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Data.Foldable

Portability | portable
Stability | experimental
Maintainer | ross@soi.city.ac.uk

Contents

Folds
  Special biased folds
  Folding actions
    Applicative actions
    Monadic actions
  Specialized folds
  Searches

Description
Class of data structures that can be folded to a summary value. Many of these functions generalize Prelude, Control.Monad and Data.List functions of the same names from lists to any `Foldable` functor. To avoid ambiguity, either import those modules hiding these names or qualify uses of these function names with an alias for this module.
Synopsis

class Foldable t where

fold :: Monoid m => t m -> m
foldMap :: Monoid m => (a -> m) -> t a -> m
foldr :: (a -> b -> b) -> b -> t a -> b
foldl :: (a -> b -> a) -> a -> t b -> a
foldr1 :: (a -> a -> a) -> t a -> a
foldl1 :: (a -> a -> a) -> t a -> a

foldr' :: Foldable t => (a -> b -> b) -> b -> t a -> b
foldl' :: Foldable t => (a -> b -> a) -> a -> t b -> a
foldrM :: (Foldable t, Monad m) => (a -> b -> m b) -> b -> t a -> m b
foldlM :: (Foldable t, Monad m) => (a -> b -> m a) -> a -> t b -> m a
traverse_ :: (Foldable t, Applicative f) => (a -> f b) -> t a -> f ()
for_ :: (Foldable t, Applicative f) => t a -> (a -> f b) -> f ()
sequenceA_ :: (Foldable t, Applicative f) => t (f a) -> f ()
asum :: (Foldable t, Alternative f) => t (f a) -> f a
mapM_ :: (Foldable t, Monad m) => (a -> m b) -> t a -> m ()
forM_ :: (Foldable t, Monad m) => t a -> (a -> m b) -> m ()
sequence_ :: (Foldable t, Monad m) => t (m a) -> m ()
msum :: (Foldable t, MonadPlus m) => t (m a) -> m a
toList :: Foldable t => t a -> [a]
concat :: Foldable t => t [a] -> [a]
concatMap :: Foldable t => (a -> [b]) -> t a -> [b]
and :: Foldable t => t Bool -> Bool
or :: Foldable t => t Bool -> Bool
any :: Foldable t => (a -> Bool) -> t a -> Bool
all :: Foldable t => (a -> Bool) -> t a -> Bool
sum :: (Foldable t, Num a) => t a -> a
product :: (Foldable t, Num a) => t a -> a
maximum :: (Foldable t, Ord a) => t a -> a
maximumBy :: Foldable t => (a -> a -> Ordering) -> t a -> a
minimum :: (Foldable t, Ord a) => t a -> a
minimumBy :: Foldable t => (a -> a -> Ordering) -> t a -> a
elem :: (Foldable t, Eq a) => a -> t a -> Bool
notElem :: (Foldable t, Eq a) => a -> t a -> Bool
find :: Foldable t => (a -> Bool) -> t a -> Maybe a

Folds
class Foldable t where

Data structures that can be folded.

Minimal complete definition: `foldMap` or `foldr`.

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
        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.

Methods

fold :: Monoid m => t m -> m
Combine the elements of a structure using a monoid.
foldMap :: Monoid m => (a -> m) -> t a -> m
Map each element of the structure to a monoid, and combine the results.
foldr :: (a -> b -> b) -> b -> t a -> b

Right-associative fold of a structure.

    foldr f z = foldr f z . toList

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

Left-associative fold of a structure.

    foldl f z = foldl f z . toList

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

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

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

    foldl1 f = foldl1 f . toList

Instances

Foldable Digit
Foldable Elem
Foldable FingerTree
Foldable IntMap
Foldable Maybe
Foldable Node
Foldable Seq
Foldable Set
Foldable Tree
Foldable ViewL
Foldable ViewR
Foldable []
Ix i => Foldable (Array i)
Foldable (Map k)

Special biased folds
foldr' :: Foldable t => (a -> b -> b) -> b -> t a -> b
Fold over the elements of a structure, associating to the right, but strictly.
foldl' :: Foldable t => (a -> b -> a) -> a -> t b -> a
Fold over the elements of a structure, associating to the left, but strictly.
foldrM :: (Foldable t, Monad m) => (a -> b -> m b) -> b -> t a -> m b
Monadic fold over the elements of a structure, associating to the right, i.e. from right to left.
foldlM :: (Foldable t, Monad m) => (a -> b -> m a) -> a -> t b -> m a
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 ()
Map each element of a structure to an action, evaluate these actions from left to right, and ignore the results.
for_ :: (Foldable t, Applicative f) => t a -> (a -> f b) -> f ()
`for_` is `traverse_` with its arguments flipped.
sequenceA_ :: (Foldable t, Applicative f) => t (f a) -> f ()
Evaluate each action in the structure from left to right, and ignore the results.
asum :: (Foldable t, Alternative f) => t (f a) -> f a
The sum of a collection of actions, generalizing `concat`.
Monadic actions
mapM_ :: (Foldable t, Monad m) => (a -> m b) -> t a -> m ()
Map each element of a structure to an monadic action, evaluate these actions from left to right, and ignore the results.
forM_ :: (Foldable t, Monad m) => t a -> (a -> m b) -> m ()
`forM_` is `mapM_` with its arguments flipped.
sequence_ :: (Foldable t, Monad m) => t (m a) -> m ()
Evaluate each monadic action in the structure from left to right, and ignore the results.
msum :: (Foldable t, MonadPlus m) => t (m a) -> m a
The sum of a collection of actions, generalizing `concat`.
Specialized folds
toList :: Foldable t => t a -> [a]
List of elements of a structure.
concat :: Foldable t => t [a] -> [a]
The concatenation of all the elements of a container of lists.
concatMap :: Foldable t => (a -> [b]) -> t a -> [b]
Map a function over all the elements of a container and concatenate the resulting lists.
and :: Foldable t => t Bool -> Bool
`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
`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
Determines whether any element of the structure satisfies the predicate.
all :: Foldable t => (a -> Bool) -> t a -> Bool
Determines whether all elements of the structure satisfy the predicate.
sum :: (Foldable t, Num a) => t a -> a
The `sum` function computes the sum of the numbers of a structure.
product :: (Foldable t, Num a) => t a -> a
The `product` function computes the product of the numbers of a structure.
maximum :: (Foldable t, Ord a) => t a -> a
The largest element of a non-empty structure.
maximumBy :: Foldable t => (a -> a -> Ordering) -> t a -> a
The largest element of a non-empty structure with respect to the given comparison function.
minimum :: (Foldable t, Ord a) => t a -> a
The least element of a non-empty structure.
minimumBy :: Foldable t => (a -> a -> Ordering) -> t a -> a
The least element of a non-empty structure with respect to the given comparison function.
Searches
elem :: (Foldable t, Eq a) => a -> t a -> Bool
Does the element occur in the structure?
notElem :: (Foldable t, Eq a) => a -> t a -> Bool
`notElem` is the negation of `elem`.
find :: Foldable t => (a -> Bool) -> t a -> Maybe a
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.
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