Safe Haskell	Safe
Language	Haskell2010

Data.Universe.Instances.Traversable

Contents

Orphan instances

Synopsis

class Foldable (t :: * -> *) where
class (Functor t, Foldable t) => Traversable (t :: * -> *) where

Documentation

A Foldable instance for functions, given the input is Finite, and a Traversable instance for functions, given the input is Ord and Finite.

class Foldable (t :: * -> *) where #

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

length = getSum . foldMap (Sum . const  1)

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 #

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.

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 #

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

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

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 #

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 #

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

toList :: t a -> [a] #

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

null :: t a -> Bool #

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 #

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 #

Does the element occur in the structure?

maximum :: Ord a => t a -> a #

The largest element of a non-empty structure.

minimum :: Ord a => t a -> a #

The least element of a non-empty structure.

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

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

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

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

Instances

Foldable []	Since: base-2.1
Instance details Defined in Data.Foldable Methods fold :: Monoid m => [m] -> m # foldMap :: Monoid m => (a -> m) -> [a] -> m # foldr :: (a -> b -> b) -> b -> [a] -> b # foldr' :: (a -> b -> b) -> b -> [a] -> b # foldl :: (b -> a -> b) -> b -> [a] -> b # foldl' :: (b -> a -> b) -> b -> [a] -> b # foldr1 :: (a -> a -> a) -> [a] -> a # foldl1 :: (a -> a -> a) -> [a] -> a # toList :: [a] -> [a] # null :: [a] -> Bool # length :: [a] -> Int # elem :: Eq a => a -> [a] -> Bool # maximum :: Ord a => [a] -> a # minimum :: Ord a => [a] -> a # sum :: Num a => [a] -> a # product :: Num a => [a] -> a #
Foldable Maybe	Since: base-2.1
Instance details Defined in Data.Foldable Methods fold :: Monoid m => Maybe m -> m # foldMap :: Monoid m => (a -> m) -> Maybe a -> m # foldr :: (a -> b -> b) -> b -> Maybe a -> b # foldr' :: (a -> b -> b) -> b -> Maybe a -> b # foldl :: (b -> a -> b) -> b -> Maybe a -> b # foldl' :: (b -> a -> b) -> b -> Maybe a -> b # foldr1 :: (a -> a -> a) -> Maybe a -> a # foldl1 :: (a -> a -> a) -> Maybe a -> a # toList :: Maybe a -> [a] # null :: Maybe a -> Bool # length :: Maybe a -> Int # elem :: Eq a => a -> Maybe a -> Bool # maximum :: Ord a => Maybe a -> a # minimum :: Ord a => Maybe a -> a # sum :: Num a => Maybe a -> a # product :: Num a => Maybe a -> a #
Foldable Par1
Instance details Defined in Data.Foldable Methods fold :: Monoid m => Par1 m -> m # foldMap :: Monoid m => (a -> m) -> Par1 a -> m # foldr :: (a -> b -> b) -> b -> Par1 a -> b # foldr' :: (a -> b -> b) -> b -> Par1 a -> b # foldl :: (b -> a -> b) -> b -> Par1 a -> b # foldl' :: (b -> a -> b) -> b -> Par1 a -> b # foldr1 :: (a -> a -> a) -> Par1 a -> a # foldl1 :: (a -> a -> a) -> Par1 a -> a # toList :: Par1 a -> [a] # null :: Par1 a -> Bool # length :: Par1 a -> Int # elem :: Eq a => a -> Par1 a -> Bool # maximum :: Ord a => Par1 a -> a # minimum :: Ord a => Par1 a -> a # sum :: Num a => Par1 a -> a # product :: Num a => Par1 a -> a #
Foldable ZipList
Instance details Defined in Control.Applicative Methods fold :: Monoid m => ZipList m -> m # foldMap :: Monoid m => (a -> m) -> ZipList a -> m # foldr :: (a -> b -> b) -> b -> ZipList a -> b # foldr' :: (a -> b -> b) -> b -> ZipList a -> b # foldl :: (b -> a -> b) -> b -> ZipList a -> b # foldl' :: (b -> a -> b) -> b -> ZipList a -> b # foldr1 :: (a -> a -> a) -> ZipList a -> a # foldl1 :: (a -> a -> a) -> ZipList a -> a # toList :: ZipList a -> [a] # null :: ZipList a -> Bool # length :: ZipList a -> Int # elem :: Eq a => a -> ZipList a -> Bool # maximum :: Ord a => ZipList a -> a # minimum :: Ord a => ZipList a -> a # sum :: Num a => ZipList a -> a # product :: Num a => ZipList a -> a #
Foldable First	Since: base-4.8.0.0
Instance details Defined in Data.Foldable Methods fold :: Monoid m => First m -> m # foldMap :: Monoid m => (a -> m) -> First a -> m # foldr :: (a -> b -> b) -> b -> First a -> b # foldr' :: (a -> b -> b) -> b -> First a -> b # foldl :: (b -> a -> b) -> b -> First a -> b # foldl' :: (b -> a -> b) -> b -> First a -> b # foldr1 :: (a -> a -> a) -> First a -> a # foldl1 :: (a -> a -> a) -> First a -> a # toList :: First a -> [a] # null :: First a -> Bool # length :: First a -> Int # elem :: Eq a => a -> First a -> Bool # maximum :: Ord a => First a -> a # minimum :: Ord a => First a -> a # sum :: Num a => First a -> a # product :: Num a => First a -> a #
Foldable Last	Since: base-4.8.0.0
Instance details Defined in Data.Foldable Methods fold :: Monoid m => Last m -> m # foldMap :: Monoid m => (a -> m) -> Last a -> m # foldr :: (a -> b -> b) -> b -> Last a -> b # foldr' :: (a -> b -> b) -> b -> Last a -> b # foldl :: (b -> a -> b) -> b -> Last a -> b # foldl' :: (b -> a -> b) -> b -> Last a -> b # foldr1 :: (a -> a -> a) -> Last a -> a # foldl1 :: (a -> a -> a) -> Last a -> a # toList :: Last a -> [a] # null :: Last a -> Bool # length :: Last a -> Int # elem :: Eq a => a -> Last a -> Bool # maximum :: Ord a => Last a -> a # minimum :: Ord a => Last a -> a # sum :: Num a => Last a -> a # product :: Num a => Last a -> a #
Foldable Dual	Since: base-4.8.0.0
