Control.Lens

This package provides lens families, setters, getters, traversals, isomorphisms, and folds that can all be composed automatically with each other (and other lenses from other van Laarhoven lens libraries) using (.) from Prelude, while reducing the complexity of the API.

For a longer description and motivation of why you should care about lens families, see http://comonad.com/reader/2012/mirrored-lenses/.

Note: If you merely want your library to provide lenses you may not have to actually import any lens library. For, say, a Simple Lens Bar Foo, just export a function with the signature:

 foo :: Functor f => (Foo -> f Foo) -> Bar -> f Bar

and then you can compose it with other lenses with (.) without needing anything from this library at all.

Usage:

You can derive lenses automatically for many data types:

 import Control.Lens.TH
 data Foo a = Foo { _fooArgs :: [String], _fooValue :: a }
 makeLenses ''Foo

This defines the following lenses:

 fooArgs :: Simple Lens (Foo a) [String]
 fooValue :: Lens (Foo a) (Foo b) a b

The combinators here have unusually specific type signatures, so for particularly tricky ones, I've tried to list the simpler type signatures you might want to pretend the combinators have.

Lenses

type Lens a b c d = forall f. Functor f => (c -> f d) -> a -> f b

A Lens is actually a lens family as described in http://comonad.com/reader/2012/mirrored-lenses/.

With great power comes great responsibility and a Lens is subject to the three common sense lens laws:

1) You get back what you put in:

 view l (set l b a)  = b

2) Putting back what you got doesn't change anything:

 set l (view l a) a  = a

3) Setting twice is the same as setting once:

 set l c (set l b a) = set l c a

These laws are strong enough that the 4 type parameters of a Lens cannot vary fully independently. For more on how they interact, read the Why is it a Lens Family? section of http://comonad.com/reader/2012/mirrored-lenses/.

Every Lens can be used directly as a Setter or Traversal.

You can also use a Lens for Getting as if it were a Fold or Getter.

 type Lens = forall f. Functor f => LensLike f a b c d

type LensLike f a b c d = (c -> f d) -> a -> f b

Many combinators that accept a Lens can also accept a Traversal in limited situations.

They do so by specializing the type of Functor that they require of the caller.

If a function accepts a LensLike f a b c d for some Functor f, then they may be passed a Lens.

Further, if f is an Applicative, they may also be passed a Traversal.

type Traversal a b c d = forall f. Applicative f => (c -> f d) -> a -> f b

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.

type Simple f a b = f a a b b

A Simple Lens, Simple Traversal, ... can be used instead of a Lens,Traversal, ... whenever the type variables don't change upon setting a value.

 imaginary :: Simple Lens (Complex a) a
 traverseHead :: Simple Traversal [a] a

Note: To use this alias in your own code with LensLike f or Setter, you may have to turn on LiberalTypeSynonyms.

type SimpleLens a b = Lens a a b b

 type SimpleLens = Simple Lens

type SimpleTraversal a b = Traversal a a b b

 type SimpleTraversal = Simple Traversal

type SimpleLensLike f a b = LensLike f a a b b

 type SimpleLensLike f = Simple (LensLike f)

(%%~) :: LensLike f a b c d -> (c -> f d) -> a -> f b

(%%~) can be used in one of two scenarios:

When applied to a Lens, it can edit the target of the Lens in a structure, extracting a functorial result.

When applied to a Traversal, it can edit the targets of the Traversals, extracting an applicative summary of its actions.

For all that the definition of this combinator is just:

 (%%~) = id

 (%%~) :: Functor f =>     Iso a b c d       -> (c -> f d) -> a -> f b
 (%%~) :: Functor f =>     Lens a b c d      -> (c -> f d) -> a -> f b
 (%%~) :: Applicative f => Traversal a b c d -> (c -> f d) -> a -> f b

It may be beneficial to think about it as if it had these even more restrictive types, however:

When applied to a Traversal, it can edit the targets of the Traversals, extracting a supplemental monoidal summary of its actions, by choosing f = ((,) m)

 (%%~) ::             Iso a b c d       -> (c -> (e, d)) -> a -> (e, b)
 (%%~) ::             Lens a b c d      -> (c -> (e, d)) -> a -> (e, b)
 (%%~) :: Monoid m => Traversal a b c d -> (c -> (m, d)) -> a -> (m, b)

(%%=) :: MonadState a m => LensLike ((,) e) a a c d -> (c -> (e, d)) -> m e

Modify the target of a Lens in the current state returning some extra information of c or modify all targets of a Traversal in the current state, extracting extra information of type c and return a monoidal summary of the changes.

 (%%=) = (state.)

It may be useful to think of (%%=), instead, as having either of the following more restricted type signatures:

 (%%=) :: MonadState a m             => Iso a a c d       -> (c -> (e, d) -> m e
 (%%=) :: MonadState a m             => Lens a a c d      -> (c -> (e, d) -> m e
 (%%=) :: (MonadState a m, Monoid e) => Traversal a a c d -> (c -> (e, d) -> m e

lens :: (a -> c) -> (a -> d -> b) -> Lens a b c d

Build a Lens from a getter and a setter.

 lens :: Functor f => (a -> c) -> (a -> d -> b) -> (c -> f d) -> a -> f b

Common Lenses

_1 :: Lens (a, c) (b, c) a b

This is a lens that can change the value (and type) of the first field of a pair.

 ghci> (1,2)^._1
 1

 ghci> _1 +~ "hello" $ (1,2)
 ("hello",2)

 _1 :: Functor f => (a -> f b) -> (a,c) -> f (a,c)

_2 :: Lens (c, a) (c, b) a b

As _1, but for the second field of a pair.

 anyOf _2 :: (c -> Bool) -> (a, c) -> Bool
 traverse._2 :: (Applicative f, Traversable t) => (a -> f b) -> t (c, a) -> f (t (c, b))
 foldMapOf (traverse._2) :: (Traversable t, Monoid m) => (c -> m) -> t (b, c) -> m

 _2 :: Functor f => (a -> f b) -> (c,a) -> f (c,b)

resultAt :: Eq e => e -> Simple Lens (e -> a) a

This lens can be used to change the result of a function but only where the arguments match the key given.

Isomorphisms

type Iso a b c d = forall k f. (Isomorphic k, Functor f) => k (c -> f d) (a -> f b)

Isomorphim families can be composed with other lenses using either' (.)' and id from the Prelude or from Control.Category. However, if you compose them with each other using '(.)' from the Prelude, they will be dumbed down to a mere Lens.

 import Control.Category
 import Prelude hiding ((.),id)

 type Iso a b c d = forall k f. (Isomorphic k, Functor f) => IsoLike k f a b c d

type SimpleIso a b = Iso a a b b

 type SimpleIso a b = Simple Iso a b

type IsoLike k f a b c d = k (c -> f d) (a -> f b)

 type LensLike f a b c d = IsoLike (->) f a b c d

type SimpleIsoLike k f a b = IsoLike k f a a b b

 type SimpleIsoLike k f a b = Simple (IsoLike k f) a b

iso :: (Isomorphic k, Functor f) => (a -> b) -> (b -> a) -> SimpleIsoLike k f a b

Build a simple isomorphism from a pair of inverse functions

 iso :: (a -> b) -> (b -> a) -> Simple Iso a b

isos :: (Isomorphic k, Functor f) => (a -> c) -> (c -> a) -> (b -> d) -> (d -> b) -> IsoLike k f a b c d

Build an isomorphism family from two pairs of inverse functions

 isos :: (a -> c) -> (c -> a) -> (b -> d) -> (d -> b) -> Iso a b c d

class Category k => Isomorphic k where

Used to provide overloading of isomorphism application

This is a Category with a canonical mapping to it from the category of isomorphisms over Haskell types.

