lens-4.15.3: 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 | Trustworthy
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
- pattern Swapped :: forall c p d. Swapped p => p d c -> p c d
- class Strict lazy strict | lazy -> strict, strict -> lazy where
- pattern Strict :: forall t s. Strict s t => t -> s
- pattern Lazy :: forall t s. Strict t s => t -> s
- lazy :: Strict lazy strict => Iso' strict lazy
- class Reversing t where
- reversed :: Reversing a => Iso' a a
- pattern Reversed :: forall t. Reversing t => t -> t
- involuted :: (a -> a) -> Iso' a a
- pattern List :: forall l. 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
- 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`.

Since every `Iso` is both a valid `Lens` and a valid `Prism`, the laws for those types imply the following laws for an `Iso` `f`:

    f . from f ≡ id
    from f . f ≡ 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.

Minimal complete definition

swapped

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 # |
Methods swapped :: (Profunctor p, Functor f) => p (Either b a) (f (Either d c)) -> p (Either a b) (f (Either c d)) #
Swapped (,) # |
Methods swapped :: (Profunctor p, Functor f) => p (b, a) (f (d, c)) -> p (a, b) (f (c, d)) #

pattern Swapped :: forall c p d. Swapped p => p d c -> p c d #

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`.

Minimal complete definition

strict

Methods

strict :: Iso' lazy strict #

Instances

Strict ByteString ByteString # |
Methods strict :: Iso' ByteString ByteString #
Strict Text Text # |
Methods strict :: Iso' Text Text #
Strict (StateT s m a) (StateT s m a) # |
Methods strict :: Iso' (StateT s m a) (StateT s m a) #
Strict (WriterT w m a) (WriterT w m a) # |
Methods strict :: Iso' (WriterT w m a) (WriterT w m a) #
Strict (RWST r w s m a) (RWST r w s m a) # |
Methods strict :: Iso' (RWST r w s m a) (RWST r w s m a) #

pattern Strict :: forall t s. Strict s t => t -> s #

pattern Lazy :: forall t s. Strict t s => t -> 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.

Minimal complete definition

reversing

Methods

reversing :: t -> t #

Instances

Reversing ByteString # |
Methods reversing :: ByteString -> ByteString #
Reversing ByteString # |
Methods reversing :: ByteString -> ByteString #
Reversing Text # |
Methods reversing :: Text -> Text #
Reversing Text # |
Methods reversing :: Text -> Text #
Reversing [a] # |
Methods reversing :: [a] -> [a] #
Reversing (NonEmpty a) # |
Methods reversing :: NonEmpty a -> NonEmpty a #
Reversing (Seq a) # |
Methods reversing :: Seq a -> Seq a #
Reversing (Vector a) # |
Methods reversing :: Vector a -> Vector a #
Storable a => Reversing (Vector a) # |
Methods reversing :: Vector a -> Vector a #
Unbox a => Reversing (Vector a) # |
Methods reversing :: Vector a -> Vector a #
Prim a => Reversing (Vector a) # |
Methods reversing :: Vector a -> Vector a #
Reversing (Deque a) # |
Methods reversing :: Deque a -> 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 :: forall t. Reversing t => t -> t #

