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 (
36 import Prelude hiding (mapM, sequence, foldr)
37 import qualified Prelude (mapM, foldr)
38 import Control.Applicative
39 import Data.Foldable (Foldable())
40 import Data.Monoid (Monoid)
43 -- | Functors representing data structures that can be traversed from
46 -- Minimal complete definition: 'traverse' or 'sequenceA'.
48 -- Instances are similar to 'Functor', e.g. given a data type
50 -- > data Tree a = Empty | Leaf a | Node (Tree a) a (Tree a)
52 -- a suitable instance would be
54 -- > instance Traversable Tree
55 -- > traverse f Empty = pure Empty
56 -- > traverse f (Leaf x) = Leaf <$> f x
57 -- > traverse f (Node l k r) = Node <$> traverse f l <*> f k <*> traverse f r
59 -- This is suitable even for abstract types, as the laws for '<*>'
60 -- imply a form of associativity.
62 -- The superclass instances should satisfy the following:
64 -- * In the 'Functor' instance, 'fmap' should be equivalent to traversal
65 -- with the identity applicative functor ('fmapDefault').
67 -- * In the 'Foldable' instance, 'Data.Foldable.foldMap' should be
68 -- equivalent to traversal with a constant applicative functor
69 -- ('foldMapDefault').
71 class (Functor t, Foldable t) => Traversable t where
72 -- | Map each element of a structure to an action, evaluate
73 -- these actions from left to right, and collect the results.
74 traverse :: Applicative f => (a -> f b) -> t a -> f (t b)
75 traverse f = sequenceA . fmap f
77 -- | Evaluate each action in the structure from left to right,
78 -- and collect the results.
79 sequenceA :: Applicative f => t (f a) -> f (t a)
80 sequenceA = traverse id
82 -- | Map each element of a structure to an monadic action, evaluate
83 -- these actions from left to right, and collect the results.
84 mapM :: Monad m => (a -> m b) -> t a -> m (t b)
85 mapM f = unwrapMonad . traverse (WrapMonad . f)
87 -- | Evaluate each monadic action in the structure from left to right,
88 -- and collect the results.
89 sequence :: Monad m => t (m a) -> m (t a)
92 -- instances for Prelude types
94 instance Traversable Maybe where
95 traverse f Nothing = pure Nothing
96 traverse f (Just x) = Just <$> f x
98 instance Traversable [] where
99 traverse f = Prelude.foldr cons_f (pure [])
100 where cons_f x ys = (:) <$> f x <*> ys
104 instance Ix i => Traversable (Array i) where
105 traverse f arr = listArray (bounds arr) <$> traverse f (elems arr)
109 -- | This function may be used as a value for `fmap` in a `Functor` instance.
110 fmapDefault :: Traversable t => (a -> b) -> t a -> t b
111 fmapDefault f = getId . traverse (Id . f)
113 -- | This function may be used as a value for `Data.Foldable.foldMap`
114 -- in a `Foldable` instance.
115 foldMapDefault :: (Traversable t, Monoid m) => (a -> m) -> t a -> m
116 foldMapDefault f = getConst . traverse (Const . f)
120 newtype Id a = Id { getId :: a }
122 instance Functor Id where
123 fmap f (Id x) = Id (f x)
125 instance Applicative Id where
127 Id f <*> Id x = Id (f x)