Instance details Defined in Data.Foldable Methods fold :: Monoid m => Dual m -> m # foldMap :: Monoid m => (a -> m) -> Dual a -> m # foldr :: (a -> b -> b) -> b -> Dual a -> b # foldr' :: (a -> b -> b) -> b -> Dual a -> b # foldl :: (b -> a -> b) -> b -> Dual a -> b # foldl' :: (b -> a -> b) -> b -> Dual a -> b # foldr1 :: (a -> a -> a) -> Dual a -> a # foldl1 :: (a -> a -> a) -> Dual a -> a # toList :: Dual a -> [a] # null :: Dual a -> Bool # length :: Dual a -> Int # elem :: Eq a => a -> Dual a -> Bool # maximum :: Ord a => Dual a -> a # minimum :: Ord a => Dual a -> a # sum :: Num a => Dual a -> a # product :: Num a => Dual a -> a #
Foldable Sum	Since: base-4.8.0.0
Instance details Defined in Data.Foldable Methods fold :: Monoid m => Sum m -> m # foldMap :: Monoid m => (a -> m) -> Sum a -> m # foldr :: (a -> b -> b) -> b -> Sum a -> b # foldr' :: (a -> b -> b) -> b -> Sum a -> b # foldl :: (b -> a -> b) -> b -> Sum a -> b # foldl' :: (b -> a -> b) -> b -> Sum a -> b # foldr1 :: (a -> a -> a) -> Sum a -> a # foldl1 :: (a -> a -> a) -> Sum a -> a # toList :: Sum a -> [a] # null :: Sum a -> Bool # length :: Sum a -> Int # elem :: Eq a => a -> Sum a -> Bool # maximum :: Ord a => Sum a -> a # minimum :: Ord a => Sum a -> a # sum :: Num a => Sum a -> a # product :: Num a => Sum a -> a #
Foldable Product	Since: base-4.8.0.0
Instance details Defined in Data.Foldable Methods fold :: Monoid m => Product m -> m # foldMap :: Monoid m => (a -> m) -> Product a -> m # foldr :: (a -> b -> b) -> b -> Product a -> b # foldr' :: (a -> b -> b) -> b -> Product a -> b # foldl :: (b -> a -> b) -> b -> Product a -> b # foldl' :: (b -> a -> b) -> b -> Product a -> b # foldr1 :: (a -> a -> a) -> Product a -> a # foldl1 :: (a -> a -> a) -> Product a -> a # toList :: Product a -> [a] # null :: Product a -> Bool # length :: Product a -> Int # elem :: Eq a => a -> Product a -> Bool # maximum :: Ord a => Product a -> a # minimum :: Ord a => Product a -> a # sum :: Num a => Product a -> a # product :: Num a => Product a -> a #
Foldable NonEmpty	Since: base-4.9.0.0
Instance details Defined in Data.Foldable Methods fold :: Monoid m => NonEmpty m -> m # foldMap :: Monoid m => (a -> m) -> NonEmpty a -> m # foldr :: (a -> b -> b) -> b -> NonEmpty a -> b # foldr' :: (a -> b -> b) -> b -> NonEmpty a -> b # foldl :: (b -> a -> b) -> b -> NonEmpty a -> b # foldl' :: (b -> a -> b) -> b -> NonEmpty a -> b # foldr1 :: (a -> a -> a) -> NonEmpty a -> a # foldl1 :: (a -> a -> a) -> NonEmpty a -> a # toList :: NonEmpty a -> [a] # null :: NonEmpty a -> Bool # length :: NonEmpty a -> Int # elem :: Eq a => a -> NonEmpty a -> Bool # maximum :: Ord a => NonEmpty a -> a # minimum :: Ord a => NonEmpty a -> a # sum :: Num a => NonEmpty a -> a # product :: Num a => NonEmpty a -> a #
Foldable (Either a)	Since: base-4.7.0.0
Instance details Defined in Data.Foldable Methods fold :: Monoid m => Either a m -> m # foldMap :: Monoid m => (a0 -> m) -> Either a a0 -> m # foldr :: (a0 -> b -> b) -> b -> Either a a0 -> b # foldr' :: (a0 -> b -> b) -> b -> Either a a0 -> b # foldl :: (b -> a0 -> b) -> b -> Either a a0 -> b # foldl' :: (b -> a0 -> b) -> b -> Either a a0 -> b # foldr1 :: (a0 -> a0 -> a0) -> Either a a0 -> a0 # foldl1 :: (a0 -> a0 -> a0) -> Either a a0 -> a0 # toList :: Either a a0 -> [a0] # null :: Either a a0 -> Bool # length :: Either a a0 -> Int # elem :: Eq a0 => a0 -> Either a a0 -> Bool # maximum :: Ord a0 => Either a a0 -> a0 # minimum :: Ord a0 => Either a a0 -> a0 # sum :: Num a0 => Either a a0 -> a0 # product :: Num a0 => Either a a0 -> a0 #
Foldable (V1 :: * -> *)
Instance details Defined in Data.Foldable Methods fold :: Monoid m => V1 m -> m # foldMap :: Monoid m => (a -> m) -> V1 a -> m # foldr :: (a -> b -> b) -> b -> V1 a -> b # foldr' :: (a -> b -> b) -> b -> V1 a -> b # foldl :: (b -> a -> b) -> b -> V1 a -> b # foldl' :: (b -> a -> b) -> b -> V1 a -> b # foldr1 :: (a -> a -> a) -> V1 a -> a # foldl1 :: (a -> a -> a) -> V1 a -> a # toList :: V1 a -> [a] # null :: V1 a -> Bool # length :: V1 a -> Int # elem :: Eq a => a -> V1 a -> Bool # maximum :: Ord a => V1 a -> a # minimum :: Ord a => V1 a -> a # sum :: Num a => V1 a -> a # product :: Num a => V1 a -> a #
Foldable (U1 :: * -> *)	Since: base-4.9.0.0
Instance details Defined in Data.Foldable Methods fold :: Monoid m => U1 m -> m # foldMap :: Monoid m => (a -> m) -> U1 a -> m # foldr :: (a -> b -> b) -> b -> U1 a -> b # foldr' :: (a -> b -> b) -> b -> U1 a -> b # foldl :: (b -> a -> b) -> b -> U1 a -> b # foldl' :: (b -> a -> b) -> b -> U1 a -> b # foldr1 :: (a -> a -> a) -> U1 a -> a # foldl1 :: (a -> a -> a) -> U1 a -> a # toList :: U1 a -> [a] # null :: U1 a -> Bool # length :: U1 a -> Int # elem :: Eq a => a -> U1 a -> Bool # maximum :: Ord a => U1 a -> a # minimum :: Ord a => U1 a -> a # sum :: Num a => U1 a -> a # product :: Num a => U1 a -> a #
Foldable ((,) a)	Since: base-4.7.0.0
Instance details Defined in Data.Foldable Methods fold :: Monoid m => (a, m) -> m # foldMap :: Monoid m => (a0 -> m) -> (a, a0) -> m # foldr :: (a0 -> b -> b) -> b -> (a, a0) -> b # foldr' :: (a0 -> b -> b) -> b -> (a, a0) -> b # foldl :: (b -> a0 -> b) -> b -> (a, a0) -> b # foldl' :: (b -> a0 -> b) -> b -> (a, a0) -> b # foldr1 :: (a0 -> a0 -> a0) -> (a, a0) -> a0 # foldl1 :: (a0 -> a0 -> a0) -> (a, a0) -> a0 # toList :: (a, a0) -> [a0] # null :: (a, a0) -> Bool # length :: (a, a0) -> Int # elem :: Eq a0 => a0 -> (a, a0) -> Bool # maximum :: Ord a0 => (a, a0) -> a0 # minimum :: Ord a0 => (a, a0) -> a0 # sum :: Num a0 => (a, a0) -> a0 # product :: Num a0 => (a, a0) -> a0 #
Foldable (Array i)	Since: base-4.8.0.0
Instance details Defined in Data.Foldable Methods fold :: Monoid m => Array i m -> m # foldMap :: Monoid m => (a -> m) -> Array i a -> m # foldr :: (a -> b -> b) -> b -> Array i a -> b # foldr' :: (a -> b -> b) -> b -> Array i a -> b # foldl :: (b -> a -> b) -> b -> Array i a -> b # foldl' :: (b -> a -> b) -> b -> Array i a -> b # foldr1 :: (a -> a -> a) -> Array i a -> a # foldl1 :: (a -> a -> a) -> Array i a -> a # toList :: Array i a -> [a] # null :: Array i a -> Bool # length :: Array i a -> Int # elem :: Eq a => a -> Array i a -> Bool # maximum :: Ord a => Array i a -> a # minimum :: Ord a => Array i a -> a # sum :: Num a => Array i a -> a # product :: Num a => Array i a -> a #
