lens-3.8.3: 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

A `Simple` `AnIso`.

# 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.

``` `under` ≡ `over` `.` `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`
```
````>>> ````(fst^.curried) 3 4
```3
```
````>>> ````view curried fst 3 4
```3
```

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`
```
````>>> ````((+)^.uncurried) (1,2)
```3
```

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

The isomorphism for flipping a function.

````>>> ````((,)^.flipped) 1 2
```(2,1)
```

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

``` `swapped` `.` `swapped` ≡ `id`
`first` f `.` `swapped` = `swapped` `.` `second` f
`second` g `.` `swapped` = `swapped` `.` `first` g
`bimap` f g `.` `swapped` = `swapped` `.` `bimap` g f
```
````>>> ````(1,2)^.swapped
```(2,1)
```

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

Formally, the class `Profunctor` represents a profunctor from `Hask` -> `Hask`.

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` `id` ≡ `id``

If you supply `lmap` and `rmap`, ensure:

``` `lmap` `id` ≡ `id`
`rmap` `id` ≡ `id`
```

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.

``rmap` ≡ `dimap` `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)