1 -----------------------------------------------------------------------------
3 -- Module : Data.Foldable
4 -- Copyright : 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 folded to a summary value.
13 -- Many of these functions generalize "Prelude", "Control.Monad" and
14 -- "Data.List" functions of the same names from lists to any 'Foldable'
15 -- functor. To avoid ambiguity, either import those modules hiding
16 -- these names or qualify uses of these function names with an alias
19 module Data.Foldable (
22 -- ** Special biased folds
28 -- *** Applicative actions
33 -- *** Monadic actions
38 -- ** Specialized folds
58 import Prelude hiding (foldl, foldr, foldl1, foldr1, mapM_, sequence_,
59 elem, notElem, concat, concatMap, and, or, any, all,
60 sum, product, maximum, minimum)
61 import qualified Prelude (foldl, foldr, foldl1, foldr1)
62 import Control.Arrow (ArrowZero(..))
63 import Control.Applicative
64 import Control.Monad (MonadPlus(..))
65 import Data.Maybe (fromMaybe, listToMaybe)
69 #ifdef __GLASGOW_HASKELL__
70 import GHC.Exts (build)
73 -- | Data structures that can be folded.
75 -- Minimal complete definition: 'foldMap' or 'foldr'.
77 -- For example, given a data type
79 -- > data Tree a = Empty | Leaf a | Node (Tree a) a (Tree a)
81 -- a suitable instance would be
83 -- > instance Foldable Tree
84 -- > foldMap f Empty = mempty
85 -- > foldMap f (Leaf x) = f x
86 -- > foldMap f (Node l k r) = foldMap f l `mappend` f k `mappend` foldMap f r
88 -- This is suitable even for abstract types, as the monoid is assumed
89 -- to satisfy the monoid laws.
91 class Foldable t where
92 -- | Combine the elements of a structure using a monoid.
93 fold :: Monoid m => t m -> m
96 -- | Map each element of the structure to a monoid,
97 -- and combine the results.
98 foldMap :: Monoid m => (a -> m) -> t a -> m
99 foldMap f = foldr (mappend . f) mempty
101 -- | Right-associative fold of a structure.
103 -- @'foldr' f z = 'Prelude.foldr' f z . 'toList'@
104 foldr :: (a -> b -> b) -> b -> t a -> b
105 foldr f z t = appEndo (foldMap (Endo . f) t) z
107 -- | Left-associative fold of a structure.
109 -- @'foldl' f z = 'Prelude.foldl' f z . 'toList'@
110 foldl :: (a -> b -> a) -> a -> t b -> a
111 foldl f z t = appEndo (getDual (foldMap (Dual . Endo . flip f) t)) z
113 -- | A variant of 'foldr' that has no base case,
114 -- and thus may only be applied to non-empty structures.
116 -- @'foldr1' f = 'Prelude.foldr1' f . 'toList'@
117 foldr1 :: (a -> a -> a) -> t a -> a
118 foldr1 f xs = fromMaybe (error "foldr1: empty structure")
119 (foldr mf Nothing xs)
120 where mf x Nothing = Just x
121 mf x (Just y) = Just (f x y)
123 -- | A variant of 'foldl' that has no base case,
124 -- and thus may only be applied to non-empty structures.
126 -- @'foldl1' f = 'Prelude.foldl1' f . 'toList'@
127 foldl1 :: (a -> a -> a) -> t a -> a
128 foldl1 f xs = fromMaybe (error "foldl1: empty structure")
129 (foldl mf Nothing xs)
130 where mf Nothing y = Just y
131 mf (Just x) y = Just (f x y)
133 -- instances for Prelude types
135 instance Foldable Maybe where
136 foldr f z Nothing = z
137 foldr f z (Just x) = f x z
139 foldl f z Nothing = z
140 foldl f z (Just x) = f z x
142 instance Foldable [] where
143 foldr = Prelude.foldr
144 foldl = Prelude.foldl
145 foldr1 = Prelude.foldr1
146 foldl1 = Prelude.foldl1
148 instance Ix i => Foldable (Array i) where
149 foldr f z = Prelude.foldr f z . elems
151 -- | Fold over the elements of a structure,
152 -- associating to the right, but strictly.
153 foldr' :: Foldable t => (a -> b -> b) -> b -> t a -> b
154 foldr' f z xs = foldl f' id xs z
155 where f' k x z = k $! f x z
157 -- | Monadic fold over the elements of a structure,
158 -- associating to the right, i.e. from right to left.
159 foldrM :: (Foldable t, Monad m) => (a -> b -> m b) -> b -> t a -> m b
160 foldrM f z xs = foldl f' return xs z
161 where f' k x z = f x z >>= k
163 -- | Fold over the elements of a structure,
164 -- associating to the left, but strictly.
165 foldl' :: Foldable t => (a -> b -> a) -> a -> t b -> a
166 foldl' f z xs = foldr f' id xs z
167 where f' x k z = k $! f z x
169 -- | Monadic fold over the elements of a structure,
170 -- associating to the left, i.e. from left to right.
