Copyright	Ross Paterson 2005
License	BSD-style (see the LICENSE file in the distribution)
Maintainer	libraries@haskell.org
Stability	experimental
Portability	portable
Safe Haskell	Trustworthy
Language	Haskell2010

Data.Foldable

Contents

Folds

Description

Class of data structures that can be folded to a summary value.

Synopsis

Folds

class Foldable t where Source

Data structures that can be folded.

For example, given a data type

data Tree a = Empty | Leaf a | Node (Tree a) a (Tree a)

a suitable instance would be

instance Foldable Tree where
   foldMap f Empty = mempty
   foldMap f (Leaf x) = f x
   foldMap f (Node l k r) = foldMap f l `mappend` f k `mappend` foldMap f r

This is suitable even for abstract types, as the monoid is assumed to satisfy the monoid laws. Alternatively, one could define foldr:

instance Foldable Tree where
   foldr f z Empty = z
   foldr f z (Leaf x) = f x z
   foldr f z (Node l k r) = foldr f (f k (foldr f z r)) l

Foldable instances are expected to satisfy the following laws:

foldr f z t = appEndo (foldMap (Endo . f) t ) z

foldl f z t = appEndo (getDual (foldMap (Dual . Endo . flip f) t)) z

fold = foldMap id

sum, product, maximum, and minimum should all be essentially equivalent to foldMap forms, such as

sum = getSum . foldMap Sum

but may be less defined.

If the type is also a Functor instance, it should satisfy

foldMap f = fold . fmap f

which implies that

foldMap f . fmap g = foldMap (f . g)

Minimal complete definition

foldMap | foldr

Methods

fold :: Monoid m => t m -> m Source

Combine the elements of a structure using a monoid.

foldMap :: Monoid m => (a -> m) -> t a -> m Source

Map each element of the structure to a monoid, and combine the results.

foldr :: (a -> b -> b) -> b -> t a -> b Source

Right-associative fold of a structure.

foldr f z = foldr f z . toList

foldr' :: (a -> b -> b) -> b -> t a -> b Source

Right-associative fold of a structure, but with strict application of the operator.

foldl :: (b -> a -> b) -> b -> t a -> b Source

Left-associative fold of a structure.

foldl f z = foldl f z . toList

foldl' :: (b -> a -> b) -> b -> t a -> b Source

Left-associative fold of a structure. but with strict application of the operator.

foldl f z = foldl' f z . toList

foldr1 :: (a -> a -> a) -> t a -> a Source

A variant of foldr that has no base case, and thus may only be applied to non-empty structures.

foldr1 f = foldr1 f . toList

foldl1 :: (a -> a -> a) -> t a -> a Source

A variant of foldl that has no base case, and thus may only be applied to non-empty structures.

foldl1 f = foldl1 f . toList

toList :: t a -> [a] Source

List of elements of a structure, from left to right.

null :: t a -> Bool Source

Test whether the structure is empty. The default implementation is optimized for structures that are similar to cons-lists, because there is no general way to do better.

length :: t a -> Int Source

Returns the size/length of a finite structure as an Int. The default implementation is optimized for structures that are similar to cons-lists, because there is no general way to do better.

elem :: Eq a => a -> t a -> Bool infix 4 Source

Does the element occur in the structure?

maximum :: forall a. Ord a => t a -> a Source

The largest element of a non-empty structure.

minimum :: forall a. Ord a => t a -> a Source

The least element of a non-empty structure.

sum :: Num a => t a -> a Source

The sum function computes the sum of the numbers of a structure.

product :: Num a => t a -> a Source

The product function computes the product of the numbers of a structure.

