1 -----------------------------------------------------------------------------
3 -- Module : Data.Traversable
4 -- Copyright : Conor McBride and Ross Paterson 2005
5 -- License : BSD-style (see the LICENSE file in the distribution)
7 -- Maintainer : ross@soi.city.ac.uk
8 -- Stability : experimental
9 -- Portability : portable
11 -- Class of data structures that can be traversed from left to right,
12 -- performing an action on each element.
16 -- * /Applicative Programming with Effects/,
17 -- by Conor McBride and Ross Paterson, online at
18 -- <http://www.soi.city.ac.uk/~ross/papers/Applicative.html>.
20 -- * /The Essence of the Iterator Pattern/,
21 -- by Jeremy Gibbons and Bruno Oliveira,
22 -- in /Mathematically-Structured Functional Programming/, 2006, and online at
23 -- <http://web.comlab.ox.ac.uk/oucl/work/jeremy.gibbons/publications/#iterator>.
25 -- Note that the functions 'mapM' and 'sequence' generalize "Prelude"
26 -- functions of the same names from lists to any 'Traversable' functor.
27 -- To avoid ambiguity, either import the "Prelude" hiding these names
28 -- or qualify uses of these function names with an alias for this module.
30 module Data.Traversable (
38 import Prelude hiding (mapM, sequence, foldr)
39 import qualified Prelude (mapM, foldr)
40 import Control.Applicative
41 import Data.Foldable (Foldable())
42 import Data.Monoid (Monoid)
45 -- | Functors representing data structures that can be traversed from
48 -- Minimal complete definition: 'traverse' or 'sequenceA'.
50 -- Instances are similar to 'Functor', e.g. given a data type
52 -- > data Tree a = Empty | Leaf a | Node (Tree a) a (Tree a)
54 -- a suitable instance would be
56 -- > instance Traversable Tree
57 -- > traverse f Empty = pure Empty
58 -- > traverse f (Leaf x) = Leaf <$> f x
59 -- > traverse f (Node l k r) = Node <$> traverse f l <*> f k <*> traverse f r
61 -- This is suitable even for abstract types, as the laws for '<*>'
62 -- imply a form of associativity.
64 -- The superclass instances should satisfy the following:
66 -- * In the 'Functor' instance, 'fmap' should be equivalent to traversal
67 -- with the identity applicative functor ('fmapDefault').
69 -- * In the 'Foldable' instance, 'Data.Foldable.foldMap' should be
70 -- equivalent to traversal with a constant applicative functor
71 -- ('foldMapDefault').
73 class (Functor t, Foldable t) => Traversable t where
74 -- | Map each element of a structure to an action, evaluate
75 -- these actions from left to right, and collect the results.
76 traverse :: Applicative f => (a -> f b) -> t a -> f (t b)
77 traverse f = sequenceA . fmap f
79 -- | Evaluate each action in the structure from left to right,
80 -- and collect the results.
81 sequenceA :: Applicative f => t (f a) -> f (t a)
82 sequenceA = traverse id
84 -- | Map each element of a structure to a monadic action, evaluate
85 -- these actions from left to right, and collect the results.
86 mapM :: Monad m => (a -> m b) -> t a -> m (t b)
87 mapM f = unwrapMonad . traverse (WrapMonad . f)
89 -- | Evaluate each monadic action in the structure from left to right,
90 -- and collect the results.
91 sequence :: Monad m => t (m a) -> m (t a)
94 -- instances for Prelude types
96 instance Traversable Maybe where
97 traverse f Nothing = pure Nothing
98 traverse f (Just x) = Just <$> f x
100 instance Traversable [] where
101 traverse f = Prelude.foldr cons_f (pure [])
102 where cons_f x ys = (:) <$> f x <*> ys
106 instance Ix i => Traversable (Array i) where
107 traverse f arr = listArray (bounds arr) <$> traverse f (elems arr)
111 -- | 'for' is 'traverse' with its arguments flipped.
112 for :: (Traversable t, Applicative f) => t a -> (a -> f b) -> f (t b)
116 -- | 'forM' is 'mapM' with its arguments flipped.
117 forM :: (Traversable t, Monad m) => t a -> (a -> m b) -> m (t b)
121 -- | This function may be used as a value for `fmap` in a `Functor` instance.
122 fmapDefault :: Traversable t => (a -> b) -> t a -> t b
123 fmapDefault f = getId . traverse (Id . f)
125 -- | This function may be used as a value for `Data.Foldable.foldMap`
126 -- in a `Foldable` instance.
127 foldMapDefault :: (Traversable t, Monoid m) => (a -> m) -> t a -> m
128 foldMapDefault f = getConst . traverse (Const . f)
132 newtype Id a = Id { getId :: a }
134 instance Functor Id where
135 fmap f (Id x) = Id (f x)
137 instance Applicative Id where
139 Id f <*> Id x = Id (f x)