Foldable (Proxy :: * -> *)	Since: base-4.7.0.0
Instance details Defined in Data.Foldable Methods fold :: Monoid m => Proxy m -> m # foldMap :: Monoid m => (a -> m) -> Proxy a -> m # foldr :: (a -> b -> b) -> b -> Proxy a -> b # foldr' :: (a -> b -> b) -> b -> Proxy a -> b # foldl :: (b -> a -> b) -> b -> Proxy a -> b # foldl' :: (b -> a -> b) -> b -> Proxy a -> b # foldr1 :: (a -> a -> a) -> Proxy a -> a # foldl1 :: (a -> a -> a) -> Proxy a -> a # toList :: Proxy a -> [a] # null :: Proxy a -> Bool # length :: Proxy a -> Int # elem :: Eq a => a -> Proxy a -> Bool # maximum :: Ord a => Proxy a -> a # minimum :: Ord a => Proxy a -> a # sum :: Num a => Proxy a -> a # product :: Num a => Proxy a -> a #
Foldable (Map k)
Instance details Defined in Data.Map.Internal Methods fold :: Monoid m => Map k m -> m # foldMap :: Monoid m => (a -> m) -> Map k a -> m # foldr :: (a -> b -> b) -> b -> Map k a -> b # foldr' :: (a -> b -> b) -> b -> Map k a -> b # foldl :: (b -> a -> b) -> b -> Map k a -> b # foldl' :: (b -> a -> b) -> b -> Map k a -> b # foldr1 :: (a -> a -> a) -> Map k a -> a # foldl1 :: (a -> a -> a) -> Map k a -> a # toList :: Map k a -> [a] # null :: Map k a -> Bool # length :: Map k a -> Int # elem :: Eq a => a -> Map k a -> Bool # maximum :: Ord a => Map k a -> a # minimum :: Ord a => Map k a -> a # sum :: Num a => Map k a -> a # product :: Num a => Map k a -> a #
Foldable f => Foldable (Rec1 f)
Instance details Defined in Data.Foldable Methods fold :: Monoid m => Rec1 f m -> m # foldMap :: Monoid m => (a -> m) -> Rec1 f a -> m # foldr :: (a -> b -> b) -> b -> Rec1 f a -> b # foldr' :: (a -> b -> b) -> b -> Rec1 f a -> b # foldl :: (b -> a -> b) -> b -> Rec1 f a -> b # foldl' :: (b -> a -> b) -> b -> Rec1 f a -> b # foldr1 :: (a -> a -> a) -> Rec1 f a -> a # foldl1 :: (a -> a -> a) -> Rec1 f a -> a # toList :: Rec1 f a -> [a] # null :: Rec1 f a -> Bool # length :: Rec1 f a -> Int # elem :: Eq a => a -> Rec1 f a -> Bool # maximum :: Ord a => Rec1 f a -> a # minimum :: Ord a => Rec1 f a -> a # sum :: Num a => Rec1 f a -> a # product :: Num a => Rec1 f a -> a #
Foldable (URec Char :: * -> *)
Instance details Defined in Data.Foldable Methods fold :: Monoid m => URec Char m -> m # foldMap :: Monoid m => (a -> m) -> URec Char a -> m # foldr :: (a -> b -> b) -> b -> URec Char a -> b # foldr' :: (a -> b -> b) -> b -> URec Char a -> b # foldl :: (b -> a -> b) -> b -> URec Char a -> b # foldl' :: (b -> a -> b) -> b -> URec Char a -> b # foldr1 :: (a -> a -> a) -> URec Char a -> a # foldl1 :: (a -> a -> a) -> URec Char a -> a # toList :: URec Char a -> [a] # null :: URec Char a -> Bool # length :: URec Char a -> Int # elem :: Eq a => a -> URec Char a -> Bool # maximum :: Ord a => URec Char a -> a # minimum :: Ord a => URec Char a -> a # sum :: Num a => URec Char a -> a # product :: Num a => URec Char a -> a #
Foldable (URec Double :: * -> *)
Instance details Defined in Data.Foldable Methods fold :: Monoid m => URec Double m -> m # foldMap :: Monoid m => (a -> m) -> URec Double a -> m # foldr :: (a -> b -> b) -> b -> URec Double a -> b # foldr' :: (a -> b -> b) -> b -> URec Double a -> b # foldl :: (b -> a -> b) -> b -> URec Double a -> b # foldl' :: (b -> a -> b) -> b -> URec Double a -> b # foldr1 :: (a -> a -> a) -> URec Double a -> a # foldl1 :: (a -> a -> a) -> URec Double a -> a # toList :: URec Double a -> [a] # null :: URec Double a -> Bool # length :: URec Double a -> Int # elem :: Eq a => a -> URec Double a -> Bool # maximum :: Ord a => URec Double a -> a # minimum :: Ord a => URec Double a -> a # sum :: Num a => URec Double a -> a # product :: Num a => URec Double a -> a #
Foldable (URec Float :: * -> *)
Instance details Defined in Data.Foldable Methods fold :: Monoid m => URec Float m -> m # foldMap :: Monoid m => (a -> m) -> URec Float a -> m # foldr :: (a -> b -> b) -> b -> URec Float a -> b # foldr' :: (a -> b -> b) -> b -> URec Float a -> b # foldl :: (b -> a -> b) -> b -> URec Float a -> b # foldl' :: (b -> a -> b) -> b -> URec Float a -> b # foldr1 :: (a -> a -> a) -> URec Float a -> a # foldl1 :: (a -> a -> a) -> URec Float a -> a # toList :: URec Float a -> [a] # null :: URec Float a -> Bool # length :: URec Float a -> Int # elem :: Eq a => a -> URec Float a -> Bool # maximum :: Ord a => URec Float a -> a # minimum :: Ord a => URec Float a -> a # sum :: Num a => URec Float a -> a # product :: Num a => URec Float a -> a #
Foldable (URec Int :: * -> *)
Instance details Defined in Data.Foldable Methods fold :: Monoid m => URec Int m -> m # foldMap :: Monoid m => (a -> m) -> URec Int a -> m # foldr :: (a -> b -> b) -> b -> URec Int a -> b # foldr' :: (a -> b -> b) -> b -> URec Int a -> b # foldl :: (b -> a -> b) -> b -> URec Int a -> b # foldl' :: (b -> a -> b) -> b -> URec Int a -> b # foldr1 :: (a -> a -> a) -> URec Int a -> a # foldl1 :: (a -> a -> a) -> URec Int a -> a # toList :: URec Int a -> [a] # null :: URec Int a -> Bool # length :: URec Int a -> Int # elem :: Eq a => a -> URec Int a -> Bool # maximum :: Ord a => URec Int a -> a # minimum :: Ord a => URec Int a -> a # sum :: Num a => URec Int a -> a # product :: Num a => URec Int a -> a #
Foldable (URec Word :: * -> *)
Instance details Defined in Data.Foldable Methods fold :: Monoid m => URec Word m -> m # foldMap :: Monoid m => (a -> m) -> URec Word a -> m # foldr :: (a -> b -> b) -> b -> URec Word a -> b # foldr' :: (a -> b -> b) -> b -> URec Word a -> b # foldl :: (b -> a -> b) -> b -> URec Word a -> b # foldl' :: (b -> a -> b) -> b -> URec Word a -> b # foldr1 :: (a -> a -> a) -> URec Word a -> a # foldl1 :: (a -> a -> a) -> URec Word a -> a # toList :: URec Word a -> [a] # null :: URec Word a -> Bool # length :: URec Word a -> Int # elem :: Eq a => a -> URec Word a -> Bool # maximum :: Ord a => URec Word a -> a # minimum :: Ord a => URec Word a -> a # sum :: Num a => URec Word a -> a # product :: Num a => URec Word a -> a #
Foldable (URec (Ptr ()) :: * -> *)
