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' = Simple Traversal

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.

 type IndexedTraversal' i = Simple (IndexedTraversal i)

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

 type ATraversal' = Simple ATraversal

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

 type AnIndexedTraversal' = Simple (AnIndexedTraversal i)

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 Gettable,
• a Lens if p is only a Functor,
• a Fold if f is Gettable and Applicative.

 type Traversing' f = Simple (Traversing f)

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

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 :: Applicative f => Traversal s t a b -> (a -> f b) -> s -> 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

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

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 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

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

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)]

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)

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)

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

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

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)

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 reflectTraversal 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 (coerced (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

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

Clone an IndexedTraversal yielding an IndexedTraversal with the same index.

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.

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

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]

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

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]

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 = childrenOf 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]

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 or a non-empty MonadicFold into an Action.

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

 singular :: Traversal s t a a          -> Lens s t a a
 singular :: Fold s a                   -> Getter s a
 singular :: MonadicFold m s a          -> Action m s a
 singular :: IndexedTraversal i s t a a -> IndexedLens i s t a a
 singular :: IndexedFold i s a          -> IndexedGetter i s a
 singular :: IndexedMonadicFold i m s a -> IndexedAction i m s a

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 or a MonadicFold into an Action.

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

 unsafeSingular :: Traversal s t a b          -> Lens s t a b
 unsafeSingular :: Fold s a                   -> Getter s a
 unsafeSingular :: MonadicFold m s a          -> Action m s a
 unsafeSingular :: IndexedTraversal i s t a b -> IndexedLens i s t a b
 unsafeSingular :: IndexedFold i s a          -> IndexedGetter i s a
 unsafeSingular :: IndexedMonadicFold i m s a -> IndexedAction i m s a

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

Minimal complete definition: traverse or sequenceA.

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).

Traverse both parts of a tuple with matching types.

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

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

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

Apply a different Traversal or Fold to each side of a tuple.

 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 :: Action m s a                     -> Action m s' a                      -> MonadicFold m (s,s') a
 beside :: MonadicFold m s a                -> MonadicFold m s' a                 -> MonadicFold m (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 :: IndexedAction i m s a            -> IndexedAction i m s' a             -> IndexedMonadicFold i m (s,s') a
 beside :: IndexedMonadicFold i m s a       -> IndexedMonadicFold i m s' a        -> IndexedMonadicFold i m (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
 beside :: IndexPreservingAction m s a      -> IndexPreservingAction m s' a       -> IndexPreservingMonadicFold m (s,s') a
 beside :: IndexPreservingMonadicFold m s a -> IndexPreservingMonadicFold m s' a  -> IndexPreservingMonadicFold m (s,s') a

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

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 -> Action m s a                     -> MonadicFold m s a
 taking :: Int -> MonadicFold m s a                -> MonadicFold m 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
 taking :: Int -> IndexedAction i m s a            -> IndexedMonadicFold i m s a
 taking :: Int -> IndexedMonadicFold i m s a       -> IndexedMonadicFold i m s 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 -> Action m s a                     -> MonadicFold m s a
 dropping :: Int -> MonadicFold m s a                -> MonadicFold m 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
 dropping :: Int -> IndexedAction i m s a            -> IndexedMonadicFold i m s a
 dropping :: Int -> IndexedMonadicFold i m s a       -> IndexedMonadicFold i m s a

This is the trivial empty Traversal.

 ignored :: IndexedTraversal i s s a b

 ignored ≡ const pure

>>> 6 & ignored %~ absurd
6

Allows IndexedTraversal the value at the smallest index.

Allows IndexedTraversal of the value at the largest index.

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

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

Traverse the nth element 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

Traverse the nth element of a Traversable container.

 element ≡ elementOf traverse

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

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

 elements ≡ elementsOf traverse

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

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

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

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

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

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

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

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)

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)

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.

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

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

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

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