    -----------------------------------------------------------------------------
    -- |
    -- Module      :  Data.Traversable
    -- Copyright   :  Conor McBride and Ross Paterson 2005
    -- License     :  BSD-style (see the LICENSE file in the distribution)
    --
    -- Maintainer  :  ross@soi.city.ac.uk
    -- 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,
    	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)

    -- | 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
    -- >	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 f Nothing = pure Nothing
    	traverse f (Just x) = Just <$> f x

    instance Traversable [] where
    	traverse f = Prelude.foldr cons_f (pure [])
    	  where cons_f x ys = (:) <$> f x <*> ys

    	mapM = Prelude.mapM

    -- 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

    -- | This function may be used as a value for `fmap` in a `Functor` instance.
    fmapDefault :: Traversable t => (a -> b) -> t a -> t b
    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)