Instance details Defined in Data.Foldable Methods fold :: Monoid m => URec (Ptr ()) m -> m # foldMap :: Monoid m => (a -> m) -> URec (Ptr ()) a -> m # foldr :: (a -> b -> b) -> b -> URec (Ptr ()) a -> b # foldr' :: (a -> b -> b) -> b -> URec (Ptr ()) a -> b # foldl :: (b -> a -> b) -> b -> URec (Ptr ()) a -> b # foldl' :: (b -> a -> b) -> b -> URec (Ptr ()) a -> b # foldr1 :: (a -> a -> a) -> URec (Ptr ()) a -> a # foldl1 :: (a -> a -> a) -> URec (Ptr ()) a -> a # toList :: URec (Ptr ()) a -> [a] # null :: URec (Ptr ()) a -> Bool # length :: URec (Ptr ()) a -> Int # elem :: Eq a => a -> URec (Ptr ()) a -> Bool # maximum :: Ord a => URec (Ptr ()) a -> a # minimum :: Ord a => URec (Ptr ()) a -> a # sum :: Num a => URec (Ptr ()) a -> a # product :: Num a => URec (Ptr ()) a -> a #
Foldable (Const m :: * -> *)	Since: base-4.7.0.0
Instance details Defined in Data.Functor.Const Methods fold :: Monoid m0 => Const m m0 -> m0 # foldMap :: Monoid m0 => (a -> m0) -> Const m a -> m0 # foldr :: (a -> b -> b) -> b -> Const m a -> b # foldr' :: (a -> b -> b) -> b -> Const m a -> b # foldl :: (b -> a -> b) -> b -> Const m a -> b # foldl' :: (b -> a -> b) -> b -> Const m a -> b # foldr1 :: (a -> a -> a) -> Const m a -> a # foldl1 :: (a -> a -> a) -> Const m a -> a # toList :: Const m a -> [a] # null :: Const m a -> Bool # length :: Const m a -> Int # elem :: Eq a => a -> Const m a -> Bool # maximum :: Ord a => Const m a -> a # minimum :: Ord a => Const m a -> a # sum :: Num a => Const m a -> a # product :: Num a => Const m a -> a #
Finite e => Foldable ((->) e :: * -> *) #
Instance details Defined in Data.Universe.Instances.Traversable Methods fold :: Monoid m => (e -> m) -> m # foldMap :: Monoid m => (a -> m) -> (e -> a) -> m # foldr :: (a -> b -> b) -> b -> (e -> a) -> b # foldr' :: (a -> b -> b) -> b -> (e -> a) -> b # foldl :: (b -> a -> b) -> b -> (e -> a) -> b # foldl' :: (b -> a -> b) -> b -> (e -> a) -> b # foldr1 :: (a -> a -> a) -> (e -> a) -> a # foldl1 :: (a -> a -> a) -> (e -> a) -> a # toList :: (e -> a) -> [a] # null :: (e -> a) -> Bool # length :: (e -> a) -> Int # elem :: Eq a => a -> (e -> a) -> Bool # maximum :: Ord a => (e -> a) -> a # minimum :: Ord a => (e -> a) -> a # sum :: Num a => (e -> a) -> a # product :: Num a => (e -> a) -> a #
Foldable (K1 i c :: * -> *)
Instance details Defined in Data.Foldable Methods fold :: Monoid m => K1 i c m -> m # foldMap :: Monoid m => (a -> m) -> K1 i c a -> m # foldr :: (a -> b -> b) -> b -> K1 i c a -> b # foldr' :: (a -> b -> b) -> b -> K1 i c a -> b # foldl :: (b -> a -> b) -> b -> K1 i c a -> b # foldl' :: (b -> a -> b) -> b -> K1 i c a -> b # foldr1 :: (a -> a -> a) -> K1 i c a -> a # foldl1 :: (a -> a -> a) -> K1 i c a -> a # toList :: K1 i c a -> [a] # null :: K1 i c a -> Bool # length :: K1 i c a -> Int # elem :: Eq a => a -> K1 i c a -> Bool # maximum :: Ord a => K1 i c a -> a # minimum :: Ord a => K1 i c a -> a # sum :: Num a => K1 i c a -> a # product :: Num a => K1 i c a -> a #
(Foldable f, Foldable g) => Foldable (f :+: g)
Instance details Defined in Data.Foldable Methods fold :: Monoid m => (f :+: g) m -> m # foldMap :: Monoid m => (a -> m) -> (f :+: g) a -> m # foldr :: (a -> b -> b) -> b -> (f :+: g) a -> b # foldr' :: (a -> b -> b) -> b -> (f :+: g) a -> b # foldl :: (b -> a -> b) -> b -> (f :+: g) a -> b # foldl' :: (b -> a -> b) -> b -> (f :+: g) a -> b # foldr1 :: (a -> a -> a) -> (f :+: g) a -> a # foldl1 :: (a -> a -> a) -> (f :+: g) a -> a # toList :: (f :+: g) a -> [a] # null :: (f :+: g) a -> Bool # length :: (f :+: g) a -> Int # elem :: Eq a => a -> (f :+: g) a -> Bool # maximum :: Ord a => (f :+: g) a -> a # minimum :: Ord a => (f :+: g) a -> a # sum :: Num a => (f :+: g) a -> a # product :: Num a => (f :+: g) a -> a #
(Foldable f, Foldable g) => Foldable (f :*: g)
Instance details Defined in Data.Foldable Methods fold :: Monoid m => (f :: g) m -> m # foldMap :: Monoid m => (a -> m) -> (f :: g) a -> m # foldr :: (a -> b -> b) -> b -> (f :: g) a -> b # foldr' :: (a -> b -> b) -> b -> (f :: g) a -> b # foldl :: (b -> a -> b) -> b -> (f :: g) a -> b # foldl' :: (b -> a -> b) -> b -> (f :: g) a -> b # foldr1 :: (a -> a -> a) -> (f :: g) a -> a # foldl1 :: (a -> a -> a) -> (f :: g) a -> a # toList :: (f :: g) a -> [a] # null :: (f :: g) a -> Bool # length :: (f :: g) a -> Int # elem :: Eq a => a -> (f :: g) a -> Bool # maximum :: Ord a => (f :: g) a -> a # minimum :: Ord a => (f :: g) a -> a # sum :: Num a => (f :: g) a -> a # product :: Num a => (f :: g) a -> a #
Foldable f => Foldable (M1 i c f)
Instance details Defined in Data.Foldable Methods fold :: Monoid m => M1 i c f m -> m # foldMap :: Monoid m => (a -> m) -> M1 i c f a -> m # foldr :: (a -> b -> b) -> b -> M1 i c f a -> b # foldr' :: (a -> b -> b) -> b -> M1 i c f a -> b # foldl :: (b -> a -> b) -> b -> M1 i c f a -> b # foldl' :: (b -> a -> b) -> b -> M1 i c f a -> b # foldr1 :: (a -> a -> a) -> M1 i c f a -> a # foldl1 :: (a -> a -> a) -> M1 i c f a -> a # toList :: M1 i c f a -> [a] # null :: M1 i c f a -> Bool # length :: M1 i c f a -> Int # elem :: Eq a => a -> M1 i c f a -> Bool # maximum :: Ord a => M1 i c f a -> a # minimum :: Ord a => M1 i c f a -> a # sum :: Num a => M1 i c f a -> a # product :: Num a => M1 i c f a -> a #
(Foldable f, Foldable g) => Foldable (f :.: g)
Instance details Defined in Data.Foldable Methods fold :: Monoid m => (f :.: g) m -> m # foldMap :: Monoid m => (a -> m) -> (f :.: g) a -> m # foldr :: (a -> b -> b) -> b -> (f :.: g) a -> b # foldr' :: (a -> b -> b) -> b -> (f :.: g) a -> b # foldl :: (b -> a -> b) -> b -> (f :.: g) a -> b # foldl' :: (b -> a -> b) -> b -> (f :.: g) a -> b # foldr1 :: (a -> a -> a) -> (f :.: g) a -> a # foldl1 :: (a -> a -> a) -> (f :.: g) a -> a # toList :: (f :.: g) a -> [a] # null :: (f :.: g) a -> Bool # length :: (f :.: g) a -> Int # elem :: Eq a => a -> (f :.: g) a -> Bool # maximum :: Ord a => (f :.: g) a -> a # minimum :: Ord a => (f :.: g) a -> a # sum :: Num a => (f :.: g) a -> a # product :: Num a => (f :.: g) a -> a #

