-- Stability : experimental
-- Portability : portable
--
--- Class of data structures that can be traversed from left to right.
+-- Class of data structures that can be traversed from left to right,
+-- performing an action on each element.
--
--- See also <http://www.soi.city.ac.uk/~ross/papers/Applicative.html>.
+-- 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(..),
- sequenceA,
- sequence,
+ for,
+ forM,
fmapDefault,
foldMapDefault,
) where
-import Prelude hiding (mapM, sequence)
-import qualified Prelude (mapM)
+import Prelude hiding (mapM, sequence, foldr)
+import qualified Prelude (mapM, foldr)
import Control.Applicative
+import Data.Foldable (Foldable())
import Data.Monoid (Monoid)
import Data.Array
-- | Functors representing data structures that can be traversed from
-- left to right.
--
--- Minimal complete definition: 'traverse'.
+-- Minimal complete definition: 'traverse' or 'sequenceA'.
--
-- Instances are similar to 'Functor', e.g. given a data type
--
-- This is suitable even for abstract types, as the laws for '<*>'
-- imply a form of associativity.
--
-class Traversable t where
+-- 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 an 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 (Just x) = Just <$> f x
instance Traversable [] where
- traverse f = foldr cons_f (pure [])
+ traverse f = Prelude.foldr cons_f (pure [])
where cons_f x ys = (:) <$> f x <*> ys
mapM = Prelude.mapM
-- general functions
--- | Evaluate each action in the structure from left to right,
--- and collect the results.
-sequenceA :: (Traversable t, Applicative f) => t (f a) -> f (t a)
-sequenceA = traverse id
+-- | '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
--- | Evaluate each monadic action in the structure from left to right,
--- and collect the results.
-sequence :: (Traversable t, Monad m) => t (m a) -> m (t a)
-sequence = mapM id
+-- | '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
--- | Any 'Traversable' can also be made an instance of 'Functor' by
--- defining 'fmap' as 'fmapDefault'.
+-- | 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)
--- | Any 'Traversable' can also be made an instance of
--- 'Data.Foldable.Foldable' by defining 'Data.Foldable.foldMap'
--- as 'foldMapDefault'.
+-- | 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)
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)