Control.Lens.Traversal

A Traversal s t a b is a generalization of traverse from Traversable. It allows you to traverse over a structure and change out its contents with monadic or Applicative side-effects. Starting from

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

we monomorphize the contents and result to obtain

type Traversal s t a b = forall f. Applicative f => (a -> f b) -> s -> f t

A Traversal can be used as a Fold. Any Traversal can be used for Getting like a Fold, because given a Monoid m, we have an Applicative for (Const m). Everything you know how to do with a Traversable container, you can with a Traversal, and here we provide combinators that generalize the usual Traversable operations.

Traversals

type Traversal s t a b = forall f. Applicative f => (a -> f b) -> s -> f t

A Traversal can be used directly as a Setter or a Fold (but not as a Lens) and provides the ability to both read and update multiple fields, subject to some relatively weak Traversal laws.

These have also been known as multilenses, but they have the signature and spirit of

traverse :: Traversable f => Traversal (f a) (f b) a b

and the more evocative name suggests their application.

Most of the time the Traversal you will want to use is just traverse, but you can also pass any Lens or Iso as a Traversal, and composition of a Traversal (or Lens or Iso) with a Traversal (or Lens or Iso) using (.) forms a valid Traversal.

The laws for a Traversal t follow from the laws for Traversable as stated in "The Essence of the Iterator Pattern".

t pure ≡ pure
fmap (t f) . t g ≡ getCompose . t (Compose . fmap f . g)

One consequence of this requirement is that a Traversal needs to leave the same number of elements as a candidate for subsequent Traversal that it started with. Another testament to the strength of these laws is that the caveat expressed in section 5.5 of the "Essence of the Iterator Pattern" about exotic Traversable instances that traverse the same entry multiple times was actually already ruled out by the second law in that same paper!

type Traversal' s a = Traversal s s a a

type Traversal' = Simple Traversal

type Traversal1 s t a b = forall f. Apply f => (a -> f b) -> s -> f t

type Traversal1' s a = Traversal1 s s a a

type IndexedTraversal i s t a b = forall p f. (Indexable i p, Applicative f) => p a (f b) -> s -> f t

Every IndexedTraversal is a valid Traversal or IndexedFold.

The Indexed constraint is used to allow an IndexedTraversal to be used directly as a Traversal.

The Traversal laws are still required to hold.

In addition, the index i should satisfy the requirement that it stays unchanged even when modifying the value a, otherwise traversals like indices break the Traversal laws.

type IndexedTraversal' i s a = IndexedTraversal i s s a a

type IndexedTraversal' i = Simple (IndexedTraversal i)

type IndexedTraversal1 i s t a b = forall p f. (Indexable i p, Apply f) => p a (f b) -> s -> f t

type IndexedTraversal1' i s a = IndexedTraversal1 i s s a a

type ATraversal s t a b = LensLike (Bazaar (->) a b) s t a b

When you see this as an argument to a function, it expects a Traversal.

type ATraversal' s a = ATraversal s s a a

type ATraversal' = Simple ATraversal

type ATraversal1 s t a b = LensLike (Bazaar1 (->) a b) s t a b

When you see this as an argument to a function, it expects a Traversal1.

type ATraversal1' s a = ATraversal1 s s a a

type ATraversal1' = Simple ATraversal1

type AnIndexedTraversal i s t a b = Over (Indexed i) (Bazaar (Indexed i) a b) s t a b

When you see this as an argument to a function, it expects an IndexedTraversal.

type AnIndexedTraversal' i s a = AnIndexedTraversal i s s a a

type AnIndexedTraversal' = Simple (AnIndexedTraversal i)

type AnIndexedTraversal1 i s t a b = Over (Indexed i) (Bazaar1 (Indexed i) a b) s t a b

When you see this as an argument to a function, it expects an IndexedTraversal1.

type AnIndexedTraversal1' i s a = AnIndexedTraversal1 i s s a a

type AnIndexedTraversal1' = Simple (AnIndexedTraversal1 i)

type Traversing p f s t a b = Over p (BazaarT p f a b) s t a b

When you see this as an argument to a function, it expects

• to be indexed if p is an instance of Indexed i,
• to be unindexed if p is (->),
• a Traversal if f is Applicative,
• a Getter if f is only a Functor and Contravariant,
• a Lens if f is only a Functor,
• a Fold if f is Applicative and Contravariant.

type Traversing' p f s a = Traversing p f s s a a

type Traversing' f = Simple (Traversing f)

type Traversing1 p f s t a b = Over p (BazaarT1 p f a b) s t a b

type Traversing1' p f s a = Traversing1 p f s s a a

Traversing and Lensing

traverseOf :: LensLike f s t a b -> (a -> f b) -> s -> f t

Map each element of a structure targeted by a Lens or Traversal, evaluate these actions from left to right, and collect the results.

This function is only provided for consistency, id is strictly more general.

>>> traverseOf each print (1,2,3)
1
2
3
((),(),())

traverseOf ≡ id
itraverseOf l ≡ traverseOf l . Indexed
itraverseOf itraversed ≡ itraverse

This yields the obvious law:

traverse ≡ traverseOf traverse

traverseOf :: Functor f     => Iso s t a b        -> (a -> f b) -> s -> f t
traverseOf :: Functor f     => Lens s t a b       -> (a -> f b) -> s -> f t
traverseOf :: Apply f       => Traversal1 s t a b -> (a -> f b) -> s -> f t
traverseOf :: Applicative f => Traversal s t a b  -> (a -> f b) -> s -> f t

forOf :: LensLike f s t a b -> s -> (a -> f b) -> f t

A version of traverseOf with the arguments flipped, such that:

>>> forOf each (1,2,3) print
1
2
3
((),(),())

This function is only provided for consistency, flip is strictly more general.

forOf ≡ flip
forOf ≡ flip . traverseOf

for ≡ forOf traverse
ifor l s ≡ for l s . Indexed

forOf :: Functor f => Iso s t a b -> s -> (a -> f b) -> f t
forOf :: Functor f => Lens s t a b -> s -> (a -> f b) -> f t
forOf :: Applicative f => Traversal s t a b -> s -> (a -> f b) -> f t

sequenceAOf :: LensLike f s t (f b) b -> s -> f t

Evaluate each action in the structure from left to right, and collect the results.

>>> sequenceAOf both ([1,2],[3,4])
[(1,3),(1,4),(2,3),(2,4)]

sequenceA ≡ sequenceAOf traverse ≡ traverse id
sequenceAOf l ≡ traverseOf l id ≡ l id

sequenceAOf :: Functor f => Iso s t (f b) b       -> s -> f t
sequenceAOf :: Functor f => Lens s t (f b) b      -> s -> f t
sequenceAOf :: Applicative f => Traversal s t (f b) b -> s -> f t

mapMOf :: LensLike (WrappedMonad m) s t a b -> (a -> m b) -> s -> m t

Map each element of a structure targeted by a Lens to a monadic action, evaluate these actions from left to right, and collect the results.

>>> mapMOf both (\x -> [x, x + 1]) (1,3)
[(1,3),(1,4),(2,3),(2,4)]

mapM ≡ mapMOf traverse
imapMOf l ≡ forM l . Indexed

mapMOf :: Monad m => Iso s t a b       -> (a -> m b) -> s -> m t
mapMOf :: Monad m => Lens s t a b      -> (a -> m b) -> s -> m t
mapMOf :: Monad m => Traversal s t a b -> (a -> m b) -> s -> m t

forMOf :: LensLike (WrappedMonad m) s t a b -> s -> (a -> m b) -> m t

forMOf is a flipped version of mapMOf, consistent with the definition of forM.

>>> forMOf both (1,3) $ \x -> [x, x + 1]
[(1,3),(1,4),(2,3),(2,4)]

forM ≡ forMOf traverse
forMOf l ≡ flip (mapMOf l)
iforMOf l s ≡ forM l s . Indexed

forMOf :: Monad m => Iso s t a b       -> s -> (a -> m b) -> m t
forMOf :: Monad m => Lens s t a b      -> s -> (a -> m b) -> m t
forMOf :: Monad m => Traversal s t a b -> s -> (a -> m b) -> m t

sequenceOf :: LensLike (WrappedMonad m) s t (m b) b -> s -> m t

Sequence the (monadic) effects targeted by a Lens in a container from left to right.

>>> sequenceOf each ([1,2],[3,4],[5,6])
[(1,3,5),(1,3,6),(1,4,5),(1,4,6),(2,3,5),(2,3,6),(2,4,5),(2,4,6)]

sequence ≡ sequenceOf traverse
sequenceOf l ≡ mapMOf l id
sequenceOf l ≡ unwrapMonad . l WrapMonad

sequenceOf :: Monad m => Iso s t (m b) b       -> s -> m t
sequenceOf :: Monad m => Lens s t (m b) b      -> s -> m t
sequenceOf :: Monad m => Traversal s t (m b) b -> s -> m t

transposeOf :: LensLike ZipList s t [a] a -> s -> [t]

This generalizes transpose to an arbitrary Traversal.

Note: transpose handles ragged inputs more intelligently, but for non-ragged inputs:

>>> transposeOf traverse [[1,2,3],[4,5,6]]
[[1,4],[2,5],[3,6]]

transpose ≡ transposeOf traverse

Since every Lens is a Traversal, we can use this as a form of monadic strength as well:

transposeOf _2 :: (b, [a]) -> [(b, a)]

mapAccumLOf :: LensLike (State acc) s t a b -> (acc -> a -> (acc, b)) -> acc -> s -> (acc, t)

This generalizes mapAccumL to an arbitrary Traversal.

mapAccumL ≡ mapAccumLOf traverse

mapAccumLOf accumulates State from left to right.

mapAccumLOf :: Iso s t a b       -> (acc -> a -> (acc, b)) -> acc -> s -> (acc, t)
mapAccumLOf :: Lens s t a b      -> (acc -> a -> (acc, b)) -> acc -> s -> (acc, t)
mapAccumLOf :: Traversal s t a b -> (acc -> a -> (acc, b)) -> acc -> s -> (acc, t)

mapAccumLOf :: LensLike (State acc) s t a b -> (acc -> a -> (acc, b)) -> acc -> s -> (acc, t)
mapAccumLOf l f acc0 s = swap (runState (l (a -> state (acc -> swap (f acc a))) s) acc0)

mapAccumROf :: LensLike (Backwards (State acc)) s t a b -> (acc -> a -> (acc, b)) -> acc -> s -> (acc, t)

This generalizes mapAccumR to an arbitrary Traversal.

mapAccumR ≡ mapAccumROf traverse

mapAccumROf accumulates State from right to left.

