace m . lower
  {-# INLINE trace #-}

-- | This is a lens that can be used to read or write from the target of 'extract'.
--
-- Using (^.) from the @lens@ package:
--
-- @foo ^. '_extract' == 'extract' foo@
--
-- For more on lenses see the @lens@ package on hackage
--
-- @'_extract' :: Lens' ('Cofree' g a) a@
_extract :: Functor f => (a -> f a) -> Cofree g a -> f (Cofree g a)
_extract f (a :< as) = (:< as) <$> f a
{-# INLINE _extract #-}

-- | This is a lens that can be used to read or write to the tails of a 'Cofree' 'Comonad'.
--
-- Using (^.) from the @lens@ package:
--
-- @foo ^. '_unwrap' == 'unwrap' foo@
--
-- For more on lenses see the @lens@ package on hackage
--
-- @'_unwrap' :: Lens' ('Cofree' g a) (g ('Cofree' g a))@
_unwrap :: Functor f => (g (Cofree g a) -> f (g (Cofree g a))) -> Cofree g a -> f (Cofree g a)
_unwrap  f (a :< as) = (a :<) <$> f as
{-# INLINE _unwrap #-}

-- | Construct an @Lens@ into a @'Cofree' g@ given a list of lenses into the base functor.
-- When the input list is empty, this is equivalent to '_extract'.
-- When the input list is non-empty, this composes the input lenses
-- with '_unwrap' to walk through the @'Cofree' g@ before using
-- '_extract' to get the element at the final location.
--
-- For more on lenses see the 'lens' package on hackage.
--
-- @telescoped :: [Lens' (g ('Cofree' g a)) ('Cofree' g a)]      -> Lens' ('Cofree' g a) a@
--
-- @telescoped :: [Traversal' (g ('Cofree' g a)) ('Cofree' g a)] -> Traversal' ('Cofree' g a) a@
--
-- @telescoped :: [Getter (g ('Cofree' g a)) ('Cofree' g a)]     -> Getter ('Cofree' g a) a@
--
-- @telescoped :: [Fold (g ('Cofree' g a)) ('Cofree' g a)]       -> Fold ('Cofree' g a) a@
--
-- @telescoped :: [Setter' (g ('Cofree' g a)) ('Cofree' g a)]    -> Setter' ('Cofree' g a) a@
telescoped :: Functor f =>
             [(Cofree g a -> f (Cofree g a)) -> g (Cofree g a) -> f (g (Cofree g a))] ->
              (a -> f a) -> Cofree g a -> f (Cofree g a)
telescoped = Prelude.foldr (\l r -> _unwrap . l . r) _extract
{-# INLINE telescoped #-}

-- not actually named 'eats'
-- | Construct an @Lens@ into a @'Cofree' g@ given a list of lenses into the base functor.
-- The only difference between this and 'telescoped' is that 'telescoped' focuses on a single value, but this focuses on the entire remaining subtree.
-- When the input list is empty, this is equivalent to 'id'.
-- When the input list is non-empty, this composes the input lenses
-- with '_unwrap' to walk through the @'Cofree' g@.
--
-- For more on lenses see the 'lens' package on hackage.
--
-- @telescoped :: [Lens' (g ('Cofree' g a)) ('Cofree' g a)]      -> Lens' ('Cofree' g a) ('Cofree' g a)@
--
-- @telescoped :: [Traversal' (g ('Cofree' g a)) ('Cofree' g a)] -> Traversal' ('Cofree' g a) ('Cofree' g a)@
--
-- @telescoped :: [Getter (g ('Cofree' g a)) ('Cofree' g a)]     -> Getter ('Cofree' g a) ('Cofree' g a)@
--
-- @telescoped :: [Fold (g ('Cofree' g a)) ('Cofree' g a)]       -> Fold ('Cofree' g a) ('Cofree' g a)@
--
-- @telescoped :: [Setter' (g ('Cofree' g a)) ('Cofree' g a)]    -> Setter' ('Cofree' g a) ('Cofree' g a)@
telescoped_ :: Functor f =>
              [(Cofree g a -> f (Cofree g a)) -> g (Cofree g a) -> f (g (Cofree g a))] ->
              (Cofree g a -> f (Cofree g a)) -> Cofree g a -> f (Cofree g a)
telescoped_ = Prelude.foldr (\l r -> _unwrap . l . r) id
{-# INLINE telescoped_ #-}

-- | A @Traversal'@ that gives access to all non-leaf @a@ elements of a
-- @'Cofree' g@ a, where non-leaf is defined as @x@ from @(x :< xs)@ where
-- @null xs@ is @False@.
--
-- Because this doesn't give access to all values in the @'Cofree' g@,
-- it cannot be used to change types.
--
-- @shoots :: Traversable g => Traversal' (Cofree g a) a@
--
-- N.B. On GHC < 7.9, this is slightly less flexible, as it has to
-- use @null (toList xs)@ instead.
shoots :: (Applicative f, Traversable g) => (a -> f a) -> Cofree g a -> f (Cofree g a)
shoots f = go
  where
    go xxs@(x :< xs) | null xs          = pure xxs
                     | otherwise        = (:<) <$> f x <*> traverse go xs
{-# INLINE shoots #-}

-- | A @Traversal'@ that gives access to all leaf @a@ elements of a
-- @'Cofree' g@ a, where leaf is defined as @x@ from @(x :< xs)@ where
-- @null xs@ is @True@.
--
-- Because this doesn't give access to all values in the @'Cofree' g@,
-- it cannot be used to change types.
--
-- @shoots :: Traversable g => Traversal' (Cofree g a) a@
--
-- N.B. On GHC < 7.9, this is slightly less flexible, as it has to
-- use @null (toList xs)@ instead.
leaves :: (Applicative f, Traversable g) => (a -> f a) -> Cofree g a -> f (Cofree g a)
leaves f = go
  where
    go (x :< xs) | null xs          = (:< xs) <$> f x
                 | otherwise        = (x :<) <$> traverse go xs
{-# INLINE leaves #-}