class (Functor t, Foldable t) => Traversable (t :: * -> *) where #

Functors representing data structures that can be traversed from left to right.

A definition of traverse must satisfy the following laws:

naturality: t . traverse f = traverse (t . f) for every applicative transformation t
identity: traverse Identity = Identity
composition: traverse (Compose . fmap g . f) = Compose . fmap (traverse g) . traverse f

A definition of sequenceA must satisfy the following laws:

naturality: t . sequenceA = sequenceA . fmap t for every applicative transformation t
identity: sequenceA . fmap Identity = Identity
composition: sequenceA . fmap Compose = Compose . fmap sequenceA . sequenceA

where an applicative transformation is a function

t :: (Applicative f, Applicative g) => f a -> g a

preserving the Applicative operations, i.e.

```
t (pure x) = pure x
```
```
t (x <*> y) = t x <*> t y
```

and the identity functor Identity and composition of functors Compose are defined as

  newtype Identity a = Identity a

  instance Functor Identity where
    fmap f (Identity x) = Identity (f x)

  instance Applicative Identity where
    pure x = Identity x
    Identity f <*> Identity x = Identity (f x)

  newtype Compose f g a = Compose (f (g a))

  instance (Functor f, Functor g) => Functor (Compose f g) where
    fmap f (Compose x) = Compose (fmap (fmap f) x)

  instance (Applicative f, Applicative g) => Applicative (Compose f g) where
    pure x = Compose (pure (pure x))
    Compose f <*> Compose x = Compose ((<*>) <$> f <*> x)

(The naturality law is implied by parametricity.)

Instances are similar to Functor, e.g. given a data type

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

a suitable instance would be

instance Traversable Tree where
   traverse f Empty = pure Empty
   traverse f (Leaf x) = Leaf <$> f x
   traverse f (Node l k r) = Node <$> traverse f l <*> f k <*> traverse f r

This is suitable even for abstract types, as the laws for <*> imply a form of associativity.

The superclass instances should satisfy the following:

In the Functor instance, fmap should be equivalent to traversal with the identity applicative functor (fmapDefault).
In the Foldable instance, foldMap should be equivalent to traversal with a constant applicative functor (foldMapDefault).

Minimal complete definition

traverse | sequenceA

Methods

traverse :: Applicative f => (a -> f b) -> t a -> f (t b) #

Map each element of a structure to an action, evaluate these actions from left to right, and collect the results. For a version that ignores the results see traverse_.

sequenceA :: Applicative f => t (f a) -> f (t a) #

Evaluate each action in the structure from left to right, and and collect the results. For a version that ignores the results see sequenceA_.

mapM :: Monad m => (a -> m b) -> t a -> m (t b) #

Map each element of a structure to a monadic action, evaluate these actions from left to right, and collect the results. For a version that ignores the results see mapM_.

sequence :: Monad m => t (m a) -> m (t a) #

Evaluate each monadic action in the structure from left to right, and collect the results. For a version that ignores the results see sequence_.

