Safe Haskell	None
Language	Haskell2010

Data.Profunctor.Optic.Affine

Contents

Affine & Ixaffine
Optics
Primitive operators
Operators
Classes

Synopsis

type Affine s t a b = forall p. (Choice p, Strong p) => Optic p s t a b
type Affine' s a = Affine s s a a
type Ixaffine i s t a b = forall p. (Choice p, Strong p) => IndexedOptic p i s t a b
type Ixaffine' i s a = Ixaffine i s s a a
affine :: (s -> t + a) -> (s -> b -> t) -> Affine s t a b
affine' :: (s -> Maybe a) -> (s -> a -> s) -> Affine' s a
iaffine :: (s -> t + (i, a)) -> (s -> b -> t) -> Ixaffine i s t a b
iaffine' :: (s -> Maybe (i, a)) -> (s -> a -> s) -> Ixaffine' i s a
affineVl :: (forall f. Functor f => (forall c. c -> f c) -> (a -> f b) -> s -> f t) -> Affine s t a b
iaffineVl :: (forall f. Functor f => (forall c. c -> f c) -> (i -> a -> f b) -> s -> f t) -> Ixaffine i s t a b
nulled :: Affine' s a
selected :: (a -> Bool) -> Affine' (a, b) b
withAffine :: AAffine s t a b -> ((s -> t + a) -> (s -> b -> t) -> r) -> r
matches :: AAffine s t a b -> s -> t + a
class Profunctor p => Strong (p :: Type -> Type -> Type) where
- first' :: p a b -> p (a, c) (b, c)
- second' :: p a b -> p (c, a) (c, b)
class Profunctor p => Choice (p :: Type -> Type -> Type) where
- left' :: p a b -> p (Either a c) (Either b c)
- right' :: p a b -> p (Either c a) (Either c b)

Affine & Ixaffine

type Affine s t a b = forall p. (Choice p, Strong p) => Optic p s t a b Source #

A Affine processes 0 or more parts of the whole, with no interactions.

\( \mathsf{Affine}\;S\;A = \exists C, D, S \cong D + C \times A \)

type Affine' s a = Affine s s a a Source #

type Ixaffine i s t a b = forall p. (Choice p, Strong p) => IndexedOptic p i s t a b Source #

type Ixaffine' i s a = Ixaffine i s s a a Source #

affine :: (s -> t + a) -> (s -> b -> t) -> Affine s t a b Source #

Create a Affine from match and constructor functions.

Caution: In order for the Affine to be well-defined, you must ensure that the input functions satisfy the following properties:

sta (sbt a s) ≡ either (Left . const a) Right (sta s)

```
either id (sbt s) (sta s) ≡ s
```
```
sbt (sbt s a1) a2 ≡ sbt s a2
```

More generally, a profunctor optic must be monoidal as a natural transformation:

```
o id ≡ id
```

o (Procompose p q) ≡ Procompose (o p) (o q)

See Property.

affine' :: (s -> Maybe a) -> (s -> a -> s) -> Affine' s a Source #

Obtain a Affine' from match and constructor functions.

iaffine :: (s -> t + (i, a)) -> (s -> b -> t) -> Ixaffine i s t a b Source #

TODO: Document

iaffine' :: (s -> Maybe (i, a)) -> (s -> a -> s) -> Ixaffine' i s a Source #

TODO: Document

affineVl :: (forall f. Functor f => (forall c. c -> f c) -> (a -> f b) -> s -> f t) -> Affine s t a b Source #

Transform a Van Laarhoven Affine into a profunctor Affine.

iaffineVl :: (forall f. Functor f => (forall c. c -> f c) -> (i -> a -> f b) -> s -> f t) -> Ixaffine i s t a b Source #

Transform an indexed Van Laarhoven Affine into an indexed profunctor Affine.

Optics

nulled :: Affine' s a Source #

TODO: Document

selected :: (a -> Bool) -> Affine' (a, b) b Source #

TODO: Document

Primitive operators

withAffine :: AAffine s t a b -> ((s -> t + a) -> (s -> b -> t) -> r) -> r Source #

TODO: Document

Operators

matches :: AAffine s t a b -> s -> t + a Source #

Test whether the optic matches or not.

>>> matches just (Just 2)
Right 2

>>> matches just (Nothing :: Maybe Int) :: Either (Maybe Bool) Int
Left Nothing

Classes

class Profunctor p => Strong (p :: Type -> Type -> Type) where #

Generalizing Star of a strong Functor

Note: Every Functor in Haskell is strong with respect to (,).