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 :: forall l. 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) # |
Methods itraverse :: Applicative f => (i -> a -> f b) -> Magma i t b a -> f (Magma i t b b) # itraversed :: (Indexable i p, Applicative f) => p a (f b) -> Magma i t b a -> f (Magma i t b b) #
FoldableWithIndex i (Magma i t b) # |
Methods ifoldMap :: Monoid m => (i -> a -> m) -> Magma i t b a -> m # ifolded :: (Indexable i p, Contravariant f, Applicative f) => p a (f a) -> Magma i t b a -> f (Magma i t b a) # ifoldr :: (i -> a -> b -> b) -> b -> Magma i t b a -> b # ifoldl :: (i -> b -> a -> b) -> b -> Magma i t b a -> b # ifoldr' :: (i -> a -> b -> b) -> b -> Magma i t b a -> b # ifoldl' :: (i -> b -> a -> b) -> b -> Magma i t b a -> b #
FunctorWithIndex i (Magma i t b) # |
Methods imap :: (i -> a -> b) -> Magma i t b a -> Magma i t b b # imapped :: (Indexable i p, Settable f) => p a (f b) -> Magma i t b a -> f (Magma i t b b) #
Functor (Magma i t b) # |
Methods fmap :: (a -> b) -> Magma i t b a -> Magma i t b b # (<$) :: a -> Magma i t b b -> Magma i t b a #
Foldable (Magma i t b) # |
Methods fold :: Monoid m => Magma i t b m -> m # foldMap :: Monoid m => (a -> m) -> Magma i t b a -> m # foldr :: (a -> b -> b) -> b -> Magma i t b a -> b # foldr' :: (a -> b -> b) -> b -> Magma i t b a -> b # foldl :: (b -> a -> b) -> b -> Magma i t b a -> b # foldl' :: (b -> a -> b) -> b -> Magma i t b a -> b # foldr1 :: (a -> a -> a) -> Magma i t b a -> a # foldl1 :: (a -> a -> a) -> Magma i t b a -> a # toList :: Magma i t b a -> [a] # null :: Magma i t b a -> Bool # length :: Magma i t b a -> Int # elem :: Eq a => a -> Magma i t b a -> Bool # maximum :: Ord a => Magma i t b a -> a # minimum :: Ord a => Magma i t b a -> a # sum :: Num a => Magma i t b a -> a # product :: Num a => Magma i t b a -> a #
Traversable (Magma i t b) # |
Methods traverse :: Applicative f => (a -> f b) -> Magma i t b a -> f (Magma i t b b) # sequenceA :: Applicative f => Magma i t b (f a) -> f (Magma i t b a) # mapM :: Monad m => (a -> m b) -> Magma i t b a -> m (Magma i t b b) # sequence :: Monad m => Magma i t b (m a) -> m (Magma i t b a) #
(Show i, Show a) => Show (Magma i t b a) # |
Methods showsPrec :: Int -> Magma i t b a -> ShowS # show :: Magma i t b a -> String # showList :: [Magma i t b a] -> ShowS #