Methods

isomorphic :: (a -> b) -> (b -> a) -> k a b

Build this morphism out of an isomorphism

The intention is that by using isomorphic, you can supply both halves of an isomorphism, but k can be instantiated to (->), so you can freely use the resulting isomorphism as a function.

isomap :: ((a -> b) -> c -> d) -> ((b -> a) -> d -> c) -> k a b -> k c d

Map a morphism in the target category using an isomorphism between morphisms in Hask.

Instances

Isomorphic (->)

Isomorphic Isomorphism

from :: Isomorphic k => Isomorphism a b -> k b a

Invert an isomorphism.

Note to compose an isomorphism and receive an isomorphism in turn you'll need to use Category

 from (from l) = l

If you imported 'Control.Category.(.)', then:

 from l . from r = from (r . l)

 from :: (a :~> b) -> (b :~> a)

Setters

type Setter a b c d = (c -> Identity d) -> a -> Identity b

The only Lens-like law that can apply to a Setter l is that

 set l c (set l b a) = set l c a

You can't view a Setter in general, so the other two laws are irrelevant.

However, two Functor laws apply to a Setter

 adjust l id = id
 adjust l f . adjust l g = adjust l (f . g)

You can compose a Setter with a Lens or a Traversal using (.) from the Prelude and the result is always only a Setter and nothing more.

 type Setter a b c d = LensLike Identity a b c d

type SimpleSetter a b = Lens a a b b

This alias is supplied for those who don't want to use LiberalTypeSynonyms with Simple.

 'SimpleSetter ' = 'Simple' 'Setter'

sets :: Isomorphic k => k ((c -> d) -> a -> b) (Setter a b c d)

Build a Setter.

 sets . adjust = id
 adjust . sets = id
 sets = from adjust
 adjust = from sets

 sets :: ((c -> d) -> a -> b) -> Setter a b c d

mapped :: Functor f => Setter (f a) (f b) a b

This setter can be used to map over all of the values in a Functor.

 fmap        = adjust mapped
 fmapDefault = adjust traverse
 (<$)        = set mapped

adjust :: Isomorphic k => k (Setter a b c d) ((c -> d) -> a -> b)

Modify the target of a Lens or all the targets of a Setter or Traversal with a function.

 fmap        = adjust mapped
 fmapDefault = adjust traverse

 sets . adjust = id
 adjust . sets = id

 adjust :: Setter a b c d -> (c -> d) -> a -> b

mapOf :: Isomorphic k => k (Setter a b c d) ((c -> d) -> a -> b)

Modify the target of a Lens or all the targets of a Setter or Traversal with a function. This is an alias for adjust that is provided for consistency.

 mapOf = adjust

 fmap        = mapOf mapped
 fmapDefault = mapOf traverse

 sets . mapOf = id
 mapOf . sets = id

 mapOf :: Setter a b c d    -> (c -> d) -> a -> b
 mapOf :: Iso a b c d       -> (c -> d) -> a -> b
 mapOf :: Lens a b c d      -> (c -> d) -> a -> b
 mapOf :: Traversal a b c d -> (c -> d) -> a -> b

set :: Setter a b c d -> d -> a -> b

Replace the target of a Lens or all of the targets of a Setter or Traversal with a constant value.

 (<$) = set mapped

 set :: Setter a b c d    -> d -> a -> b
 set :: Iso a b c d       -> d -> a -> b
 set :: Lens a b c d      -> d -> a -> b
 set :: Traversal a b c d -> d -> a -> b

whisper :: (MonadWriter b m, Monoid a) => Setter a b c d -> d -> m ()

Tell a part of a value to a MonadWriter, filling in the rest from mempty

 whisper l d = tell (set l d mempty)

(^~) :: Setter a b c d -> d -> a -> b

Replace the target of a Lens or all of the targets of a Setter or Traversal with a constant value.

This is an infix version of set

 f <$ a = mapped ^~ f $ a

 ghci> bitAt 0 ^~ True $ 0
 1

 (^~) :: Setter a b c d    -> d -> a -> b
 (^~) :: Iso a b c d       -> d -> a -> b
 (^~) :: Lens a b c d      -> d -> a -> b
 (^~) :: Traversal a b c d -> d -> a -> b

(%~) :: Setter a b c d -> (c -> d) -> a -> b

Modifies the target of a Lens or all of the targets of a Setter or Traversal with a user supplied function.

This is an infix version of adjust

 fmap f = mapped %~ f
 fmapDefault f = traverse %~ f

 ghci> _2 %~ length $ (1,"hello")
 (1,5)

 (%~) :: Setter a b c d    -> (c -> d) -> a -> b
 (%~) :: Iso a b c d       -> (c -> d) -> a -> b
 (%~) :: Lens a b c d      -> (c -> d) -> a -> b
 (%~) :: Traversal a b c d -> (c -> d) -> a -> b

(^=) :: MonadState a m => Setter a a c d -> d -> m ()

Replace the target of a Lens or all of the targets of a Setter or Traversal in our monadic state with a new value, irrespective of the old.

 (^=) :: MonadState a m => Iso a a c d       -> d -> m ()
 (^=) :: MonadState a m => Lens a a c d      -> d -> m ()
 (^=) :: MonadState a m => Traversal a a c d -> d -> m ()
 (^=) :: MonadState a m => Setter a a c d    -> d -> m ()

(%=) :: MonadState a m => Setter a a c d -> (c -> d) -> m ()

Map over the target of a Lens or all of the targets of a Setter or 'Traversal in our monadic state.

 (%=) :: MonadState a m => Iso a a c d       -> (c -> d) -> m ()
 (%=) :: MonadState a m => Lens a a c d      -> (c -> d) -> m ()
 (%=) :: MonadState a m => Traversal a a c d -> (c -> d) -> m ()
 (%=) :: MonadState a m => Setter a a c d    -> (c -> d) -> m ()

Getters and Folds

type Getter a c = forall r b d. (c -> Const r d) -> a -> Const r b

A Getter describes how to retrieve a single value in a way that can be composed with other lens-like constructions.

Unlike a Lens a Getter is read-only. Since a Getter cannot be used to write back there are no lens laws that can be applied to it.

Moreover, a Getter can be used directly as a Fold, since it just ignores the Monoid.

In practice the b and d are left dangling and unused, and as such is no real point in using a Simple Getter.

type Getter a c = forall r. LensLike (Const r) a b c d

type Fold a c = forall m b d. Monoid m => (c -> Const m d) -> a -> Const m b

A Fold describes how to retrieve multiple values in a way that can be composed with other lens-like constructions.