mapAccumROf :: Iso s t a b       -> (acc -> a -> (acc, b)) -> acc -> s -> (acc, t)
mapAccumROf :: Lens s t a b      -> (acc -> a -> (acc, b)) -> acc -> s -> (acc, t)
mapAccumROf :: Traversal s t a b -> (acc -> a -> (acc, b)) -> acc -> s -> (acc, t)

mapAccumROf :: LensLike (Backwards (State acc)) s t a b -> (acc -> a -> (acc, b)) -> acc -> s -> (acc, t)

scanr1Of :: LensLike (Backwards (State (Maybe a))) s t a a -> (a -> a -> a) -> s -> t

This permits the use of scanr1 over an arbitrary Traversal or Lens.

scanr1 ≡ scanr1Of traverse

scanr1Of :: Iso s t a a       -> (a -> a -> a) -> s -> t
scanr1Of :: Lens s t a a      -> (a -> a -> a) -> s -> t
scanr1Of :: Traversal s t a a -> (a -> a -> a) -> s -> t

scanl1Of :: LensLike (State (Maybe a)) s t a a -> (a -> a -> a) -> s -> t

This permits the use of scanl1 over an arbitrary Traversal or Lens.

scanl1 ≡ scanl1Of traverse

scanl1Of :: Iso s t a a       -> (a -> a -> a) -> s -> t
scanl1Of :: Lens s t a a      -> (a -> a -> a) -> s -> t
scanl1Of :: Traversal s t a a -> (a -> a -> a) -> s -> t

failover :: Alternative m => LensLike ((,) Any) s t a b -> (a -> b) -> s -> m t

Try to map a function over this Traversal, failing if the Traversal has no targets.

>>> failover (element 3) (*2) [1,2] :: Maybe [Int]
Nothing

>>> failover _Left (*2) (Right 4) :: Maybe (Either Int Int)
Nothing

>>> failover _Right (*2) (Right 4) :: Maybe (Either Int Int)
Just (Right 8)

failover :: Alternative m => Traversal s t a b -> (a -> b) -> s -> m t

ifailover :: Alternative m => Over (Indexed i) ((,) Any) s t a b -> (i -> a -> b) -> s -> m t

Try to map a function which uses the index over this IndexedTraversal, failing if the IndexedTraversal has no targets.

ifailover :: Alternative m => IndexedTraversal i s t a b -> (i -> a -> b) -> s -> m t

Monomorphic Traversals

cloneTraversal :: ATraversal s t a b -> Traversal s t a b

A Traversal is completely characterized by its behavior on a Bazaar.

Cloning a Traversal is one way to make sure you aren't given something weaker, such as a Fold and can be used as a way to pass around traversals that have to be monomorphic in f.

Note: This only accepts a proper Traversal (or Lens). To clone a Lens as such, use cloneLens.

Note: It is usually better to use ReifiedTraversal and runTraversal than to cloneTraversal. The former can execute at full speed, while the latter needs to round trip through the Bazaar.

>>> let foo l a = (view (getting (cloneTraversal l)) a, set (cloneTraversal l) 10 a)
>>> foo both ("hello","world")
("helloworld",(10,10))

cloneTraversal :: LensLike (Bazaar (->) a b) s t a b -> Traversal s t a b

cloneIndexPreservingTraversal :: ATraversal s t a b -> IndexPreservingTraversal s t a b

Clone a Traversal yielding an IndexPreservingTraversal that passes through whatever index it is composed with.

cloneIndexedTraversal :: AnIndexedTraversal i s t a b -> IndexedTraversal i s t a b

Clone an IndexedTraversal yielding an IndexedTraversal with the same index.

cloneTraversal1 :: ATraversal1 s t a b -> Traversal1 s t a b

A Traversal1 is completely characterized by its behavior on a Bazaar1.

cloneIndexPreservingTraversal1 :: ATraversal1 s t a b -> IndexPreservingTraversal1 s t a b

Clone a Traversal1 yielding an IndexPreservingTraversal1 that passes through whatever index it is composed with.

cloneIndexedTraversal1 :: AnIndexedTraversal1 i s t a b -> IndexedTraversal1 i s t a b

Clone an IndexedTraversal1 yielding an IndexedTraversal1 with the same index.

Parts and Holes

partsOf :: Functor f => Traversing (->) f s t a a -> LensLike f s t [a] [a]

partsOf turns a Traversal into a Lens that resembles an early version of the uniplate (or biplate) type.

Note: You should really try to maintain the invariant of the number of children in the list.

>>> (a,b,c) & partsOf each .~ [x,y,z]
(x,y,z)

Any extras will be lost. If you do not supply enough, then the remainder will come from the original structure.

>>> (a,b,c) & partsOf each .~ [w,x,y,z]
(w,x,y)

>>> (a,b,c) & partsOf each .~ [x,y]
(x,y,c)

>>> ('b', 'a', 'd', 'c') & partsOf each %~ sort
('a','b','c','d')

So technically, this is only a Lens if you do not change the number of results it returns.

When applied to a Fold the result is merely a Getter.

partsOf :: Iso' s a       -> Lens' s [a]
partsOf :: Lens' s a      -> Lens' s [a]
partsOf :: Traversal' s a -> Lens' s [a]
partsOf :: Fold s a       -> Getter s [a]
partsOf :: Getter s a     -> Getter s [a]

partsOf' :: ATraversal s t a a -> Lens s t [a] [a]

A type-restricted version of partsOf that can only be used with a Traversal.

unsafePartsOf :: Functor f => Traversing (->) f s t a b -> LensLike f s t [a] [b]

unsafePartsOf turns a Traversal into a uniplate (or biplate) family.

If you do not need the types of s and t to be different, it is recommended that you use partsOf.

It is generally safer to traverse with the Bazaar rather than use this combinator. However, it is sometimes convenient.

This is unsafe because if you don't supply at least as many b's as you were given a's, then the reconstruction of t will result in an error!

When applied to a Fold the result is merely a Getter (and becomes safe).

unsafePartsOf :: Iso s t a b       -> Lens s t [a] [b]
unsafePartsOf :: Lens s t a b      -> Lens s t [a] [b]
unsafePartsOf :: Traversal s t a b -> Lens s t [a] [b]
unsafePartsOf :: Fold s a          -> Getter s [a]
unsafePartsOf :: Getter s a        -> Getter s [a]

unsafePartsOf' :: ATraversal s t a b -> Lens s t [a] [b]

holesOf :: Conjoined p => Over p (Bazaar p a a) s t a a -> s -> [Pretext p a a t]

The one-level version of contextsOf. This extracts a list of the immediate children according to a given Traversal as editable contexts.

Given a context you can use pos to see the values, peek at what the structure would be like with an edited result, or simply extract the original structure.

propChildren l x = toListOf l x == map pos (holesOf l x)
propId l x = all (== x) [extract w | w <- holesOf l x]

holesOf :: Iso' s a                -> s -> [Pretext' (->) a s]
holesOf :: Lens' s a               -> s -> [Pretext' (->) a s]
holesOf :: Traversal' s a          -> s -> [Pretext' (->) a s]
holesOf :: IndexedLens' i s a      -> s -> [Pretext' (Indexed i) a s]
holesOf :: IndexedTraversal' i s a -> s -> [Pretext' (Indexed i) a s]

singular :: (HasCallStack, Conjoined p, Functor f) => Traversing p f s t a a -> Over p f s t a a

This converts a Traversal that you "know" will target one or more elements to a Lens. It can also be used to transform a non-empty Fold into a Getter.

The resulting Lens or Getter will be partial if the supplied Traversal returns no results.

>>> [1,2,3] ^. singular _head
1

>>> Left (ErrorCall "singular: empty traversal") <- try (evaluate ([] ^. singular _head)) :: IO (Either ErrorCall ())

>>> Left 4 ^. singular _Left
4

>>> [1..10] ^. singular (ix 7)
8

>>> [] & singular traverse .~ 0
[]

singular :: Traversal s t a a          -> Lens s t a a
singular :: Fold s a                   -> Getter s a
singular :: IndexedTraversal i s t a a -> IndexedLens i s t a a
singular :: IndexedFold i s a          -> IndexedGetter i s a

unsafeSingular :: (HasCallStack, Conjoined p, Functor f) => Traversing p f s t a b -> Over p f s t a b

This converts a Traversal that you "know" will target only one element to a Lens. It can also be used to transform a Fold into a Getter.

The resulting Lens or Getter will be partial if the Traversal targets nothing or more than one element.

>>> Left (ErrorCall "unsafeSingular: empty traversal") <- try (evaluate ([] & unsafeSingular traverse .~ 0)) :: IO (Either ErrorCall [Integer])

unsafeSingular :: Traversal s t a b          -> Lens s t a b
unsafeSingular :: Fold s a                   -> Getter s a
unsafeSingular :: IndexedTraversal i s t a b -> IndexedLens i s t a b
unsafeSingular :: IndexedFold i s a          -> IndexedGetter i s a

Common Traversals

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_.

Instances

Traversable []

Since: base-2.1

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

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

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 Complex

Defined in Data.Complex

Methods

traverse :: Applicative f => (a -> f b) -> Complex a -> f (Complex b)

sequenceA :: Applicative f => Complex (f a) -> f (Complex a)

mapM :: Monad m => (a -> m b) -> Complex a -> m (Complex b)

sequence :: Monad m => Complex (m a) -> m (Complex a)

Traversable Min

Since: base-4.9.0.0

Defined in Data.Semigroup

Methods

traverse :: Applicative f => (a -> f b) -> Min a -> f (Min b)

sequenceA :: Applicative f => Min (f a) -> f (Min a)

mapM :: Monad m => (a -> m b) -> Min a -> m (Min b)

sequence :: Monad m => Min (m a) -> m (Min a)

Traversable Max

Since: base-4.9.0.0

Defined in Data.Semigroup

Methods

traverse :: Applicative f => (a -> f b) -> Max a -> f (Max b)

sequenceA :: Applicative f => Max (f a) -> f (Max a)

mapM :: Monad m => (a -> m b) -> Max a -> m (Max b)

sequence :: Monad m => Max (m a) -> m (Max a)

Traversable First

Since: base-4.9.0.0

Defined in Data.Semigroup

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-4.9.0.0

Defined in Data.Semigroup

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 Option

Since: base-4.9.0.0

Defined in Data.Semigroup

Methods

traverse :: Applicative f => (a -> f b) -> Option a -> f (Option b)

sequenceA :: Applicative f => Option (f a) -> f (Option a)

mapM :: Monad m => (a -> m b) -> Option a -> m (Option b)

sequence :: Monad m => Option (m a) -> m (Option a)

