-- Hoogle documentation, generated by Haddock -- See Hoogle, http://www.haskell.org/hoogle/ -- | Profunctors -- -- Profunctors @package profunctors @version 4.0.4 module Data.Profunctor.Trace -- | Coend of Profunctor from Hask -> Hask. data Trace f Trace :: f a a -> Trace f -- | For a good explanation of profunctors in Haskell see Dan Piponi's -- article: -- -- http://blog.sigfpe.com/2011/07/profunctors-in-haskell.html -- -- This module includes unsafe composition operators that are -- useful in practice when it comes to generating optimal core in GHC. -- -- If you import this module you are taking upon yourself the obligation -- that you will only call the operators with # in their names -- with functions that are operationally identity such as -- newtype constructors or the field accessor of a -- newtype. -- -- If you are ever in doubt, use rmap or lmap. module Data.Profunctor.Unsafe -- | 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
--

--   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
--

class Profunctor p where dimap f g = lmap f . rmap g lmap f = dimap f id rmap = dimap id #. = \ f -> \ p -> p `seq` rmap f p .# = \ p -> p `seq` \ f -> lmap f p dimap :: Profunctor p => (a -> b) -> (c -> d) -> p b c -> p a d lmap :: Profunctor p => (a -> b) -> p b c -> p a c rmap :: Profunctor p => (b -> c) -> p a b -> p a c -- | Generalizing UpStar of a strong Functor -- -- Minimal complete definition: first' or second' -- -- Note: Every Functor in Haskell is strong. -- -- http://takeichi.ipl-lab.org/~asada/papers/arrStrMnd.pdf class Profunctor p => Strong p where first' = dimap swap swap . second' second' = dimap swap swap . first' first' :: Strong p => p a b -> p (a, c) (b, c) second' :: Strong p => p a b -> p (c, a) (c, b) -- | The generalization of DownStar of a "costrong" Functor -- -- Minimal complete definition: left' or right' -- -- Note: We use traverse and extract as approximate -- costrength as needed. class Profunctor p => Choice p where left' = dimap (either Right Left) (either Right Left) . right' right' = dimap (either Right Left) (either Right Left) . left' left' :: Choice p => p a b -> p (Either a c) (Either b c) right' :: Choice p => p a b -> p (Either c a) (Either c b) -- | Lift a Functor into a Profunctor (forwards). newtype UpStar f d c UpStar :: (d -> f c) -> UpStar f d c runUpStar :: UpStar f d c -> d -> f c -- | Lift a Functor into a Profunctor (backwards). newtype DownStar f d c DownStar :: (f d -> c) -> DownStar f d c runDownStar :: DownStar f d c -> f d -> c -- | Wrap an arrow for use as a Profunctor. newtype WrappedArrow p a b WrapArrow :: p a b -> WrappedArrow p a b unwrapArrow :: WrappedArrow p a b -> p a b newtype Forget r a b Forget :: (a -> r) -> Forget r a b runForget :: Forget r a b -> a -> r instance Monoid r => Choice (Forget r) instance ArrowChoice p => Choice (WrappedArrow p) instance Choice (Tagged *) instance Traversable w => Choice (DownStar w) instance Comonad w => Choice (Cokleisli w) instance Applicative f => Choice (UpStar f) instance Monad m => Choice (Kleisli m) instance Choice (->) instance Strong (Forget r) instance Arrow p => Strong (WrappedArrow p) instance Functor m => Strong (UpStar m) instance Monad m => Strong (Kleisli m) instance Strong (->) instance Traversable (Forget r a) instance Foldable (Forget r a) instance Functor (Forget r a) instance Profunctor (Forget r) instance Arrow p => Profunctor (WrappedArrow p) instance ArrowLoop p => ArrowLoop (WrappedArrow p) instance ArrowApply p => ArrowApply (WrappedArrow p) instance ArrowChoice p => ArrowChoice (WrappedArrow p) instance ArrowZero p => ArrowZero (WrappedArrow p) instance Arrow p => Arrow (WrappedArrow p) instance Category p => Category (WrappedArrow p) instance Functor (DownStar f a) instance Functor f => Profunctor (DownStar f) instance Functor f => Functor (UpStar f a) instance Functor f => Profunctor (UpStar f) module Data.Profunctor.Rep -- | A Profunctor p is Representable if there exists -- a Functor f such that p d c is isomorphic to -- d -> f c. class (Functor (Rep p), Profunctor p) => Representable p where type family Rep p :: * -> * tabulate :: Representable p => (d -> Rep p c) -> p d c rep :: Representable p => p d c -> d -> Rep p c -- | tabulate and rep form two halves of an isomorphism. -- -- This can be used with the combinators from the lens package. -- --