This describes profunctor strength with respect to the product structure of Hask.

http://www.riec.tohoku.ac.jp/~asada/papers/arrStrMnd.pdf

Minimal complete definition

first' | second'

Methods

first' :: p a b -> p (a, c) (b, c) #

Laws:

first' ≡ dimap swap swap . second'
lmap fst ≡ rmap fst . first'
lmap (second' f) . first' ≡ rmap (second' f) . first'
first' . first' ≡ dimap assoc unassoc . first' where
  assoc ((a,b),c) = (a,(b,c))
  unassoc (a,(b,c)) = ((a,b),c)

second' :: p a b -> p (c, a) (c, b) #

Laws:

second' ≡ dimap swap swap . first'
lmap snd ≡ rmap snd . second'
lmap (first' f) . second' ≡ rmap (first' f) . second'
second' . second' ≡ dimap unassoc assoc . second' where
  assoc ((a,b),c) = (a,(b,c))
  unassoc (a,(b,c)) = ((a,b),c)

Instances

Monad m => Strong (Kleisli m)
Instance details Defined in Data.Profunctor.Strong Methods first' :: Kleisli m a b -> Kleisli m (a, c) (b, c) # second' :: Kleisli m a b -> Kleisli m (c, a) (c, b) #
Strong (Pastro p)
Instance details Defined in Data.Profunctor.Strong Methods first' :: Pastro p a b -> Pastro p (a, c) (b, c) # second' :: Pastro p a b -> Pastro p (c, a) (c, b) #
Strong p => Strong (Coyoneda p)
Instance details Defined in Data.Profunctor.Yoneda Methods first' :: Coyoneda p a b -> Coyoneda p (a, c) (b, c) # second' :: Coyoneda p a b -> Coyoneda p (c, a) (c, b) #
Strong p => Strong (Yoneda p)
Instance details Defined in Data.Profunctor.Yoneda Methods first' :: Yoneda p a b -> Yoneda p (a, c) (b, c) # second' :: Yoneda p a b -> Yoneda p (c, a) (c, b) #
Strong p => Strong (Closure p)
Instance details Defined in Data.Profunctor.Closed Methods first' :: Closure p a b -> Closure p (a, c) (b, c) # second' :: Closure p a b -> Closure p (c, a) (c, b) #
Profunctor p => Strong (Tambara p)
Instance details Defined in Data.Profunctor.Strong Methods first' :: Tambara p a b -> Tambara p (a, c) (b, c) # second' :: Tambara p a b -> Tambara p (c, a) (c, b) #
Functor m => Strong (Star m)
Instance details Defined in Data.Profunctor.Strong Methods first' :: Star m a b -> Star m (a, c) (b, c) # second' :: Star m a b -> Star m (c, a) (c, b) #
Arrow p => Strong (WrappedArrow p)	`Arrow` is `Strong` `Category`
Instance details Defined in Data.Profunctor.Strong Methods first' :: WrappedArrow p a b -> WrappedArrow p (a, c) (b, c) # second' :: WrappedArrow p a b -> WrappedArrow p (c, a) (c, b) #
Strong (Forget r)
Instance details Defined in Data.Profunctor.Strong Methods first' :: Forget r a b -> Forget r (a, c) (b, c) # second' :: Forget r a b -> Forget r (c, a) (c, b) #
Strong (Conjoin j) Source #
Instance details Defined in Data.Profunctor.Optic.Index Methods first' :: Conjoin j a b -> Conjoin j (a, c) (b, c) # second' :: Conjoin j a b -> Conjoin j (c, a) (c, b) #
Strong (OptionRep r) Source #
Instance details Defined in Data.Profunctor.Optic.Carrier Methods first' :: OptionRep r a b -> OptionRep r (a, c) (b, c) # second' :: OptionRep r a b -> OptionRep r (c, a) (c, b) #
Strong ((->) :: Type -> Type -> Type)
Instance details Defined in Data.Profunctor.Strong Methods first' :: (a -> b) -> (a, c) -> (b, c) # second' :: (a -> b) -> (c, a) -> (c, b) #
Strong (AffineRep a b) Source #
Instance details Defined in Data.Profunctor.Optic.Carrier Methods first' :: AffineRep a b a0 b0 -> AffineRep a b (a0, c) (b0, c) # second' :: AffineRep a b a0 b0 -> AffineRep a b (c, a0) (c, b0) #
Strong (LensRep a b) Source #
Instance details Defined in Data.Profunctor.Optic.Carrier Methods first' :: LensRep a b a0 b0 -> LensRep a b (a0, c) (b0, c) # second' :: LensRep a b a0 b0 -> LensRep a b (c, a0) (c, b0) #
Contravariant f => Strong (Clown f :: Type -> Type -> Type)
Instance details Defined in Data.Profunctor.Strong Methods first' :: Clown f a b -> Clown f (a, c) (b, c) # second' :: Clown f a b -> Clown f (c, a) (c, b) #
Costrong p => Strong (Re p s t) Source #
Instance details Defined in Data.Profunctor.Optic.Types Methods first' :: Re p s t a b -> Re p s t (a, c) (b, c) # second' :: Re p s t a b -> Re p s t (c, a) (c, b) #
Strong (IxlensRep i a b) Source #
Instance details Defined in Data.Profunctor.Optic.Carrier Methods first' :: IxlensRep i a b a0 b0 -> IxlensRep i a b (a0, c) (b0, c) # second' :: IxlensRep i a b a0 b0 -> IxlensRep i a b (c, a0) (c, b0) #
(Strong p, Strong q) => Strong (Sum p q)
Instance details Defined in Data.Profunctor.Strong Methods first' :: Sum p q a b -> Sum p q (a, c) (b, c) # second' :: Sum p q a b -> Sum p q (c, a) (c, b) #
(Strong p, Strong q) => Strong (Product p q)
Instance details Defined in Data.Profunctor.Strong Methods first' :: Product p q a b -> Product p q (a, c) (b, c) # second' :: Product p q a b -> Product p q (c, a) (c, b) #
(Functor f, Strong p) => Strong (Tannen f p)
Instance details Defined in Data.Profunctor.Strong Methods first' :: Tannen f p a b -> Tannen f p (a, c) (b, c) # second' :: Tannen f p a b -> Tannen f p (c, a) (c, b) #