Instances

Foldable [] Source
Methods fold :: Monoid m => [m] -> m Source foldMap :: Monoid m => (a -> m) -> [a] -> m Source foldr :: (a -> b -> b) -> b -> [a] -> b Source foldr' :: (a -> b -> b) -> b -> [a] -> b Source foldl :: (b -> a -> b) -> b -> [a] -> b Source foldl' :: (b -> a -> b) -> b -> [a] -> b Source foldr1 :: (a -> a -> a) -> [a] -> a Source foldl1 :: (a -> a -> a) -> [a] -> a Source toList :: [a] -> [a] Source null :: [a] -> Bool Source length :: [a] -> Int Source elem :: Eq a => a -> [a] -> Bool Source maximum :: Ord a => [a] -> a Source minimum :: Ord a => [a] -> a Source sum :: Num a => [a] -> a Source product :: Num a => [a] -> a Source
Foldable Maybe Source
Methods fold :: Monoid m => Maybe m -> m Source foldMap :: Monoid m => (a -> m) -> Maybe a -> m Source foldr :: (a -> b -> b) -> b -> Maybe a -> b Source foldr' :: (a -> b -> b) -> b -> Maybe a -> b Source foldl :: (b -> a -> b) -> b -> Maybe a -> b Source foldl' :: (b -> a -> b) -> b -> Maybe a -> b Source foldr1 :: (a -> a -> a) -> Maybe a -> a Source foldl1 :: (a -> a -> a) -> Maybe a -> a Source toList :: Maybe a -> [a] Source null :: Maybe a -> Bool Source length :: Maybe a -> Int Source elem :: Eq a => a -> Maybe a -> Bool Source maximum :: Ord a => Maybe a -> a Source minimum :: Ord a => Maybe a -> a Source sum :: Num a => Maybe a -> a Source product :: Num a => Maybe a -> a Source
Foldable Identity Source
Methods fold :: Monoid m => Identity m -> m Source foldMap :: Monoid m => (a -> m) -> Identity a -> m Source foldr :: (a -> b -> b) -> b -> Identity a -> b Source foldr' :: (a -> b -> b) -> b -> Identity a -> b Source foldl :: (b -> a -> b) -> b -> Identity a -> b Source foldl' :: (b -> a -> b) -> b -> Identity a -> b Source foldr1 :: (a -> a -> a) -> Identity a -> a Source foldl1 :: (a -> a -> a) -> Identity a -> a Source toList :: Identity a -> [a] Source null :: Identity a -> Bool Source length :: Identity a -> Int Source elem :: Eq a => a -> Identity a -> Bool Source maximum :: Ord a => Identity a -> a Source minimum :: Ord a => Identity a -> a Source sum :: Num a => Identity a -> a Source product :: Num a => Identity a -> a Source
Foldable (Either a) Source
Methods fold :: Monoid m => Either a m -> m Source foldMap :: Monoid m => (b -> m) -> Either a b -> m Source foldr :: (b -> c -> c) -> c -> Either a b -> c Source foldr' :: (b -> c -> c) -> c -> Either a b -> c Source foldl :: (b -> c -> b) -> b -> Either a c -> b Source foldl' :: (b -> c -> b) -> b -> Either a c -> b Source foldr1 :: (b -> b -> b) -> Either a b -> b Source foldl1 :: (b -> b -> b) -> Either a b -> b Source toList :: Either a b -> [b] Source null :: Either a b -> Bool Source length :: Either a b -> Int Source elem :: Eq b => b -> Either a b -> Bool Source maximum :: Ord b => Either a b -> b Source minimum :: Ord b => Either a b -> b Source sum :: Num b => Either a b -> b Source product :: Num b => Either a b -> b Source
Foldable ((,) a) Source
Methods fold :: Monoid m => (a, m) -> m Source foldMap :: Monoid m => (b -> m) -> (a, b) -> m Source foldr :: (b -> c -> c) -> c -> (a, b) -> c Source foldr' :: (b -> c -> c) -> c -> (a, b) -> c Source foldl :: (b -> c -> b) -> b -> (a, c) -> b Source foldl' :: (b -> c -> b) -> b -> (a, c) -> b Source foldr1 :: (b -> b -> b) -> (a, b) -> b Source foldl1 :: (b -> b -> b) -> (a, b) -> b Source toList :: (a, b) -> [b] Source null :: (a, b) -> Bool Source length :: (a, b) -> Int Source elem :: Eq b => b -> (a, b) -> Bool Source maximum :: Ord b => (a, b) -> b Source minimum :: Ord b => (a, b) -> b Source sum :: Num b => (a, b) -> b Source product :: Num b => (a, b) -> b Source
Foldable (Proxy *) Source
Methods fold :: Monoid m => Proxy * m -> m Source foldMap :: Monoid m => (a -> m) -> Proxy * a -> m Source foldr :: (a -> b -> b) -> b -> Proxy * a -> b Source foldr' :: (a -> b -> b) -> b -> Proxy * a -> b Source foldl :: (b -> a -> b) -> b -> Proxy * a -> b Source foldl' :: (b -> a -> b) -> b -> Proxy * a -> b Source foldr1 :: (a -> a -> a) -> Proxy * a -> a Source foldl1 :: (a -> a -> a) -> Proxy * a -> a Source toList :: Proxy * a -> [a] Source null :: Proxy * a -> Bool Source length :: Proxy * a -> Int Source elem :: Eq a => a -> Proxy * a -> Bool Source maximum :: Ord a => Proxy * a -> a Source minimum :: Ord a => Proxy * a -> a Source sum :: Num a => Proxy * a -> a Source product :: Num a => Proxy * a -> a Source
Foldable (Const m) Source
Methods fold :: Monoid a => Const m a -> a Source foldMap :: Monoid b => (a -> b) -> Const m a -> b Source foldr :: (a -> b -> b) -> b -> Const m a -> b Source foldr' :: (a -> b -> b) -> b -> Const m a -> b Source foldl :: (b -> a -> b) -> b -> Const m a -> b Source foldl' :: (b -> a -> b) -> b -> Const m a -> b Source foldr1 :: (a -> a -> a) -> Const m a -> a Source foldl1 :: (a -> a -> a) -> Const m a -> a Source toList :: Const m a -> [a] Source null :: Const m a -> Bool Source length :: Const m a -> Int Source elem :: Eq a => a -> Const m a -> Bool Source maximum :: Ord a => Const m a -> a Source minimum :: Ord a => Const m a -> a Source sum :: Num a => Const m a -> a Source product :: Num a => Const m a -> a Source