--   tabulated :: Representable p => Iso' (d -> Rep p c) (p d c)
--

tabulated :: (Representable p, Representable q) => Iso (d -> Rep p c) (d' -> Rep q c') (p d c) (q d' c') -- | A Profunctor p is Corepresentable if there -- exists a Functor f such that p d c is -- isomorphic to f d -> c. class (Functor (Corep p), Profunctor p) => Corepresentable p where type family Corep p :: * -> * cotabulate :: Corepresentable p => (Corep p d -> c) -> p d c corep :: Corepresentable p => p d c -> Corep p d -> c -- | cotabulate and corep form two halves of an isomorphism. -- -- This can be used with the combinators from the lens package. -- --

--   tabulated :: Corep f p => Iso' (f d -> c) (p d c)
--

cotabulated :: (Corepresentable p, Corepresentable q) => Iso (Corep p d -> c) (Corep q d' -> c') (p d c) (q d' c') instance Functor f => Corepresentable (DownStar f) instance Corepresentable (Tagged *) instance Functor w => Corepresentable (Cokleisli w) instance Corepresentable (->) instance Functor f => Representable (UpStar f) instance (Monad m, Functor m) => Representable (Kleisli m) instance Representable (->) module Data.Profunctor.Composition -- | Procompose p q is the Profunctor composition of -- the Profunctors p and q. -- -- For a good explanation of Profunctor composition in Haskell see -- Dan Piponi's article: -- -- http://blog.sigfpe.com/2011/07/profunctors-in-haskell.html data Procompose p q d c Procompose :: p d a -> q a c -> Procompose p q d c procomposed :: Category p => Procompose p p a b -> p a b -- | (->) functions as a lax identity for Profunctor -- composition. -- -- This provides an Iso for the lens package that -- witnesses the isomorphism between Procompose (->) q d -- c and q d c, which is the left identity law. -- --

--   idl :: Profunctor q => Iso' (Procompose (->) q d c) (q d c)
--

idl :: Profunctor q => Iso (Procompose (->) q d c) (Procompose (->) r d' c') (q d c) (r d' c') -- | (->) functions as a lax identity for Profunctor -- composition. -- -- This provides an Iso for the lens package that -- witnesses the isomorphism between Procompose q (->) d -- c and q d c, which is the right identity law. -- --

--   idr :: Profunctor q => Iso' (Procompose q (->) d c) (q d c)
--

idr :: Profunctor q => Iso (Procompose q (->) d c) (Procompose r (->) d' c') (q d c) (r d' c') -- | The associator for Profunctor composition. -- -- This provides an Iso for the lens package that -- witnesses the isomorphism between Procompose p -- (Procompose q r) a b and Procompose -- (Procompose p q) r a b, which arises because Prof -- is only a bicategory, rather than a strict 2-category. assoc :: Iso (Procompose p (Procompose q r) a b) (Procompose x (Procompose y z) a b) (Procompose (Procompose p q) r a b) (Procompose (Procompose x y) z a b) -- | Profunctor composition generalizes Functor composition -- in two ways. -- -- This is the first, which shows that exists b. (a -> f b, b -- -> g c) is isomorphic to a -> f (g c). -- --

--   upstars :: Functor f => Iso' (Procompose (UpStar f) (UpStar g) d c) (UpStar (Compose f g) d c)
--

upstars :: Functor f => Iso (Procompose (UpStar f) (UpStar g) d c) (Procompose (UpStar f') (UpStar g') d' c') (UpStar (Compose f g) d c) (UpStar (Compose f' g') d' c') -- | This is a variant on upstars that uses Kleisli instead -- of UpStar. -- --