Traversable ZipList

Since: base-4.9.0.0

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

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

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

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

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

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

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

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 IntMap

Defined in Data.IntMap.Internal

Methods

traverse :: Applicative f => (a -> f b) -> IntMap a -> f (IntMap b)

sequenceA :: Applicative f => IntMap (f a) -> f (IntMap a)

mapM :: Monad m => (a -> m b) -> IntMap a -> m (IntMap b)

sequence :: Monad m => IntMap (m a) -> m (IntMap a)

Traversable Tree

Defined in Data.Tree

Methods

traverse :: Applicative f => (a -> f b) -> Tree a -> f (Tree b)

sequenceA :: Applicative f => Tree (f a) -> f (Tree a)

mapM :: Monad m => (a -> m b) -> Tree a -> m (Tree b)

sequence :: Monad m => Tree (m a) -> m (Tree a)

Traversable Seq

Defined in Data.Sequence.Internal

Methods

traverse :: Applicative f => (a -> f b) -> Seq a -> f (Seq b)

sequenceA :: Applicative f => Seq (f a) -> f (Seq a)

mapM :: Monad m => (a -> m b) -> Seq a -> m (Seq b)

sequence :: Monad m => Seq (m a) -> m (Seq a)

Traversable FingerTree

Defined in Data.Sequence.Internal

Methods

traverse :: Applicative f => (a -> f b) -> FingerTree a -> f (FingerTree b)

sequenceA :: Applicative f => FingerTree (f a) -> f (FingerTree a)

mapM :: Monad m => (a -> m b) -> FingerTree a -> m (FingerTree b)

sequence :: Monad m => FingerTree (m a) -> m (FingerTree a)

Traversable Digit

Defined in Data.Sequence.Internal

Methods

traverse :: Applicative f => (a -> f b) -> Digit a -> f (Digit b)

sequenceA :: Applicative f => Digit (f a) -> f (Digit a)

mapM :: Monad m => (a -> m b) -> Digit a -> m (Digit b)

sequence :: Monad m => Digit (m a) -> m (Digit a)

Traversable Node

Defined in Data.Sequence.Internal

Methods

traverse :: Applicative f => (a -> f b) -> Node a -> f (Node b)

sequenceA :: Applicative f => Node (f a) -> f (Node a)

mapM :: Monad m => (a -> m b) -> Node a -> m (Node b)

sequence :: Monad m => Node (m a) -> m (Node a)

Traversable Elem

Defined in Data.Sequence.Internal

Methods

traverse :: Applicative f => (a -> f b) -> Elem a -> f (Elem b)

sequenceA :: Applicative f => Elem (f a) -> f (Elem a)

mapM :: Monad m => (a -> m b) -> Elem a -> m (Elem b)

sequence :: Monad m => Elem (m a) -> m (Elem a)

Traversable ViewL

Defined in Data.Sequence.Internal

Methods

traverse :: Applicative f => (a -> f b) -> ViewL a -> f (ViewL b)

sequenceA :: Applicative f => ViewL (f a) -> f (ViewL a)

mapM :: Monad m => (a -> m b) -> ViewL a -> m (ViewL b)

sequence :: Monad m => ViewL (m a) -> m (ViewL a)

Traversable ViewR

Defined in Data.Sequence.Internal

Methods

traverse :: Applicative f => (a -> f b) -> ViewR a -> f (ViewR b)

sequenceA :: Applicative f => ViewR (f a) -> f (ViewR a)

mapM :: Monad m => (a -> m b) -> ViewR a -> m (ViewR b)

sequence :: Monad m => ViewR (m a) -> m (ViewR a)

Traversable SmallArray

Defined in Data.Primitive.SmallArray

Methods

traverse :: Applicative f => (a -> f b) -> SmallArray a -> f (SmallArray b)

sequenceA :: Applicative f => SmallArray (f a) -> f (SmallArray a)

mapM :: Monad m => (a -> m b) -> SmallArray a -> m (SmallArray b)

sequence :: Monad m => SmallArray (m a) -> m (SmallArray a)

Traversable Array

Defined in Data.Primitive.Array

Methods

traverse :: Applicative f => (a -> f b) -> Array a -> f (Array b)

sequenceA :: Applicative f => Array (f a) -> f (Array a)

mapM :: Monad m => (a -> m b) -> Array a -> m (Array b)

sequence :: Monad m => Array (m a) -> m (Array a)

Traversable Vector

Defined in Data.Vector

Methods

traverse :: Applicative f => (a -> f b) -> Vector a -> f (Vector b)

sequenceA :: Applicative f => Vector (f a) -> f (Vector a)

mapM :: Monad m => (a -> m b) -> Vector a -> m (Vector b)

sequence :: Monad m => Vector (m a) -> m (Vector a)

Traversable Deque

Defined in Control.Lens.Internal.Deque

Methods

traverse :: Applicative f => (a -> f b) -> Deque a -> f (Deque b)

sequenceA :: Applicative f => Deque (f a) -> f (Deque a)

mapM :: Monad m => (a -> m b) -> Deque a -> m (Deque b)

sequence :: Monad m => Deque (m a) -> m (Deque a)

Traversable (Either a)

Since: base-4.7.0.0

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 :: * -> *)

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

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

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

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 (Arg a)

Since: base-4.9.0.0

Defined in Data.Semigroup

Methods

traverse :: Applicative f => (a0 -> f b) -> Arg a a0 -> f (Arg a b)

sequenceA :: Applicative f => Arg a (f a0) -> f (Arg a a0)

mapM :: Monad m => (a0 -> m b) -> Arg a a0 -> m (Arg a b)

sequence :: Monad m => Arg a (m a0) -> m (Arg a a0)

Traversable (Proxy :: * -> *)

Since: base-4.7.0.0

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)

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 (MaybeT f)

Defined in Control.Monad.Trans.Maybe

Methods

traverse :: Applicative f0 => (a -> f0 b) -> MaybeT f a -> f0 (MaybeT f b)

sequenceA :: Applicative f0 => MaybeT f (f0 a) -> f0 (MaybeT f a)

mapM :: Monad m => (a -> m b) -> MaybeT f a -> m (MaybeT f b)

sequence :: Monad m => MaybeT f (m a) -> m (MaybeT f a)

(Monad m, Traversable m) => Traversable (CatchT m)

Defined in Control.Monad.Catch.Pure

Methods

traverse :: Applicative f => (a -> f b) -> CatchT m a -> f (CatchT m b)

sequenceA :: Applicative f => CatchT m (f a) -> f (CatchT m a)

mapM :: Monad m0 => (a -> m0 b) -> CatchT m a -> m0 (CatchT m b)

sequence :: Monad m0 => CatchT m (m0 a) -> m0 (CatchT m a)

Traversable f => Traversable (Cofree f)

Defined in Control.Comonad.Cofree

Methods

traverse :: Applicative f0 => (a -> f0 b) -> Cofree f a -> f0 (Cofree f b)

sequenceA :: Applicative f0 => Cofree f (f0 a) -> f0 (Cofree f a)

mapM :: Monad m => (a -> m b) -> Cofree f a -> m (Cofree f b)

sequence :: Monad m => Cofree f (m a) -> m (Cofree f a)

Traversable w => Traversable (CoiterT w)

Defined in Control.Comonad.Trans.Coiter

Methods

traverse :: Applicative f => (a -> f b) -> CoiterT w a -> f (CoiterT w b)

sequenceA :: Applicative f => CoiterT w (f a) -> f (CoiterT w a)

mapM :: Monad m => (a -> m b) -> CoiterT w a -> m (CoiterT w b)

sequence :: Monad m => CoiterT w (m a) -> m (CoiterT w a)

Traversable f => Traversable (F f)

Defined in Control.Monad.Free.Church

Methods

traverse :: Applicative f0 => (a -> f0 b) -> F f a -> f0 (F f b)

sequenceA :: Applicative f0 => F f (f0 a) -> f0 (F f a)

mapM :: Monad m => (a -> m b) -> F f a -> m (F f b)

sequence :: Monad m => F f (m a) -> m (F f a)

Traversable f => Traversable (Free f)

Defined in Control.Monad.Free

Methods

traverse :: Applicative f0 => (a -> f0 b) -> Free f a -> f0 (Free f b)

sequenceA :: Applicative f0 => Free f (f0 a) -> f0 (Free f a)

mapM :: Monad m => (a -> m b) -> Free f a -> m (Free f b)

sequence :: Monad m => Free f (m a) -> m (Free f a)

(Monad m, Traversable m) => Traversable (IterT m)

Defined in Control.Monad.Trans.Iter

Methods

traverse :: Applicative f => (a -> f b) -> IterT m a -> f (IterT m b)

sequenceA :: Applicative f => IterT m (f a) -> f (IterT m a)

mapM :: Monad m0 => (a -> m0 b) -> IterT m a -> m0 (IterT m b)

sequence :: Monad m0 => IterT m (m0 a) -> m0 (IterT m a)

Traversable f => Traversable (Yoneda f)

Defined in Data.Functor.Yoneda

Methods

traverse :: Applicative f0 => (a -> f0 b) -> Yoneda f a -> f0 (Yoneda f b)

sequenceA :: Applicative f0 => Yoneda f (f0 a) -> f0 (Yoneda f a)

mapM :: Monad m => (a -> m b) -> Yoneda f a -> m (Yoneda f b)

sequence :: Monad m => Yoneda f (m a) -> m (Yoneda f a)

Traversable f => Traversable (ListT f)

Defined in Control.Monad.Trans.List

Methods

traverse :: Applicative f0 => (a -> f0 b) -> ListT f a -> f0 (ListT f b)

sequenceA :: Applicative f0 => ListT f (f0 a) -> f0 (ListT f a)

mapM :: Monad m => (a -> m b) -> ListT f a -> m (ListT f b)

sequence :: Monad m => ListT f (m a) -> m (ListT f a)

Traversable (HashMap k)

Defined in Data.HashMap.Base

Methods

traverse :: Applicative f => (a -> f b) -> HashMap k a -> f (HashMap k b)

sequenceA :: Applicative f => HashMap k (f a) -> f (HashMap k a)

mapM :: Monad m => (a -> m b) -> HashMap k a -> m (HashMap k b)

sequence :: Monad m => HashMap k (m a) -> m (HashMap k a)

Traversable f => Traversable (Lift f)

Defined in Control.Applicative.Lift

Methods

traverse :: Applicative f0 => (a -> f0 b) -> Lift f a -> f0 (Lift f b)

sequenceA :: Applicative f0 => Lift f (f0 a) -> f0 (Lift f a)

mapM :: Monad m => (a -> m b) -> Lift f a -> m (Lift f b)

sequence :: Monad m => Lift f (m a) -> m (Lift f a)