A Fold a c provides a structure with operations very similar to those of the Foldable typeclass, see foldMapOf and the other Fold combinators.

By convention, if there exists a foo method that expects a Foldable (f c), then there should be a fooOf method that takes a Fold a c and a value of type a.

A Getter is a legal Fold that just ignores the supplied Monoid

Unlike a Traversal a Fold is read-only. Since a Fold cannot be used to write back there are no lens laws that apply.

 type Fold a c = forall m b d. Monoid m => Getting m a b c d

type Getting r a b c d = (c -> Const r d) -> a -> Const r b

Most Getter combinators are able to be used with both a Getter or a Fold in limited situations, to do so, they need to be monomorphic in what we are going to extract with Const. To be compatible with Lens, Traversal and Iso we also restricted choices of the irrelevant b and d parameters.

If a function accepts a Getting r a b c d, then when r is a Monoid, you can pass a Fold (or Traversal), otherwise you can only pass this a Getter or Lens.

 type Getting r a b c d = LensLike (Const r) a b c d

to :: (a -> c) -> Getter a c

Build a Getter from an arbitrary Haskell function.

 to f . to g = to (g . f)
 to = from view

 to . from = id

folds :: Isomorphic k => k ((c -> m) -> a -> m) (Getting m a b c d)

Build a Getter or Fold from a foldMap-like function.

 folds :: ((c -> m) -> a -> m) -> (c -> Const m d) -> a -> Const m b
 folds :: ((c -> m) -> a -> m) -> Getting m a b c d

folding :: Foldable f => (a -> f c) -> Fold a c

Obtain a Fold by lifting an operation that returns a foldable result.

This can be useful to lift operations from Data.List and elsewhere into a Fold.

folded :: Foldable f => Fold (f c) c

Obtain a Fold from any Foldable.

 folded = folds foldMap

filtered :: Monoid m => (c -> Bool) -> Getting m a b c d -> Getting m a b c d

Obtain a Fold by filtering a Lens, Iso, 'Getter, Fold or Traversal.

reversed :: Getting (Dual m) a b c d -> Getting m a b c d

Obtain a Fold by reversing the order of traversal for a Lens, Iso, Getter, Fold or Traversal.

Of course, reversing a Fold or Getter has no effect.

takingWhile :: Monoid m => (c -> Bool) -> Getting (Endo m) a b c d -> Getting m a b c d

Obtain a Fold by taking elements from another Fold, Lens, Iso, Getter or Traversal while a predicate holds.

 takeWhile p = toListOf (takingWhile p folded)

 ghci> toList (takingWhile (<=3) folded) [1..]
 [1,2,3]

droppingWhile :: Monoid m => (c -> Bool) -> Getting (Endo m) a b c d -> Getting m a b c d

Obtain a Fold by dropping elements from another Fold, Lens, Iso, Getter or Traversal while a predicate holds.

 dropWhile p = toListOf (droppingWhile p folded)

 ghci> toList (dropWhile (<=3) folded) [1..6]
 [4,5,6]

view :: Getting c a b c d -> a -> c

View the value pointed to by a Getter, Iso or Lens or the result of folding over all the results of a Fold or Traversal that points at a monoidal values.

It may be useful to think of view as having these more restrictive signatures:

 view ::             Getter a c        -> a -> c
 view :: Monoid m => Fold a m          -> a -> m
 view ::             Iso a b c d       -> a -> c
 view ::             Lens a b c d      -> a -> c
 view :: Monoid m => Traversal a b m d -> a -> m

views :: Isomorphic k => k (Getting m a b c d) ((c -> m) -> a -> m)

View the value of a Getter, Iso, Lens or the result of folding over the result of mapping the targets of a Fold or Traversal.

It may be useful to think of views as having these more restrictive signatures:

 views ::             Getter a c        -> (c -> d) -> a -> d
 views :: Monoid m => Fold a c          -> (c -> m) -> a -> m
 views ::             Iso a b c d       -> (c -> d) -> a -> d
 views ::             Lens a b c d      -> (c -> d) -> a -> d
 views :: Monoid m => Traversal a b c d -> (c -> m) -> a -> m

 views :: ((c -> Const m d) -> a -> Const m b) -> (c -> m) -> a -> m

(^.) :: a -> Getting c a b c d -> c

View the value pointed to by a Getter or Lens or the result of folding over all the results of a Fold or Traversal that points at a monoidal values.

This is the same operation as view with the arguments flipped.

The fixity and semantics are such that subsequent field accesses can be performed with (Prelude..)

 ghci> ((0, 1 :+ 2), 3)^._1._2.to magnitude
 2.23606797749979

 (^.) ::             a -> Getter a c        -> c
 (^.) :: Monoid m => a -> Fold a m          -> m
 (^.) ::             a -> Iso a b c d       -> c
 (^.) ::             a -> Lens a b c d      -> c
 (^.) :: Monoid m => a -> Traversal a b m d -> m

 (^.) :: a -> ((c -> Const c d) -> a -> Const c b) -> c

(^$) :: Getting c a b c d -> a -> c

View the value pointed to by a Getter, Iso or Lens or the result of folding over all the results of a Fold or Traversal that points at a monoidal values.

This is the same operation as view, only infix.

 (^$) ::             Getter a c        -> a -> c
 (^$) :: Monoid m => Fold a m          -> a -> m
 (^$) ::             Iso a b c d       -> a -> c
 (^$) ::             Lens a b c d      -> a -> c
 (^$) :: Monoid m => Traversal a b m d -> a -> m

 (^$) :: ((c -> Const c d) -> a -> Const c b) -> a -> c

use :: MonadState a m => Getting c a b c d -> m c

Use the target of a Lens, Iso, or Getter in the current state, or use a summary of a Fold or Traversal that points to a monoidal value.

 use :: MonadState a m             => Getter a c        -> m c
 use :: (MonadState a m, Monoid r) => Fold a r          -> m r
 use :: MonadState a m             => Iso a b c d       -> m c
 use :: MonadState a m             => Lens a b c d      -> m c
 use :: (MonadState a m, Monoid r) => Traversal a b r d -> m r

 use :: MonadState a m => ((c -> Const c d) -> a -> Const c b) -> m c

uses :: MonadState a m => Getting e a b c d -> (c -> e) -> m e

Use the target of a Lens, Iso or Getter in the current state, or use a summary of a Fold or Traversal that points to a monoidal value.

 uses :: MonadState a m             => Getter a c        -> (c -> e) -> m e
 uses :: (MonadState a m, Monoid r) => Fold a c          -> (c -> r) -> m r
 uses :: MonadState a m             => Lens a b c d      -> (c -> e) -> m e
 uses :: MonadState a m             => Iso a b c d       -> (c -> e) -> m e
 uses :: (MonadState a m, Monoid r) => Traversal a b c d -> (c -> r) -> m r

 uses :: MonadState a m => ((c -> Const e d) -> a -> Const e b) -> (c -> e) -> m e

query :: MonadReader a m => Getting c a b c d -> m c

Query the target of a Lens, Iso or Getter in the current state, or use a summary of a Fold or Traversal that points to a monoidal value.

