content: {-# LANGUAGE Trustworthy #-}
  {-# LANGUAGE CPP #-}
  
  -----------------------------------------------------------------------------
  -- |
  -- 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, 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, and online at
  --    <http://web.comlab.ox.ac.uk/oucl/work/jeremy.gibbons/publications/#iterator>.
  --
  -- Note that the functions 'mapM' and 'sequence' generalize "Prelude"
  -- functions of the same names from lists to any 'Traversable' functor.
  -- To avoid ambiguity, either import the "Prelude" hiding these names
  -- or qualify uses of these function names with an alias for this module.
  --
  -----------------------------------------------------------------------------
  
  module Data.Traversable (
      Traversable(..),
      for,
      forM,
      mapAccumL,
      mapAccumR,
      fmapDefault,
      foldMapDefault,
      ) where
  
  import Prelude hiding (mapM, sequence, foldr)
  import qualified Prelude (mapM, foldr)
  import Control.Applicative
  import Data.Foldable (Foldable())
  import Data.Monoid (Monoid)
  
  #if defined(__GLASGOW_HASKELL__)
  import GHC.Arr
  #elif defined(__HUGS__)
  import Hugs.Array
  #elif defined(__NHC__)
  import Array
  #endif
  
  -- | Functors representing data structures that can be traversed from
  -- left to right.
  --
  -- Minimal complete definition: 'traverse' or 'sequenceA'.
  --
  -- 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
      -- | Map each element of a structure to an action, evaluate
      -- these actions from left to right, and collect the results.
      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 collect the results.
      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.
      mapM :: Monad m => (a -> m b) -> t a -> m (t b)
      mapM f = unwrapMonad . traverse (WrapMonad . f)
  
      -- | Evaluate each monadic action in the structure from left to right,
      -- and collect the results.
      sequence :: Monad m => t (m a) -> m (t a)
      sequence = mapM id
  
  -- 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 = Prelude.foldr cons_f (pure [])
        where cons_f x ys = (:) <$> f x <*> ys
  
      mapM = Prelude.mapM
  
  instance Ix i => Traversable (Array i) where
      traverse f arr = listArray (bounds arr) `fmap` traverse f (elems arr)
  
  -- general functions
  
  -- | 'for' is 'traverse' with its arguments flipped.
  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.
  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)
  
  