Traversable (Level i)

Defined in Control.Lens.Internal.Level

Methods

traverse :: Applicative f => (a -> f b) -> Level i a -> f (Level i b)

sequenceA :: Applicative f => Level i (f a) -> f (Level i a)

mapM :: Monad m => (a -> m b) -> Level i a -> m (Level i b)

sequence :: Monad m => Level i (m a) -> m (Level i a)

Traversable f => Traversable (Rec1 f)

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 :: * -> *)

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 :: * -> *)

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 :: * -> *)

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 :: * -> *)

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 :: * -> *)

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 ()) :: * -> *)

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

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)

Bitraversable p => Traversable (Join p)

Defined in Data.Bifunctor.Join

Methods

traverse :: Applicative f => (a -> f b) -> Join p a -> f (Join p b)

sequenceA :: Applicative f => Join p (f a) -> f (Join p a)

mapM :: Monad m => (a -> m b) -> Join p a -> m (Join p b)

sequence :: Monad m => Join p (m a) -> m (Join p a)

Bitraversable p => Traversable (Fix p)

Defined in Data.Bifunctor.Fix

Methods

traverse :: Applicative f => (a -> f b) -> Fix p a -> f (Fix p b)

sequenceA :: Applicative f => Fix p (f a) -> f (Fix p a)

mapM :: Monad m => (a -> m b) -> Fix p a -> m (Fix p b)

sequence :: Monad m => Fix p (m a) -> m (Fix p a)

Traversable w => Traversable (EnvT e w)

Defined in Control.Comonad.Trans.Env

Methods

traverse :: Applicative f => (a -> f b) -> EnvT e w a -> f (EnvT e w b)

sequenceA :: Applicative f => EnvT e w (f a) -> f (EnvT e w a)

mapM :: Monad m => (a -> m b) -> EnvT e w a -> m (EnvT e w b)

sequence :: Monad m => EnvT e w (m a) -> m (EnvT e w a)

Traversable f => Traversable (IdentityT f)

Defined in Control.Monad.Trans.Identity

Methods

traverse :: Applicative f0 => (a -> f0 b) -> IdentityT f a -> f0 (IdentityT f b)

sequenceA :: Applicative f0 => IdentityT f (f0 a) -> f0 (IdentityT f a)

mapM :: Monad m => (a -> m b) -> IdentityT f a -> m (IdentityT f b)

sequence :: Monad m => IdentityT f (m a) -> m (IdentityT f a)

Traversable f => Traversable (ExceptT e f)

Defined in Control.Monad.Trans.Except

Methods

traverse :: Applicative f0 => (a -> f0 b) -> ExceptT e f a -> f0 (ExceptT e f b)

sequenceA :: Applicative f0 => ExceptT e f (f0 a) -> f0 (ExceptT e f a)

mapM :: Monad m => (a -> m b) -> ExceptT e f a -> m (ExceptT e f b)

sequence :: Monad m => ExceptT e f (m a) -> m (ExceptT e f a)

Traversable f => Traversable (FreeF f a)

Defined in Control.Monad.Trans.Free

Methods

traverse :: Applicative f0 => (a0 -> f0 b) -> FreeF f a a0 -> f0 (FreeF f a b)

sequenceA :: Applicative f0 => FreeF f a (f0 a0) -> f0 (FreeF f a a0)

mapM :: Monad m => (a0 -> m b) -> FreeF f a a0 -> m (FreeF f a b)

sequence :: Monad m => FreeF f a (m a0) -> m (FreeF f a a0)

(Monad m, Traversable m, Traversable f) => Traversable (FreeT f m)

Defined in Control.Monad.Trans.Free

Methods

traverse :: Applicative f0 => (a -> f0 b) -> FreeT f m a -> f0 (FreeT f m b)

sequenceA :: Applicative f0 => FreeT f m (f0 a) -> f0 (FreeT f m a)

mapM :: Monad m0 => (a -> m0 b) -> FreeT f m a -> m0 (FreeT f m b)

sequence :: Monad m0 => FreeT f m (m0 a) -> m0 (FreeT f m a)

Traversable f => Traversable (CofreeF f a)

Defined in Control.Comonad.Trans.Cofree

Methods

traverse :: Applicative f0 => (a0 -> f0 b) -> CofreeF f a a0 -> f0 (CofreeF f a b)

sequenceA :: Applicative f0 => CofreeF f a (f0 a0) -> f0 (CofreeF f a a0)

mapM :: Monad m => (a0 -> m b) -> CofreeF f a a0 -> m (CofreeF f a b)

sequence :: Monad m => CofreeF f a (m a0) -> m (CofreeF f a a0)

(Traversable f, Traversable w) => Traversable (CofreeT f w)

Defined in Control.Comonad.Trans.Cofree

Methods

traverse :: Applicative f0 => (a -> f0 b) -> CofreeT f w a -> f0 (CofreeT f w b)

sequenceA :: Applicative f0 => CofreeT f w (f0 a) -> f0 (CofreeT f w a)

mapM :: Monad m => (a -> m b) -> CofreeT f w a -> m (CofreeT f w b)

sequence :: Monad m => CofreeT f w (m a) -> m (CofreeT f w a)

Traversable f => Traversable (ErrorT e f)

Defined in Control.Monad.Trans.Error

Methods

traverse :: Applicative f0 => (a -> f0 b) -> ErrorT e f a -> f0 (ErrorT e f b)

sequenceA :: Applicative f0 => ErrorT e f (f0 a) -> f0 (ErrorT e f a)

mapM :: Monad m => (a -> m b) -> ErrorT e f a -> m (ErrorT e f b)

sequence :: Monad m => ErrorT e f (m a) -> m (ErrorT e f a)

Traversable f => Traversable (WriterT w f)

Defined in Control.Monad.Trans.Writer.Lazy

Methods

traverse :: Applicative f0 => (a -> f0 b) -> WriterT w f a -> f0 (WriterT w f b)

sequenceA :: Applicative f0 => WriterT w f (f0 a) -> f0 (WriterT w f a)

mapM :: Monad m => (a -> m b) -> WriterT w f a -> m (WriterT w f b)

sequence :: Monad m => WriterT w f (m a) -> m (WriterT w f a)

Traversable f => Traversable (WriterT w f)

Defined in Control.Monad.Trans.Writer.Strict

Methods

traverse :: Applicative f0 => (a -> f0 b) -> WriterT w f a -> f0 (WriterT w f b)

sequenceA :: Applicative f0 => WriterT w f (f0 a) -> f0 (WriterT w f a)

mapM :: Monad m => (a -> m b) -> WriterT w f a -> m (WriterT w f b)

sequence :: Monad m => WriterT w f (m a) -> m (WriterT w f a)

Traversable (Forget r a)

Defined in Data.Profunctor.Types

Methods

traverse :: Applicative f => (a0 -> f b) -> Forget r a a0 -> f (Forget r a b)

sequenceA :: Applicative f => Forget r a (f a0) -> f (Forget r a a0)

mapM :: Monad m => (a0 -> m b) -> Forget r a a0 -> m (Forget r a b)

sequence :: Monad m => Forget r a (m a0) -> m (Forget r a a0)

Traversable (Tagged s)

Defined in Data.Tagged

Methods

traverse :: Applicative f => (a -> f b) -> Tagged s a -> f (Tagged s b)

sequenceA :: Applicative f => Tagged s (f a) -> f (Tagged s a)

mapM :: Monad m => (a -> m b) -> Tagged s a -> m (Tagged s b)

sequence :: Monad m => Tagged s (m a) -> m (Tagged s a)

Traversable f => Traversable (Reverse f)

Traverse from right to left.

Defined in Data.Functor.Reverse

Methods

traverse :: Applicative f0 => (a -> f0 b) -> Reverse f a -> f0 (Reverse f b)

sequenceA :: Applicative f0 => Reverse f (f0 a) -> f0 (Reverse f a)

mapM :: Monad m => (a -> m b) -> Reverse f a -> m (Reverse f b)

sequence :: Monad m => Reverse f (m a) -> m (Reverse f a)

Traversable (Constant a :: * -> *)

Defined in Data.Functor.Constant

Methods

traverse :: Applicative f => (a0 -> f b) -> Constant a a0 -> f (Constant a b)

sequenceA :: Applicative f => Constant a (f a0) -> f (Constant a a0)

mapM :: Monad m => (a0 -> m b) -> Constant a a0 -> m (Constant a b)

sequence :: Monad m => Constant a (m a0) -> m (Constant a a0)

Traversable f => Traversable (Backwards f)

Derived instance.

Defined in Control.Applicative.Backwards

Methods

traverse :: Applicative f0 => (a -> f0 b) -> Backwards f a -> f0 (Backwards f b)

sequenceA :: Applicative f0 => Backwards f (f0 a) -> f0 (Backwards f a)

mapM :: Monad m => (a -> m b) -> Backwards f a -> m (Backwards f b)

sequence :: Monad m => Backwards f (m a) -> m (Backwards f a)

Traversable f => Traversable (AlongsideRight f a)

Defined in Control.Lens.Internal.Getter

Methods

traverse :: Applicative f0 => (a0 -> f0 b) -> AlongsideRight f a a0 -> f0 (AlongsideRight f a b)

sequenceA :: Applicative f0 => AlongsideRight f a (f0 a0) -> f0 (AlongsideRight f a a0)

mapM :: Monad m => (a0 -> m b) -> AlongsideRight f a a0 -> m (AlongsideRight f a b)

sequence :: Monad m => AlongsideRight f a (m a0) -> m (AlongsideRight f a a0)

Traversable f => Traversable (AlongsideLeft f b)

Defined in Control.Lens.Internal.Getter

Methods

traverse :: Applicative f0 => (a -> f0 b0) -> AlongsideLeft f b a -> f0 (AlongsideLeft f b b0)

sequenceA :: Applicative f0 => AlongsideLeft f b (f0 a) -> f0 (AlongsideLeft f b a)

mapM :: Monad m => (a -> m b0) -> AlongsideLeft f b a -> m (AlongsideLeft f b b0)

sequence :: Monad m => AlongsideLeft f b (m a) -> m (AlongsideLeft f b a)

Traversable (K1 i c :: * -> *)

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)

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)

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 (Product f g)

Since: base-4.9.0.0

Defined in Data.Functor.Product

Methods

traverse :: Applicative f0 => (a -> f0 b) -> Product f g a -> f0 (Product f g b)

sequenceA :: Applicative f0 => Product f g (f0 a) -> f0 (Product f g a)

mapM :: Monad m => (a -> m b) -> Product f g a -> m (Product f g b)

sequence :: Monad m => Product f g (m a) -> m (Product f g a)

