lens-4.11: Lenses, Folds and Traversals

Control.Lens.Traversal

Description

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.

Synopsis

Traversals

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

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

type Traversal1' s a = Traversal1 s s a a Source

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Traversing and Lensing

traverseOf :: Over p f s t a b -> p a (f b) -> s -> f t Source

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
((),(),())
traverseOfid
itraverseOf l ≡ traverseOf l . Indexed
itraverseOf itraverseditraverse

This yields the obvious law:

traversetraverseOf 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 Source

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.

forOfflip
forOfflip . traverseOf
forforOf 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 Source

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)]
sequenceAsequenceAOf traversetraverse 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 Source

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)]
mapMmapMOf 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 Source

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)]
forMforMOf 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 Source

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)]
sequencesequenceOf traverse
sequenceOf l ≡ mapMOf l id
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] Source

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]]
transposetransposeOf 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) Source

This generalizes mapAccumL to an arbitrary Traversal.

mapAccumLmapAccumLOf 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) Source

This generalizes mapAccumR to an arbitrary Traversal.

mapAccumRmapAccumROf 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 Source

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

scanr1scanr1Of 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 Source

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

scanl1scanl1Of 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 Source

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 Source

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 Source

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 (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 Source

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 Source

Clone an IndexedTraversal yielding an IndexedTraversal with the same index.

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

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

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

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 Source

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] Source

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] Source

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] Source

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] Source

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

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 Source

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
*** Exception: singular: empty traversal
>>> 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 :: (Conjoined p, Functor f) => Traversing p f s t a b -> Over p f s t a b Source

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 traverse .~ 0
*** Exception: unsafeSingular: empty traversal
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.

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.

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 Indentity 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:

Minimal complete definition

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 Digit Traversable Node Traversable Elem Traversable FingerTree Traversable IntMap Traversable Tree Traversable Seq Traversable ViewL Traversable ViewR Traversable Min Traversable Max Traversable First Traversable Last Traversable Option Traversable NonEmpty Traversable Identity Traversable Vector Traversable Deque Traversable (Either a) Traversable ((,) a) Ix i => Traversable (Array i) Traversable (Const m) Traversable (Proxy *) Bitraversable p => Traversable (Join p) Traversable (Map k) Traversable (Arg a) Traversable f => Traversable (IdentityT f) Traversable f => Traversable (ListT f) Traversable f => Traversable (MaybeT f) Traversable f => Traversable (Reverse f) Traversable (Constant a) Traversable f => Traversable (Backwards f) Traversable (HashMap k) Traversable f => Traversable (Lift f) Traversable (Level i) Traversable f => Traversable (Yoneda f) Traversable f => Traversable (Cofree f) Traversable f => Traversable (Free f) (Traversable f, Traversable g) => Traversable (Compose f g) Traversable (Tagged k s) Traversable f => Traversable (WriterT w f) Traversable f => Traversable (WriterT w f) Traversable f => Traversable (ErrorT e f) Traversable f => Traversable (ExceptT e f) (Traversable f, Traversable g) => Traversable (Sum f g) (Traversable f, Traversable g) => Traversable (Product f g) Traversable w => Traversable (EnvT e w) Traversable (Forget r a) Traversable f => Traversable (AlongsideRight f a) Traversable f => Traversable (AlongsideLeft f b) (Traversable f, Traversable g) => Traversable (Coproduct f g) (Monad m, Traversable m, Traversable f) => Traversable (FreeT f m) Traversable f => Traversable (FreeF f a) Typeable ((* -> *) -> Constraint) Traversable Traversable (Magma i t b)

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 Tree Traversable1 NonEmpty Traversable1 Identity Traversable1 ((,) a) Bitraversable1 p => Traversable1 (Join p) Traversable1 f => Traversable1 (IdentityT f) Traversable1 f => Traversable1 (Reverse f) Traversable1 f => Traversable1 (Backwards f) Traversable1 f => Traversable1 (Lift f) Traversable1 f => Traversable1 (Yoneda f) Traversable1 f => Traversable1 (Cofree f) Traversable1 f => Traversable1 (Free f) Traversable1 g => Traversable1 (Joker g a) (Traversable1 f, Traversable1 g) => Traversable1 (Compose f g) (Traversable1 f, Traversable1 g) => Traversable1 (Sum f g) (Traversable1 f, Traversable1 g) => Traversable1 (Product f g) Traversable1 f => Traversable1 (AlongsideRight f a) Traversable1 f => Traversable1 (AlongsideLeft f b) (Traversable1 f, Traversable1 g) => Traversable1 (Coproduct f g)

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

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 Source

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 Source

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 Source

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 Source

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 Source

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 Source

This is the trivial empty Traversal.

ignored :: IndexedTraversal i s s a b
ignoredconst pure
>>> 6 & ignored %~ absurd
6

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

Allows IndexedTraversal the value at the smallest index.

Methods

traverseMin :: IndexedTraversal' k (m v) v Source

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 Source

Allows IndexedTraversal of the value at the largest index.

Methods

traverseMax :: IndexedTraversal' k (m v) v Source

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 Source

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 Source

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 Source

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 Source

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 Source

Traverse the nth element of a Traversable container.

elementelementOf traverse

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

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 Source

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

elementselementsOf 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] Source

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] Source

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] Source

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] Source

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

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!

itraverseOfwithIndex
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 Source

Traverse with an index (and the arguments flipped).

forOf l a ≡ iforOf l a . const
iforOfflip . 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 Source

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 Source

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
iforMOfflip . 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) Source

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) Source

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 Source

This is used to characterize a Traversal.

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

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 FieldsrunBazaar :: 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) => ComonadApply (Bazaar p a b) ((~) * a b, Conjoined p) => Comonad (Bazaar p a b) Apply (Bazaar p a b)

type Bazaar' p a = Bazaar p a a Source

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 Source

This is used to characterize a Traversal.

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

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 FieldsrunBazaar1 :: 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) => ComonadApply (Bazaar1 p a b) ((~) * a b, Conjoined p) => Comonad (Bazaar1 p a b) Apply (Bazaar1 p a b)

type Bazaar1' p a = Bazaar1 p a a Source

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 Source

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 Source

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

Fusion

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

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 Rift) 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