## 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 (->) |
Methods dimap :: (a -> b) -> (c -> d) -> (b -> c) -> a -> d # lmap :: (a -> b) -> (b -> c) -> a -> c # rmap :: (b -> c) -> (a -> b) -> a -> c # (#.) :: Coercible * c b => (b -> c) -> (a -> b) -> a -> c # (.#) :: Coercible * b a => (b -> c) -> (a -> b) -> a -> c #
Profunctor ReifiedFold # |
Methods dimap :: (a -> b) -> (c -> d) -> ReifiedFold b c -> ReifiedFold a d # lmap :: (a -> b) -> ReifiedFold b c -> ReifiedFold a c # rmap :: (b -> c) -> ReifiedFold a b -> ReifiedFold a c # (#.) :: Coercible * c b => (b -> c) -> ReifiedFold a b -> ReifiedFold a c # (.#) :: Coercible * b a => ReifiedFold b c -> (a -> b) -> ReifiedFold a c #
Profunctor ReifiedGetter # |
Methods dimap :: (a -> b) -> (c -> d) -> ReifiedGetter b c -> ReifiedGetter a d # lmap :: (a -> b) -> ReifiedGetter b c -> ReifiedGetter a c # rmap :: (b -> c) -> ReifiedGetter a b -> ReifiedGetter a c # (#.) :: Coercible * c b => (b -> c) -> ReifiedGetter a b -> ReifiedGetter a c # (.#) :: Coercible * b a => ReifiedGetter b c -> (a -> b) -> ReifiedGetter a c #
Monad m => Profunctor (Kleisli m) |
Methods dimap :: (a -> b) -> (c -> d) -> Kleisli m b c -> Kleisli m a d # lmap :: (a -> b) -> Kleisli m b c -> Kleisli m a c # rmap :: (b -> c) -> Kleisli m a b -> Kleisli m a c # (#.) :: Coercible * c b => (b -> c) -> Kleisli m a b -> Kleisli m a c # (.#) :: Coercible * b a => Kleisli m b c -> (a -> b) -> Kleisli m a c #
Functor w => Profunctor (Cokleisli w) |
Methods dimap :: (a -> b) -> (c -> d) -> Cokleisli w b c -> Cokleisli w a d # lmap :: (a -> b) -> Cokleisli w b c -> Cokleisli w a c # rmap :: (b -> c) -> Cokleisli w a b -> Cokleisli w a c # (#.) :: Coercible * c b => (b -> c) -> Cokleisli w a b -> Cokleisli w a c # (.#) :: Coercible * b a => Cokleisli w b c -> (a -> b) -> Cokleisli w a c #
Profunctor p => Profunctor (Closure p) |
Methods dimap :: (a -> b) -> (c -> d) -> Closure p b c -> Closure p a d # lmap :: (a -> b) -> Closure p b c -> Closure p a c # rmap :: (b -> c) -> Closure p a b -> Closure p a c # (#.) :: Coercible * c b => (b -> c) -> Closure p a b -> Closure p a c # (.#) :: Coercible * b a => Closure p b c -> (a -> b) -> Closure p a c #
Profunctor (Environment p) |
Methods dimap :: (a -> b) -> (c -> d) -> Environment p b c -> Environment p a d # lmap :: (a -> b) -> Environment p b c -> Environment p a c # rmap :: (b -> c) -> Environment p a b -> Environment p a c # (#.) :: Coercible * c b => (b -> c) -> Environment p a b -> Environment p a c # (.#) :: Coercible * b a => Environment p b c -> (a -> b) -> Environment p a c #
Profunctor p => Profunctor (TambaraSum p) |
Methods dimap :: (a -> b) -> (c -> d) -> TambaraSum p b c -> TambaraSum p a d # lmap :: (a -> b) -> TambaraSum p b c -> TambaraSum p a c # rmap :: (b -> c) -> TambaraSum p a b -> TambaraSum p a c # (#.) :: Coercible * c b => (b -> c) -> TambaraSum p a b -> TambaraSum p a c # (.#) :: Coercible * b a => TambaraSum p b c -> (a -> b) -> TambaraSum p a c #
Profunctor (PastroSum p) |
Methods dimap :: (a -> b) -> (c -> d) -> PastroSum p b c -> PastroSum p a d # lmap :: (a -> b) -> PastroSum p b c -> PastroSum p a c # rmap :: (b -> c) -> PastroSum p a b -> PastroSum p a c # (#.) :: Coercible * c b => (b -> c) -> PastroSum p a b -> PastroSum p a c # (.#) :: Coercible * b a => PastroSum p b c -> (a -> b) -> PastroSum p a c #
Profunctor (CotambaraSum p) |
Methods dimap :: (a -> b) -> (c -> d) -> CotambaraSum p b c -> CotambaraSum p a d # lmap :: (a -> b) -> CotambaraSum p b c -> CotambaraSum p a c # rmap :: (b -> c) -> CotambaraSum p a b -> CotambaraSum p a c # (#.) :: Coercible * c b => (b -> c) -> CotambaraSum p a b -> CotambaraSum p a c # (.#) :: Coercible * b a => CotambaraSum p b c -> (a -> b) -> CotambaraSum p a c #
Profunctor (CopastroSum p) |
Methods dimap :: (a -> b) -> (c -> d) -> CopastroSum p b c -> CopastroSum p a d # lmap :: (a -> b) -> CopastroSum p b c -> CopastroSum p a c # rmap :: (b -> c) -> CopastroSum p a b -> CopastroSum p a c # (#.) :: Coercible * c b => (b -> c) -> CopastroSum p a b -> CopastroSum p a c # (.#) :: Coercible * b a => CopastroSum p b c -> (a -> b) -> CopastroSum p a c #
