lens-4.13: Lenses, Folds and Traversals

Copyright | (C) 2012-15 Edward Kmett
License | BSD-style (see the file LICENSE)
Maintainer | Edward Kmett <ekmett@gmail.com>
Stability | provisional
Portability | non-portable
Safe Haskell | Trustworthy
Language | Haskell98

Control.Lens.Prism

Contents

- Prisms
- Constructing Prisms
- Consuming Prisms
- Common Prisms
- Prismatic profunctors

Description

Synopsis

- type Prism s t a b = forall p f. (Choice p, Applicative f) => p a (f b) -> p s (f t)
- type Prism' s a = Prism s s a a
- type APrism s t a b = Market a b a (Identity b) -> Market a b s (Identity t)
- type APrism' s a = APrism s s a a
- prism :: (b -> t) -> (s -> Either t a) -> Prism s t a b
- prism' :: (b -> s) -> (s -> Maybe a) -> Prism s s a b
- withPrism :: APrism s t a b -> ((b -> t) -> (s -> Either t a) -> r) -> r
- clonePrism :: APrism s t a b -> Prism s t a b
- outside :: Representable p => APrism s t a b -> Lens (p t r) (p s r) (p b r) (p a r)
- aside :: APrism s t a b -> Prism (e, s) (e, t) (e, a) (e, b)
- without :: APrism s t a b -> APrism u v c d -> Prism (Either s u) (Either t v) (Either a c) (Either b d)
- below :: Traversable f => APrism' s a -> Prism' (f s) (f a)
- isn't :: APrism s t a b -> s -> Bool
- matching :: APrism s t a b -> s -> Either t a
- _Left :: Prism (Either a c) (Either b c) a b
- _Right :: Prism (Either c a) (Either c b) a b
- _Just :: Prism (Maybe a) (Maybe b) a b
- _Nothing :: Prism' (Maybe a) ()
- _Void :: Prism s s a Void
- _Show :: (Read a, Show a) => Prism' String a
- only :: Eq a => a -> Prism' a ()
- nearly :: a -> (a -> Bool) -> Prism' a ()
- class Profunctor p => Choice p where
  - left' :: p a b -> p (Either a c) (Either b c)
  - right' :: p a b -> p (Either c a) (Either c b)

# Prisms

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

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 Source

A `Simple` `Prism`.

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

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 Source

    type APrism' = Simple APrism

# Constructing Prisms

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

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 Source

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 Source

Convert `APrism` to the pair of functions that characterize it.

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

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) Source

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) Source

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) Source

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) Source

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

    >>> [Left 1, Right "foo", Left 4, Right "woot"]^..below _Right
    []

    >>> [Right "hail hydra!", Right "foo", Right "blah", Right "woot"]^..below _Right
    [["hail hydra!","foo","blah","woot"]]

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

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 Source

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 Source

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 Source

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 Source

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) () Source

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 Source

`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 Source

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 () Source

This `Prism` compares for exact equality with a given value.

    >>> only 4 # ()
    4

    >>> 5 ^? only 4
    Nothing

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

This `Prism` compares for approximate equality with a given value and a predicate for testing, an example where the value is the empty list and the predicate checks that a list is empty (same as `_Empty` with the `AsEmpty` list instance):

    >>> nearly [] null # ()
    []
    >>> [1,2,3,4] ^? nearly [] null
    Nothing

    nearly [] null :: Prism' [a] ()

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 `Costar` of `Functor` that is strong with respect to `Either`.

Note: This is also a notion of strength, except with regards to another monoidal structure that we can choose to equip Hask with: the cocartesian coproduct.

Minimal complete definition

left' | right'

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 (Star f) |
Traversable w => Choice (Costar w) |
ArrowChoice p => Choice (WrappedArrow p) |
Monoid r => Choice (Forget r) |
Choice (Tagged *) |
Choice (Indexed i) |
Choice (Market a b) |