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.

In the case of lists, foldr, when applied to a binary operator, a starting value (typically the right-identity of the operator), and a list, reduces the list using the binary operator, from right to left:

foldr f z [x1, x2, ..., xn] == x1 `f` (x2 `f` ... (xn `f` z)...)

Note that, since the head of the resulting expression is produced by an application of the operator to the first element of the list, foldr can produce a terminating expression from an infinite list.

For a general Foldable structure this should be semantically identical to,

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.

In the case of lists, foldl, when applied to a binary operator, a starting value (typically the left-identity of the operator), and a list, reduces the list using the binary operator, from left to right:

foldl f z [x1, x2, ..., xn] == (...((z `f` x1) `f` x2) `f`...) `f` xn

Note that to produce the outermost application of the operator the entire input list must be traversed. This means that foldl' will diverge if given an infinite list.

Also note that if you want an efficient left-fold, you probably want to use foldl' instead of foldl. The reason for this is that latter does not force the "inner" results (e.g. z f x1 in the above example) before applying them to the operator (e.g. to (f x2)). This results in a thunk chain O(n) elements long, which then must be evaluated from the outside-in.

For a general Foldable structure this should be semantically identical to,

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.

This ensures that each step of the fold is forced to weak head normal form before being applied, avoiding the collection of thunks that would otherwise occur. This is often what you want to strictly reduce a finite list to a single, monolithic result (e.g. length).