Profunctor p => Profunctor (Tambara p) |
Methods dimap :: (a -> b) -> (c -> d) -> Tambara p b c -> Tambara p a d # lmap :: (a -> b) -> Tambara p b c -> Tambara p a c # rmap :: (b -> c) -> Tambara p a b -> Tambara p a c # (#.) :: Coercible * c b => (b -> c) -> Tambara p a b -> Tambara p a c # (.#) :: Coercible * b a => Tambara p b c -> (a -> b) -> Tambara p a c #
Profunctor (Pastro p) |
Methods dimap :: (a -> b) -> (c -> d) -> Pastro p b c -> Pastro p a d # lmap :: (a -> b) -> Pastro p b c -> Pastro p a c # rmap :: (b -> c) -> Pastro p a b -> Pastro p a c # (#.) :: Coercible * c b => (b -> c) -> Pastro p a b -> Pastro p a c # (.#) :: Coercible * b a => Pastro p b c -> (a -> b) -> Pastro p a c #
Profunctor (Cotambara p) |
Methods dimap :: (a -> b) -> (c -> d) -> Cotambara p b c -> Cotambara p a d # lmap :: (a -> b) -> Cotambara p b c -> Cotambara p a c # rmap :: (b -> c) -> Cotambara p a b -> Cotambara p a c # (#.) :: Coercible * c b => (b -> c) -> Cotambara p a b -> Cotambara p a c # (.#) :: Coercible * b a => Cotambara p b c -> (a -> b) -> Cotambara p a c #
Profunctor (Copastro p) |
Methods dimap :: (a -> b) -> (c -> d) -> Copastro p b c -> Copastro p a d # lmap :: (a -> b) -> Copastro p b c -> Copastro p a c # rmap :: (b -> c) -> Copastro p a b -> Copastro p a c # (#.) :: Coercible * c b => (b -> c) -> Copastro p a b -> Copastro p a c # (.#) :: Coercible * b a => Copastro p b c -> (a -> b) -> Copastro p a c #
Functor f => Profunctor (Star f) |
Methods dimap :: (a -> b) -> (c -> d) -> Star f b c -> Star f a d # lmap :: (a -> b) -> Star f b c -> Star f a c # rmap :: (b -> c) -> Star f a b -> Star f a c # (#.) :: Coercible * c b => (b -> c) -> Star f a b -> Star f a c # (.#) :: Coercible * b a => Star f b c -> (a -> b) -> Star f a c #
Functor f => Profunctor (Costar f) |
Methods dimap :: (a -> b) -> (c -> d) -> Costar f b c -> Costar f a d # lmap :: (a -> b) -> Costar f b c -> Costar f a c # rmap :: (b -> c) -> Costar f a b -> Costar f a c # (#.) :: Coercible * c b => (b -> c) -> Costar f a b -> Costar f a c # (.#) :: Coercible * b a => Costar f b c -> (a -> b) -> Costar f a c #
Arrow p => Profunctor (WrappedArrow p) |
Methods dimap :: (a -> b) -> (c -> d) -> WrappedArrow p b c -> WrappedArrow p a d # lmap :: (a -> b) -> WrappedArrow p b c -> WrappedArrow p a c # rmap :: (b -> c) -> WrappedArrow p a b -> WrappedArrow p a c # (#.) :: Coercible * c b => (b -> c) -> WrappedArrow p a b -> WrappedArrow p a c # (.#) :: Coercible * b a => WrappedArrow p b c -> (a -> b) -> WrappedArrow p a c #
Profunctor (Forget r) |
Methods dimap :: (a -> b) -> (c -> d) -> Forget r b c -> Forget r a d # lmap :: (a -> b) -> Forget r b c -> Forget r a c # rmap :: (b -> c) -> Forget r a b -> Forget r a c # (#.) :: Coercible * c b => (b -> c) -> Forget r a b -> Forget r a c # (.#) :: Coercible * b a => Forget r b c -> (a -> b) -> Forget r a c #
Profunctor (Tagged *) |
Methods dimap :: (a -> b) -> (c -> d) -> Tagged * b c -> Tagged * a d # lmap :: (a -> b) -> Tagged * b c -> Tagged * a c # rmap :: (b -> c) -> Tagged * a b -> Tagged * a c # (#.) :: Coercible * c b => (b -> c) -> Tagged * a b -> Tagged * a c # (.#) :: Coercible * b a => Tagged * b c -> (a -> b) -> Tagged * a c #
Profunctor (Indexed i) # |
Methods dimap :: (a -> b) -> (c -> d) -> Indexed i b c -> Indexed i a d # lmap :: (a -> b) -> Indexed i b c -> Indexed i a c # rmap :: (b -> c) -> Indexed i a b -> Indexed i a c # (#.) :: Coercible * c b => (b -> c) -> Indexed i a b -> Indexed i a c # (.#) :: Coercible * b a => Indexed i b c -> (a -> b) -> Indexed i a c #
Profunctor (ReifiedIndexedFold i) # |
Methods dimap :: (a -> b) -> (c -> d) -> ReifiedIndexedFold i b c -> ReifiedIndexedFold i a d # lmap :: (a -> b) -> ReifiedIndexedFold i b c -> ReifiedIndexedFold i a c # rmap :: (b -> c) -> ReifiedIndexedFold i a b -> ReifiedIndexedFold i a c # (#.) :: Coercible * c b => (b -> c) -> ReifiedIndexedFold i a b -> ReifiedIndexedFold i a c # (.#) :: Coercible * b a => ReifiedIndexedFold i b c -> (a -> b) -> ReifiedIndexedFold i a c #