Instances

Traversable []	Since: base-2.1
Instance details Defined in Data.Traversable Methods traverse :: Applicative f => (a -> f b) -> [a] -> f [b] # sequenceA :: Applicative f => [f a] -> f [a] # mapM :: Monad m => (a -> m b) -> [a] -> m [b] # sequence :: Monad m => [m a] -> m [a] #
Traversable Maybe	Since: base-2.1
Instance details Defined in Data.Traversable Methods traverse :: Applicative f => (a -> f b) -> Maybe a -> f (Maybe b) # sequenceA :: Applicative f => Maybe (f a) -> f (Maybe a) # mapM :: Monad m => (a -> m b) -> Maybe a -> m (Maybe b) # sequence :: Monad m => Maybe (m a) -> m (Maybe a) #
Traversable Par1
Instance details Defined in Data.Traversable Methods traverse :: Applicative f => (a -> f b) -> Par1 a -> f (Par1 b) # sequenceA :: Applicative f => Par1 (f a) -> f (Par1 a) # mapM :: Monad m => (a -> m b) -> Par1 a -> m (Par1 b) # sequence :: Monad m => Par1 (m a) -> m (Par1 a) #
Traversable ZipList	Since: base-4.9.0.0
Instance details Defined in Data.Traversable Methods traverse :: Applicative f => (a -> f b) -> ZipList a -> f (ZipList b) # sequenceA :: Applicative f => ZipList (f a) -> f (ZipList a) # mapM :: Monad m => (a -> m b) -> ZipList a -> m (ZipList b) # sequence :: Monad m => ZipList (m a) -> m (ZipList a) #
Traversable Identity
Instance details Defined in Data.Traversable Methods traverse :: Applicative f => (a -> f b) -> Identity a -> f (Identity b) # sequenceA :: Applicative f => Identity (f a) -> f (Identity a) # mapM :: Monad m => (a -> m b) -> Identity a -> m (Identity b) # sequence :: Monad m => Identity (m a) -> m (Identity a) #
Traversable First	Since: base-4.8.0.0
Instance details Defined in Data.Traversable Methods traverse :: Applicative f => (a -> f b) -> First a -> f (First b) # sequenceA :: Applicative f => First (f a) -> f (First a) # mapM :: Monad m => (a -> m b) -> First a -> m (First b) # sequence :: Monad m => First (m a) -> m (First a) #
Traversable Last	Since: base-4.8.0.0
Instance details Defined in Data.Traversable Methods traverse :: Applicative f => (a -> f b) -> Last a -> f (Last b) # sequenceA :: Applicative f => Last (f a) -> f (Last a) # mapM :: Monad m => (a -> m b) -> Last a -> m (Last b) # sequence :: Monad m => Last (m a) -> m (Last a) #
Traversable Dual	Since: base-4.8.0.0
Instance details Defined in Data.Traversable Methods traverse :: Applicative f => (a -> f b) -> Dual a -> f (Dual b) # sequenceA :: Applicative f => Dual (f a) -> f (Dual a) # mapM :: Monad m => (a -> m b) -> Dual a -> m (Dual b) # sequence :: Monad m => Dual (m a) -> m (Dual a) #
Traversable Sum	Since: base-4.8.0.0
Instance details Defined in Data.Traversable Methods traverse :: Applicative f => (a -> f b) -> Sum a -> f (Sum b) # sequenceA :: Applicative f => Sum (f a) -> f (Sum a) # mapM :: Monad m => (a -> m b) -> Sum a -> m (Sum b) # sequence :: Monad m => Sum (m a) -> m (Sum a) #
Traversable Product	Since: base-4.8.0.0
Instance details Defined in Data.Traversable Methods traverse :: Applicative f => (a -> f b) -> Product a -> f (Product b) # sequenceA :: Applicative f => Product (f a) -> f (Product a) # mapM :: Monad m => (a -> m b) -> Product a -> m (Product b) # sequence :: Monad m => Product (m a) -> m (Product a) #
Traversable NonEmpty	Since: base-4.9.0.0
Instance details Defined in Data.Traversable Methods traverse :: Applicative f => (a -> f b) -> NonEmpty a -> f (NonEmpty b) # sequenceA :: Applicative f => NonEmpty (f a) -> f (NonEmpty a) # mapM :: Monad m => (a -> m b) -> NonEmpty a -> m (NonEmpty b) # sequence :: Monad m => NonEmpty (m a) -> m (NonEmpty a) #
Traversable (Either a)	Since: base-4.7.0.0
Instance details Defined in Data.Traversable Methods traverse :: Applicative f => (a0 -> f b) -> Either a a0 -> f (Either a b) # sequenceA :: Applicative f => Either a (f a0) -> f (Either a a0) # mapM :: Monad m => (a0 -> m b) -> Either a a0 -> m (Either a b) # sequence :: Monad m => Either a (m a0) -> m (Either a a0) #
Traversable (V1 :: * -> *)
Instance details Defined in Data.Traversable Methods traverse :: Applicative f => (a -> f b) -> V1 a -> f (V1 b) # sequenceA :: Applicative f => V1 (f a) -> f (V1 a) # mapM :: Monad m => (a -> m b) -> V1 a -> m (V1 b) # sequence :: Monad m => V1 (m a) -> m (V1 a) #
Traversable (U1 :: * -> *)	Since: base-4.9.0.0
Instance details Defined in Data.Traversable Methods traverse :: Applicative f => (a -> f b) -> U1 a -> f (U1 b) # sequenceA :: Applicative f => U1 (f a) -> f (U1 a) # mapM :: Monad m => (a -> m b) -> U1 a -> m (U1 b) # sequence :: Monad m => U1 (m a) -> m (U1 a) #
Traversable ((,) a)	Since: base-4.7.0.0
Instance details Defined in Data.Traversable Methods traverse :: Applicative f => (a0 -> f b) -> (a, a0) -> f (a, b) # sequenceA :: Applicative f => (a, f a0) -> f (a, a0) # mapM :: Monad m => (a0 -> m b) -> (a, a0) -> m (a, b) # sequence :: Monad m => (a, m a0) -> m (a, a0) #
Ix i => Traversable (Array i)	Since: base-2.1
Instance details Defined in Data.Traversable Methods traverse :: Applicative f => (a -> f b) -> Array i a -> f (Array i b) # sequenceA :: Applicative f => Array i (f a) -> f (Array i a) # mapM :: Monad m => (a -> m b) -> Array i a -> m (Array i b) # sequence :: Monad m => Array i (m a) -> m (Array i a) #
Traversable (Proxy :: * -> *)	Since: base-4.7.0.0
Instance details Defined in Data.Traversable Methods traverse :: Applicative f => (a -> f b) -> Proxy a -> f (Proxy b) # sequenceA :: Applicative f => Proxy (f a) -> f (Proxy a) # mapM :: Monad m => (a -> m b) -> Proxy a -> m (Proxy b) # sequence :: Monad m => Proxy (m a) -> m (Proxy a) #