For a general Foldable structure this should be semantically identical to,

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 []
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
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 Last
Methods fold :: Monoid m => Last m -> m Source foldMap :: Monoid m => (a -> m) -> Last a -> m Source foldr :: (a -> b -> b) -> b -> Last a -> b Source foldr' :: (a -> b -> b) -> b -> Last a -> b Source foldl :: (b -> a -> b) -> b -> Last a -> b Source foldl' :: (b -> a -> b) -> b -> Last a -> b Source foldr1 :: (a -> a -> a) -> Last a -> a Source foldl1 :: (a -> a -> a) -> Last a -> a Source toList :: Last a -> [a] Source null :: Last a -> Bool Source length :: Last a -> Int Source elem :: Eq a => a -> Last a -> Bool Source maximum :: Ord a => Last a -> a Source minimum :: Ord a => Last a -> a Source sum :: Num a => Last a -> a Source product :: Num a => Last a -> a Source
Foldable First
Methods fold :: Monoid m => First m -> m Source foldMap :: Monoid m => (a -> m) -> First a -> m Source foldr :: (a -> b -> b) -> b -> First a -> b Source foldr' :: (a -> b -> b) -> b -> First a -> b Source foldl :: (b -> a -> b) -> b -> First a -> b Source foldl' :: (b -> a -> b) -> b -> First a -> b Source foldr1 :: (a -> a -> a) -> First a -> a Source foldl1 :: (a -> a -> a) -> First a -> a Source toList :: First a -> [a] Source null :: First a -> Bool Source length :: First a -> Int Source elem :: Eq a => a -> First a -> Bool Source maximum :: Ord a => First a -> a Source minimum :: Ord a => First a -> a Source sum :: Num a => First a -> a Source product :: Num a => First a -> a Source
Foldable Product
Methods fold :: Monoid m => Product m -> m Source foldMap :: Monoid m => (a -> m) -> Product a -> m Source foldr :: (a -> b -> b) -> b -> Product a -> b Source foldr' :: (a -> b -> b) -> b -> Product a -> b Source foldl :: (b -> a -> b) -> b -> Product a -> b Source foldl' :: (b -> a -> b) -> b -> Product a -> b Source foldr1 :: (a -> a -> a) -> Product a -> a Source foldl1 :: (a -> a -> a) -> Product a -> a Source toList :: Product a -> [a] Source null :: Product a -> Bool Source length :: Product a -> Int Source elem :: Eq a => a -> Product a -> Bool Source maximum :: Ord a => Product a -> a Source minimum :: Ord a => Product a -> a Source sum :: Num a => Product a -> a Source product :: Num a => Product a -> a Source
Foldable Sum
Methods fold :: Monoid m => Sum m -> m Source foldMap :: Monoid m => (a -> m) -> Sum a -> m Source foldr :: (a -> b -> b) -> b -> Sum a -> b Source foldr' :: (a -> b -> b) -> b -> Sum a -> b Source foldl :: (b -> a -> b) -> b -> Sum a -> b Source foldl' :: (b -> a -> b) -> b -> Sum a -> b Source foldr1 :: (a -> a -> a) -> Sum a -> a Source foldl1 :: (a -> a -> a) -> Sum a -> a Source toList :: Sum a -> [a] Source null :: Sum a -> Bool Source length :: Sum a -> Int Source elem :: Eq a => a -> Sum a -> Bool Source maximum :: Ord a => Sum a -> a Source minimum :: Ord a => Sum a -> a Source sum :: Num a => Sum a -> a Source product :: Num a => Sum a -> a Source
Foldable Dual
Methods fold :: Monoid m => Dual m -> m Source foldMap :: Monoid m => (a -> m) -> Dual a -> m Source foldr :: (a -> b -> b) -> b -> Dual a -> b Source foldr' :: (a -> b -> b) -> b -> Dual a -> b Source foldl :: (b -> a -> b) -> b -> Dual a -> b Source foldl' :: (b -> a -> b) -> b -> Dual a -> b Source foldr1 :: (a -> a -> a) -> Dual a -> a Source foldl1 :: (a -> a -> a) -> Dual a -> a Source toList :: Dual a -> [a] Source null :: Dual a -> Bool Source length :: Dual a -> Int Source elem :: Eq a => a -> Dual a -> Bool Source maximum :: Ord a => Dual a -> a Source minimum :: Ord a => Dual a -> a Source sum :: Num a => Dual a -> a Source product :: Num a => Dual a -> a Source