(Traversable f, Traversable g) => Traversable (Sum f g)

Since: base-4.9.0.0

Defined in Data.Functor.Sum

Methods

traverse :: Applicative f0 => (a -> f0 b) -> Sum f g a -> f0 (Sum f g b)

sequenceA :: Applicative f0 => Sum f g (f0 a) -> f0 (Sum f g a)

mapM :: Monad m => (a -> m b) -> Sum f g a -> m (Sum f g b)

sequence :: Monad m => Sum f g (m a) -> m (Sum f g a)

Traversable (Magma i t b)

Defined in Control.Lens.Internal.Magma

Methods

traverse :: Applicative f => (a -> f b0) -> Magma i t b a -> f (Magma i t b b0)

sequenceA :: Applicative f => Magma i t b (f a) -> f (Magma i t b a)

mapM :: Monad m => (a -> m b0) -> Magma i t b a -> m (Magma i t b b0)

sequence :: Monad m => Magma i t b (m a) -> m (Magma i t b a)

Traversable f => Traversable (M1 i c f)

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)

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 (Compose f g)

Since: base-4.9.0.0

Defined in Data.Functor.Compose

Methods

traverse :: Applicative f0 => (a -> f0 b) -> Compose f g a -> f0 (Compose f g b)

sequenceA :: Applicative f0 => Compose f g (f0 a) -> f0 (Compose f g a)

mapM :: Monad m => (a -> m b) -> Compose f g a -> m (Compose f g b)

sequence :: Monad m => Compose f g (m a) -> m (Compose f g a)

Bitraversable p => Traversable (WrappedBifunctor p a)

Defined in Data.Bifunctor.Wrapped

Methods

traverse :: Applicative f => (a0 -> f b) -> WrappedBifunctor p a a0 -> f (WrappedBifunctor p a b)

sequenceA :: Applicative f => WrappedBifunctor p a (f a0) -> f (WrappedBifunctor p a a0)

mapM :: Monad m => (a0 -> m b) -> WrappedBifunctor p a a0 -> m (WrappedBifunctor p a b)

sequence :: Monad m => WrappedBifunctor p a (m a0) -> m (WrappedBifunctor p a a0)

Traversable g => Traversable (Joker g a)

Defined in Data.Bifunctor.Joker

Methods

traverse :: Applicative f => (a0 -> f b) -> Joker g a a0 -> f (Joker g a b)

sequenceA :: Applicative f => Joker g a (f a0) -> f (Joker g a a0)

mapM :: Monad m => (a0 -> m b) -> Joker g a a0 -> m (Joker g a b)

sequence :: Monad m => Joker g a (m a0) -> m (Joker g a a0)

Bitraversable p => Traversable (Flip p a)

Defined in Data.Bifunctor.Flip

Methods

traverse :: Applicative f => (a0 -> f b) -> Flip p a a0 -> f (Flip p a b)

sequenceA :: Applicative f => Flip p a (f a0) -> f (Flip p a a0)

mapM :: Monad m => (a0 -> m b) -> Flip p a a0 -> m (Flip p a b)

sequence :: Monad m => Flip p a (m a0) -> m (Flip p a a0)

Traversable (Clown f a :: * -> *)

Defined in Data.Bifunctor.Clown

Methods

traverse :: Applicative f0 => (a0 -> f0 b) -> Clown f a a0 -> f0 (Clown f a b)

sequenceA :: Applicative f0 => Clown f a (f0 a0) -> f0 (Clown f a a0)

mapM :: Monad m => (a0 -> m b) -> Clown f a a0 -> m (Clown f a b)

sequence :: Monad m => Clown f a (m a0) -> m (Clown f a a0)

(Traversable f, Bitraversable p) => Traversable (Tannen f p a)

Defined in Data.Bifunctor.Tannen

Methods

traverse :: Applicative f0 => (a0 -> f0 b) -> Tannen f p a a0 -> f0 (Tannen f p a b)

sequenceA :: Applicative f0 => Tannen f p a (f0 a0) -> f0 (Tannen f p a a0)

mapM :: Monad m => (a0 -> m b) -> Tannen f p a a0 -> m (Tannen f p a b)

sequence :: Monad m => Tannen f p a (m a0) -> m (Tannen f p a a0)

(Bitraversable p, Traversable g) => Traversable (Biff p f g a)

Defined in Data.Bifunctor.Biff

Methods

traverse :: Applicative f0 => (a0 -> f0 b) -> Biff p f g a a0 -> f0 (Biff p f g a b)

sequenceA :: Applicative f0 => Biff p f g a (f0 a0) -> f0 (Biff p f g a a0)

mapM :: Monad m => (a0 -> m b) -> Biff p f g a a0 -> m (Biff p f g a b)

sequence :: Monad m => Biff p f g a (m a0) -> m (Biff p f g a a0)

class (Foldable1 t, Traversable t) => Traversable1 (t :: * -> *) where

Minimal complete definition

traverse1 | sequence1

Methods

traverse1 :: Apply f => (a -> f b) -> t a -> f (t b)

Instances

Traversable1 Par1

Defined in Data.Semigroup.Traversable.Class

Methods

traverse1 :: Apply f => (a -> f b) -> Par1 a -> f (Par1 b)

sequence1 :: Apply f => Par1 (f b) -> f (Par1 b)

Traversable1 Complex

Defined in Data.Semigroup.Traversable.Class

Methods

traverse1 :: Apply f => (a -> f b) -> Complex a -> f (Complex b)

sequence1 :: Apply f => Complex (f b) -> f (Complex b)

Traversable1 Min

Defined in Data.Semigroup.Traversable.Class

Methods

traverse1 :: Apply f => (a -> f b) -> Min a -> f (Min b)

sequence1 :: Apply f => Min (f b) -> f (Min b)

Traversable1 Max

Defined in Data.Semigroup.Traversable.Class

Methods

traverse1 :: Apply f => (a -> f b) -> Max a -> f (Max b)

sequence1 :: Apply f => Max (f b) -> f (Max b)

Traversable1 First

Defined in Data.Semigroup.Traversable.Class

Methods

traverse1 :: Apply f => (a -> f b) -> First a -> f (First b)

sequence1 :: Apply f => First (f b) -> f (First b)

Traversable1 Last

Defined in Data.Semigroup.Traversable.Class

Methods

traverse1 :: Apply f => (a -> f b) -> Last a -> f (Last b)

sequence1 :: Apply f => Last (f b) -> f (Last b)

Traversable1 Identity

Defined in Data.Semigroup.Traversable.Class

Methods

traverse1 :: Apply f => (a -> f b) -> Identity a -> f (Identity b)

sequence1 :: Apply f => Identity (f b) -> f (Identity b)

Traversable1 Dual

Defined in Data.Semigroup.Traversable.Class

Methods

traverse1 :: Apply f => (a -> f b) -> Dual a -> f (Dual b)

sequence1 :: Apply f => Dual (f b) -> f (Dual b)

Traversable1 Sum

Defined in Data.Semigroup.Traversable.Class

Methods

traverse1 :: Apply f => (a -> f b) -> Sum a -> f (Sum b)

sequence1 :: Apply f => Sum (f b) -> f (Sum b)

Traversable1 Product

Defined in Data.Semigroup.Traversable.Class

Methods

traverse1 :: Apply f => (a -> f b) -> Product a -> f (Product b)

sequence1 :: Apply f => Product (f b) -> f (Product b)

Traversable1 NonEmpty

Defined in Data.Semigroup.Traversable.Class

Methods

traverse1 :: Apply f => (a -> f b) -> NonEmpty a -> f (NonEmpty b)

sequence1 :: Apply f => NonEmpty (f b) -> f (NonEmpty b)

Traversable1 Tree

Defined in Data.Semigroup.Traversable.Class

Methods

traverse1 :: Apply f => (a -> f b) -> Tree a -> f (Tree b)

sequence1 :: Apply f => Tree (f b) -> f (Tree b)

Traversable1 (V1 :: * -> *)

Defined in Data.Semigroup.Traversable.Class

Methods

traverse1 :: Apply f => (a -> f b) -> V1 a -> f (V1 b)

sequence1 :: Apply f => V1 (f b) -> f (V1 b)

Traversable1 ((,) a)

Defined in Data.Semigroup.Traversable.Class

Methods

traverse1 :: Apply f => (a0 -> f b) -> (a, a0) -> f (a, b)

sequence1 :: Apply f => (a, f b) -> f (a, b)

Traversable1 f => Traversable1 (Cofree f)

Defined in Control.Comonad.Cofree

Methods

traverse1 :: Apply f0 => (a -> f0 b) -> Cofree f a -> f0 (Cofree f b)

sequence1 :: Apply f0 => Cofree f (f0 b) -> f0 (Cofree f b)

Traversable1 f => Traversable1 (F f)

Defined in Control.Monad.Free.Church

Methods

traverse1 :: Apply f0 => (a -> f0 b) -> F f a -> f0 (F f b)

sequence1 :: Apply f0 => F f (f0 b) -> f0 (F f b)

Traversable1 f => Traversable1 (Free f)

Defined in Control.Monad.Free

Methods

traverse1 :: Apply f0 => (a -> f0 b) -> Free f a -> f0 (Free f b)

sequence1 :: Apply f0 => Free f (f0 b) -> f0 (Free f b)

(Monad m, Traversable1 m) => Traversable1 (IterT m)

Defined in Control.Monad.Trans.Iter

Methods

traverse1 :: Apply f => (a -> f b) -> IterT m a -> f (IterT m b)

sequence1 :: Apply f => IterT m (f b) -> f (IterT m b)

Traversable1 f => Traversable1 (Yoneda f)

Defined in Data.Functor.Yoneda

Methods

traverse1 :: Apply f0 => (a -> f0 b) -> Yoneda f a -> f0 (Yoneda f b)

sequence1 :: Apply f0 => Yoneda f (f0 b) -> f0 (Yoneda f b)

Traversable1 f => Traversable1 (Lift f)

Defined in Data.Semigroup.Traversable.Class

Methods

traverse1 :: Apply f0 => (a -> f0 b) -> Lift f a -> f0 (Lift f b)

sequence1 :: Apply f0 => Lift f (f0 b) -> f0 (Lift f b)

Traversable1 f => Traversable1 (Rec1 f)

Defined in Data.Semigroup.Traversable.Class

Methods

traverse1 :: Apply f0 => (a -> f0 b) -> Rec1 f a -> f0 (Rec1 f b)

sequence1 :: Apply f0 => Rec1 f (f0 b) -> f0 (Rec1 f b)