class Profunctor p => Choice (p :: Type -> Type -> Type) where #

The generalization of Costar of Functor that is strong with respect to Either.

Note: This is also a notion of strength, except with regards to another monoidal structure that we can choose to equip Hask with: the cocartesian coproduct.

Minimal complete definition

left' | right'

Methods

left' :: p a b -> p (Either a c) (Either b c) #

Laws:

left' ≡ dimap swapE swapE . right' where
  swapE :: Either a b -> Either b a
  swapE = either Right Left
rmap Left ≡ lmap Left . left'
lmap (right f) . left' ≡ rmap (right f) . left'
left' . left' ≡ dimap assocE unassocE . left' where
  assocE :: Either (Either a b) c -> Either a (Either b c)
  assocE (Left (Left a)) = Left a
  assocE (Left (Right b)) = Right (Left b)
  assocE (Right c) = Right (Right c)
  unassocE :: Either a (Either b c) -> Either (Either a b) c
  unassocE (Left a) = Left (Left a)
  unassocE (Right (Left b)) = Left (Right b)
  unassocE (Right (Right c)) = Right c

right' :: p a b -> p (Either c a) (Either c b) #

Laws:

right' ≡ dimap swapE swapE . left' where
  swapE :: Either a b -> Either b a
  swapE = either Right Left
rmap Right ≡ lmap Right . right'
lmap (left f) . right' ≡ rmap (left f) . right'
right' . right' ≡ dimap unassocE assocE . right' where
  assocE :: Either (Either a b) c -> Either a (Either b c)
  assocE (Left (Left a)) = Left a
  assocE (Left (Right b)) = Right (Left b)
  assocE (Right c) = Right (Right c)
  unassocE :: Either a (Either b c) -> Either (Either a b) c
  unassocE (Left a) = Left (Left a)
  unassocE (Right (Left b)) = Left (Right b)
  unassocE (Right (Right c)) = Right c