Traversable (Map k)
Instance details Defined in Data.Map.Internal Methods traverse :: Applicative f => (a -> f b) -> Map k a -> f (Map k b) # sequenceA :: Applicative f => Map k (f a) -> f (Map k a) # mapM :: Monad m => (a -> m b) -> Map k a -> m (Map k b) # sequence :: Monad m => Map k (m a) -> m (Map k a) #
Traversable f => Traversable (Rec1 f)
Instance details Defined in Data.Traversable Methods traverse :: Applicative f0 => (a -> f0 b) -> Rec1 f a -> f0 (Rec1 f b) # sequenceA :: Applicative f0 => Rec1 f (f0 a) -> f0 (Rec1 f a) # mapM :: Monad m => (a -> m b) -> Rec1 f a -> m (Rec1 f b) # sequence :: Monad m => Rec1 f (m a) -> m (Rec1 f a) #
Traversable (URec Char :: * -> *)
Instance details Defined in Data.Traversable Methods traverse :: Applicative f => (a -> f b) -> URec Char a -> f (URec Char b) # sequenceA :: Applicative f => URec Char (f a) -> f (URec Char a) # mapM :: Monad m => (a -> m b) -> URec Char a -> m (URec Char b) # sequence :: Monad m => URec Char (m a) -> m (URec Char a) #
Traversable (URec Double :: * -> *)
Instance details Defined in Data.Traversable Methods traverse :: Applicative f => (a -> f b) -> URec Double a -> f (URec Double b) # sequenceA :: Applicative f => URec Double (f a) -> f (URec Double a) # mapM :: Monad m => (a -> m b) -> URec Double a -> m (URec Double b) # sequence :: Monad m => URec Double (m a) -> m (URec Double a) #
Traversable (URec Float :: * -> *)
Instance details Defined in Data.Traversable Methods traverse :: Applicative f => (a -> f b) -> URec Float a -> f (URec Float b) # sequenceA :: Applicative f => URec Float (f a) -> f (URec Float a) # mapM :: Monad m => (a -> m b) -> URec Float a -> m (URec Float b) # sequence :: Monad m => URec Float (m a) -> m (URec Float a) #
Traversable (URec Int :: * -> *)
Instance details Defined in Data.Traversable Methods traverse :: Applicative f => (a -> f b) -> URec Int a -> f (URec Int b) # sequenceA :: Applicative f => URec Int (f a) -> f (URec Int a) # mapM :: Monad m => (a -> m b) -> URec Int a -> m (URec Int b) # sequence :: Monad m => URec Int (m a) -> m (URec Int a) #
Traversable (URec Word :: * -> *)
Instance details Defined in Data.Traversable Methods traverse :: Applicative f => (a -> f b) -> URec Word a -> f (URec Word b) # sequenceA :: Applicative f => URec Word (f a) -> f (URec Word a) # mapM :: Monad m => (a -> m b) -> URec Word a -> m (URec Word b) # sequence :: Monad m => URec Word (m a) -> m (URec Word a) #
Traversable (URec (Ptr ()) :: * -> *)
Instance details Defined in Data.Traversable Methods traverse :: Applicative f => (a -> f b) -> URec (Ptr ()) a -> f (URec (Ptr ()) b) # sequenceA :: Applicative f => URec (Ptr ()) (f a) -> f (URec (Ptr ()) a) # mapM :: Monad m => (a -> m b) -> URec (Ptr ()) a -> m (URec (Ptr ()) b) # sequence :: Monad m => URec (Ptr ()) (m a) -> m (URec (Ptr ()) a) #
Traversable (Const m :: * -> *)	Since: base-4.7.0.0
Instance details Defined in Data.Traversable Methods traverse :: Applicative f => (a -> f b) -> Const m a -> f (Const m b) # sequenceA :: Applicative f => Const m (f a) -> f (Const m a) # mapM :: Monad m0 => (a -> m0 b) -> Const m a -> m0 (Const m b) # sequence :: Monad m0 => Const m (m0 a) -> m0 (Const m a) #
(Ord e, Finite e) => Traversable ((->) e :: * -> *) #
Instance details Defined in Data.Universe.Instances.Traversable Methods traverse :: Applicative f => (a -> f b) -> (e -> a) -> f (e -> b) # sequenceA :: Applicative f => (e -> f a) -> f (e -> a) # mapM :: Monad m => (a -> m b) -> (e -> a) -> m (e -> b) # sequence :: Monad m => (e -> m a) -> m (e -> a) #
Traversable (K1 i c :: * -> *)
Instance details Defined in Data.Traversable Methods traverse :: Applicative f => (a -> f b) -> K1 i c a -> f (K1 i c b) # sequenceA :: Applicative f => K1 i c (f a) -> f (K1 i c a) # mapM :: Monad m => (a -> m b) -> K1 i c a -> m (K1 i c b) # sequence :: Monad m => K1 i c (m a) -> m (K1 i c a) #
(Traversable f, Traversable g) => Traversable (f :+: g)
Instance details Defined in Data.Traversable Methods traverse :: Applicative f0 => (a -> f0 b) -> (f :+: g) a -> f0 ((f :+: g) b) # sequenceA :: Applicative f0 => (f :+: g) (f0 a) -> f0 ((f :+: g) a) # mapM :: Monad m => (a -> m b) -> (f :+: g) a -> m ((f :+: g) b) # sequence :: Monad m => (f :+: g) (m a) -> m ((f :+: g) a) #
(Traversable f, Traversable g) => Traversable (f :*: g)
Instance details Defined in Data.Traversable Methods traverse :: Applicative f0 => (a -> f0 b) -> (f :: g) a -> f0 ((f :: g) b) # sequenceA :: Applicative f0 => (f :: g) (f0 a) -> f0 ((f :: g) a) # mapM :: Monad m => (a -> m b) -> (f :: g) a -> m ((f :: g) b) # sequence :: Monad m => (f :: g) (m a) -> m ((f :: g) a) #
Traversable f => Traversable (M1 i c f)
Instance details Defined in Data.Traversable Methods traverse :: Applicative f0 => (a -> f0 b) -> M1 i c f a -> f0 (M1 i c f b) # sequenceA :: Applicative f0 => M1 i c f (f0 a) -> f0 (M1 i c f a) # mapM :: Monad m => (a -> m b) -> M1 i c f a -> m (M1 i c f b) # sequence :: Monad m => M1 i c f (m a) -> m (M1 i c f a) #
(Traversable f, Traversable g) => Traversable (f :.: g)
Instance details Defined in Data.Traversable Methods traverse :: Applicative f0 => (a -> f0 b) -> (f :.: g) a -> f0 ((f :.: g) b) # sequenceA :: Applicative f0 => (f :.: g) (f0 a) -> f0 ((f :.: g) a) # mapM :: Monad m => (a -> m b) -> (f :.: g) a -> m ((f :.: g) b) # sequence :: Monad m => (f :.: g) (m a) -> m ((f :.: g) a) #