 query :: MonadReader a m             => Getter a c        -> m c
 query :: (MonadReader a m, Monoid c) => Fold a c          -> m c
 query :: MonadReader a m             => Iso a b c d       -> m c
 query :: MonadReader a m             => Lens a b c d      -> m c
 query :: (MonadReader a m, Monoid c) => Traversal a b c d -> m c

 query :: MonadReader a m => ((c -> Const c d) -> a -> Const c b) -> m c

queries :: MonadReader a m => Getting e a b c d -> (c -> e) -> m e

Use the target of a Lens, Iso or Getter in the current state, or use a summary of a Fold or Traversal that points to a monoidal value.

 queries :: MonadReader a m             => Getter a c        -> (c -> e) -> m e
 queries :: (MonadReader a m, Monoid c) => Fold a c          -> (c -> e) -> m e
 queries :: MonadReader a m             => Iso a b c d       -> (c -> e) -> m e
 queries :: MonadReader a m             => Lens a b c d      -> (c -> e) -> m e
 queries :: (MonadReader a m, Monoid c) => Traversal a b c d -> (c -> e) -> m e

 queries :: MonadReader a m => ((c -> Const e d) -> a -> Const e b) -> (c -> e) -> m e

Getting and Folding

foldMapOf :: Isomorphic k => k (Getting m a b c d) ((c -> m) -> a -> m)

 foldMap = foldMapOf folded

 foldMapOf = views
 foldMapOf = from folds

 foldMapOf ::             Getter a c        -> (c -> m) -> a -> m
 foldMapOf :: Monoid m => Fold a c          -> (c -> m) -> a -> m
 foldMapOf ::             Lens a b c d      -> (c -> m) -> a -> m
 foldMapOf ::             Iso a b c d       -> (c -> m) -> a -> m
 foldMapOf :: Monoid m => Traversal a b c d -> (c -> m) -> a -> m

 foldMapOf :: Getting m a b c d -> (c -> m) -> a -> m

foldOf :: Getting m a b m d -> a -> m

 fold = foldOf folded

 foldOf = view

 foldOf ::             Getter a m        -> a -> m
 foldOf :: Monoid m => Fold a m          -> a -> m
 foldOf ::             Lens a b m d      -> a -> m
 foldOf ::             Iso a b m d       -> a -> m
 foldOf :: Monoid m => Traversal a b m d -> a -> m

foldrOf :: Getting (Endo e) a b c d -> (c -> e -> e) -> e -> a -> e

Right-associative fold of parts of a structure that are viewed through a Lens, Getter, Fold or Traversal.

 foldr = foldrOf folded

 foldrOf :: Getter a c        -> (c -> e -> e) -> e -> a -> e
 foldrOf :: Fold a c          -> (c -> e -> e) -> e -> a -> e
 foldrOf :: Lens a b c d      -> (c -> e -> e) -> e -> a -> e
 foldrOf :: Iso a b c d       -> (c -> e -> e) -> e -> a -> e
 foldrOf :: Traversal a b c d -> (c -> e -> e) -> e -> a -> e

foldlOf :: Getting (Dual (Endo e)) a b c d -> (e -> c -> e) -> e -> a -> e

Left-associative fold of the parts of a structure that are viewed through a Lens, Getter, Fold or Traversal.

 foldl = foldlOf folded

 foldlOf :: Getter a c        -> (e -> c -> e) -> e -> a -> e
 foldlOf :: Fold a c          -> (e -> c -> e) -> e -> a -> e
 foldlOf :: Lens a b c d      -> (e -> c -> e) -> e -> a -> e
 foldlOf :: Iso a b c d       -> (e -> c -> e) -> e -> a -> e
 foldlOf :: Traversal a b c d -> (e -> c -> e) -> e -> a -> e

toListOf :: Getting [c] a b c d -> a -> [c]

 toList = toListOf folded

 toListOf :: Getter a c        -> a -> [c]
 toListOf :: Fold a c          -> a -> [c]
 toListOf :: Lens a b c d      -> a -> [c]
 toListOf :: Iso a b c d       -> a -> [c]
 toListOf :: Traversal a b c d -> a -> [c]

anyOf :: Getting Any a b c d -> (c -> Bool) -> a -> Bool

 any = anyOf folded

 anyOf :: Getter a c        -> (c -> Bool) -> a -> Bool
 anyOf :: Fold a c          -> (c -> Bool) -> a -> Bool
 anyOf :: Lens a b c d      -> (c -> Bool) -> a -> Bool
 anyOf :: Iso a b c d       -> (c -> Bool) -> a -> Bool
 anyOf :: Traversal a b c d -> (c -> Bool) -> a -> Bool

allOf :: Getting All a b c d -> (c -> Bool) -> a -> Bool

 all = allOf folded

 allOf :: Getter a c        -> (c -> Bool) -> a -> Bool
 allOf :: Fold a c          -> (c -> Bool) -> a -> Bool
 allOf :: Lens a b c d      -> (c -> Bool) -> a -> Bool
 allOf :: Iso a b c d       -> (c -> Bool) -> a -> Bool
 allOf :: Traversal a b c d -> (c -> Bool) -> a -> Bool

andOf :: Getting All a b Bool d -> a -> Bool

 and = andOf folded

 andOf :: Getter a Bool       -> a -> Bool
 andOf :: Fold a Bool         -> a -> Bool
 andOf :: Lens a b Bool d     -> a -> Bool
 andOf :: Iso a b Bool d      -> a -> Bool
 andOf :: Traversl a b Bool d -> a -> Bool

orOf :: Getting Any a b Bool d -> a -> Bool

 or = orOf folded

 orOf :: Getter a Bool        -> a -> Bool
 orOf :: Fold a Bool          -> a -> Bool
 orOf :: Lens a b Bool d      -> a -> Bool
 orOf :: Iso a b Bool d       -> a -> Bool
 orOf :: Traversal a b Bool d -> a -> Bool

productOf :: Getting (Product c) a b c d -> a -> c

 product = productOf folded

 productOf ::          Getter a c        -> a -> c
 productOf :: Num c => Fold a c          -> a -> c
 productOf ::          Lens a b c d      -> a -> c
 productOf ::          Iso a b c d       -> a -> c
 productOf :: Num c => Traversal a b c d -> a -> c

sumOf :: Getting (Sum c) a b c d -> a -> c

 sum = sumOf folded

 sumOf _1 :: (a, b) -> a
 sumOf (folded._1) :: (Foldable f, Num a) => f (a, b) -> a

 sumOf ::          Getter a c        -> a -> c
 sumOf :: Num c => Fold a c          -> a -> c
 sumOf ::          Lens a b c d      -> a -> c
 sumOf ::          Iso a b c d       -> a -> c
 sumOf :: Num c => Traversal a b c d -> a -> c

traverseOf_ :: Functor f => Getting (Traversed f) a b c d -> (c -> f e) -> a -> f ()

When passed a Getter, traverseOf_ can work over a Functor.

