Control.Lens.Iso

lens-4.13.2.1: Lenses, Folds and Traversals

Copyright | (C) 2012-16 Edward Kmett
License | BSD-style (see the file LICENSE)
Maintainer | Edward Kmett <ekmett@gmail.com>
Stability | provisional
Portability | Rank2Types
Safe Haskell | None
Language | Haskell98

Control.Lens.Iso

Contents

- Isomorphism Lenses
- Isomorphism Construction
- Consuming Isomorphisms
- Working with isomorphisms
  - Common Isomorphisms
  - Uncommon Isomorphisms
  - Contravariant functors
- Profunctors
- Bifunctors
- Coercions

Description

Synopsis

- type Iso s t a b = forall p f. (Profunctor p, Functor f) => p a (f b) -> p s (f t)
- type Iso' s a = Iso s s a a
- type AnIso s t a b = Exchange a b a (Identity b) -> Exchange a b s (Identity t)
- type AnIso' s a = AnIso s s a a
- iso :: (s -> a) -> (b -> t) -> Iso s t a b
- from :: AnIso s t a b -> Iso b a t s
- cloneIso :: AnIso s t a b -> Iso s t a b
- withIso :: AnIso s t a b -> ((s -> a) -> (b -> t) -> r) -> r
- au :: Functor f => AnIso s t a b -> ((b -> t) -> f s) -> f a
- auf :: Optic (Costar f) g s t a b -> (f a -> g b) -> f s -> g t
- under :: AnIso s t a b -> (t -> s) -> b -> a
- mapping :: (Functor f, Functor g) => AnIso s t a b -> Iso (f s) (g t) (f a) (g b)
- simple :: Equality' a a
- non :: Eq a => a -> Iso' (Maybe a) a
- non' :: APrism' a () -> Iso' (Maybe a) a
- anon :: a -> (a -> Bool) -> Iso' (Maybe a) a
- enum :: Enum a => Iso' Int a
- curried :: Iso ((a, b) -> c) ((d, e) -> f) (a -> b -> c) (d -> e -> f)
- uncurried :: Iso (a -> b -> c) (d -> e -> f) ((a, b) -> c) ((d, e) -> f)
- flipped :: Iso (a -> b -> c) (a' -> b' -> c') (b -> a -> c) (b' -> a' -> c')
- class Bifunctor p => Swapped p where
  - swapped :: Iso (p a b) (p c d) (p b a) (p d c)
- pattern Swapped :: () => Swapped p => p b a -> p a b
- class Strict lazy strict | lazy -> strict, strict -> lazy where
  - strict :: Iso' lazy strict
- pattern Strict :: () => Strict s strict => strict -> s
- pattern Lazy :: () => Strict lazy s => lazy -> s
- lazy :: Strict lazy strict => Iso' strict lazy
- class Reversing t where
  - reversing :: t -> t
- reversed :: Reversing a => Iso' a a
- pattern Reversed :: () => Reversing s => s -> s
- involuted :: (a -> a) -> Iso' a a
- pattern List :: () => IsList l => [Item l] -> l
- magma :: LensLike (Mafic a b) s t a b -> Iso s u (Magma Int t b a) (Magma j u c c)
- imagma :: Over (Indexed i) (Molten i a b) s t a b -> Iso s t' (Magma i t b a) (Magma j t' c c)
- data Magma i t b a
- contramapping :: Contravariant f => AnIso s t a b -> Iso (f a) (f b) (f s) (f t)
- class Profunctor p where
  - dimap :: (a -> b) -> (c -> d) -> p b c -> p a d
  - lmap :: (a -> b) -> p b c -> p a c
  - rmap :: (b -> c) -> p a b -> p a c
- dimapping :: (Profunctor p, Profunctor q) => AnIso s t a b -> AnIso s' t' a' b' -> Iso (p a s') (q b t') (p s a') (q t b')
- lmapping :: (Profunctor p, Profunctor q) => AnIso s t a b -> Iso (p a x) (q b y) (p s x) (q t y)
- rmapping :: (Profunctor p, Profunctor q) => AnIso s t a b -> Iso (p x s) (q y t) (p x a) (q y b)
- bimapping :: (Bifunctor f, Bifunctor g) => AnIso s t a b -> AnIso s' t' a' b' -> Iso (f s s') (g t t') (f a a') (g b b')
- firsting :: (Bifunctor f, Bifunctor g) => AnIso s t a b -> Iso (f s x) (g t y) (f a x) (g b y)
- seconding :: (Bifunctor f, Bifunctor g) => AnIso s t a b -> Iso (f x s) (g y t) (f x a) (g y b)
- coerced :: forall s t a b. (Coercible s a, Coercible t b) => Iso s t a b

# Isomorphism Lenses

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

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 a

    type Iso' = Simple Iso

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

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

type AnIso' s a = AnIso s s a a

A `Simple` `AnIso`.