Foldable ZipList
Methods fold :: Monoid m => ZipList m -> m Source foldMap :: Monoid m => (a -> m) -> ZipList a -> m Source foldr :: (a -> b -> b) -> b -> ZipList a -> b Source foldr' :: (a -> b -> b) -> b -> ZipList a -> b Source foldl :: (b -> a -> b) -> b -> ZipList a -> b Source foldl' :: (b -> a -> b) -> b -> ZipList a -> b Source foldr1 :: (a -> a -> a) -> ZipList a -> a Source foldl1 :: (a -> a -> a) -> ZipList a -> a Source toList :: ZipList a -> [a] Source null :: ZipList a -> Bool Source length :: ZipList a -> Int Source elem :: Eq a => a -> ZipList a -> Bool Source maximum :: Ord a => ZipList a -> a Source minimum :: Ord a => ZipList a -> a Source sum :: Num a => ZipList a -> a Source product :: Num a => ZipList a -> a Source
Foldable Complex
Methods fold :: Monoid m => Complex m -> m Source foldMap :: Monoid m => (a -> m) -> Complex a -> m Source foldr :: (a -> b -> b) -> b -> Complex a -> b Source foldr' :: (a -> b -> b) -> b -> Complex a -> b Source foldl :: (b -> a -> b) -> b -> Complex a -> b Source foldl' :: (b -> a -> b) -> b -> Complex a -> b Source foldr1 :: (a -> a -> a) -> Complex a -> a Source foldl1 :: (a -> a -> a) -> Complex a -> a Source toList :: Complex a -> [a] Source null :: Complex a -> Bool Source length :: Complex a -> Int Source elem :: Eq a => a -> Complex a -> Bool Source maximum :: Ord a => Complex a -> a Source minimum :: Ord a => Complex a -> a Source sum :: Num a => Complex a -> a Source product :: Num a => Complex a -> a Source
Foldable NonEmpty
Methods fold :: Monoid m => NonEmpty m -> m Source foldMap :: Monoid m => (a -> m) -> NonEmpty a -> m Source foldr :: (a -> b -> b) -> b -> NonEmpty a -> b Source foldr' :: (a -> b -> b) -> b -> NonEmpty a -> b Source foldl :: (b -> a -> b) -> b -> NonEmpty a -> b Source foldl' :: (b -> a -> b) -> b -> NonEmpty a -> b Source foldr1 :: (a -> a -> a) -> NonEmpty a -> a Source foldl1 :: (a -> a -> a) -> NonEmpty a -> a Source toList :: NonEmpty a -> [a] Source null :: NonEmpty a -> Bool Source length :: NonEmpty a -> Int Source elem :: Eq a => a -> NonEmpty a -> Bool Source maximum :: Ord a => NonEmpty a -> a Source minimum :: Ord a => NonEmpty a -> a Source sum :: Num a => NonEmpty a -> a Source product :: Num a => NonEmpty a -> a Source
Foldable Option
Methods fold :: Monoid m => Option m -> m Source foldMap :: Monoid m => (a -> m) -> Option a -> m Source foldr :: (a -> b -> b) -> b -> Option a -> b Source foldr' :: (a -> b -> b) -> b -> Option a -> b Source foldl :: (b -> a -> b) -> b -> Option a -> b Source foldl' :: (b -> a -> b) -> b -> Option a -> b Source foldr1 :: (a -> a -> a) -> Option a -> a Source foldl1 :: (a -> a -> a) -> Option a -> a Source toList :: Option a -> [a] Source null :: Option a -> Bool Source length :: Option a -> Int Source elem :: Eq a => a -> Option a -> Bool Source maximum :: Ord a => Option a -> a Source minimum :: Ord a => Option a -> a Source sum :: Num a => Option a -> a Source product :: Num a => Option a -> a Source
Foldable Last