When passed a Fold, traverseOf_ requires an Applicative.

 traverse_ = traverseOf_ folded

 traverseOf_ _2 :: Functor f => (c -> f e) -> (c1, c) -> f ()
 traverseOf_ traverseLeft :: Applicative f => (a -> f b) -> Either a c -> f ()

The rather specific signature of traverseOf_ allows it to be used as if the signature was either:

 traverseOf_ :: Functor f     => Getter a c        -> (c -> f e) -> a -> f ()
 traverseOf_ :: Applicative f => Fold a c          -> (c -> f e) -> a -> f ()
 traverseOf_ :: Functor f     => Lens a b c d      -> (c -> f e) -> a -> f ()
 traverseOf_ :: Functor f     => Iso a b c d       -> (c -> f e) -> a -> f ()
 traverseOf_ :: Applicative f => Traversal a b c d -> (c -> f e) -> a -> f ()

forOf_ :: Functor f => Getting (Traversed f) a b c d -> a -> (c -> f e) -> f ()

 for_ = forOf_ folded

 forOf_ :: Functor f     => Getter a c        -> a -> (c -> f e) -> f ()
 forOf_ :: Applicative f => Fold a c          -> a -> (c -> f e) -> f ()
 forOf_ :: Functor f     => Lens a b c d      -> a -> (c -> f e) -> f ()
 forOf_ :: Functor f     => Iso a b c d       -> a -> (c -> f e) -> f ()
 forOf_ :: Applicative f => Traversal a b c d -> a -> (c -> f e) -> f ()

sequenceAOf_ :: Functor f => Getting (Traversed f) a b (f ()) d -> a -> f ()

 sequenceA_ = sequenceAOf_ folded

 sequenceAOf_ :: Functor f     => Getter a (f ())        -> a -> f ()
 sequenceAOf_ :: Applicative f => Fold a (f ())          -> a -> f ()
 sequenceAOf_ :: Functor f     => Lens a b (f ()) d      -> a -> f ()
 sequenceAOf_ :: Functor f     => Iso a b (f ()) d       -> a -> f ()
 sequenceAOf_ :: Applicative f => Traversal a b (f ()) d -> a -> f ()

mapMOf_ :: Monad m => Getting (Action m) a b c d -> (c -> m e) -> a -> m ()

 mapM_ = mapMOf_ folded

 mapMOf_ :: Monad m => Getter a c        -> (c -> m e) -> a -> m ()
 mapMOf_ :: Monad m => Fold a c          -> (c -> m e) -> a -> m ()
 mapMOf_ :: Monad m => Lens a b c d      -> (c -> m e) -> a -> m ()
 mapMOf_ :: Monad m => Iso a b c d       -> (c -> m e) -> a -> m ()
 mapMOf_ :: Monad m => Traversal a b c d -> (c -> m e) -> a -> m ()

forMOf_ :: Monad m => Getting (Action m) a b c d -> a -> (c -> m e) -> m ()

 forM_ = forMOf_ folded

 forMOf_ :: Monad m => Getter a c        -> a -> (c -> m e) -> m ()
 forMOf_ :: Monad m => Fold a c          -> a -> (c -> m e) -> m ()
 forMOf_ :: Monad m => Lens a b c d      -> a -> (c -> m e) -> m ()
 forMOf_ :: Monad m => Iso a b c d       -> a -> (c -> m e) -> m ()
 forMOf_ :: Monad m => Traversal a b c d -> a -> (c -> m e) -> m ()

sequenceOf_ :: Monad m => Getting (Action m) a b (m c) d -> a -> m ()

 sequence_ = sequenceOf_ folded

 sequenceOf_ :: Monad m => Getter a (m b)        -> a -> m ()
 sequenceOf_ :: Monad m => Fold a (m b)          -> a -> m ()
 sequenceOf_ :: Monad m => Lens a b (m b) d      -> a -> m ()
 sequenceOf_ :: Monad m => Iso a b (m b) d       -> a -> m ()
 sequenceOf_ :: Monad m => Traversal a b (m b) d -> a -> m ()

asumOf :: Alternative f => Getting (Endo (f c)) a b (f c) d -> a -> f c

The sum of a collection of actions, generalizing concatOf.

 asum = asumOf folded

 asumOf :: Alternative f => Getter a c        -> a -> f c
 asumOf :: Alternative f => Fold a c          -> a -> f c
 asumOf :: Alternative f => Lens a b c d      -> a -> f c
 asumOf :: Alternative f => Iso a b c d       -> a -> f c
 asumOf :: Alternative f => Traversal a b c d -> a -> f c

msumOf :: MonadPlus m => Getting (Endo (m c)) a b (m c) d -> a -> m c

The sum of a collection of actions, generalizing concatOf.

 msum = msumOf folded

 msumOf :: MonadPlus m => Getter a c        -> a -> m c
 msumOf :: MonadPlus m => Fold a c          -> a -> m c
 msumOf :: MonadPlus m => Lens a b c d      -> a -> m c
 msumOf :: MonadPlus m => Iso a b c d       -> a -> m c
 msumOf :: MonadPlus m => Traversal a b c d -> a -> m c

concatMapOf :: Getting [e] a b c d -> (c -> [e]) -> a -> [e]

 concatMap = concatMapOf folded

 concatMapOf :: Getter a c        -> (c -> [e]) -> a -> [e]
 concatMapOf :: Fold a c          -> (c -> [e]) -> a -> [e]
 concatMapOf :: Lens a b c d      -> (c -> [e]) -> a -> [e]
 concatMapOf :: Iso a b c d       -> (c -> [e]) -> a -> [e]
 concatMapOf :: Traversal a b c d -> (c -> [e]) -> a -> [e]

concatOf :: Getting [e] a b [e] d -> a -> [e]

 concat = concatOf folded

 concatOf :: Getter a [e]        -> a -> [e]
 concatOf :: Fold a [e]          -> a -> [e]
 concatOf :: Iso a b [e] d       -> a -> [e]
 concatOf :: Lens a b [e] d      -> a -> [e]
 concatOf :: Traversal a b [e] d -> a -> [e]

elemOf :: Eq c => Getting Any a b c d -> c -> a -> Bool

 elem = elemOf folded

 elemOf :: Eq c => Getter a c        -> c -> a -> Bool
 elemOf :: Eq c => Fold a c          -> c -> a -> Bool
 elemOf :: Eq c => Lens a b c d      -> c -> a -> Bool
 elemOf :: Eq c => Iso a b c d       -> c -> a -> Bool
 elemOf :: Eq c => Traversal a b c d -> c -> a -> Bool

notElemOf :: Eq c => Getting All a b c d -> c -> a -> Bool

 notElem = notElemOf folded

 notElemOf :: Eq c => Getter a c        -> c -> a -> Bool
 notElemOf :: Eq c => Fold a c          -> c -> a -> Bool
 notElemOf :: Eq c => Iso a b c d       -> c -> a -> Bool
 notElemOf :: Eq c => Lens a b c d      -> c -> a -> Bool
 notElemOf :: Eq c => Traversal a b c d -> c -> a -> Bool

lengthOf :: Getting (Sum Int) a b c d -> a -> Int

Note: this can be rather inefficient for large containers.

 length = lengthOf folded

 lengthOf _1 :: (a, b) -> Int
 lengthOf _1 = 1
 lengthOf (folded.folded) :: Foldable f => f (g a) -> Int

 lengthOf :: Getter a c        -> a -> Int
 lengthOf :: Fold a c          -> a -> Int
 lengthOf :: Lens a b c d      -> a -> Int
 lengthOf :: Iso a b c d       -> a -> Int
 lengthOf :: Traversal a b c d -> a -> Int

nullOf :: Getting All a b c d -> a -> Bool

Returns True if this Fold or Traversal has no targets in the given container.

