Control.Lens.Iso

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 (p :: * -> * -> *) c 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 (ST s a) (ST s a)

Methods

strict :: Iso' (ST s a) (ST s a)

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 s t. 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

Unbox a => Reversing (Vector a)

Methods

reversing :: Vector a -> Vector a

Storable a => Reversing (Vector a)

Methods

reversing :: Vector a -> Vector a

Prim a => Reversing (Vector a)

Methods

reversing :: Vector a -> Vector 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 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 (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 (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 (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

Profunctor ((->) LiftedRep LiftedRep)

Methods

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

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

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

(#.) :: Coercible * c b => (b -> c) -> (LiftedRep -> LiftedRep) a b -> (LiftedRep -> LiftedRep) a c

(.#) :: Coercible * b a => (LiftedRep -> LiftedRep) b c -> (a -> b) -> (LiftedRep -> LiftedRep) 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 p, Profunctor q) => Profunctor (Procompose p q)

Methods

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

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

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

(#.) :: Coercible * c b => (b -> c) -> Procompose p q a b -> Procompose p q a c

(.#) :: Coercible * b a => Procompose p q b c -> (a -> b) -> Procompose p q a c

(Profunctor p, Profunctor q) => Profunctor (Rift p q)

Methods

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

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

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

(#.) :: Coercible * c b => (b -> c) -> Rift p q a b -> Rift p q a c

(.#) :: Coercible * b a => Rift p q b c -> (a -> b) -> Rift p q 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 Isos 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 Isos 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