Methods fold :: Monoid m => Last m -> m Source foldMap :: Monoid m => (a -> m) -> Last a -> m Source foldr :: (a -> b -> b) -> b -> Last a -> b Source foldr' :: (a -> b -> b) -> b -> Last a -> b Source foldl :: (b -> a -> b) -> b -> Last a -> b Source foldl' :: (b -> a -> b) -> b -> Last a -> b Source foldr1 :: (a -> a -> a) -> Last a -> a Source foldl1 :: (a -> a -> a) -> Last a -> a Source toList :: Last a -> [a] Source null :: Last a -> Bool Source length :: Last a -> Int Source elem :: Eq a => a -> Last a -> Bool Source maximum :: Ord a => Last a -> a Source minimum :: Ord a => Last a -> a Source sum :: Num a => Last a -> a Source product :: Num a => Last a -> a Source
Foldable First
Methods fold :: Monoid m => First m -> m Source foldMap :: Monoid m => (a -> m) -> First a -> m Source foldr :: (a -> b -> b) -> b -> First a -> b Source foldr' :: (a -> b -> b) -> b -> First a -> b Source foldl :: (b -> a -> b) -> b -> First a -> b Source foldl' :: (b -> a -> b) -> b -> First a -> b Source foldr1 :: (a -> a -> a) -> First a -> a Source foldl1 :: (a -> a -> a) -> First a -> a Source toList :: First a -> [a] Source null :: First a -> Bool Source length :: First a -> Int Source elem :: Eq a => a -> First a -> Bool Source maximum :: Ord a => First a -> a Source minimum :: Ord a => First a -> a Source sum :: Num a => First a -> a Source product :: Num a => First a -> a Source
Foldable Max
Methods fold :: Monoid m => Max m -> m Source foldMap :: Monoid m => (a -> m) -> Max a -> m Source foldr :: (a -> b -> b) -> b -> Max a -> b Source foldr' :: (a -> b -> b) -> b -> Max a -> b Source foldl :: (b -> a -> b) -> b -> Max a -> b Source foldl' :: (b -> a -> b) -> b -> Max a -> b Source foldr1 :: (a -> a -> a) -> Max a -> a Source foldl1 :: (a -> a -> a) -> Max a -> a Source toList :: Max a -> [a] Source null :: Max a -> Bool Source length :: Max a -> Int Source elem :: Eq a => a -> Max a -> Bool Source maximum :: Ord a => Max a -> a Source minimum :: Ord a => Max a -> a Source sum :: Num a => Max a -> a Source product :: Num a => Max a -> a Source
Foldable Min
Methods fold :: Monoid m => Min m -> m Source foldMap :: Monoid m => (a -> m) -> Min a -> m Source foldr :: (a -> b -> b) -> b -> Min a -> b Source foldr' :: (a -> b -> b) -> b -> Min a -> b Source foldl :: (b -> a -> b) -> b -> Min a -> b Source foldl' :: (b -> a -> b) -> b -> Min a -> b Source foldr1 :: (a -> a -> a) -> Min a -> a Source foldl1 :: (a -> a -> a) -> Min a -> a Source toList :: Min a -> [a] Source null :: Min a -> Bool Source length :: Min a -> Int Source elem :: Eq a => a -> Min a -> Bool Source maximum :: Ord a => Min a -> a Source minimum :: Ord a => Min a -> a Source sum :: Num a => Min a -> a Source product :: Num a => Min a -> a Source
Foldable Identity
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)
Methods fold :: Monoid m => Either a m -> m Source foldMap :: Monoid m => (a -> m) -> Either a a -> m Source foldr :: (a -> b -> b) -> b -> Either a a -> b Source foldr' :: (a -> b -> b) -> b -> Either a a -> b Source foldl :: (b -> a -> b) -> b -> Either a a -> b Source foldl' :: (b -> a -> b) -> b -> Either a a -> b Source foldr1 :: (a -> a -> a) -> Either a a -> a Source foldl1 :: (a -> a -> a) -> Either a a -> a Source toList :: Either a a -> [a] Source null :: Either a a -> Bool Source length :: Either a a -> Int Source elem :: Eq a => a -> Either a a -> Bool Source maximum :: Ord a => Either a a -> a Source minimum :: Ord a => Either a a -> a Source sum :: Num a => Either a a -> a Source product :: Num a => Either a a -> a Source