Traversable1 f => Traversable1 (Alt f)

Defined in Data.Semigroup.Traversable.Class

Methods

traverse1 :: Apply f0 => (a -> f0 b) -> Alt f a -> f0 (Alt f b)

sequence1 :: Apply f0 => Alt f (f0 b) -> f0 (Alt f b)

Bitraversable1 p => Traversable1 (Join p)

Defined in Data.Semigroup.Traversable.Class

Methods

traverse1 :: Apply f => (a -> f b) -> Join p a -> f (Join p b)

sequence1 :: Apply f => Join p (f b) -> f (Join p b)

Traversable1 f => Traversable1 (IdentityT f)

Defined in Data.Semigroup.Traversable.Class

Methods

traverse1 :: Apply f0 => (a -> f0 b) -> IdentityT f a -> f0 (IdentityT f b)

sequence1 :: Apply f0 => IdentityT f (f0 b) -> f0 (IdentityT f b)

Traversable1 (Tagged a)

Defined in Data.Semigroup.Traversable.Class

Methods

traverse1 :: Apply f => (a0 -> f b) -> Tagged a a0 -> f (Tagged a b)

sequence1 :: Apply f => Tagged a (f b) -> f (Tagged a b)

Traversable1 f => Traversable1 (Reverse f)

Defined in Data.Semigroup.Traversable.Class

Methods

traverse1 :: Apply f0 => (a -> f0 b) -> Reverse f a -> f0 (Reverse f b)

sequence1 :: Apply f0 => Reverse f (f0 b) -> f0 (Reverse f b)

Traversable1 f => Traversable1 (Backwards f)

Defined in Data.Semigroup.Traversable.Class

Methods

traverse1 :: Apply f0 => (a -> f0 b) -> Backwards f a -> f0 (Backwards f b)

sequence1 :: Apply f0 => Backwards f (f0 b) -> f0 (Backwards f b)

Traversable1 f => Traversable1 (AlongsideRight f a)

Defined in Control.Lens.Internal.Getter

Methods

traverse1 :: Apply f0 => (a0 -> f0 b) -> AlongsideRight f a a0 -> f0 (AlongsideRight f a b)

sequence1 :: Apply f0 => AlongsideRight f a (f0 b) -> f0 (AlongsideRight f a b)

Traversable1 f => Traversable1 (AlongsideLeft f b)

Defined in Control.Lens.Internal.Getter

Methods

traverse1 :: Apply f0 => (a -> f0 b0) -> AlongsideLeft f b a -> f0 (AlongsideLeft f b b0)

sequence1 :: Apply f0 => AlongsideLeft f b (f0 b0) -> f0 (AlongsideLeft f b b0)

(Traversable1 f, Traversable1 g) => Traversable1 (f :+: g)

Defined in Data.Semigroup.Traversable.Class

Methods

traverse1 :: Apply f0 => (a -> f0 b) -> (f :+: g) a -> f0 ((f :+: g) b)

sequence1 :: Apply f0 => (f :+: g) (f0 b) -> f0 ((f :+: g) b)

(Traversable1 f, Traversable1 g) => Traversable1 (f :*: g)

Defined in Data.Semigroup.Traversable.Class

Methods

traverse1 :: Apply f0 => (a -> f0 b) -> (f :*: g) a -> f0 ((f :*: g) b)

sequence1 :: Apply f0 => (f :*: g) (f0 b) -> f0 ((f :*: g) b)

(Traversable1 f, Traversable1 g) => Traversable1 (Product f g)

Defined in Data.Semigroup.Traversable.Class

Methods

traverse1 :: Apply f0 => (a -> f0 b) -> Product f g a -> f0 (Product f g b)

sequence1 :: Apply f0 => Product f g (f0 b) -> f0 (Product f g b)

(Traversable1 f, Traversable1 g) => Traversable1 (Sum f g)

Defined in Data.Semigroup.Traversable.Class

Methods

traverse1 :: Apply f0 => (a -> f0 b) -> Sum f g a -> f0 (Sum f g b)

sequence1 :: Apply f0 => Sum f g (f0 b) -> f0 (Sum f g b)

Traversable1 f => Traversable1 (M1 i c f)

Defined in Data.Semigroup.Traversable.Class

Methods

traverse1 :: Apply f0 => (a -> f0 b) -> M1 i c f a -> f0 (M1 i c f b)

sequence1 :: Apply f0 => M1 i c f (f0 b) -> f0 (M1 i c f b)

(Traversable1 f, Traversable1 g) => Traversable1 (f :.: g)

Defined in Data.Semigroup.Traversable.Class

Methods

traverse1 :: Apply f0 => (a -> f0 b) -> (f :.: g) a -> f0 ((f :.: g) b)

sequence1 :: Apply f0 => (f :.: g) (f0 b) -> f0 ((f :.: g) b)

(Traversable1 f, Traversable1 g) => Traversable1 (Compose f g)

Defined in Data.Semigroup.Traversable.Class

Methods

traverse1 :: Apply f0 => (a -> f0 b) -> Compose f g a -> f0 (Compose f g b)

sequence1 :: Apply f0 => Compose f g (f0 b) -> f0 (Compose f g b)

Traversable1 g => Traversable1 (Joker g a)

Defined in Data.Semigroup.Traversable.Class

Methods

traverse1 :: Apply f => (a0 -> f b) -> Joker g a a0 -> f (Joker g a b)

sequence1 :: Apply f => Joker g a (f b) -> f (Joker g a b)

both :: Bitraversable r => Traversal (r a a) (r b b) a b

Traverse both parts of a Bitraversable container with matching types.

Usually that type will be a pair.

>>> (1,2) & both *~ 10
(10,20)

>>> over both length ("hello","world")
(5,5)

>>> ("hello","world")^.both
"helloworld"

both :: Traversal (a, a)       (b, b)       a b
both :: Traversal (Either a a) (Either b b) a b

both1 :: Bitraversable1 r => Traversal1 (r a a) (r b b) a b

Traverse both parts of a Bitraversable1 container with matching types.

Usually that type will be a pair.

both1 :: Traversal1 (a, a)       (b, b)       a b
both1 :: Traversal1 (Either a a) (Either b b) a b

beside :: (Representable q, Applicative (Rep q), Applicative f, Bitraversable r) => Optical p q f s t a b -> Optical p q f s' t' a b -> Optical p q f (r s s') (r t t') a b

Apply a different Traversal or Fold to each side of a Bitraversable container.

beside :: Traversal s t a b                -> Traversal s' t' a b                -> Traversal (r s s') (r t t') a b
beside :: IndexedTraversal i s t a b       -> IndexedTraversal i s' t' a b       -> IndexedTraversal i (r s s') (r t t') a b
beside :: IndexPreservingTraversal s t a b -> IndexPreservingTraversal s' t' a b -> IndexPreservingTraversal (r s s') (r t t') a b

beside :: Traversal s t a b                -> Traversal s' t' a b                -> Traversal (s,s') (t,t') a b
beside :: Lens s t a b                     -> Lens s' t' a b                     -> Traversal (s,s') (t,t') a b
beside :: Fold s a                         -> Fold s' a                          -> Fold (s,s') a
beside :: Getter s a                       -> Getter s' a                        -> Fold (s,s') a

beside :: IndexedTraversal i s t a b       -> IndexedTraversal i s' t' a b       -> IndexedTraversal i (s,s') (t,t') a b
beside :: IndexedLens i s t a b            -> IndexedLens i s' t' a b            -> IndexedTraversal i (s,s') (t,t') a b
beside :: IndexedFold i s a                -> IndexedFold i s' a                 -> IndexedFold i (s,s') a
beside :: IndexedGetter i s a              -> IndexedGetter i s' a               -> IndexedFold i (s,s') a

beside :: IndexPreservingTraversal s t a b -> IndexPreservingTraversal s' t' a b -> IndexPreservingTraversal (s,s') (t,t') a b
beside :: IndexPreservingLens s t a b      -> IndexPreservingLens s' t' a b      -> IndexPreservingTraversal (s,s') (t,t') a b
beside :: IndexPreservingFold s a          -> IndexPreservingFold s' a           -> IndexPreservingFold (s,s') a
beside :: IndexPreservingGetter s a        -> IndexPreservingGetter s' a         -> IndexPreservingFold (s,s') a

>>> ("hello",["world","!!!"])^..beside id traverse
["hello","world","!!!"]

taking :: (Conjoined p, Applicative f) => Int -> Traversing p f s t a a -> Over p f s t a a

Visit the first n targets of a Traversal, Fold, Getter or Lens.

>>> [("hello","world"),("!!!","!!!")]^.. taking 2 (traverse.both)
["hello","world"]

>>> timingOut $ [1..] ^.. taking 3 traverse
[1,2,3]

>>> over (taking 5 traverse) succ "hello world"
"ifmmp world"

taking :: Int -> Traversal' s a                   -> Traversal' s a
taking :: Int -> Lens' s a                        -> Traversal' s a
taking :: Int -> Iso' s a                         -> Traversal' s a
taking :: Int -> Prism' s a                       -> Traversal' s a
taking :: Int -> Getter s a                       -> Fold s a
taking :: Int -> Fold s a                         -> Fold s a
taking :: Int -> IndexedTraversal' i s a          -> IndexedTraversal' i s a
taking :: Int -> IndexedLens' i s a               -> IndexedTraversal' i s a
taking :: Int -> IndexedGetter i s a              -> IndexedFold i s a
taking :: Int -> IndexedFold i s a                -> IndexedFold i s a

dropping :: (Conjoined p, Applicative f) => Int -> Over p (Indexing f) s t a a -> Over p f s t a a

Visit all but the first n targets of a Traversal, Fold, Getter or Lens.

>>> ("hello","world") ^? dropping 1 both
Just "world"

Dropping works on infinite traversals as well:

>>> [1..] ^? dropping 1 folded
Just 2

dropping :: Int -> Traversal' s a                   -> Traversal' s a
dropping :: Int -> Lens' s a                        -> Traversal' s a
dropping :: Int -> Iso' s a                         -> Traversal' s a
dropping :: Int -> Prism' s a                       -> Traversal' s a
dropping :: Int -> Getter s a                       -> Fold s a
dropping :: Int -> Fold s a                         -> Fold s a
dropping :: Int -> IndexedTraversal' i s a          -> IndexedTraversal' i s a
dropping :: Int -> IndexedLens' i s a               -> IndexedTraversal' i s a
dropping :: Int -> IndexedGetter i s a              -> IndexedFold i s a
dropping :: Int -> IndexedFold i s a                -> IndexedFold i s a

failing :: (Conjoined p, Applicative f) => Traversing p f s t a b -> Over p f s t a b -> Over p f s t a b infixl 5

Try the first Traversal (or Fold), falling back on the second Traversal (or Fold) if it returns no entries.