# Isomorphism Construction

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

Build a simple isomorphism from a pair of inverse functions.

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

# Consuming Isomorphisms

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

Invert an isomorphism.

    from (from l) ≡ l

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

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.

withIso :: AnIso s t a b -> ((s -> a) -> (b -> t) -> r) -> r

Extract the two functions, one from `s -> a` and one from `b -> t` that characterize an `Iso`.

# Working with isomorphisms

au :: Functor f => AnIso s t a b -> ((b -> t) -> f s) -> f a

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

You may want to think of this combinator as having the following, simpler type:

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

auf :: Optic (Costar f) g s t a b -> (f a -> g b) -> f s -> g t

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`.

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

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

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

but the signature is general.

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

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

    under ≡ over . from

    under :: Iso s t a b -> (t -> s) -> b -> a

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

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

## Common Isomorphisms

simple :: Equality' a a

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) a

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`.

    non ≡ non' . only

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 []

It can also be used in reverse to exclude a given value:

    >>> non 0 # rem 10 4
    Just 2

    >>> non 0 # rem 10 5
    Nothing

non' :: APrism' a () -> Iso' (Maybe a) a

``non'` p` generalizes ``non` (p # ())` to take any unit `Prism`

This function generates an isomorphism between ``Maybe` (a | `isn't` p a)` and `a`.

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

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

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

``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 a

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)

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)

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')

The isomorphism for flipping a function.

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

class Bifunctor p => Swapped p where

This class provides for symmetric bifunctors.

Methods

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

    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 (,) |

pattern Swapped :: () => Swapped p => p b a -> p a b

class Strict lazy strict | lazy -> strict, strict -> lazy where

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

Methods

strict :: Iso' lazy strict

Instances

Strict ByteString ByteString |
Strict Text Text |
Strict (StateT s m a) (StateT s m a) |
Strict (WriterT w m a) (WriterT w m a) |
Strict (RWST r w s m a) (RWST r w s m a) |

pattern Strict :: () => Strict s strict => strict -> s

pattern Lazy :: () => Strict lazy s => lazy -> s

lazy :: Strict lazy strict => Iso' strict lazy

An `Iso` between the strict variant of a structure and its lazy counterpart.

    lazy = from strict

See http://hackage.haskell.org/package/strict-base-types for an example use.

class Reversing t where

This class provides a generalized notion of list reversal extended to other containers.

Methods

reversing :: t -> t

Instances

Reversing ByteString |
Reversing ByteString |
Reversing Text |
Reversing Text |
Reversing [a] |
Reversing (Seq a) |
Reversing (NonEmpty a) |
Reversing (Vector a) |
Unbox a => Reversing (Vector a) |
Storable a => Reversing (Vector a) |
Prim a => Reversing (Vector a) |
Reversing (Deque a) |

reversed :: Reversing a => Iso' a a

An `Iso` between a list, `ByteString`, `Text` fragment, etc. and its reversal.

    >>> "live" ^. reversed
    "evil"

    >>> "live" & reversed %~ ('d':)
    "lived"

pattern Reversed :: () => Reversing s => s -> s

involuted :: (a -> a) -> Iso' a a

Given a function that is its own inverse, this gives you an `Iso` using it in both directions.

    involuted ≡ join iso

    >>> "live" ^. involuted reverse
    "evil"

    >>> "live" & involuted reverse %~ ('d':)
    "lived"

pattern List :: () => IsList l => [Item l] -> l

## Uncommon Isomorphisms

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

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

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

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

Instances

TraversableWithIndex i (Magma i t b) |
FoldableWithIndex i (Magma i t b) |
FunctorWithIndex i (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) |

## Contravariant functors

contramapping :: Contravariant f => AnIso s t a b -> Iso (f a) (f b) (f s) (f t)

Lift an `Iso` into a `Contravariant` functor.

    contramapping :: Contravariant f => Iso s t a b -> Iso (f a) (f b) (f s) (f t)
    contramapping :: Contravariant f => Iso' s a -> Iso' (f a) (f s)

