lens-3.8.0.2: Lenses, Folds and Traversals

Portability Rank2Types provisional Edward Kmett Trustworthy

Control.Lens.Iso

Description

Synopsis

# Isomorphism Lenses

type Iso s t a b = forall p f. (Profunctor p, Functor f) => p a (f b) -> p s (f t)Source

Isomorphism families can be composed with another Lens using (.) and id.

Note: Composition with an Iso is index- and measure- preserving.

type Iso' s a = Iso s s a aSource

type Iso' = Simple Iso

type AnIso s t a b = Exchange a b a (Mutator b) -> Exchange a b s (Mutator t)Source

When you see this as an argument to a function, it expects an Iso.

type AnIso' s a = AnIso s s a aSource

# Isomorphism Construction

iso :: (s -> a) -> (b -> t) -> Iso s t a bSource

Build a simple isomorphism from a pair of inverse functions.

view (iso f g) ≡ f
view (from (iso f g)) ≡ g
set (iso f g) h ≡ g . h . f
set (from (iso f g)) h ≡ f . h . g

# Consuming Isomorphisms

from :: AnIso s t a b -> Iso b a t sSource

Invert an isomorphism.

from (from l) ≡ l

cloneIso :: AnIso s t a b -> Iso s t a bSource

Convert from AnIso back to any Iso.

This is useful when you need to store an isomorphism as a data type inside a container and later reconstitute it as an overloaded function.

See cloneLens or cloneTraversal for more information on why you might want to do this.

# Working with isomorphisms

au :: AnIso s t a b -> ((s -> a) -> e -> b) -> e -> tSource

Based on ala from Conor McBride's work on Epigram.

This version is generalized to accept any Iso, not just a newtype.

>>> au (wrapping Sum) foldMap [1,2,3,4]
10

auf :: AnIso s t a b -> ((r -> a) -> e -> b) -> (r -> s) -> e -> tSource

Based on ala' from Conor McBride's work on Epigram.

This version is generalized to accept any Iso, not just a newtype.

For a version you pass the name of the newtype constructor to, see alaf.

Mnemonically, the German auf plays a similar role to à la, and the combinator is au with an extra function argument.

>>> auf (wrapping Sum) (foldMapOf both) Prelude.length ("hello","world")
10

under :: AnIso s t a b -> (t -> s) -> b -> aSource

The opposite of working over a Setter is working under an isomorphism.

underover . from
under :: Iso s t a b -> (s -> t) -> a -> b

mapping :: Functor f => AnIso s t a b -> Iso (f s) (f t) (f a) (f b)Source

This can be used to lift any Iso into an arbitrary Functor.

## Common Isomorphisms

simple :: Iso' a aSource

Composition with this isomorphism is occasionally useful when your Lens, Traversal or Iso has a constraint on an unused argument to force that argument to agree with the type of a used argument and avoid ScopedTypeVariables or other ugliness.

non :: Eq a => a -> Iso' (Maybe a) aSource

If v is an element of a type a, and a' is a sans the element v, then non v is an isomorphism from Maybe a' to a.

Keep in mind this is only a real isomorphism if you treat the domain as being Maybe (a sans v).

This is practically quite useful when you want to have a Map where all the entries should have non-zero values.

>>> Map.fromList [("hello",1)] & at "hello" . non 0 +~ 2
fromList [("hello",3)]
>>> Map.fromList [("hello",1)] & at "hello" . non 0 -~ 1
fromList []
>>> Map.fromList [("hello",1)] ^. at "hello" . non 0
1
>>> Map.fromList [] ^. at "hello" . non 0
0

This combinator is also particularly useful when working with nested maps.

e.g. When you want to create the nested Map when it is missing:

>>> Map.empty & at "hello" . non Map.empty . at "world" ?~ "!!!"
fromList [("hello",fromList [("world","!!!")])]

and when have deleting the last entry from the nested Map mean that we should delete its entry from the surrounding one:

>>> fromList [("hello",fromList [("world","!!!")])] & at "hello" . non Map.empty . at "world" .~ Nothing
fromList []

anon :: a -> (a -> Bool) -> Iso' (Maybe a) aSource

anon a p generalizes non a to take any value and a predicate.

This function assumes that p a holds True and generates an isomorphism between Maybe (a | not (p a)) and a.