This is only a valid Traversal if the second Traversal is disjoint from the result of the first or returns exactly the same results. These conditions are trivially met when given a Lens, Iso, Getter, Prism or "affine" Traversal -- one that has 0 or 1 target.

Mutatis mutandis for Fold.

>>> [0,1,2,3] ^? failing (ix 1) (ix 2)
Just 1

>>> [0,1,2,3] ^? failing (ix 42) (ix 2)
Just 2

failing :: Traversal s t a b -> Traversal s t a b -> Traversal s t a b
failing :: Prism s t a b     -> Prism s t a b     -> Traversal s t a b
failing :: Fold s a          -> Fold s a          -> Fold s a

These cases are also supported, trivially, but are boring, because the left hand side always succeeds.

failing :: Lens s t a b      -> Traversal s t a b -> Traversal s t a b
failing :: Iso s t a b       -> Traversal s t a b -> Traversal s t a b
failing :: Equality s t a b  -> Traversal s t a b -> Traversal s t a b
failing :: Getter s a        -> Fold s a          -> Fold s a

If both of the inputs are indexed, the result is also indexed, so you can apply this to a pair of indexed traversals or indexed folds, obtaining an indexed traversal or indexed fold.

failing :: IndexedTraversal i s t a b -> IndexedTraversal i s t a b -> IndexedTraversal i s t a b
failing :: IndexedFold i s a          -> IndexedFold i s a          -> IndexedFold i s a

These cases are also supported, trivially, but are boring, because the left hand side always succeeds.

failing :: IndexedLens i s t a b      -> IndexedTraversal i s t a b -> IndexedTraversal i s t a b
failing :: IndexedGetter i s a        -> IndexedGetter i s a        -> IndexedFold i s a

deepOf :: (Conjoined p, Applicative f) => LensLike f s t s t -> Traversing p f s t a b -> Over p f s t a b

Try the second traversal. If it returns no entries, try again with all entries from the first traversal, recursively.

deepOf :: Fold s s          -> Fold s a                   -> Fold s a
deepOf :: Traversal' s s    -> Traversal' s a             -> Traversal' s a
deepOf :: Traversal s t s t -> Traversal s t a b          -> Traversal s t a b
deepOf :: Fold s s          -> IndexedFold i s a          -> IndexedFold i s a
deepOf :: Traversal s t s t -> IndexedTraversal i s t a b -> IndexedTraversal i s t a b

Indexed Traversals

Common

ignored :: Applicative f => pafb -> s -> f s

This is the trivial empty Traversal.

ignored :: IndexedTraversal i s s a b

ignored ≡ const pure

>>> 6 & ignored %~ absurd
6

class Ord k => TraverseMin k m | m -> k where

Allows IndexedTraversal the value at the smallest index.

Minimal complete definition

traverseMin

Methods

traverseMin :: IndexedTraversal' k (m v) v

IndexedTraversal of the element with the smallest index.

Instances

TraverseMin Int IntMap

Defined in Control.Lens.Traversal

Methods

traverseMin :: (Indexable Int p, Applicative f) => p v (f v) -> IntMap v -> f (IntMap v)

Ord k => TraverseMin k (Map k)

Defined in Control.Lens.Traversal

Methods

traverseMin :: (Indexable k p, Applicative f) => p v (f v) -> Map k v -> f (Map k v)

class Ord k => TraverseMax k m | m -> k where

Allows IndexedTraversal of the value at the largest index.

Minimal complete definition

traverseMax

Methods

traverseMax :: IndexedTraversal' k (m v) v

IndexedTraversal of the element at the largest index.

Instances

TraverseMax Int IntMap

Defined in Control.Lens.Traversal

Methods

traverseMax :: (Indexable Int p, Applicative f) => p v (f v) -> IntMap v -> f (IntMap v)

Ord k => TraverseMax k (Map k)

Defined in Control.Lens.Traversal

Methods

traverseMax :: (Indexable k p, Applicative f) => p v (f v) -> Map k v -> f (Map k v)

traversed :: Traversable f => IndexedTraversal Int (f a) (f b) a b

Traverse any Traversable container. This is an IndexedTraversal that is indexed by ordinal position.

traversed1 :: Traversable1 f => IndexedTraversal1 Int (f a) (f b) a b

Traverse any Traversable1 container. This is an IndexedTraversal1 that is indexed by ordinal position.

traversed64 :: Traversable f => IndexedTraversal Int64 (f a) (f b) a b

Traverse any Traversable container. This is an IndexedTraversal that is indexed by ordinal position.

elementOf :: Applicative f => LensLike (Indexing f) s t a a -> Int -> IndexedLensLike Int f s t a a

Traverse the nth elementOf a Traversal, Lens or Iso if it exists.

>>> [[1],[3,4]] & elementOf (traverse.traverse) 1 .~ 5
[[1],[5,4]]

>>> [[1],[3,4]] ^? elementOf (folded.folded) 1
Just 3

>>> timingOut $ ['a'..] ^?! elementOf folded 5
'f'

>>> timingOut $ take 10 $ elementOf traverse 3 .~ 16 $ [0..]
[0,1,2,16,4,5,6,7,8,9]

elementOf :: Traversal' s a -> Int -> IndexedTraversal' Int s a
elementOf :: Fold s a       -> Int -> IndexedFold Int s a

element :: Traversable t => Int -> IndexedTraversal' Int (t a) a

Traverse the nth element of a Traversable container.

element ≡ elementOf traverse

elementsOf :: Applicative f => LensLike (Indexing f) s t a a -> (Int -> Bool) -> IndexedLensLike Int f s t a a

Traverse (or fold) selected elements of a Traversal (or Fold) where their ordinal positions match a predicate.

elementsOf :: Traversal' s a -> (Int -> Bool) -> IndexedTraversal' Int s a
elementsOf :: Fold s a       -> (Int -> Bool) -> IndexedFold Int s a

elements :: Traversable t => (Int -> Bool) -> IndexedTraversal' Int (t a) a

Traverse elements of a Traversable container where their ordinal positions match a predicate.

elements ≡ elementsOf traverse

Combinators

ipartsOf :: forall i p f s t a. (Indexable [i] p, Functor f) => Traversing (Indexed i) f s t a a -> Over p f s t [a] [a]

An indexed version of partsOf that receives the entire list of indices as its index.

ipartsOf' :: forall i p f s t a. (Indexable [i] p, Functor f) => Over (Indexed i) (Bazaar' (Indexed i) a) s t a a -> Over p f s t [a] [a]

A type-restricted version of ipartsOf that can only be used with an IndexedTraversal.

iunsafePartsOf :: forall i p f s t a b. (Indexable [i] p, Functor f) => Traversing (Indexed i) f s t a b -> Over p f s t [a] [b]

An indexed version of unsafePartsOf that receives the entire list of indices as its index.

iunsafePartsOf' :: forall i s t a b. Over (Indexed i) (Bazaar (Indexed i) a b) s t a b -> IndexedLens [i] s t [a] [b]

itraverseOf :: (Indexed i a (f b) -> s -> f t) -> (i -> a -> f b) -> s -> f t

Traversal with an index.

NB: When you don't need access to the index then you can just apply your IndexedTraversal directly as a function!

itraverseOf ≡ withIndex
traverseOf l = itraverseOf l . const = id

itraverseOf :: Functor f     => IndexedLens i s t a b       -> (i -> a -> f b) -> s -> f t
itraverseOf :: Applicative f => IndexedTraversal i s t a b  -> (i -> a -> f b) -> s -> f t
itraverseOf :: Apply f       => IndexedTraversal1 i s t a b -> (i -> a -> f b) -> s -> f t

iforOf :: (Indexed i a (f b) -> s -> f t) -> s -> (i -> a -> f b) -> f t

Traverse with an index (and the arguments flipped).

forOf l a ≡ iforOf l a . const
iforOf ≡ flip . itraverseOf

iforOf :: Functor f     => IndexedLens i s t a b       -> s -> (i -> a -> f b) -> f t
iforOf :: Applicative f => IndexedTraversal i s t a b  -> s -> (i -> a -> f b) -> f t
iforOf :: Apply f       => IndexedTraversal1 i s t a b -> s -> (i -> a -> f b) -> f t

imapMOf :: Over (Indexed i) (WrappedMonad m) s t a b -> (i -> a -> m b) -> s -> m t

Map each element of a structure targeted by a Lens to a monadic action, evaluate these actions from left to right, and collect the results, with access its position.

When you don't need access to the index mapMOf is more liberal in what it can accept.

mapMOf l ≡ imapMOf l . const

imapMOf :: Monad m => IndexedLens       i s t a b -> (i -> a -> m b) -> s -> m t
imapMOf :: Monad m => IndexedTraversal  i s t a b -> (i -> a -> m b) -> s -> m t
imapMOf :: Bind  m => IndexedTraversal1 i s t a b -> (i -> a -> m b) -> s -> m t

iforMOf :: (Indexed i a (WrappedMonad m b) -> s -> WrappedMonad m t) -> s -> (i -> a -> m b) -> m t

Map each element of a structure targeted by a Lens to a monadic action, evaluate these actions from left to right, and collect the results, with access its position (and the arguments flipped).

forMOf l a ≡ iforMOf l a . const
iforMOf ≡ flip . imapMOf

iforMOf :: Monad m => IndexedLens i s t a b      -> s -> (i -> a -> m b) -> m t
iforMOf :: Monad m => IndexedTraversal i s t a b -> s -> (i -> a -> m b) -> m t

imapAccumROf :: Over (Indexed i) (Backwards (State acc)) s t a b -> (i -> acc -> a -> (acc, b)) -> acc -> s -> (acc, t)

Generalizes mapAccumR to an arbitrary IndexedTraversal with access to the index.

imapAccumROf accumulates state from right to left.

mapAccumROf l ≡ imapAccumROf l . const

imapAccumROf :: IndexedLens i s t a b      -> (i -> acc -> a -> (acc, b)) -> acc -> s -> (acc, t)
imapAccumROf :: IndexedTraversal i s t a b -> (i -> acc -> a -> (acc, b)) -> acc -> s -> (acc, t)

imapAccumLOf :: Over (Indexed i) (State acc) s t a b -> (i -> acc -> a -> (acc, b)) -> acc -> s -> (acc, t)

Generalizes mapAccumL to an arbitrary IndexedTraversal with access to the index.

imapAccumLOf accumulates state from left to right.

mapAccumLOf l ≡ imapAccumLOf l . const

imapAccumLOf :: IndexedLens i s t a b      -> (i -> acc -> a -> (acc, b)) -> acc -> s -> (acc, t)
imapAccumLOf :: IndexedTraversal i s t a b -> (i -> acc -> a -> (acc, b)) -> acc -> s -> (acc, t)

Reflection

traverseBy :: Traversable t => (forall x. x -> f x) -> (forall x y. f (x -> y) -> f x -> f y) -> (a -> f b) -> t a -> f (t b)

Traverse a container using its Traversable instance using explicitly provided Applicative operations. This is like traverse where the Applicative instance can be manually specified.

traverseByOf :: Traversal s t a b -> (forall x. x -> f x) -> (forall x y. f (x -> y) -> f x -> f y) -> (a -> f b) -> s -> f t

Traverse a container using a specified Applicative.