Note: nullOf on a valid Iso, Lens or Getter should always return False

 null = nullOf folded

This may be rather inefficient compared to the null check of many containers.

 nullOf _1 :: (a, b) -> Int
 nullOf _1 = False
 nullOf (folded._1.folded) :: Foldable f => f (g a, b) -> Bool

 nullOf :: Getter a c        -> a -> Bool
 nullOf :: Fold a c          -> a -> Bool
 nullOf :: Iso a b c d       -> a -> Bool
 nullOf :: Lens a b c d      -> a -> Bool
 nullOf :: Traversal a b c d -> a -> Bool

headOf :: Getting (First c) a b c d -> a -> Maybe c

Perform a safe head of a Fold or Traversal or retrieve Just the result from a Getter or Lens.

 listToMaybe . toList = headOf folded

 headOf :: Getter a c        -> a -> Maybe c
 headOf :: Fold a c          -> a -> Maybe c
 headOf :: Lens a b c d      -> a -> Maybe c
 headOf :: Iso a b c d       -> a -> Maybe c
 headOf :: Traversal a b c d -> a -> Maybe c

lastOf :: Getting (Last c) a b c d -> a -> Maybe c

Perform a safe last of a Fold or Traversal or retrieve Just the result from a Getter or Lens.

 lastOf :: Getter a c        -> a -> Maybe c
 lastOf :: Fold a c          -> a -> Maybe c
 lastOf :: Lens a b c d      -> a -> Maybe c
 lastOf :: Iso a b c d       -> a -> Maybe c
 lastOf :: Traversal a b c d -> a -> Maybe c

maximumOf :: Getting (Max c) a b c d -> a -> Maybe c

Obtain the maximum element (if any) targeted by a Fold or Traversal

Note: maximumOf on a valid Iso, Lens or Getter will always return Just a value.

 maximum = fromMaybe (error "empty") . maximumOf folded

 maximumOf ::          Getter a c        -> a -> Maybe c
 maximumOf :: Ord c => Fold a c          -> a -> Maybe c
 maximumOf ::          Iso a b c d       -> a -> Maybe c
 maximumOf ::          Lens a b c d      -> a -> Maybe c
 maximumOf :: Ord c => Traversal a b c d -> a -> Maybe c

minimumOf :: Getting (Min c) a b c d -> a -> Maybe c

Obtain the minimum element (if any) targeted by a Fold or Traversal

Note: minimumOf on a valid Iso, Lens or Getter will always return Just a value.

 minimum = fromMaybe (error "empty") . minimumOf folded

 minimumOf ::          Getter a c        -> a -> Maybe c
 minimumOf :: Ord c => Fold a c          -> a -> Maybe c
 minimumOf ::          Iso a b c d       -> a -> Maybe c
 minimumOf ::          Lens a b c d      -> a -> Maybe c
 minimumOf :: Ord c => Traversal a b c d -> a -> Maybe c

maximumByOf :: Getting (Endo (Maybe c)) a b c d -> (c -> c -> Ordering) -> a -> Maybe c

Obtain the maximum element (if any) targeted by a Fold, Traversal, Lens, Iso, or Getter according to a user supplied ordering.

 maximumBy cmp = fromMaybe (error "empty") . maximumByOf folded cmp

 maximumByOf :: Getter a c        -> (c -> c -> Ordering) -> a -> Maybe c
 maximumByOf :: Fold a c          -> (c -> c -> Ordering) -> a -> Maybe c
 maximumByOf :: Iso a b c d       -> (c -> c -> Ordering) -> a -> Maybe c
 maximumByOf :: Lens a b c d      -> (c -> c -> Ordering) -> a -> Maybe c
 maximumByOf :: Traversal a b c d -> (c -> c -> Ordering) -> a -> Maybe c

minimumByOf :: Getting (Endo (Maybe c)) a b c d -> (c -> c -> Ordering) -> a -> Maybe c

Obtain the minimum element (if any) targeted by a Fold, Traversal, Lens, Iso or Getter according to a user supplied ordering.

 minimumBy cmp = fromMaybe (error "empty") . minimumByOf folded cmp

 minimumByOf :: Getter a c        -> (c -> c -> Ordering) -> a -> Maybe c
 minimumByOf :: Fold a c          -> (c -> c -> Ordering) -> a -> Maybe c
 minimumByOf :: Iso a b c d       -> (c -> c -> Ordering) -> a -> Maybe c
 minimumByOf :: Lens a b c d      -> (c -> c -> Ordering) -> a -> Maybe c
 minimumByOf :: Traversal a b c d -> (c -> c -> Ordering) -> a -> Maybe c

findOf :: Getting (First c) a b c d -> (c -> Bool) -> a -> Maybe c

The findOf function takes a lens (or , getter, iso, fold, or traversal), a predicate and a structure and returns the leftmost element of the structure matching the predicate, or Nothing if there is no such element.

 findOf :: Getter a c        -> (c -> Bool) -> a -> Maybe c
 findOf :: Fold a c          -> (c -> Bool) -> a -> Maybe c
 findOf :: Iso a b c d       -> (c -> Bool) -> a -> Maybe c
 findOf :: Lens a b c d      -> (c -> Bool) -> a -> Maybe c
 findOf :: Traversal a b c d -> (c -> Bool) -> a -> Maybe c

foldrOf' :: Getting (Dual (Endo (e -> e))) a b c d -> (c -> e -> e) -> e -> a -> e

Strictly fold right over the elements of a structure.

 foldr' = foldrOf' folded

 foldrOf' :: Getter a c        -> (c -> e -> e) -> e -> a -> e
 foldrOf' :: Fold a c          -> (c -> e -> e) -> e -> a -> e
 foldrOf' :: Iso a b c d       -> (c -> e -> e) -> e -> a -> e
 foldrOf' :: Lens a b c d      -> (c -> e -> e) -> e -> a -> e
 foldrOf' :: Traversal a b c d -> (c -> e -> e) -> e -> a -> e

foldlOf' :: Getting (Endo (e -> e)) a b c d -> (e -> c -> e) -> e -> a -> e

Fold over the elements of a structure, associating to the left, but strictly.

 foldl' = foldlOf' folded

 foldlOf' :: Getter a c          -> (e -> c -> e) -> e -> a -> e
 foldlOf' :: Fold a c            -> (e -> c -> e) -> e -> a -> e
 foldlOf' :: Iso a b c d         -> (e -> c -> e) -> e -> a -> e
 foldlOf' :: Lens a b c d        -> (e -> c -> e) -> e -> a -> e
 foldlOf' :: Traversal a b c d   -> (e -> c -> e) -> e -> a -> e

foldr1Of :: Getting (Endo (Maybe c)) a b c d -> (c -> c -> c) -> a -> c

A variant of foldrOf that has no base case and thus may only be applied to lenses and structures such that the lens views at least one element of the structure.