Foldable ((,) a)
Methods fold :: Monoid m => (a, m) -> m Source foldMap :: Monoid m => (a -> m) -> (a, a) -> m Source foldr :: (a -> b -> b) -> b -> (a, a) -> b Source foldr' :: (a -> b -> b) -> b -> (a, a) -> b Source foldl :: (b -> a -> b) -> b -> (a, a) -> b Source foldl' :: (b -> a -> b) -> b -> (a, a) -> b Source foldr1 :: (a -> a -> a) -> (a, a) -> a Source foldl1 :: (a -> a -> a) -> (a, a) -> a Source toList :: (a, a) -> [a] Source null :: (a, a) -> Bool Source length :: (a, a) -> Int Source elem :: Eq a => a -> (a, a) -> Bool Source maximum :: Ord a => (a, a) -> a Source minimum :: Ord a => (a, a) -> a Source sum :: Num a => (a, a) -> a Source product :: Num a => (a, a) -> a Source
Foldable (Proxy (TYPE Lifted))
Methods fold :: Monoid m => Proxy (TYPE Lifted) m -> m Source foldMap :: Monoid m => (a -> m) -> Proxy (TYPE Lifted) a -> m Source foldr :: (a -> b -> b) -> b -> Proxy (TYPE Lifted) a -> b Source foldr' :: (a -> b -> b) -> b -> Proxy (TYPE Lifted) a -> b Source foldl :: (b -> a -> b) -> b -> Proxy (TYPE Lifted) a -> b Source foldl' :: (b -> a -> b) -> b -> Proxy (TYPE Lifted) a -> b Source foldr1 :: (a -> a -> a) -> Proxy (TYPE Lifted) a -> a Source foldl1 :: (a -> a -> a) -> Proxy (TYPE Lifted) a -> a Source toList :: Proxy (TYPE Lifted) a -> [a] Source null :: Proxy (TYPE Lifted) a -> Bool Source length :: Proxy (TYPE Lifted) a -> Int Source elem :: Eq a => a -> Proxy (TYPE Lifted) a -> Bool Source maximum :: Ord a => Proxy (TYPE Lifted) a -> a Source minimum :: Ord a => Proxy (TYPE Lifted) a -> a Source sum :: Num a => Proxy (TYPE Lifted) a -> a Source product :: Num a => Proxy (TYPE Lifted) a -> a Source
Foldable (Arg a)
Methods fold :: Monoid m => Arg a m -> m Source foldMap :: Monoid m => (a -> m) -> Arg a a -> m Source foldr :: (a -> b -> b) -> b -> Arg a a -> b Source foldr' :: (a -> b -> b) -> b -> Arg a a -> b Source foldl :: (b -> a -> b) -> b -> Arg a a -> b Source foldl' :: (b -> a -> b) -> b -> Arg a a -> b Source foldr1 :: (a -> a -> a) -> Arg a a -> a Source foldl1 :: (a -> a -> a) -> Arg a a -> a Source toList :: Arg a a -> [a] Source null :: Arg a a -> Bool Source length :: Arg a a -> Int Source elem :: Eq a => a -> Arg a a -> Bool Source maximum :: Ord a => Arg a a -> a Source minimum :: Ord a => Arg a a -> a Source sum :: Num a => Arg a a -> a Source product :: Num a => Arg a a -> a Source
Foldable (Const (TYPE Lifted) m)
Methods fold :: Monoid m => Const (TYPE Lifted) m m -> m Source foldMap :: Monoid m => (a -> m) -> Const (TYPE Lifted) m a -> m Source foldr :: (a -> b -> b) -> b -> Const (TYPE Lifted) m a -> b Source foldr' :: (a -> b -> b) -> b -> Const (TYPE Lifted) m a -> b Source foldl :: (b -> a -> b) -> b -> Const (TYPE Lifted) m a -> b Source foldl' :: (b -> a -> b) -> b -> Const (TYPE Lifted) m a -> b Source foldr1 :: (a -> a -> a) -> Const (TYPE Lifted) m a -> a Source foldl1 :: (a -> a -> a) -> Const (TYPE Lifted) m a -> a Source toList :: Const (TYPE Lifted) m a -> [a] Source null :: Const (TYPE Lifted) m a -> Bool Source length :: Const (TYPE Lifted) m a -> Int Source elem :: Eq a => a -> Const (TYPE Lifted) m a -> Bool Source maximum :: Ord a => Const (TYPE Lifted) m a -> a Source minimum :: Ord a => Const (TYPE Lifted) m a -> a Source sum :: Num a => Const (TYPE Lifted) m a -> a Source product :: Num a => Const (TYPE Lifted) m a -> a Source
