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

While a Traversal isn't quite a Fold, it _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 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 Gettable,
• a Lens if p is only a Functor,
• a Fold if f is Gettable and Applicative.

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

forOf :: Over p f s t a b -> s -> p 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 :: Profunctor p => Over p (WrappedMonad m) s t a b -> p 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 :: Profunctor p => Over p (WrappedMonad m) s t a b -> s -> p 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 :: Conjoined p => Over p (State acc) s t a b -> p 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 :: Conjoined p => Over p (Backwards (State acc)) s t a b -> p 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 :: (Profunctor p, Alternative m) => Over p ((,) Any) s t a b -> p 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 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

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)

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 :: forall p s t a. 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 = 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]

singular :: (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

>>> [] ^. singular _head
*** Exception: singular: empty traversal

>>> Left 4 ^. singular _Left
4

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

 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 :: (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.

 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.

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).

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.

Instances

Traversable []

Traversable Maybe

Traversable Identity

Traversable Digit

Traversable Node

Traversable Elem

Traversable FingerTree

Traversable Tree

Traversable Seq

Traversable ViewL

Traversable ViewR

Traversable IntMap

Traversable Min

Traversable Max

Traversable First

Traversable Last

Traversable Option

Traversable NonEmpty

Traversable Vector

Traversable Deque

Traversable (Either a)

Traversable ((,) b)

Ix i => Traversable (Array i)

Traversable (Const m)

Traversable f => Traversable (IdentityT f)

Traversable (Map k)

Traversable f => Traversable (Cofree f)

Traversable f => Traversable (Free f)

Traversable f => Traversable (ListT f)

Traversable (Proxy *)

Traversable f => Traversable (Reverse f)

Traverse from right to left.

Traversable f => Traversable (Backwards f)

Derived instance.

Traversable f => Traversable (MaybeT f)

Traversable (Constant a)

Traversable (HashMap k)

Traversable (Level i)

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

Traversable w => Traversable (EnvT e w)

Traversable f => Traversable (FreeF f a)

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

Traversable f => Traversable (ExceptT e f)

Traversable f => Traversable (ErrorT e f)

Traversable f => Traversable (WriterT w f)

Traversable f => Traversable (WriterT w f)

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

Traversable (Forget r a)

Traversable (Tagged k s)

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

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

Traversable f => Traversable (AlongsideRight f a)

Traversable f => Traversable (AlongsideLeft f b)

Traversable (Magma i t b)

class (Foldable1 t, Traversable t) => Traversable1 t where

Methods

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

Instances

Traversable1 Identity

Traversable1 Tree

Traversable1 NonEmpty

Traversable1 ((,) a)

Traversable1 f => Traversable1 (IdentityT f)

Traversable1 f => Traversable1 (Cofree f)

Traversable1 f => Traversable1 (Free f)

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

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

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

Traversable1 f => Traversable1 (AlongsideRight f a)

Traversable1 f => Traversable1 (AlongsideLeft f 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

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 :: 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","!!!"]

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 -> Traversing p f s t a b -> Over p f s t a b

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.

 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 for 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.

Methods

traverseMin :: IndexedTraversal' k (m v) v

IndexedTraversal of the element with the smallest index.

Instances

TraverseMin Int IntMap

Ord k => TraverseMin k (Map k)

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

Allows IndexedTraversal of the value at the largest index.

Methods

traverseMax :: IndexedTraversal' k (m v) v

IndexedTraversal of the element at the largest index.

Instances

TraverseMax Int IntMap

Ord k => TraverseMax k (Map k)

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 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

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

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)

Profunctor p => Bizarre p (Bazaar p)

Conjoined p => IndexedComonad (Bazaar p)

IndexedFunctor (Bazaar p)

Functor (Bazaar p a b)

Applicative (Bazaar p a b)

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

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

Apply (Bazaar p a b)

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)

Profunctor p => Bizarre1 p (Bazaar1 p)

Conjoined p => IndexedComonad (Bazaar1 p)

IndexedFunctor (Bazaar1 p)

Functor (Bazaar1 p a b)

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

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

Apply (Bazaar1 p a b)

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.