 foldr1Of l f = Prelude.foldr1 f . toListOf l

 foldr1 = foldr1Of folded

 foldr1Of :: Getter a c        -> (c -> c -> c) -> a -> c
 foldr1Of :: Fold a c          -> (c -> c -> c) -> a -> c
 foldr1Of :: Iso a b c d       -> (c -> c -> c) -> a -> c
 foldr1Of :: Lens a b c d      -> (c -> c -> c) -> a -> c
 foldr1Of :: Traversal a b c d -> (c -> c -> c) -> a -> c

foldl1Of :: Getting (Dual (Endo (Maybe c))) a b c d -> (c -> c -> c) -> a -> c

A variant of foldlOf that has no base case and thus may only be applied to lenses and strutures such that the lens views at least one element of the structure.

 foldl1Of l f = Prelude.foldl1Of l f . toList

 foldl1 = foldl1Of folded

 foldl1Of :: Getter a c        -> (c -> c -> c) -> a -> c
 foldl1Of :: Fold a c          -> (c -> c -> c) -> a -> c
 foldl1Of :: Iso a b c d       -> (c -> c -> c) -> a -> c
 foldl1Of :: Lens a b c d      -> (c -> c -> c) -> a -> c
 foldl1Of :: Traversal a b c d -> (c -> c -> c) -> a -> c

foldrMOf :: Monad m => Getting (Dual (Endo (e -> m e))) a b c d -> (c -> e -> m e) -> e -> a -> m e

Monadic fold over the elements of a structure, associating to the right, i.e. from right to left.

 foldrM = foldrMOf folded

 foldrMOf :: Monad m => Getter a c        -> (c -> e -> m e) -> e -> a -> m e
 foldrMOf :: Monad m => Fold a c          -> (c -> e -> m e) -> e -> a -> m e
 foldrMOf :: Monad m => Iso a b c d       -> (c -> e -> m e) -> e -> a -> m e
 foldrMOf :: Monad m => Lens a b c d      -> (c -> e -> m e) -> e -> a -> m e
 foldrMOf :: Monad m => Traversal a b c d -> (c -> e -> m e) -> e -> a -> m e

foldlMOf :: Monad m => Getting (Endo (e -> m e)) a b c d -> (e -> c -> m e) -> e -> a -> m e

Monadic fold over the elements of a structure, associating to the left, i.e. from left to right.

 foldlM = foldlMOf folded

 foldlMOf :: Monad m => Getter a c        -> (e -> c -> m e) -> e -> a -> m e
 foldlMOf :: Monad m => Fold a c          -> (e -> c -> m e) -> e -> a -> m e
 foldlMOf :: Monad m => Iso a b c d       -> (e -> c -> m e) -> e -> a -> m e
 foldlMOf :: Monad m => Lens a b c d      -> (e -> c -> m e) -> e -> a -> m e
 foldlMOf :: Monad m => Traversal a b c d -> (e -> c -> m e) -> e -> a -> m e

Setting

(+~) :: Num c => Setter a b c c -> c -> a -> b

Increment the target(s) of a numerically valued Lens, Setter' or Traversal

 ghci> _1 +~ 1 $ (1,2)
 (2,2)

(-~) :: Num c => Setter a b c c -> c -> a -> b

Decrement the target(s) of a numerically valued Lens, Iso, Setter or Traversal

 ghci> _1 -~ 2 $ (1,2)
 (-1,2)

(*~) :: Num c => Setter a b c c -> c -> a -> b

Multiply the target(s) of a numerically valued Lens, Iso, Setter or Traversal

 ghci> _2 *~ 4 $ (1,2)
 (1,8)

(//~) :: Fractional c => Setter a b c c -> c -> a -> b

Divide the target(s) of a numerically valued Lens, Iso, Setter or Traversal

(||~) :: Setter a b Bool Bool -> Bool -> a -> b

Logically || the target(s) of a Bool-valued Lens or Setter

(&&~) :: Setter a b Bool Bool -> Bool -> a -> b

Logically && the target(s) of a Bool-valued Lens or Setter

(<>~) :: Monoid c => Setter a b c c -> c -> a -> b

Modify the target of a monoidally valued by mappending another value.

(+=) :: (MonadState a m, Num b) => Simple Setter a b -> b -> m ()

Modify the target(s) of a Simple Lens, Iso, Setter or Traversal by adding a value

Example:

 fresh = do
   id += 1
   access id

(-=) :: (MonadState a m, Num b) => Simple Setter a b -> b -> m ()

Modify the target(s) of a Simple Lens, Iso, Setter or Traversal by subtracting a value

(*=) :: (MonadState a m, Num b) => Simple Setter a b -> b -> m ()

Modify the target(s) of a Simple Lens, Iso, Setter or Traversal by multiplying by value

(//=) :: (MonadState a m, Fractional b) => Simple Setter a b -> b -> m ()

Modify the target(s) of a Simple Lens, Iso, Setter or Traversal by dividing by a value

(||=) :: MonadState a m => Simple Setter a Bool -> Bool -> m ()

Modify the target(s) of a Simple Lens, 'Iso, Setter or Traversal by taking their logical || with a value

(&&=) :: MonadState a m => Simple Setter a Bool -> Bool -> m ()

Modify the target(s) of a Simple Lens, Iso, Setter or Traversal by taking their logical && with a value

(<>=) :: (MonadState a m, Monoid b) => Simple Setter a b -> b -> m ()

Modify the target(s) of a Simple Lens, Iso, Setter or Traversal by mappending a value.

Traversing and Lensing

class Focus st where

This class allows us to use focus on a number of different monad transformers.

Methods

focus :: Monad m => LensLike (Focusing m c) a a b b -> st b m c -> st a m c

Run a monadic action in a larger context than it was defined in, using a Simple Lens or Simple Traversal.

This is commonly used to lift actions in a simpler state monad into a state monad with a larger state type.

When applied to a 'Simple Traversal over multiple values, the actions for each target are executed sequentially and the results are aggregated monoidally and a monoidal summary of the result is given.

 focus :: Monad m             => Simple Iso a b       -> st b m c -> st a m c
 focus :: Monad m             => Simple Lens a b      -> st b m c -> st a m c
 focus :: (Monad m, Monoid c) => Simple Traversal a b -> st b m c -> st a m c

focus_ :: Monad m => LensLike (Focusing m ()) a a b b -> st b m c -> st a m ()

Like focus, but discarding any accumulated results as you go.

 focus_ :: Monad m             => Simple Iso a b       -> st b m c -> st a m ()
 focus_ :: Monad m             => Simple Lens a b      -> st b m c -> st a m ()
 focus_ :: (Monad m, Monoid c) => Simple Traversal a b -> st b m c -> st a m ()

setFocus :: Simple Setter a b -> st b Identity c -> st a Identity ()

A much more limited version of focus that can work with a Setter.