Instances

Monad m => Choice (Kleisli m)
Instance details Defined in Data.Profunctor.Choice Methods left' :: Kleisli m a b -> Kleisli m (Either a c) (Either b c) # right' :: Kleisli m a b -> Kleisli m (Either c a) (Either c b) #
Choice (PastroSum p)
Instance details Defined in Data.Profunctor.Choice Methods left' :: PastroSum p a b -> PastroSum p (Either a c) (Either b c) # right' :: PastroSum p a b -> PastroSum p (Either c a) (Either c b) #
Choice p => Choice (Coyoneda p)
Instance details Defined in Data.Profunctor.Yoneda Methods left' :: Coyoneda p a b -> Coyoneda p (Either a c) (Either b c) # right' :: Coyoneda p a b -> Coyoneda p (Either c a) (Either c b) #
Choice p => Choice (Yoneda p)
Instance details Defined in Data.Profunctor.Yoneda Methods left' :: Yoneda p a b -> Yoneda p (Either a c) (Either b c) # right' :: Yoneda p a b -> Yoneda p (Either c a) (Either c b) #
Profunctor p => Choice (TambaraSum p)
Instance details Defined in Data.Profunctor.Choice Methods left' :: TambaraSum p a b -> TambaraSum p (Either a c) (Either b c) # right' :: TambaraSum p a b -> TambaraSum p (Either c a) (Either c b) #
Choice p => Choice (Tambara p)
Instance details Defined in Data.Profunctor.Choice Methods left' :: Tambara p a b -> Tambara p (Either a c) (Either b c) # right' :: Tambara p a b -> Tambara p (Either c a) (Either c b) #
Applicative f => Choice (Star f)
Instance details Defined in Data.Profunctor.Choice Methods left' :: Star f a b -> Star f (Either a c) (Either b c) # right' :: Star f a b -> Star f (Either c a) (Either c b) #
Coapplicative f => Choice (Costar f) Source #
Instance details Defined in Data.Profunctor.Optic.Types Methods left' :: Costar f a b -> Costar f (Either a c) (Either b c) # right' :: Costar f a b -> Costar f (Either c a) (Either c b) #
ArrowChoice p => Choice (WrappedArrow p)
Instance details Defined in Data.Profunctor.Choice Methods left' :: WrappedArrow p a b -> WrappedArrow p (Either a c) (Either b c) # right' :: WrappedArrow p a b -> WrappedArrow p (Either c a) (Either c b) #
Monoid r => Choice (Forget r)
Instance details Defined in Data.Profunctor.Choice Methods left' :: Forget r a b -> Forget r (Either a c) (Either b c) # right' :: Forget r a b -> Forget r (Either c a) (Either c b) #
Choice (Tagged :: Type -> Type -> Type)
Instance details Defined in Data.Profunctor.Choice Methods left' :: Tagged a b -> Tagged (Either a c) (Either b c) # right' :: Tagged a b -> Tagged (Either c a) (Either c b) #
Choice (Conjoin j) Source #
Instance details Defined in Data.Profunctor.Optic.Index Methods left' :: Conjoin j a b -> Conjoin j (Either a c) (Either b c) # right' :: Conjoin j a b -> Conjoin j (Either c a) (Either c b) #
Choice (OptionRep r) Source #
Instance details Defined in Data.Profunctor.Optic.Carrier Methods left' :: OptionRep r a b -> OptionRep r (Either a c) (Either b c) # right' :: OptionRep r a b -> OptionRep r (Either c a) (Either c b) #
Choice ((->) :: Type -> Type -> Type)
Instance details Defined in Data.Profunctor.Choice Methods left' :: (a -> b) -> Either a c -> Either b c # right' :: (a -> b) -> Either c a -> Either c b #
Comonad w => Choice (Cokleisli w)	`extract` approximates `costrength`
Instance details Defined in Data.Profunctor.Choice Methods left' :: Cokleisli w a b -> Cokleisli w (Either a c) (Either b c) # right' :: Cokleisli w a b -> Cokleisli w (Either c a) (Either c b) #
Choice (GrismRep a b) Source #
Instance details Defined in Data.Profunctor.Optic.Carrier Methods left' :: GrismRep a b a0 b0 -> GrismRep a b (Either a0 c) (Either b0 c) # right' :: GrismRep a b a0 b0 -> GrismRep a b (Either c a0) (Either c b0) #
Choice (AffineRep a b) Source #
Instance details Defined in Data.Profunctor.Optic.Carrier Methods left' :: AffineRep a b a0 b0 -> AffineRep a b (Either a0 c) (Either b0 c) # right' :: AffineRep a b a0 b0 -> AffineRep a b (Either c a0) (Either c b0) #
Choice (PrismRep a b) Source #
Instance details Defined in Data.Profunctor.Optic.Carrier Methods left' :: PrismRep a b a0 b0 -> PrismRep a b (Either a0 c) (Either b0 c) # right' :: PrismRep a b a0 b0 -> PrismRep a b (Either c a0) (Either c b0) #
Functor f => Choice (Joker f :: Type -> Type -> Type)
Instance details Defined in Data.Profunctor.Choice Methods left' :: Joker f a b -> Joker f (Either a c) (Either b c) # right' :: Joker f a b -> Joker f (Either c a) (Either c b) #
Cochoice p => Choice (Re p s t) Source #
Instance details Defined in Data.Profunctor.Optic.Types Methods left' :: Re p s t a b -> Re p s t (Either a c) (Either b c) # right' :: Re p s t a b -> Re p s t (Either c a) (Either c b) #
(Choice p, Choice q) => Choice (Sum p q)
Instance details Defined in Data.Profunctor.Choice Methods left' :: Sum p q a b -> Sum p q (Either a c) (Either b c) # right' :: Sum p q a b -> Sum p q (Either c a) (Either c b) #
(Choice p, Choice q) => Choice (Product p q)
Instance details Defined in Data.Profunctor.Choice Methods left' :: Product p q a b -> Product p q (Either a c) (Either b c) # right' :: Product p q a b -> Product p q (Either c a) (Either c b) #
(Functor f, Choice p) => Choice (Tannen f p)
Instance details Defined in Data.Profunctor.Choice Methods left' :: Tannen f p a b -> Tannen f p (Either a c) (Either b c) # right' :: Tannen f p a b -> Tannen f p (Either c a) (Either c b) #