data Foo a = Foo Int Int aYou can define lenses such as
-- bar :: Lens' (Foo a) Int bar :: Functor f => (Int -> f Int) -> Foo a -> f (Foo a) bar f (Foo a b c) = fmap (\a' -> Foo a' b c) (f a)
-- quux :: Lens (Foo a) (Foo b) a b quux :: Functor f => (a -> f b) -> Foo a -> f (Foo b) quux f (Foo a b c) = fmap (Foo a b) (f c)without the need to use any type that isn't already defined in the Prelude. And you can define a traversal of multiple fields with Control.Applicative.Applicative:
-- traverseBarAndBaz :: Traversal' (Foo a) Int traverseBarAndBaz :: Applicative f => (Int -> f Int) -> Foo a -> f (Foo a) traverseBarAndBaz f (Foo a b c) = Foo <$> f a <*> f b <*> pure cWhat is provided in this library is a number of stock lenses and traversals for common haskell types, a wide array of combinators for working them, and more exotic functionality, (e.g. getters, setters, indexed folds, isomorphisms).
import Control.Lens data FooBar a = Foo { _x :: [Int], _y :: a } | Bar { _x :: [Int] } makeLenses ''FooBarThis defines the following lenses:
x :: Lens' (FooBar a) [Int] y :: Traversal (FooBar a) (FooBar b) a bYou can then access the value of _x with (^.), the value of _y – with (^?) or (^?!) (since it can fail), set the values with (.~), modify them with (%~), and use almost any other combinator that is re-exported here on those fields. The combinators here have unusually specific type signatures, so for particularly tricky ones, the simpler type signatures you might want to pretend the combinators have are specified as well. More information on how to use lenses is available on the lens wiki: http://github.com/ekmett/lens/wiki
view l (set l v s) ≡ v2) Putting back what you got doesn't change anything:
set l (view l s) s ≡ s3) Setting twice is the same as setting once:
set l v' (set l v s) ≡ set l v' sThese 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/. There are some emergent properties of these laws: 1) set l s must be injective for every s This is a consequence of law #1 2) set l must be surjective, because of law #2, which indicates that it is possible to obtain any v from some s such that set s v = s 3) Given just the first two laws you can prove a weaker form of law #3 where the values v that you are setting match:
set l v (set l v s) ≡ set l v sEvery 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. Since every Lens is a valid Traversal, the Traversal laws are required of any Lens you create:
l pure ≡ pure fmap (l f) . l g ≡ getCompose . l (Compose . fmap f . g)
type Lens s t a b = forall f. Functor f => LensLike f s t a b
type Lens s t a b = forall f. Functor f => (a -> f b) -> s -> f tEvery Lens is a valid Setter. Every Lens can be used for Getting like a Fold that doesn't use the Applicative or Contravariant. Every Lens is a valid Traversal that only uses the Functor part of the Applicative it is supplied. Every Lens can be used for Getting like a valid Getter. Since every Lens can be used for Getting like a valid Getter it follows that it must view exactly one element in the structure. The Lens laws follow from this property and the desire for it to act like a Traversable when used as a Traversal. In the examples below, getter and setter are supplied as example getters and setters, and are not actual functions supplied by this package.
>>> [0..10] ^? ix 4 Just 4
>>> [0..5] & ix 4 .~ 2 [0,1,2,3,2,5]
>>> [0..10] ^? ix 14 Nothing
>>> [0..5] & ix 14 .~ 2 [0,1,2,3,4,5]The Cons and AsEmpty classes provide Prisms for list constructors.
>>> [1..10] ^? _Cons Just (1,[2,3,4,5,6,7,8,9,10])
>>> [] ^? _Cons Nothing
>>> [] ^? _Empty Just ()
>>> _Cons # (1, _Empty # ()) :: [Int] [1]Additionally, Snoc provides a Prism for accessing the end of a list. Note that this Prism always will need to traverse the whole list.
>>> [1..5] ^? _Snoc Just ([1,2,3,4],5)
>>> _Snoc # ([1,2],5) [1,2,5]An instance of Plated allows for finding locations in the list where a traversal matches.
>>> [Nothing, Just 7, Just 3, Nothing] & deep (ix 0 . _Just) +~ 10 [Nothing,Just 17,Just 3,Nothing]An instance of Reversing provides an Iso between a list and its reverse.
>>> "live" & reversed %~ ('d':) "lived"It's possible to work under a prefix or suffix of a list using Prefixed and Suffixed.
>>> "preview" ^? prefixed "pre" Just "view"
>>> suffixed ".o" # "hello" "hello.o"At present, Data.List.Lens re-exports Prefixed and Suffixed for backwards compatibility, as prefixed and suffixed used to be top-level functions defined in this module. This may change in a future major release of lens. Finally, it's possible to traverse, fold over, and map over index-value pairs thanks to instances of TraversableWithIndex, FoldableWithIndex, and FunctorWithIndex.
>>> imap (,) "Hello" [(0,'H'),(1,'e'),(2,'l'),(3,'l'),(4,'o')]
>>> ifoldMap replicate "Hello" "ellllloooo"
>>> itraverse_ (curry print) "Hello" (0,'H') (1,'e') (2,'l') (3,'l') (4,'o')
>>> Map.fromList [(1, "world")] ^.at 1 Just "world"
>>> at 1 .~ Just "world" $ Map.empty fromList [(1,"world")]
>>> at 0 ?~ "hello" $ Map.empty fromList [(0,"hello")]We can traverse, fold over, and map over key-value pairs in a Map, thanks to its TraversableWithIndex, FoldableWithIndex, and FunctorWithIndex instances.
>>> imap const $ Map.fromList [(1, "Venus")] fromList [(1,1)]
>>> ifoldMap (\i _ -> Sum i) $ Map.fromList [(2, "Earth"), (3, "Mars")] Sum {getSum = 5}
>>> itraverse_ (curry print) $ Map.fromList [(4, "Jupiter")] (4,"Jupiter")
>>> itoList $ Map.fromList [(5, "Saturn")] [(5,"Saturn")]A related class, Ixed, allows us to use ix to traverse a value at a particular key.
>>> ix 2 %~ ("New " ++) $ Map.fromList [(2, "Earth")] fromList [(2,"New Earth")]
>>> preview (ix 8) $ Map.empty NothingAdditionally, Map has TraverseMin and TraverseMax instances, which let us traverse over the value at the least and greatest keys, respectively.
>>> preview traverseMin $ Map.fromList [(5, "Saturn"), (6, "Uranus")] Just "Saturn"
>>> preview traverseMax $ Map.fromList [(5, "Saturn"), (6, "Uranus")] Just "Uranus"