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 |

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

- 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
- foldr' :: (a -> b -> b) -> b -> t a -> b
- foldl :: (b -> a -> b) -> b -> t a -> b
- foldl' :: (b -> a -> b) -> b -> t a -> b
- foldr1 :: (a -> a -> a) -> t a -> a
- foldl1 :: (a -> a -> a) -> t a -> a
- toList :: t a -> [a]
- null :: t a -> Bool
- length :: t a -> Int
- elem :: Eq a => a -> t a -> Bool
- maximum :: forall a. Ord a => t a -> a
- minimum :: forall a. Ord a => t a -> a
- sum :: Num a => t a -> a
- product :: Num a => t a -> a

- foldrM :: (Foldable t, Monad m) => (a -> b -> m b) -> b -> t a -> m b
- foldlM :: (Foldable t, Monad m) => (b -> a -> m b) -> b -> t a -> m b
- 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
- 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
- maximumBy :: Foldable t => (a -> a -> Ordering) -> t a -> a
- minimumBy :: Foldable t => (a -> a -> Ordering) -> t a -> a
- notElem :: (Foldable t, Eq a) => a -> t a -> Bool
- find :: Foldable t => (a -> Bool) -> t a -> Maybe a

# Folds

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)

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

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

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`

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

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.

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.

## 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

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

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`

.

## 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.

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.