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{-# LANGUAGE Trustworthy #-}
-----------------------------------------------------------------------------
-- |
-- 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>.
--
-- 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 (
-- * The 'Traversable' class
Traversable(..),
-- * Utility functions
for,
forM,
mapAccumL,
mapAccumR,
-- * General definitions for superclass methods
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)
import Data.Proxy
import GHC.Arr
-- | Functors representing data structures that can be traversed from
-- left to right.
--
-- Minimal complete definition: 'traverse' or 'sequenceA'.
--
-- 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 <$> traverse f l <*> f k <*> traverse f r
--
-- This is suitable even for abstract types, as the laws for '<*>'
-- imply a form of associativity.
--
-- The superclass instances should satisfy the following:
--
-- * In the 'Functor' instance, 'fmap' should be equivalent to traversal
-- with the identity applicative functor ('fmapDefault').
--
-- * In the 'Foldable' instance, 'Data.Foldable.foldMap' should be
-- equivalent to traversal with a constant applicative functor
-- ('foldMapDefault').
--
class (Functor t, Foldable t) => Traversable t where
-- | Map each element of a structure to an action, evaluate
-- these actions from left to right, and collect the results.
traverse :: Applicative f => (a -> f b) -> t a -> f (t b)
traverse f = sequenceA . fmap f
-- | Evaluate each action in the structure from left to right,
-- and collect the results.
sequenceA :: Applicative f => t (f a) -> f (t a)
sequenceA = traverse id
-- | Map each element of a structure to a monadic action, evaluate
-- these actions from left to right, and collect the results.
mapM :: Monad m => (a -> m b) -> t a -> m (t b)
mapM f = unwrapMonad . traverse (WrapMonad . f)
-- | Evaluate each monadic action in the structure from left to right,
-- and collect the results.
sequence :: Monad m => t (m a) -> m (t a)
sequence = mapM id
{-# MINIMAL traverse | sequenceA #-}
-- instances for Prelude types
instance Traversable Maybe where
traverse _ Nothing = pure Nothing
traverse f (Just x) = Just <$> f x
instance Traversable [] where
{-# INLINE traverse #-} -- so that traverse can fuse
traverse f = Prelude.foldr cons_f (pure [])
where cons_f x ys = (:) <$> f x <*> ys
mapM = Prelude.mapM
instance Traversable (Either a) where
traverse _ (Left x) = pure (Left x)
traverse f (Right y) = Right <$> 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 :: (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)