Special biased folds

foldrM :: (Foldable t, Monad m) => (a -> b -> m b) -> b -> t a -> m b Source

Monadic fold over the elements of a structure, associating to the right, i.e. from right to left.

foldlM :: (Foldable t, Monad m) => (b -> a -> m b) -> b -> t a -> m b Source

Monadic fold over the elements of a structure, associating to the left, i.e. from left to right.

Folding actions

Applicative actions

traverse_ :: (Foldable t, Applicative f) => (a -> f b) -> t a -> f () Source

Map each element of a structure to an action, evaluate these actions from left to right, and ignore the results. For a version that doesn't ignore the results see traverse.

for_ :: (Foldable t, Applicative f) => t a -> (a -> f b) -> f () Source

for_ is traverse_ with its arguments flipped. For a version that doesn't ignore the results see for.

>>> for_ [1..4] print
1
2
3
4

sequenceA_ :: (Foldable t, Applicative f) => t (f a) -> f () Source

Evaluate each action in the structure from left to right, and ignore the results. For a version that doesn't ignore the results see sequenceA.

asum :: (Foldable t, Alternative f) => t (f a) -> f a Source

The sum of a collection of actions, generalizing concat.

Monadic actions

mapM_ :: (Foldable t, Monad m) => (a -> m b) -> t a -> m () Source

Map each element of a structure to a monadic action, evaluate these actions from left to right, and ignore the results. For a version that doesn't ignore the results see mapM.

As of base 4.8.0.0, mapM_ is just traverse_, specialized to Monad.

forM_ :: (Foldable t, Monad m) => t a -> (a -> m b) -> m () Source

forM_ is mapM_ with its arguments flipped. For a version that doesn't ignore the results see forM.

As of base 4.8.0.0, forM_ is just for_, specialized to Monad.

sequence_ :: (Foldable t, Monad m) => t (m a) -> m () Source

Evaluate each monadic action in the structure from left to right, and ignore the results. For a version that doesn't ignore the results see sequence.

As of base 4.8.0.0, sequence_ is just sequenceA_, specialized to Monad.

msum :: (Foldable t, MonadPlus m) => t (m a) -> m a Source

The sum of a collection of actions, generalizing concat. As of base 4.8.0.0, msum is just asum, specialized to MonadPlus.

Specialized folds

concat :: Foldable t => t [a] -> [a] Source

The concatenation of all the elements of a container of lists.

concatMap :: Foldable t => (a -> [b]) -> t a -> [b] Source

Map a function over all the elements of a container and concatenate the resulting lists.

and :: Foldable t => t Bool -> Bool Source

and returns the conjunction of a container of Bools. For the result to be True, the container must be finite; False, however, results from a False value finitely far from the left end.

or :: Foldable t => t Bool -> Bool Source

or returns the disjunction of a container of Bools. For the result to be False, the container must be finite; True, however, results from a True value finitely far from the left end.

any :: Foldable t => (a -> Bool) -> t a -> Bool Source

Determines whether any element of the structure satisfies the predicate.

all :: Foldable t => (a -> Bool) -> t a -> Bool Source

Determines whether all elements of the structure satisfy the predicate.

maximumBy :: Foldable t => (a -> a -> Ordering) -> t a -> a Source

The largest element of a non-empty structure with respect to the given comparison function.

minimumBy :: Foldable t => (a -> a -> Ordering) -> t a -> a Source

The least element of a non-empty structure with respect to the given comparison function.

Searches

notElem :: (Foldable t, Eq a) => a -> t a -> Bool infix 4 Source

notElem is the negation of elem.

find :: Foldable t => (a -> Bool) -> t a -> Maybe a Source

The find function takes a predicate and a structure and returns the leftmost element of the structure matching the predicate, or Nothing if there is no such element.