content: {-# LANGUAGE Trustworthy #-}
  {-# LANGUAGE NoImplicitPrelude #-}
  
  -----------------------------------------------------------------------------
  -- |
  -- Module      :  Data.Traversable
  -- Copyright   :  Conor McBride and Ross Paterson 2005
  -- License     :  BSD-style (see the LICENSE file in the distribution)
  --
  -- Maintainer  :  libraries@haskell.org
  -- Stability   :  experimental
  -- Portability :  portable
  --
  -- Class of data structures that can be traversed from left to right,
  -- performing an action on each element.
  --
  -- See also
  --
  --  * \"Applicative Programming with Effects\",
  --    by Conor McBride and Ross Paterson,
  --    /Journal of Functional Programming/ 18:1 (2008) 1-13, online at
  --    <http://www.soi.city.ac.uk/~ross/papers/Applicative.html>.
  --
  --  * \"The Essence of the Iterator Pattern\",
  --    by Jeremy Gibbons and Bruno Oliveira,
  --    in /Mathematically-Structured Functional Programming/, 2006, online at
  --    <http://web.comlab.ox.ac.uk/oucl/work/jeremy.gibbons/publications/#iterator>.
  --
  --  * \"An Investigation of the Laws of Traversals\",
  --    by Mauro Jaskelioff and Ondrej Rypacek,
  --    in /Mathematically-Structured Functional Programming/, 2012, online at
  --    <http://arxiv.org/pdf/1202.2919>.
  --
  -----------------------------------------------------------------------------
  
  module Data.Traversable (
      -- * The 'Traversable' class
      Traversable(..),
      -- * Utility functions
      for,
      forM,
      mapAccumL,
      mapAccumR,
      -- * General definitions for superclass methods
      fmapDefault,
      foldMapDefault,
      ) where
  
  import Control.Applicative ( Const(..) )
  import Data.Either ( Either(..) )
  import Data.Foldable ( Foldable )
  import Data.Functor
  import Data.Proxy ( Proxy(..) )
  
  import GHC.Arr
  import GHC.Base ( Applicative(..), Monad(..), Monoid, Maybe(..),
                    ($), (.), id, flip )
  import qualified GHC.Base as Monad ( mapM )
  import qualified GHC.List as List ( foldr )
  
  -- | Functors representing data structures that can be traversed from
  -- left to right.
  --
  -- A definition of 'traverse' must satisfy the following laws:
  --
  -- [/naturality/]
  --   @t . 'traverse' f = 'traverse' (t . f)@
  --   for every applicative transformation @t@
  --
  -- [/identity/]
  --   @'traverse' Identity = Identity@
  --
  -- [/composition/]
  --   @'traverse' (Compose . 'fmap' g . f) = Compose . 'fmap' ('traverse' g) . 'traverse' f@
  --
  -- A definition of 'sequenceA' must satisfy the following laws:
  --
  -- [/naturality/]
  --   @t . 'sequenceA' = 'sequenceA' . 'fmap' t@
  --   for every applicative transformation @t@
  --
  -- [/identity/]
  --   @'sequenceA' . 'fmap' Identity = Identity@
  --
  -- [/composition/]
  --   @'sequenceA' . 'fmap' Compose = Compose . 'fmap' 'sequenceA' . 'sequenceA'@
  --
  -- where an /applicative transformation/ is a function
  --
  -- @t :: (Applicative f, Applicative g) => f a -> g a@
  --
  -- preserving the 'Applicative' operations, i.e.
  --
  --  * @t ('pure' x) = 'pure' x@
  --
  --  * @t (x '<*>' y) = t x '<*>' t y@
  --
  -- and the identity functor @Identity@ and composition of functors @Compose@
  -- are defined as
  --
  -- >   newtype Identity a = Identity a
  -- >
  -- >   instance Functor Identity where
  -- >     fmap f (Identity x) = Identity (f x)
  -- >
  -- >   instance Applicative Indentity where
  -- >     pure x = Identity x
  -- >     Identity f <*> Identity x = Identity (f x)
  -- >
  -- >   newtype Compose f g a = Compose (f (g a))
  -- >
  -- >   instance (Functor f, Functor g) => Functor (Compose f g) where
  -- >     fmap f (Compose x) = Compose (fmap (fmap f) x)
  -- >
  -- >   instance (Applicative f, Applicative g) => Applicative (Compose f g) where
  -- >     pure x = Compose (pure (pure x))
  -- >     Compose f <*> Compose x = Compose ((<*>) <$> f <*> x)
  --
  -- (The naturality law is implied by parametricity.)
  --
  -- Instances are similar to 'Functor', e.g. given a data type
  --
  -- > data Tree a = Empty | Leaf a | Node (Tree a) a (Tree a)
  --
  -- a suitable instance would be
  --
  -- > instance Traversable Tree where
  -- >    traverse f Empty = pure Empty
  -- >    traverse f (Leaf x) = Leaf <$> f x
  -- >    traverse f (Node l k r) = Node <$> traverse f l <*> f k <*> traverse f r
  --
  -- This is suitable even for abstract types, as the laws for '<*>'
  -- imply a form of associativity.
  --
  -- The superclass instances should satisfy the following:
  --
  --  * In the 'Functor' instance, 'fmap' should be equivalent to traversal
  --    with the identity applicative functor ('fmapDefault').
  --
  --  * In the 'Foldable' instance, 'Data.Foldable.foldMap' should be
  --    equivalent to traversal with a constant applicative functor
  --    ('foldMapDefault').
  --
  class (Functor t, Foldable t) => Traversable t where
      {-# MINIMAL traverse | sequenceA #-}
  