This is like traverseBy where the Traversable instance can be specified by any Traversal

traverseByOf traverse ≡ traverseBy

sequenceBy :: Traversable t => (forall x. x -> f x) -> (forall x y. f (x -> y) -> f x -> f y) -> t (f a) -> f (t a)

Sequence a container using its Traversable instance using explicitly provided Applicative operations. This is like sequence where the Applicative instance can be manually specified.

sequenceByOf :: Traversal s t (f b) b -> (forall x. x -> f x) -> (forall x y. f (x -> y) -> f x -> f y) -> s -> f t

Sequence a container using a specified Applicative.

This is like traverseBy where the Traversable instance can be specified by any Traversal

sequenceByOf traverse ≡ sequenceBy

Implementation Details

newtype Bazaar p a b t

This is used to characterize a Traversal.

a.k.a. indexed Cartesian store comonad, indexed Kleene store comonad, or an indexed FunList.

http://twanvl.nl/blog/haskell/non-regular1

A Bazaar is like a Traversal that has already been applied to some structure.

Where a Context a b t holds an a and a function from b to t, a Bazaar a b t holds N as and a function from N bs to t, (where N might be infinite).

Mnemonically, a Bazaar holds many stores and you can easily add more.

This is a final encoding of Bazaar.

Constructors

Bazaar

Fields

• runBazaar :: forall f. Applicative f => p a (f b) -> f t

Instances

Corepresentable p => Sellable p (Bazaar p)

Defined in Control.Lens.Internal.Bazaar

Methods

sell :: p a (Bazaar p a b b)

Profunctor p => Bizarre p (Bazaar p)

Defined in Control.Lens.Internal.Bazaar

Methods

bazaar :: Applicative f => p a (f b) -> Bazaar p a b t -> f t

Conjoined p => IndexedComonad (Bazaar p)

Defined in Control.Lens.Internal.Bazaar

Methods

iextract :: Bazaar p a a t -> t

iduplicate :: Bazaar p a c t -> Bazaar p a b (Bazaar p b c t)

iextend :: (Bazaar p b c t -> r) -> Bazaar p a c t -> Bazaar p a b r

IndexedFunctor (Bazaar p)

Defined in Control.Lens.Internal.Bazaar

Methods

ifmap :: (s -> t) -> Bazaar p a b s -> Bazaar p a b t

Functor (Bazaar p a b)

Defined in Control.Lens.Internal.Bazaar

Methods

fmap :: (a0 -> b0) -> Bazaar p a b a0 -> Bazaar p a b b0

(<$) :: a0 -> Bazaar p a b b0 -> Bazaar p a b a0

Applicative (Bazaar p a b)

Defined in Control.Lens.Internal.Bazaar

Methods

pure :: a0 -> Bazaar p a b a0

(<*>) :: Bazaar p a b (a0 -> b0) -> Bazaar p a b a0 -> Bazaar p a b b0

liftA2 :: (a0 -> b0 -> c) -> Bazaar p a b a0 -> Bazaar p a b b0 -> Bazaar p a b c

(*>) :: Bazaar p a b a0 -> Bazaar p a b b0 -> Bazaar p a b b0

(<*) :: Bazaar p a b a0 -> Bazaar p a b b0 -> Bazaar p a b a0

(a ~ b, Conjoined p) => Comonad (Bazaar p a b)

Defined in Control.Lens.Internal.Bazaar

Methods

extract :: Bazaar p a b a0 -> a0

duplicate :: Bazaar p a b a0 -> Bazaar p a b (Bazaar p a b a0)

extend :: (Bazaar p a b a0 -> b0) -> Bazaar p a b a0 -> Bazaar p a b b0

(a ~ b, Conjoined p) => ComonadApply (Bazaar p a b)

Defined in Control.Lens.Internal.Bazaar

Methods

(<@>) :: Bazaar p a b (a0 -> b0) -> Bazaar p a b a0 -> Bazaar p a b b0

(@>) :: Bazaar p a b a0 -> Bazaar p a b b0 -> Bazaar p a b b0

(<@) :: Bazaar p a b a0 -> Bazaar p a b b0 -> Bazaar p a b a0

Apply (Bazaar p a b)

Defined in Control.Lens.Internal.Bazaar

Methods

(<.>) :: Bazaar p a b (a0 -> b0) -> Bazaar p a b a0 -> Bazaar p a b b0

(.>) :: Bazaar p a b a0 -> Bazaar p a b b0 -> Bazaar p a b b0

(<.) :: Bazaar p a b a0 -> Bazaar p a b b0 -> Bazaar p a b a0

liftF2 :: (a0 -> b0 -> c) -> Bazaar p a b a0 -> Bazaar p a b b0 -> Bazaar p a b c

type Bazaar' p a = Bazaar p a a

This alias is helpful when it comes to reducing repetition in type signatures.

type Bazaar' p a t = Bazaar p a a t

newtype Bazaar1 p a b t

This is used to characterize a Traversal.

a.k.a. indexed Cartesian store comonad, indexed Kleene store comonad, or an indexed FunList.

http://twanvl.nl/blog/haskell/non-regular1

A Bazaar1 is like a Traversal that has already been applied to some structure.

Where a Context a b t holds an a and a function from b to t, a Bazaar1 a b t holds N as and a function from N bs to t, (where N might be infinite).

Mnemonically, a Bazaar1 holds many stores and you can easily add more.

This is a final encoding of Bazaar1.

Constructors

Bazaar1

Fields

• runBazaar1 :: forall f. Apply f => p a (f b) -> f t

Instances

Corepresentable p => Sellable p (Bazaar1 p)

Defined in Control.Lens.Internal.Bazaar

Methods

sell :: p a (Bazaar1 p a b b)

Profunctor p => Bizarre1 p (Bazaar1 p)

Defined in Control.Lens.Internal.Bazaar

Methods

bazaar1 :: Apply f => p a (f b) -> Bazaar1 p a b t -> f t

Conjoined p => IndexedComonad (Bazaar1 p)

Defined in Control.Lens.Internal.Bazaar

Methods

iextract :: Bazaar1 p a a t -> t

iduplicate :: Bazaar1 p a c t -> Bazaar1 p a b (Bazaar1 p b c t)

iextend :: (Bazaar1 p b c t -> r) -> Bazaar1 p a c t -> Bazaar1 p a b r

IndexedFunctor (Bazaar1 p)

Defined in Control.Lens.Internal.Bazaar

Methods

ifmap :: (s -> t) -> Bazaar1 p a b s -> Bazaar1 p a b t

Functor (Bazaar1 p a b)

Defined in Control.Lens.Internal.Bazaar

Methods

fmap :: (a0 -> b0) -> Bazaar1 p a b a0 -> Bazaar1 p a b b0

(<$) :: a0 -> Bazaar1 p a b b0 -> Bazaar1 p a b a0

(a ~ b, Conjoined p) => Comonad (Bazaar1 p a b)

Defined in Control.Lens.Internal.Bazaar

Methods

extract :: Bazaar1 p a b a0 -> a0

duplicate :: Bazaar1 p a b a0 -> Bazaar1 p a b (Bazaar1 p a b a0)

extend :: (Bazaar1 p a b a0 -> b0) -> Bazaar1 p a b a0 -> Bazaar1 p a b b0

(a ~ b, Conjoined p) => ComonadApply (Bazaar1 p a b)

Defined in Control.Lens.Internal.Bazaar

Methods

(<@>) :: Bazaar1 p a b (a0 -> b0) -> Bazaar1 p a b a0 -> Bazaar1 p a b b0

(@>) :: Bazaar1 p a b a0 -> Bazaar1 p a b b0 -> Bazaar1 p a b b0

(<@) :: Bazaar1 p a b a0 -> Bazaar1 p a b b0 -> Bazaar1 p a b a0

Apply (Bazaar1 p a b)

Defined in Control.Lens.Internal.Bazaar

Methods

(<.>) :: Bazaar1 p a b (a0 -> b0) -> Bazaar1 p a b a0 -> Bazaar1 p a b b0

(.>) :: Bazaar1 p a b a0 -> Bazaar1 p a b b0 -> Bazaar1 p a b b0

(<.) :: Bazaar1 p a b a0 -> Bazaar1 p a b b0 -> Bazaar1 p a b a0

liftF2 :: (a0 -> b0 -> c) -> Bazaar1 p a b a0 -> Bazaar1 p a b b0 -> Bazaar1 p a b c

type Bazaar1' p a = Bazaar1 p a a

This alias is helpful when it comes to reducing repetition in type signatures.

type Bazaar1' p a t = Bazaar1 p a a t

loci :: Traversal (Bazaar (->) a c s) (Bazaar (->) b c s) a b

This Traversal allows you to traverse the individual stores in a Bazaar.

iloci :: IndexedTraversal i (Bazaar (Indexed i) a c s) (Bazaar (Indexed i) b c s) a b

This IndexedTraversal allows you to traverse the individual stores in a Bazaar with access to their indices.

Fusion

confusing :: Applicative f => LensLike (Curried (Yoneda f) (Yoneda f)) s t a b -> LensLike f s t a b

Fuse a Traversal by reassociating all of the (<*>) operations to the left and fusing all of the fmap calls into one. This is particularly useful when constructing a Traversal using operations from GHC.Generics.

Given a pair of Traversals foo and bar,

confusing (foo.bar) = foo.bar

However, foo and bar are each going to use the Applicative they are given.

confusing exploits the Yoneda lemma to merge their separate uses of fmap into a single fmap. and it further exploits an interesting property of the right Kan lift (or Curried) to left associate all of the uses of (<*>) to make it possible to fuse together more fmaps.

This is particularly effective when the choice of functor f is unknown at compile time or when the Traversal foo.bar in the above description is recursive or complex enough to prevent inlining.

fusing is a version of this combinator suitable for fusing lenses.

confusing :: Traversal s t a b -> Traversal s t a b