Instances

Focus ReaderT

Focus StateT

Focus StateT

traverseOf :: Category k => k (LensLike f a b c d) ((c -> f d) -> a -> f b)

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

 traverseOf = id

 traverse = traverseOf traverse

 traverseOf :: Iso a b c d       -> (c -> f d) -> a -> f b
 traverseOf :: Lens a b c d      -> (c -> f d) -> a -> f b
 traverseOf :: Traversal a b c d -> (c -> f d) -> a -> f b

forOf :: Isomorphic k => k (LensLike f a b c d) (a -> (c -> f d) -> f b)

 forOf l = flip (traverseOf l)

 for = forOf traverse
 forOf = morphism flip flip

 forOf :: Lens a b c d -> a -> (c -> f d) -> f b

sequenceAOf :: LensLike f a b (f c) c -> a -> f b

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

 sequenceA = sequenceAOf traverse
 sequenceAOf l = traverseOf l id
 sequenceAOf l = l id

 sequenceAOf ::                  Iso a b (f c) c       -> a -> f b
 sequenceAOf ::                  Lens a b (f c) c      -> a -> f b
 sequenceAOf :: Applicative f => Traversal a b (f c) c -> a -> f b

mapMOf :: LensLike (WrappedMonad m) a b c d -> (c -> m d) -> a -> m b

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 a b c d       -> (c -> m d) -> a -> m b
 mapMOf ::            Lens a b c d      -> (c -> m d) -> a -> m b
 mapMOf :: Monad m => Traversal a b c d -> (c -> m d) -> a -> m b

forMOf :: LensLike (WrappedMonad m) a b c d -> a -> (c -> m d) -> m b

 forM = forMOf traverse
 forMOf l = flip (mapMOf l)

 forMOf ::            Iso a b c d       -> a -> (c -> m d) -> m b
 forMOf ::            Lens a b c d      -> a -> (c -> m d) -> m b
 forMOf :: Monad m => Traversal a b c d -> a -> (c -> m d) -> m b

sequenceOf :: LensLike (WrappedMonad m) a b (m c) c -> a -> m b

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

 sequenceOf ::            Iso a b (m c) c       -> a -> m b
 sequenceOf ::            Lens a b (m c) c      -> a -> m b
 sequenceOf :: Monad m => Traversal a b (m c) c -> a -> m b

transposeOf :: LensLike ZipList a b [c] c -> a -> [b]

This generalizes transpose to an arbitrary Traversal.

 transpose = transposeOf traverse

 ghci> 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.

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

mapAccumLOf :: LensLike (Backwards (State s)) a b c d -> (s -> c -> (s, d)) -> s -> a -> (s, b)

Generalized mapAccumL to an arbitrary Traversal.

 mapAccumL = mapAccumLOf traverse

mapAccumLOf accumulates state from left to right.

 mapAccumLOf :: Iso a b c d       -> (s -> c -> (s, d)) -> s -> a -> (s, b)
 mapAccumLOf :: Lens a b c d      -> (s -> c -> (s, d)) -> s -> a -> (s, b)
 mapAccumLOf :: Traversal a b c d -> (s -> c -> (s, d)) -> s -> a -> (s, b)

mapAccumROf :: LensLike (State s) a b c d -> (s -> c -> (s, d)) -> s -> a -> (s, b)

Generalizes mapAccumR to an arbitrary Traversal.

 mapAccumR = mapAccumROf traverse

mapAccumROf accumulates state from right to left.

 mapAccumROf :: Iso a b c d       -> (s -> c -> (s, d)) -> s -> a -> (s, b)
 mapAccumROf :: Lens a b c d      -> (s -> c -> (s, d)) -> s -> a -> (s, b)
 mapAccumROf :: Traversal a b c d -> (s -> c -> (s, d)) -> s -> a -> (s, b)

scanr1Of :: LensLike (State (Maybe c)) a b c c -> (c -> c -> c) -> a -> b

Permit the use of scanr1 over an arbitrary Traversal or Lens.

 scanr1 = scanr1Of traverse

 scanr1Of :: Iso a b c c       -> (c -> c -> c) -> a -> b
 scanr1Of :: Lens a b c c      -> (c -> c -> c) -> a -> b
 scanr1Of :: Traversal a b c c -> (c -> c -> c) -> a -> b

scanl1Of :: LensLike (Backwards (State (Maybe c))) a b c c -> (c -> c -> c) -> a -> b

Permit the use of scanl1 over an arbitrary Traversal or Lens.

 scanl1 = scanl1Of traverse

 scanr1Of :: Iso a b c c       -> (c -> c -> c) -> a -> b
 scanr1Of :: Lens a b c c      -> (c -> c -> c) -> a -> b
 scanr1Of :: Traversal a b c c -> (c -> c -> c) -> a -> b

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 Tree

Traversable Seq

Traversable ViewL

Traversable ViewR

Traversable IntMap

Traversable Identity

Traversable Node

Traversable Digit

Traversable FingerTree

Traversable Elem

Ix i => Traversable (Array i)

Traversable (Map k)

Traversable f => Traversable (ListT f)

Traversable f => Traversable (Backwards f)

Derived instance.

Traversable f => Traversable (MaybeT f)

Traversable f => Traversable (IdentityT f)

Traversable f => Traversable (ErrorT e f)

Traversable f => Traversable (WriterT w f)

Traversable f => Traversable (WriterT w f)

traverseNothing :: Traversal a a c d

This is the traversal that never succeeds at returning any values

 traverseNothing :: Applicative f => (c -> f d) -> a -> f a

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

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

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

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

traverse the right-hand value in an Either:

 traverseRight = traverse

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

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

traverseValue :: (k -> Bool) -> Simple Traversal (k, v) v

This provides a Traversal that checks a predicate on a key before allowing you to traverse into a value.

Transforming Traversals

backwards :: Isomorphic k => IsoLike k (Backwards f) a b c d -> IsoLike k f a b c d

This allows you to traverse the elements of a Traversal in the opposite order.

Note: reversed is similar, but is able to accept a Fold (or Getter) and produce a Fold (or Getter).

This requires at least a Traversal (or Lens) and can produce a Traversal (or Lens) in turn.

A backwards Iso is the same Iso. If you reverse the direction of the isomorphism use from instead.

Cloning Lenses

clone :: Functor f => LensLike (IndexedStore c d) a b c d -> (c -> f d) -> a -> f b

Cloning a Lens is one way to make sure you arent given something weaker, such as a Traversal and can be used as a way to pass around lenses that have to be monomorphic in f.

Note: This only accepts a proper Lens, because IndexedStore lacks its (admissable) Applicative instance.

merged :: Functor f => LensLike f a b c c -> LensLike f a' b' c c -> LensLike f (Either a a') (Either b b') c c

Merge two lenses, getters, setters, folds or traversals.

bothLenses :: Lens a b c d -> Lens a' b' c' d' -> Lens (a, a') (b, b') (c, c') (d, d')

bothLenses makes a lens from two other lenses (or isomorphisms)

Common Isomorphisms

identity :: Iso a b (Identity a) (Identity b)

This isomorphism can be used to wrap or unwrap a value in Identity.

 x^.identity = Identity x
 Identity x^.from identity = x

konst :: Iso a b (Const a c) (Const b d)

This isomorphism can be used to wrap or unwrap a value in Const

 x^.konst = Const x
 Const x^.from konst = x