--   kleislis :: Monad f => Iso' (Procompose (Kleisli f) (Kleisli g) d c) (Kleisli (Compose f g) d c)
--

kleislis :: Monad f => Iso (Procompose (Kleisli f) (Kleisli g) d c) (Procompose (Kleisli f') (Kleisli g') d' c') (Kleisli (Compose f g) d c) (Kleisli (Compose f' g') d' c') -- | Profunctor composition generalizes Functor composition -- in two ways. -- -- This is the second, which shows that exists b. (f a -> b, g b -- -> c) is isomorphic to g (f a) -> c. -- --

--   downstars :: Functor f => Iso' (Procompose (DownStar f) (DownStar g) d c) (DownStar (Compose g f) d c)
--

downstars :: Functor g => Iso (Procompose (DownStar f) (DownStar g) d c) (Procompose (DownStar f') (DownStar g') d' c') (DownStar (Compose g f) d c) (DownStar (Compose g' f') d' c') -- | This is a variant on downstars that uses Cokleisli -- instead of DownStar. -- --

--   cokleislis :: Functor f => Iso' (Procompose (Cokleisli f) (Cokleisli g) d c) (Cokleisli (Compose g f) d c)
--

cokleislis :: Functor g => Iso (Procompose (Cokleisli f) (Cokleisli g) d c) (Procompose (Cokleisli f') (Cokleisli g') d' c') (Cokleisli (Compose g f) d c) (Cokleisli (Compose g' f') d' c') instance (Choice p, Choice q) => Choice (Procompose p q) instance (Strong p, Strong q) => Strong (Procompose p q) instance (Corepresentable p, Corepresentable q) => Corepresentable (Procompose p q) instance (Representable p, Representable q) => Representable (Procompose p q) instance Profunctor q => Functor (Procompose p q a) instance (Profunctor p, Profunctor q) => Profunctor (Procompose p q) module Data.Profunctor.Lift -- | This represents the left Kan lift of a Profunctor q -- along a Profunctor p in a limited version of the -- 2-category of Profunctors where the only object is the category Hask, -- 1-morphisms are profunctors composed and compose with Profunctor -- composition, and 2-morphisms are just natural transformations. newtype Lift p q a b Lift :: (forall x. p b x -> q a x) -> Lift p q a b runLift :: Lift p q a b -> forall x. p b x -> q a x -- | The 2-morphism that defines a left Kan lift. -- -- Note: When p is right adjoint to Lift p -- (->) then decomposeLift is the counit of the -- adjunction. decomposeLift :: Procompose (Lift p q) p a b -> q a b instance p ~ q => Category (Lift p q) instance Profunctor p => Functor (Lift p q a) instance (Profunctor p, Profunctor q) => Profunctor (Lift p q) module Data.Profunctor.Rift -- | This represents the right Kan lift of a Profunctor q -- along a Profunctor p in a limited version of the -- 2-category of Profunctors where the only object is the category Hask, -- 1-morphisms are profunctors composed and compose with Profunctor -- composition, and 2-morphisms are just natural transformations. newtype Rift p q a b Rift :: (forall x. p x a -> q x b) -> Rift p q a b runRift :: Rift p q a b -> forall x. p x a -> q x b -- | The 2-morphism that defines a right Kan lift. -- -- Note: When q is left adjoint to Rift q -- (->) then decomposeRift is the counit of the -- adjunction. decomposeRift :: Procompose q (Rift q p) a b -> p a b precomposeRift :: Profunctor q => Procompose (Rift p (->)) q a b -> Rift p q a b instance p ~ q => Category (Rift p q) instance Profunctor q => Functor (Rift p q a) instance (Profunctor p, Profunctor q) => Profunctor (Rift p q) module Data.Profunctor.Collage -- | The cograph of a Profunctor. data Collage k b a L :: (b -> b') -> Collage k (L b) (L b') R :: (a -> a') -> Collage k (R a) (R a') C :: k b a -> Collage k (L b) (R a) instance Profunctor k => Ob (Collage k) (R a) instance Profunctor k => Ob (Collage k) (L a) instance Profunctor k => Semigroupoid (Collage k)