Control.Lens.Prism

Prisms

type Prism s t a b = forall p f. (Choice p, Applicative f) => p a (f b) -> p s (f t)

A Prism l is a Traversal that can also be turned around with re to obtain a Getter in the opposite direction.

There are two laws that a Prism should satisfy:

First, if I re or review a value with a Prism and then preview or use (^?), I will get it back:

 preview l (review l b) ≡ Just b

Second, if you can extract a value a using a Prism l from a value s, then the value s is completely described by l and a:

If preview l s ≡ Just a then review l a ≡ s

These two laws imply that the Traversal laws hold for every Prism and that we traverse at most 1 element:

 lengthOf l x <= 1

It may help to think of this as a Iso that can be partial in one direction.

Every Prism is a valid Traversal.

Every Iso is a valid Prism.

For example, you might have a Prism' Integer Natural allows you to always go from a Natural to an Integer, and provide you with tools to check if an Integer is a Natural and/or to edit one if it is.

 nat :: Prism' Integer Natural
 nat = prism toInteger $ \ i ->
    if i < 0
    then Left i
    else Right (fromInteger i)

Now we can ask if an Integer is a Natural.

>>> 5^?nat
Just 5

>>> (-5)^?nat
Nothing

We can update the ones that are:

>>> (-3,4) & both.nat *~ 2
(-3,8)

And we can then convert from a Natural to an Integer.

>>> 5 ^. re nat -- :: Natural
5

Similarly we can use a Prism to traverse the Left half of an Either:

>>> Left "hello" & _Left %~ length
Left 5

or to construct an Either:

>>> 5^.re _Left
Left 5

such that if you query it with the Prism, you will get your original input back.

>>> 5^.re _Left ^? _Left
Just 5

Another interesting way to think of a Prism is as the categorical dual of a Lens -- a co-Lens, so to speak. This is what permits the construction of outside.

Note: Composition with a Prism is index-preserving.

type Prism' s a = Prism s s a a

A Simple Prism.

type APrism s t a b = Market a b a (Identity b) -> Market a b s (Identity t)

If you see this in a signature for a function, the function is expecting a Prism.

type APrism' s a = APrism s s a a

 type APrism' = Simple APrism

Constructing Prisms

prism :: (b -> t) -> (s -> Either t a) -> Prism s t a b

Build a Prism.

Either t a is used instead of Maybe a to permit the types of s and t to differ.

prism' :: (b -> s) -> (s -> Maybe a) -> Prism s s a b

This is usually used to build a Prism', when you have to use an operation like cast which already returns a Maybe.

Consuming Prisms

withPrism :: APrism s t a b -> ((b -> t) -> (s -> Either t a) -> r) -> r

Convert APrism to the pair of functions that characterize it.

clonePrism :: APrism s t a b -> Prism s t a b

Clone a Prism so that you can reuse the same monomorphically typed Prism for different purposes.

See cloneLens and cloneTraversal for examples of why you might want to do this.

outside :: Representable p => APrism s t a b -> Lens (p t r) (p s r) (p b r) (p a r)

Use a Prism as a kind of first-class pattern.

outside :: Prism s t a b -> Lens (t -> r) (s -> r) (b -> r) (a -> r)

aside :: APrism s t a b -> Prism (e, s) (e, t) (e, a) (e, b)

Use a Prism to work over part of a structure.

without :: APrism s t a b -> APrism u v c d -> Prism (Either s u) (Either t v) (Either a c) (Either b d)

Given a pair of prisms, project sums.

Viewing a Prism as a co-Lens, this combinator can be seen to be dual to alongside.

below :: Traversable f => APrism' s a -> Prism' (f s) (f a)

lift a Prism through a Traversable functor, giving a Prism that matches only if all the elements of the container match the Prism.

isn't :: APrism s t a b -> s -> Bool

Check to see if this Prism doesn't match.

>>> isn't _Left (Right 12)
True

>>> isn't _Left (Left 12)
False

>>> isn't _Empty []
False

matching :: APrism s t a b -> s -> Either t a

Retrieve the value targeted by a Prism or return the original value while allowing the type to change if it does not match.

>>> matching _Just (Just 12)
Right 12

>>> matching _Just (Nothing :: Maybe Int) :: Either (Maybe Bool) Int
Left Nothing

Common Prisms

_Left :: Prism (Either a c) (Either b c) a b

This Prism provides a Traversal for tweaking the Left half 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"

It also can be turned around to obtain the embedding into the Left half of an Either:

>>> _Left # 5
Left 5

>>> 5^.re _Left
Left 5

_Right :: Prism (Either c a) (Either c b) a b

This Prism provides a Traversal for tweaking the Right half of an Either:

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

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

>>> Right "hello" ^._Right
"hello"

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

It also can be turned around to obtain the embedding into the Right half of an Either:

>>> _Right # 5
Right 5

>>> 5^.re _Right
Right 5

_Just :: Prism (Maybe a) (Maybe b) a b

This Prism provides a Traversal for tweaking the target of the value of Just in a Maybe.

>>> over _Just (+1) (Just 2)
Just 3

Unlike traverse this is a Prism, and so you can use it to inject as well:

>>> _Just # 5
Just 5

>>> 5^.re _Just
Just 5

Interestingly,

 m ^? _Just ≡ m

>>> Just x ^? _Just
Just x

>>> Nothing ^? _Just
Nothing

_Nothing :: Prism' (Maybe a) ()

This Prism provides the Traversal of a Nothing in a Maybe.

>>> Nothing ^? _Nothing
Just ()

>>> Just () ^? _Nothing
Nothing

But you can turn it around and use it to construct Nothing as well:

>>> _Nothing # ()
Nothing

_Void :: Prism s s a Void

Void is a logically uninhabited data type.

This is a Prism that will always fail to match.

_Show :: (Read a, Show a) => Prism' String a

This is an improper prism for text formatting based on Read and Show.

This Prism is "improper" in the sense that it normalizes the text formatting, but round tripping is idempotent given sane 'Read'/'Show' instances.

>>> _Show # 2
"2"

>>> "EQ" ^? _Show :: Maybe Ordering
Just EQ

 _Show ≡ prism' show readMaybe

only :: Eq a => a -> Prism' a ()

This Prism compares for exact equality with a given value.

>>> only 4 # ()
4

>>> 5 ^? only 4
Nothing

nearly :: a -> (a -> Bool) -> Prism' a ()

This Prism compares for approximate equality with a given value and a predicate for testing.

To comply with the Prism laws the arguments you supply to nearly a p are somewhat constrained.

We assume p x holds iff x ≡ a. Under that assumption then this is a valid Prism.

This is useful when working with a type where you can test equality for only a subset of its values, and the prism selects such a value.

Prismatic profunctors

class Profunctor p => Choice p where

The generalization of DownStar of a "costrong" Functor

Minimal complete definition: left' or right'

Note: We use traverse and extract as approximate costrength as needed.

Methods

left' :: p a b -> p (Either a c) (Either b c)

right' :: p a b -> p (Either c a) (Either c b)

Instances

Choice (->)

Choice ReifiedFold

Choice ReifiedGetter

Monad m => Choice (Kleisli m)

Comonad w => Choice (Cokleisli w)

extract approximates costrength

Applicative f => Choice (UpStar f)

Traversable w => Choice (DownStar w)

sequence approximates costrength

ArrowChoice p => Choice (WrappedArrow p)

Monoid r => Choice (Forget r)

Choice (Tagged *)

Choice (Indexed i)

Choice (ReifiedMonadicFold m)

Choice (Market a b)