Profunctor (ReifiedIndexedGetter i) # |
Methods dimap :: (a -> b) -> (c -> d) -> ReifiedIndexedGetter i b c -> ReifiedIndexedGetter i a d # lmap :: (a -> b) -> ReifiedIndexedGetter i b c -> ReifiedIndexedGetter i a c # rmap :: (b -> c) -> ReifiedIndexedGetter i a b -> ReifiedIndexedGetter i a c # (#.) :: Coercible * c b => (b -> c) -> ReifiedIndexedGetter i a b -> ReifiedIndexedGetter i a c # (.#) :: Coercible * b a => ReifiedIndexedGetter i b c -> (a -> b) -> ReifiedIndexedGetter i a c #
(Functor f, Profunctor p) => Profunctor (Cayley f p) |
Methods dimap :: (a -> b) -> (c -> d) -> Cayley f p b c -> Cayley f p a d # lmap :: (a -> b) -> Cayley f p b c -> Cayley f p a c # rmap :: (b -> c) -> Cayley f p a b -> Cayley f p a c # (#.) :: Coercible * c b => (b -> c) -> Cayley f p a b -> Cayley f p a c # (.#) :: Coercible * b a => Cayley f p b c -> (a -> b) -> Cayley f p a c #
Profunctor (Exchange a b) # |
Methods dimap :: (a -> b) -> (c -> d) -> Exchange a b b c -> Exchange a b a d # lmap :: (a -> b) -> Exchange a b b c -> Exchange a b a c # rmap :: (b -> c) -> Exchange a b a b -> Exchange a b a c # (#.) :: Coercible * c b => (b -> c) -> Exchange a b a b -> Exchange a b a c # (.#) :: Coercible * b a => Exchange a b b c -> (a -> b) -> Exchange a b a c #
Profunctor (Market a b) # |
Methods dimap :: (a -> b) -> (c -> d) -> Market a b b c -> Market a b a d # lmap :: (a -> b) -> Market a b b c -> Market a b a c # rmap :: (b -> c) -> Market a b a b -> Market a b a c # (#.) :: Coercible * c b => (b -> c) -> Market a b a b -> Market a b a c # (.#) :: Coercible * b a => Market a b b c -> (a -> b) -> Market a b a c #
Functor f => Profunctor (Joker * * f) |
Methods dimap :: (a -> b) -> (c -> d) -> Joker * * f b c -> Joker * * f a d # lmap :: (a -> b) -> Joker * * f b c -> Joker * * f a c # rmap :: (b -> c) -> Joker * * f a b -> Joker * * f a c # (#.) :: Coercible * c b => (b -> c) -> Joker * * f a b -> Joker * * f a c # (.#) :: Coercible * b a => Joker * * f b c -> (a -> b) -> Joker * * f a c #
Contravariant f => Profunctor (Clown * * f) |
Methods dimap :: (a -> b) -> (c -> d) -> Clown * * f b c -> Clown * * f a d # lmap :: (a -> b) -> Clown * * f b c -> Clown * * f a c # rmap :: (b -> c) -> Clown * * f a b -> Clown * * f a c # (#.) :: Coercible * c b => (b -> c) -> Clown * * f a b -> Clown * * f a c # (.#) :: Coercible * b a => Clown * * f b c -> (a -> b) -> Clown * * f a c #
(Profunctor p, Profunctor q) => Profunctor (Product * * p q) |
Methods dimap :: (a -> b) -> (c -> d) -> Product * * p q b c -> Product * * p q a d # lmap :: (a -> b) -> Product * * p q b c -> Product * * p q a c # rmap :: (b -> c) -> Product * * p q a b -> Product * * p q a c # (#.) :: Coercible * c b => (b -> c) -> Product * * p q a b -> Product * * p q a c # (.#) :: Coercible * b a => Product * * p q b c -> (a -> b) -> Product * * p q a c #
(Functor f, Profunctor p) => Profunctor (Tannen * * * f p) |
Methods dimap :: (a -> b) -> (c -> d) -> Tannen * * * f p b c -> Tannen * * * f p a d # lmap :: (a -> b) -> Tannen * * * f p b c -> Tannen * * * f p a c # rmap :: (b -> c) -> Tannen * * * f p a b -> Tannen * * * f p a c # (#.) :: Coercible * c b => (b -> c) -> Tannen * * * f p a b -> Tannen * * * f p a c # (.#) :: Coercible * b a => Tannen * * * f p b c -> (a -> b) -> Tannen * * * f p a c #
(Profunctor p, Functor f, Functor g) => Profunctor (Biff * * * * p f g) |
Methods dimap :: (a -> b) -> (c -> d) -> Biff * * * * p f g b c -> Biff * * * * p f g a d # lmap :: (a -> b) -> Biff * * * * p f g b c -> Biff * * * * p f g a c # rmap :: (b -> c) -> Biff * * * * p f g a b -> Biff * * * * p f g a c # (#.) :: Coercible * c b => (b -> c) -> Biff * * * * p f g a b -> Biff * * * * p f g a c # (.#) :: Coercible * b a => Biff * * * * p f g b c -> (a -> b) -> Biff * * * * p f g 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') #

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 7.8+

Since: 4.13