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.

Lenses

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!

Traversing and Lensing

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

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

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

traverseOf ≡ id

This yields the obvious law:

traverse ≡ traverseOf traverse

 traverseOf :: Iso s t a b       -> (a -> f b) -> s -> f t
 traverseOf :: Lens s t a b      -> (a -> f b) -> s -> f t
 traverseOf :: Traversal s t a b -> (a -> f b) -> s -> f t

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

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

forOf l ≡ flip (traverseOf l)

 for ≡ forOf traverse

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

 forOf ≡ flip

 forOf :: Iso s t a b -> s -> (a -> f b) -> f t
 forOf :: Lens s t a b -> s -> (a -> f b) -> f t
 forOf :: 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.

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

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

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

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

mapM ≡ mapMOf traverse

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

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

forMOf is a flipped version of mapMOf, consistent with the definition of forM. forM ≡ forMOf traverse forMOf l ≡ flip (mapMOf l)

 forMOf ::            Iso s t a b       -> s -> (a -> m b) -> m t
 forMOf ::            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.

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

 sequenceOf ::            Iso s t (m b) b       -> s -> m t
 sequenceOf ::            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:

transpose ≡ transposeOf traverse

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

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

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

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

This generalizes mapAccumL to an arbitrary Traversal.

mapAccumL ≡ mapAccumLOf traverse

mapAccumLOf accumulates state from left to right.

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

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

This generalizes mapAccumR to an arbitrary Traversal.

mapAccumR ≡ mapAccumROf traverse

mapAccumROf accumulates state from right to left.

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

scanr1Of :: LensLike (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 (Backwards (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

 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

Parts and Holes

partsOf :: Functor f => LensLike (BazaarT a a 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.

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

partsOf' :: LensLike (Bazaar a a) 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 => LensLike (BazaarT a b 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' :: LensLike (Bazaar a b) s t a b -> Lens s t [a] [b]

holesOf :: LensLike (Bazaar a a) s t a a -> s -> [Context 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 :: Simple Iso s a       -> s -> [Context a a s]
 holesOf :: Simple Lens s a      -> s -> [Context a a s]
 holesOf :: Simple Traversal s a -> s -> [Context a a s]

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 NonEmpty

Traversable Vector

Traversable Level

(Functor (Array i), Foldable (Array i), Ix i) => Traversable (Array i)

(Functor (IdentityT f), Foldable (IdentityT f), Traversable f) => Traversable (IdentityT f)

(Functor (Map k), Foldable (Map k)) => Traversable (Map k)

(Functor (ListT f), Foldable (ListT f), Traversable f) => Traversable (ListT f)

(Functor (Backwards f), Foldable (Backwards f), Traversable f) => Traversable (Backwards f)

Derived instance.

(Functor (MaybeT f), Foldable (MaybeT f), Traversable f) => Traversable (MaybeT f)

(Functor (HashMap k), Foldable (HashMap k)) => Traversable (HashMap k)

(Functor (EnvT e w), Foldable (EnvT e w), Traversable w) => Traversable (EnvT e w)

(Functor (ErrorT e f), Foldable (ErrorT e f), Traversable f) => Traversable (ErrorT e f)

(Functor (WriterT w f), Foldable (WriterT w f), Traversable f) => Traversable (WriterT w f)

(Functor (WriterT w f), Foldable (WriterT w f), Traversable f) => Traversable (WriterT w f)

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

_left :: Traversal (Either a c) (Either b c) a b

A traversal for tweaking the left-hand value of an Either:

>>> over _left (+1) (Left 2)
Left 3

>>> over _left (+1) (Right 2)
Right 2

>>> Right 42 ^._left :: String
""

>>> Left "hello" ^._left
"hello"

_left :: Applicative f => (a -> f b) -> Either a c -> f (Either b c)

_right :: Traversal (Either c a) (Either c b) a b

traverse the right-hand value of an Either:

_right ≡ traverse

Unfortunately the instance for Traversable (Either c) is still missing from base, so this can't just be traverse

>>> over _right (+1) (Left 2)
Left 2

>>> over _right (+1) (Right 2)
Right 3

>>> Right "hello" ^._right
"hello"

>>> Left "hello" ^._right :: [Double]
[]

_right :: Applicative f => (a -> f b) -> Either c a -> f (Either c a)

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

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"

beside :: Applicative f => LensLike f s t a b -> LensLike f s' t' a b -> LensLike f (s, s') (t, t') a b

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

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

taking :: Applicative f => Int -> SimpleLensLike (BazaarT a a f) s a -> SimpleLensLike f s a

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

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

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

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

dropping :: Applicative f => Int -> SimpleLensLike (Indexing f) s a -> SimpleLensLike f 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

Cloning Traversals

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

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 ReifyTraversal and use 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 (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

data ReifiedTraversal s t a b

A form of Traversal that can be stored monomorphically in a container.

Constructors

ReifyTraversal

Fields

reflectTraversal :: Traversal s t a b

Simple

type SimpleTraversal s a = Traversal s s a a

type SimpleTraversal = Simple Traversal

type SimpleReifiedTraversal s a = ReifiedTraversal s s a a

type SimpleReifiedTraversal = Simple ReifiedTraversal

Exposed Implementation Details

newtype Bazaar 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

Bazaar a b t is isomorphic to data Bazaar a b t = Buy t | Trade (Bazaar a b (b -> t)) a, and to exists s. (s, Traversal s t a b).

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.

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 => (a -> f b) -> f t

Instances

Functor (Bazaar a b)

Functor (Bazaar a b) => Applicative (Bazaar a b)

(Functor (Bazaar a b), ~ * a b) => Comonad (Bazaar a b)

(Comonad (Bazaar a b), ~ * a b) => ComonadApply (Bazaar a b)