# 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

Minimal complete definition

dimap | lmap, rmap

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 ReifiedFold |
Profunctor ReifiedGetter |
Monad m => Profunctor (Kleisli m) |
Functor w => Profunctor (Cokleisli w) |
Profunctor p => Profunctor (TambaraSum p) |
Profunctor (PastroSum p) |
Profunctor (CotambaraSum p) |
Profunctor (CopastroSum p) |
Profunctor p => Profunctor (Closure p) |
Profunctor (Environment p) |
Profunctor p => Profunctor (Tambara p) |
Profunctor (Pastro p) |
Profunctor (Cotambara p) |
Profunctor (Copastro p) |
Functor f => Profunctor (Star f) |
Functor f => Profunctor (Costar f) |
Arrow p => Profunctor (WrappedArrow p) |
Profunctor (Forget r) |
Profunctor (Tagged *) |
Profunctor (Indexed i) |
Profunctor (ReifiedIndexedFold i) |
Profunctor (ReifiedIndexedGetter i) |
Profunctor (Market a b) |
Profunctor (Exchange a b) |
Functor f => Profunctor (Joker * * f) |
Contravariant f => Profunctor (Clown * * f) |
(Profunctor p, Profunctor q) => Profunctor (Product * * p q) |
(Functor f, Profunctor p) => Profunctor (Tannen * * * f p) |
(Profunctor p, Functor f, Functor g) => Profunctor (Biff * * * * p f g) |

dimapping :: (Profunctor p, Profunctor q) => AnIso s t a b -> AnIso s' t' a' b' -> Iso (p a s') (q b t') (p s a') (q t b')

Lift two `Iso`s into both arguments of a `Profunctor` simultaneously.

    dimapping :: Profunctor p => Iso s t a b -> Iso s' t' a' b' -> Iso (p a s') (p b t') (p s a') (p t b')
    dimapping :: Profunctor p => Iso' s a -> Iso' s' a' -> Iso' (p a s') (p s a')

lmapping :: (Profunctor p, Profunctor q) => AnIso s t a b -> Iso (p a x) (q b y) (p s x) (q t y)

Lift an `Iso` contravariantly into the left argument of a `Profunctor`.

    lmapping :: Profunctor p => Iso s t a b -> Iso (p a x) (p b y) (p s x) (p t y)
    lmapping :: Profunctor p => Iso' s a -> Iso' (p a x) (p s x)

rmapping :: (Profunctor p, Profunctor q) => AnIso s t a b -> Iso (p x s) (q y t) (p x a) (q y b)

Lift an `Iso` covariantly into the right argument of a `Profunctor`.

    rmapping :: Profunctor p => Iso s t a b -> Iso (p x s) (p y t) (p x a) (p y b)
    rmapping :: Profunctor p => Iso' s a -> Iso' (p x s) (p x a)

# Bifunctors

bimapping :: (Bifunctor f, Bifunctor g) => AnIso s t a b -> AnIso s' t' a' b' -> Iso (f s s') (g t t') (f a a') (g b b')

Lift two `Iso`s into both arguments of a `Bifunctor`.

    bimapping :: Bifunctor p => Iso s t a b -> Iso s' t' a' b' -> Iso (p s s') (p t t') (p a a') (p b b')
    bimapping :: Bifunctor p => Iso' s a -> Iso' s' a' -> Iso' (p s s') (p a a')

firsting :: (Bifunctor f, Bifunctor g) => AnIso s t a b -> Iso (f s x) (g t y) (f a x) (g b y)

Lift an `Iso` into the first argument of a `Bifunctor`.

    firsting :: Bifunctor p => Iso s t a b -> Iso (p s x) (p t y) (p a x) (p b y)
    firsting :: Bifunctor p => Iso' s a -> Iso' (p s x) (p a x)

seconding :: (Bifunctor f, Bifunctor g) => AnIso s t a b -> Iso (f x s) (g y t) (f x a) (g y b)

Lift an `Iso` into the second argument of a `Bifunctor`. This is essentially the same as `mapping`, but it takes a 'Bifunctor p' constraint instead of a 'Functor (p a)' one.

    seconding :: Bifunctor p => Iso s t a b -> Iso (p x s) (p y t) (p x a) (p y b)
    seconding :: Bifunctor p => Iso' s a -> Iso' (p x s) (p x a)

# Coercions

coerced :: forall s t a b. (Coercible s a, Coercible t b) => Iso s t a b

Data types that are representationally equal are isomorphic.

This is only available on GHC 7.10+

Since: 4.13