>>> Map.empty & at "hello" . anon Map.empty Map.null . at "world" ?~ "!!!"
fromList [("hello",fromList [("world","!!!")])]
>>> fromList [("hello",fromList [("world","!!!")])] & at "hello" . anon Map.empty Map.null . at "world" .~ Nothing
fromList []

enum :: Enum a => Iso' Int aSource

This isomorphism can be used to convert to or from an instance of Enum.

>>> LT^.from enum
0
>>> 97^.enum :: Char
'a'

Note: this is only an isomorphism from the numeric range actually used and it is a bit of a pleasant fiction, since there are questionable Enum instances for Double, and Float that exist solely for [1.0 .. 4.0] sugar and the instances for those and Integer don't cover all values in their range.

curried :: Iso ((a, b) -> c) ((d, e) -> f) (a -> b -> c) (d -> e -> f)Source

The canonical isomorphism for currying and uncurrying a function.

curried = iso curry uncurry

uncurried :: Iso (a -> b -> c) (d -> e -> f) ((a, b) -> c) ((d, e) -> f)Source

The canonical isomorphism for uncurrying and currying a function.

uncurried = iso uncurry curry
uncurried = from curried

flipped :: Iso (a -> b -> c) (a' -> b' -> c') (b -> a -> c) (b' -> a' -> c')Source

The isomorphism for flipping a function.

class Bifunctor p => Swapped p whereSource

This class provides for symmetric bifunctors.

Methods

swapped :: Iso (p a b) (p c d) (p b a) (p d c)Source

Instances

 Swapped Either Swapped (,)

class Strict s a | s -> a, a -> s whereSource

Ad hoc conversion between "strict" and "lazy" versions of a structure, such as Text or ByteString.

Methods

strict :: Iso' s aSource

Instances

 Strict ByteString ByteString Strict Text Text

## Uncommon Isomorphisms

magma :: LensLike (Mafic a b) s t a b -> Iso s u (Magma Int t b a) (Magma j u c c)Source

This isomorphism can be used to inspect a Traversal to see how it associates the structure and it can also be used to bake the Traversal into a Magma so that you can traverse over it multiple times.

imagma :: Overloading (Indexed i) (->) (Molten i a b) s t a b -> Iso s t' (Magma i t b a) (Magma j t' c c)Source

This isomorphism can be used to inspect an IndexedTraversal to see how it associates the structure and it can also be used to bake the IndexedTraversal into a Magma so that you can traverse over it multiple times with access to the original indices.

data Magma i t b a Source

This provides a way to peek at the internal structure of a Traversal or IndexedTraversal

Instances

 (FunctorWithIndex i (Magma i t b), FoldableWithIndex i (Magma i t b), Traversable (Magma i t b)) => TraversableWithIndex i (Magma i t b) Foldable (Magma i t b) => FoldableWithIndex i (Magma i t b) Functor (Magma i t b) => FunctorWithIndex i (Magma i t b) Functor (Magma i t b) Foldable (Magma i t b) (Functor (Magma i t b), Foldable (Magma i t b)) => Traversable (Magma i t b) (Show i, Show a) => Show (Magma i t b a)

# Profunctors

class Profunctor p where

Intuitively it is a bifunctor where the first argument is contravariant and the second argument is covariant.

You can define a Profunctor by either defining dimap or by defining both lmap and rmap.

If you supply dimap, you should ensure that:

dimap id idid

If you supply lmap and rmap, ensure:

lmap idid
rmap idid

If you supply both, you should also ensure:

dimap f g ≡ lmap f . rmap g

These ensure by parametricity:

dimap (f . g) (h . i) ≡ dimap g h . dimap f i
lmap (f . g) ≡ lmap g . lmap f
rmap (f . g) ≡ rmap f . rmap g

Methods

dimap :: (a -> b) -> (c -> d) -> p b c -> p a d

Map over both arguments at the same time.

dimap f g ≡ lmap f . rmap g

lmap :: (a -> b) -> p b c -> p a c

Map the first argument contravariantly.

lmap f ≡ dimap f id

rmap :: (b -> c) -> p a b -> p a c

Map the second argument covariantly.

rmapdimap id

Instances

 Profunctor (->) Profunctor Reviewed Monad m => Profunctor (Kleisli m) Functor w => Profunctor (Cokleisli w) Functor f => Profunctor (DownStar f) Functor f => Profunctor (UpStar f) Arrow p => Profunctor (WrappedArrow p) Profunctor (Tagged *) Profunctor (Indexed i) Profunctor (Market a b) Profunctor (Exchange a b)