      -- | Map each element of a structure to an action, evaluate these
      -- these actions from left to right, and collect the results.
      -- actions from left to right, and collect the results. For a
      -- version that ignores the results see 'Data.Foldable.traverse_'.
      traverse :: Applicative f => (a -> f b) -> t a -> f (t b)
      traverse f = sequenceA . fmap f
  
      -- | Evaluate each action in the structure from left to right, and
      -- and collect the results. For a version that ignores the results
      -- see 'Data.Foldable.sequenceA_'.
      sequenceA :: Applicative f => t (f a) -> f (t a)
      sequenceA = traverse id
  
      -- | Map each element of a structure to a monadic action, evaluate
      -- these actions from left to right, and collect the results. For
      -- a version that ignores the results see 'Data.Foldable.mapM_'.
      mapM :: Monad m => (a -> m b) -> t a -> m (t b)
      mapM = traverse
  
      -- | Evaluate each monadic action in the structure from left to
      -- right, and collect the results. For a version that ignores the
      -- results see 'Data.Foldable.sequence_'.
      sequence :: Monad m => t (m a) -> m (t a)
      sequence = sequenceA
  
  -- instances for Prelude types
  
  instance Traversable Maybe where
      traverse _ Nothing = pure Nothing
      traverse f (Just x) = Just <$> f x
  
  instance Traversable [] where
      {-# INLINE traverse #-} -- so that traverse can fuse
      traverse f = List.foldr cons_f (pure [])
        where cons_f x ys = (:) <$> f x <*> ys
  
      mapM = Monad.mapM
  
  instance Traversable (Either a) where
      traverse _ (Left x) = pure (Left x)
      traverse f (Right y) = Right <$> f y
  
  instance Traversable ((,) a) where
      traverse f (x, y) = (,) x <$> f y
  
  instance Ix i => Traversable (Array i) where
      traverse f arr = listArray (bounds arr) `fmap` traverse f (elems arr)
  
  instance Traversable Proxy where
      traverse _ _ = pure Proxy
      {-# INLINE traverse #-}
      sequenceA _ = pure Proxy
      {-# INLINE sequenceA #-}
      mapM _ _ = return Proxy
      {-# INLINE mapM #-}
      sequence _ = return Proxy
      {-# INLINE sequence #-}
  
  instance Traversable (Const m) where
      traverse _ (Const m) = pure $ Const m
  
  -- general functions
  
  -- | 'for' is 'traverse' with its arguments flipped. For a version
  -- that ignores the results see 'Data.Foldable.for_'.
  for :: (Traversable t, Applicative f) => t a -> (a -> f b) -> f (t b)
  {-# INLINE for #-}
  for = flip traverse
  
  -- | 'forM' is 'mapM' with its arguments flipped. For a version that
  -- ignores the results see 'Data.Foldable.forM_'.
  forM :: (Traversable t, Monad m) => t a -> (a -> m b) -> m (t b)
  {-# INLINE forM #-}
  forM = flip mapM
  
  -- left-to-right state transformer
  newtype StateL s a = StateL { runStateL :: s -> (s, a) }
  
  instance Functor (StateL s) where
      fmap f (StateL k) = StateL $ \ s -> let (s', v) = k s in (s', f v)
  
  instance Applicative (StateL s) where
      pure x = StateL (\ s -> (s, x))
      StateL kf <*> StateL kv = StateL $ \ s ->
          let (s', f) = kf s
              (s'', v) = kv s'
          in (s'', f v)
  
  -- |The 'mapAccumL' function behaves like a combination of 'fmap'
  -- and 'foldl'; it applies a function to each element of a structure,
  -- passing an accumulating parameter from left to right, and returning
  -- a final value of this accumulator together with the new structure.
  mapAccumL :: Traversable t => (a -> b -> (a, c)) -> a -> t b -> (a, t c)
  mapAccumL f s t = runStateL (traverse (StateL . flip f) t) s
  
  -- right-to-left state transformer
  newtype StateR s a = StateR { runStateR :: s -> (s, a) }
  
  instance Functor (StateR s) where
      fmap f (StateR k) = StateR $ \ s -> let (s', v) = k s in (s', f v)
  
  instance Applicative (StateR s) where
      pure x = StateR (\ s -> (s, x))
      StateR kf <*> StateR kv = StateR $ \ s ->
          let (s', v) = kv s
              (s'', f) = kf s'
          in (s'', f v)
  
  -- |The 'mapAccumR' function behaves like a combination of 'fmap'
  -- and 'foldr'; it applies a function to each element of a structure,
  -- passing an accumulating parameter from right to left, and returning
  -- a final value of this accumulator together with the new structure.
  mapAccumR :: Traversable t => (a -> b -> (a, c)) -> a -> t b -> (a, t c)
  mapAccumR f s t = runStateR (traverse (StateR . flip f) t) s
  
  -- | This function may be used as a value for `fmap` in a `Functor`
  --   instance, provided that 'traverse' is defined. (Using
  --   `fmapDefault` with a `Traversable` instance defined only by
  --   'sequenceA' will result in infinite recursion.)
  fmapDefault :: Traversable t => (a -> b) -> t a -> t b
  {-# INLINE fmapDefault #-}
  fmapDefault f = getId . traverse (Id . f)
  
  -- | This function may be used as a value for `Data.Foldable.foldMap`
  -- in a `Foldable` instance.
  foldMapDefault :: (Traversable t, Monoid m) => (a -> m) -> t a -> m
  foldMapDefault f = getConst . traverse (Const . f)
  
  -- local instances
  
  newtype Id a = Id { getId :: a }
  
  instance Functor Id where
      fmap f (Id x) = Id (f x)
  
  instance Applicative Id where
      pure = Id
      Id f <*> Id x = Id (f x)
  
  