171 foldlM :: (Foldable t, Monad m) => (a -> b -> m a) -> a -> t b -> m a
172 foldlM f z xs = foldr f' return xs z
173 where f' x k z = f z x >>= k
175 -- | Map each element of a structure to an action, evaluate
176 -- these actions from left to right, and ignore the results.
177 traverse_ :: (Foldable t, Applicative f) => (a -> f b) -> t a -> f ()
178 traverse_ f = foldr ((*>) . f) (pure ())
180 -- | 'for_' is 'traverse_' with its arguments flipped.
181 for_ :: (Foldable t, Applicative f) => t a -> (a -> f b) -> f ()
183 for_ = flip traverse_
185 -- | Map each element of a structure to an monadic action, evaluate
186 -- these actions from left to right, and ignore the results.
187 mapM_ :: (Foldable t, Monad m) => (a -> m b) -> t a -> m ()
188 mapM_ f = foldr ((>>) . f) (return ())
190 -- | 'forM_' is 'mapM_' with its arguments flipped.
191 forM_ :: (Foldable t, Monad m) => t a -> (a -> m b) -> m ()
195 -- | Evaluate each action in the structure from left to right,
196 -- and ignore the results.
197 sequenceA_ :: (Foldable t, Applicative f) => t (f a) -> f ()
198 sequenceA_ = foldr (*>) (pure ())
200 -- | Evaluate each monadic action in the structure from left to right,
201 -- and ignore the results.
202 sequence_ :: (Foldable t, Monad m) => t (m a) -> m ()
203 sequence_ = foldr (>>) (return ())
205 -- | The sum of a collection of actions, generalizing 'concat'.
206 asum :: (Foldable t, Alternative f) => t (f a) -> f a
208 asum = foldr (<|>) empty
210 -- | The sum of a collection of actions, generalizing 'concat'.
211 msum :: (Foldable t, MonadPlus m) => t (m a) -> m a
213 msum = foldr mplus mzero
215 -- These use foldr rather than foldMap to avoid repeated concatenation.
217 -- | List of elements of a structure.
218 toList :: Foldable t => t a -> [a]
219 #ifdef __GLASGOW_HASKELL__
220 toList t = build (\ c n -> foldr c n t)
222 toList = foldr (:) []
225 -- | The concatenation of all the elements of a container of lists.
226 concat :: Foldable t => t [a] -> [a]
229 -- | Map a function over all the elements of a container and concatenate
230 -- the resulting lists.
231 concatMap :: Foldable t => (a -> [b]) -> t a -> [b]
234 -- | 'and' returns the conjunction of a container of Bools. For the
235 -- result to be 'True', the container must be finite; 'False', however,
236 -- results from a 'False' value finitely far from the left end.
237 and :: Foldable t => t Bool -> Bool
238 and = getAll . foldMap All
240 -- | 'or' returns the disjunction of a container of Bools. For the
241 -- result to be 'False', the container must be finite; 'True', however,
242 -- results from a 'True' value finitely far from the left end.
243 or :: Foldable t => t Bool -> Bool
244 or = getAny . foldMap Any
246 -- | Determines whether any element of the structure satisfies the predicate.
247 any :: Foldable t => (a -> Bool) -> t a -> Bool
248 any p = getAny . foldMap (Any . p)
250 -- | Determines whether all elements of the structure satisfy the predicate.
251 all :: Foldable t => (a -> Bool) -> t a -> Bool
252 all p = getAll . foldMap (All . p)
254 -- | The 'sum' function computes the sum of the numbers of a structure.
255 sum :: (Foldable t, Num a) => t a -> a
256 sum = getSum . foldMap Sum
258 -- | The 'product' function computes the product of the numbers of a structure.
259 product :: (Foldable t, Num a) => t a -> a
260 product = getProduct . foldMap Product
262 -- | The largest element of a non-empty structure.
263 maximum :: (Foldable t, Ord a) => t a -> a
266 -- | The largest element of a non-empty structure with respect to the
267 -- given comparison function.
268 maximumBy :: Foldable t => (a -> a -> Ordering) -> t a -> a
269 maximumBy cmp = foldr1 max'
270 where max' x y = case cmp x y of
274 -- | The least element of a non-empty structure.
275 minimum :: (Foldable t, Ord a) => t a -> a
278 -- | The least element of a non-empty structure with respect to the
279 -- given comparison function.
280 minimumBy :: Foldable t => (a -> a -> Ordering) -> t a -> a
281 minimumBy cmp = foldr1 min'
282 where min' x y = case cmp x y of
286 -- | Does the element occur in the structure?
287 elem :: (Foldable t, Eq a) => a -> t a -> Bool
290 -- | 'notElem' is the negation of 'elem'.
291 notElem :: (Foldable t, Eq a) => a -> t a -> Bool
292 notElem x = not . elem x
294 -- | The 'find' function takes a predicate and a structure and returns
295 -- the leftmost element of the structure matching the predicate, or
296 -- 'Nothing' if there is no such element.
297 find :: Foldable t => (a -> Bool) -> t a -> Maybe a
298 find p = listToMaybe . concatMap (\ x -> if p x then [x] else [])