Data/Traversable.hs

{-# 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)