Orphan instances

Finite e => Foldable ((->) e :: * -> *) #
Instance details Methods fold :: Monoid m => (e -> m) -> m # foldMap :: Monoid m => (a -> m) -> (e -> a) -> m # foldr :: (a -> b -> b) -> b -> (e -> a) -> b # foldr' :: (a -> b -> b) -> b -> (e -> a) -> b # foldl :: (b -> a -> b) -> b -> (e -> a) -> b # foldl' :: (b -> a -> b) -> b -> (e -> a) -> b # foldr1 :: (a -> a -> a) -> (e -> a) -> a # foldl1 :: (a -> a -> a) -> (e -> a) -> a # toList :: (e -> a) -> [a] # null :: (e -> a) -> Bool # length :: (e -> a) -> Int # elem :: Eq a => a -> (e -> a) -> Bool # maximum :: Ord a => (e -> a) -> a # minimum :: Ord a => (e -> a) -> a # sum :: Num a => (e -> a) -> a # product :: Num a => (e -> a) -> a #
(Ord e, Finite e) => Traversable ((->) e :: * -> *) #
Instance details Methods traverse :: Applicative f => (a -> f b) -> (e -> a) -> f (e -> b) # sequenceA :: Applicative f => (e -> f a) -> f (e -> a) # mapM :: Monad m => (a -> m b) -> (e -> a) -> m (e -> b) # sequence :: Monad m => (e -> m a) -> m (e -> a) #