(Foldable f, Foldable g) => Foldable (Product (TYPE Lifted) f g)
Methods fold :: Monoid m => Product (TYPE Lifted) f g m -> m Source foldMap :: Monoid m => (a -> m) -> Product (TYPE Lifted) f g a -> m Source foldr :: (a -> b -> b) -> b -> Product (TYPE Lifted) f g a -> b Source foldr' :: (a -> b -> b) -> b -> Product (TYPE Lifted) f g a -> b Source foldl :: (b -> a -> b) -> b -> Product (TYPE Lifted) f g a -> b Source foldl' :: (b -> a -> b) -> b -> Product (TYPE Lifted) f g a -> b Source foldr1 :: (a -> a -> a) -> Product (TYPE Lifted) f g a -> a Source foldl1 :: (a -> a -> a) -> Product (TYPE Lifted) f g a -> a Source toList :: Product (TYPE Lifted) f g a -> [a] Source null :: Product (TYPE Lifted) f g a -> Bool Source length :: Product (TYPE Lifted) f g a -> Int Source elem :: Eq a => a -> Product (TYPE Lifted) f g a -> Bool Source maximum :: Ord a => Product (TYPE Lifted) f g a -> a Source minimum :: Ord a => Product (TYPE Lifted) f g a -> a Source sum :: Num a => Product (TYPE Lifted) f g a -> a Source product :: Num a => Product (TYPE Lifted) f g a -> a Source
(Foldable f, Foldable g) => Foldable (Sum (TYPE Lifted) f g)
Methods fold :: Monoid m => Sum (TYPE Lifted) f g m -> m Source foldMap :: Monoid m => (a -> m) -> Sum (TYPE Lifted) f g a -> m Source foldr :: (a -> b -> b) -> b -> Sum (TYPE Lifted) f g a -> b Source foldr' :: (a -> b -> b) -> b -> Sum (TYPE Lifted) f g a -> b Source foldl :: (b -> a -> b) -> b -> Sum (TYPE Lifted) f g a -> b Source foldl' :: (b -> a -> b) -> b -> Sum (TYPE Lifted) f g a -> b Source foldr1 :: (a -> a -> a) -> Sum (TYPE Lifted) f g a -> a Source foldl1 :: (a -> a -> a) -> Sum (TYPE Lifted) f g a -> a Source toList :: Sum (TYPE Lifted) f g a -> [a] Source null :: Sum (TYPE Lifted) f g a -> Bool Source length :: Sum (TYPE Lifted) f g a -> Int Source elem :: Eq a => a -> Sum (TYPE Lifted) f g a -> Bool Source maximum :: Ord a => Sum (TYPE Lifted) f g a -> a Source minimum :: Ord a => Sum (TYPE Lifted) f g a -> a Source sum :: Num a => Sum (TYPE Lifted) f g a -> a Source product :: Num a => Sum (TYPE Lifted) f g a -> a Source
(Foldable f, Foldable g) => Foldable (Compose (TYPE Lifted) (TYPE Lifted) f g)
Methods fold :: Monoid m => Compose (TYPE Lifted) (TYPE Lifted) f g m -> m Source foldMap :: Monoid m => (a -> m) -> Compose (TYPE Lifted) (TYPE Lifted) f g a -> m Source foldr :: (a -> b -> b) -> b -> Compose (TYPE Lifted) (TYPE Lifted) f g a -> b Source foldr' :: (a -> b -> b) -> b -> Compose (TYPE Lifted) (TYPE Lifted) f g a -> b Source foldl :: (b -> a -> b) -> b -> Compose (TYPE Lifted) (TYPE Lifted) f g a -> b Source foldl' :: (b -> a -> b) -> b -> Compose (TYPE Lifted) (TYPE Lifted) f g a -> b Source foldr1 :: (a -> a -> a) -> Compose (TYPE Lifted) (TYPE Lifted) f g a -> a Source foldl1 :: (a -> a -> a) -> Compose (TYPE Lifted) (TYPE Lifted) f g a -> a Source toList :: Compose (TYPE Lifted) (TYPE Lifted) f g a -> [a] Source null :: Compose (TYPE Lifted) (TYPE Lifted) f g a -> Bool Source length :: Compose (TYPE Lifted) (TYPE Lifted) f g a -> Int Source elem :: Eq a => a -> Compose (TYPE Lifted) (TYPE Lifted) f g a -> Bool Source maximum :: Ord a => Compose (TYPE Lifted) (TYPE Lifted) f g a -> a Source minimum :: Ord a => Compose (TYPE Lifted) (TYPE Lifted) f g a -> a Source sum :: Num a => Compose (TYPE Lifted) (TYPE Lifted) f g a -> a Source product :: Num a => Compose (TYPE Lifted) (TYPE Lifted) f g 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.