contravariant-1.4.1: Contravariant functors

Data.Functor.Contravariant.Divisible

Description

This module supplies contravariant analogues to the Applicative and Alternative classes.

Synopsis

# Contravariant Applicative

class Contravariant f => Divisible f where Source #

A Divisible contravariant functor is the contravariant analogue of Applicative.

Continuing the intuition that Contravariant functors consume input, a Divisible contravariant functor also has the ability to be composed "beside" another contravariant functor.

Serializers provide a good example of Divisible contravariant functors. To begin let's start with the type of serializers for specific types:

newtype Serializer a = Serializer { runSerializer :: a -> ByteString }


This is a contravariant functor:

instance Contravariant Serializer where
contramap f s = Serializer (runSerializer s . f)


That is, given a serializer for a (s :: Serializer a), and a way to turn bs into as (a mapping f :: b -> a), we have a serializer for b: contramap f s :: Serializer b.

Divisible gives us a way to combine two serializers that focus on different parts of a structure. If we postulate the existance of two primitive serializers - string :: Serializer String and int :: Serializer Int, we would like to be able to combine these into a serializer for pairs of Strings and Ints. How can we do this? Simply run both serializer and combine their output!

data StringAndInt = StringAndInt String Int

stringAndInt :: Serializer StringAndInt
stringAndInt = Serializer $(StringAndInt s i) -> let sBytes = runSerializer string s iBytes = runSerializer int i in sBytes <> iBytes  divide is a generalization by also taking a contramap like function to split any a into a pair. This conveniently allows you to target fields of a record, for instance, by extracting the values under two fields and combining them into a tuple. To complete the example, here is how to write stringAndInt using a Divisible instance: instance Divisible Serializer where conquer = Serializer (const mempty) divide toBC bSerializer cSerializer = Serializer$ a ->
case toBC a of
(b, c) ->
let bBytes = runSerializer bSerializer b
cBytes = runSerializer cSerializer c
in bBytes <> cBytes

stringAndInt :: Serializer StringAndInt
stringAndInt =
divide ((StringAndInt s i) -> (s, i)) string int


Minimal complete definition

Methods

divide :: (a -> (b, c)) -> f b -> f c -> f a Source #

conquer :: f a Source #

Conquer acts as an identity for combining Divisible functors.

Instances

 Source # Methodsdivide :: (a -> (b, c)) -> SettableStateVar b -> SettableStateVar c -> SettableStateVar a Source # Source # Methodsdivide :: (a -> (b, c)) -> Equivalence b -> Equivalence c -> Equivalence a Source # Source # Methodsdivide :: (a -> (b, c)) -> Comparison b -> Comparison c -> Comparison a Source # Source # Methodsdivide :: (a -> (b, c)) -> Predicate b -> Predicate c -> Predicate a Source # Source # Methodsdivide :: (a -> (b, c)) -> U1 * b -> U1 * c -> U1 * a Source # Source # Methodsdivide :: (a -> (b, c)) -> Proxy * b -> Proxy * c -> Proxy * a Source # Divisible m => Divisible (MaybeT m) Source # Methodsdivide :: (a -> (b, c)) -> MaybeT m b -> MaybeT m c -> MaybeT m a Source #conquer :: MaybeT m a Source # Divisible m => Divisible (ListT m) Source # Methodsdivide :: (a -> (b, c)) -> ListT m b -> ListT m c -> ListT m a Source #conquer :: ListT m a Source # Monoid r => Divisible (Op r) Source # Methodsdivide :: (a -> (b, c)) -> Op r b -> Op r c -> Op r a Source #conquer :: Op r a Source # Divisible f => Divisible (Rec1 * f) Source # Methodsdivide :: (a -> (b, c)) -> Rec1 * f b -> Rec1 * f c -> Rec1 * f a Source #conquer :: Rec1 * f a Source # Monoid m => Divisible (Const * m) Source # Methodsdivide :: (a -> (b, c)) -> Const * m b -> Const * m c -> Const * m a Source #conquer :: Const * m a Source # Divisible f => Divisible (Alt * f) Source # Methodsdivide :: (a -> (b, c)) -> Alt * f b -> Alt * f c -> Alt * f a Source #conquer :: Alt * f a Source # Divisible f => Divisible (Reverse * f) Source # Methodsdivide :: (a -> (b, c)) -> Reverse * f b -> Reverse * f c -> Reverse * f a Source # Monoid m => Divisible (Constant * m) Source # Methodsdivide :: (a -> (b, c)) -> Constant * m b -> Constant * m c -> Constant * m a Source # Divisible m => Divisible (WriterT w m) Source # Methodsdivide :: (a -> (b, c)) -> WriterT w m b -> WriterT w m c -> WriterT w m a Source #conquer :: WriterT w m a Source # Divisible m => Divisible (WriterT w m) Source # Methodsdivide :: (a -> (b, c)) -> WriterT w m b -> WriterT w m c -> WriterT w m a Source #conquer :: WriterT w m a Source # Divisible m => Divisible (StateT s m) Source # Methodsdivide :: (a -> (b, c)) -> StateT s m b -> StateT s m c -> StateT s m a Source #conquer :: StateT s m a Source # Divisible m => Divisible (StateT s m) Source # Methodsdivide :: (a -> (b, c)) -> StateT s m b -> StateT s m c -> StateT s m a Source #conquer :: StateT s m a Source # Divisible f => Divisible (IdentityT * f) Source # Methodsdivide :: (a -> (b, c)) -> IdentityT * f b -> IdentityT * f c -> IdentityT * f a Source # Divisible m => Divisible (ExceptT e m) Source # Methodsdivide :: (a -> (b, c)) -> ExceptT e m b -> ExceptT e m c -> ExceptT e m a Source #conquer :: ExceptT e m a Source # Divisible m => Divisible (ErrorT e m) Source # Methodsdivide :: (a -> (b, c)) -> ErrorT e m b -> ErrorT e m c -> ErrorT e m a Source #conquer :: ErrorT e m a Source # Divisible f => Divisible (Backwards * f) Source # Methodsdivide :: (a -> (b, c)) -> Backwards * f b -> Backwards * f c -> Backwards * f a Source # (Divisible f, Applicative g) => Divisible (ComposeCF f g) Source # Methodsdivide :: (a -> (b, c)) -> ComposeCF f g b -> ComposeCF f g c -> ComposeCF f g a Source #conquer :: ComposeCF f g a Source # (Applicative f, Divisible g) => Divisible (ComposeFC f g) Source # Methodsdivide :: (a -> (b, c)) -> ComposeFC f g b -> ComposeFC f g c -> ComposeFC f g a Source #conquer :: ComposeFC f g a Source # (Divisible f, Divisible g) => Divisible ((:*:) * f g) Source # Methodsdivide :: (a -> (b, c)) -> (* :*: f) g b -> (* :*: f) g c -> (* :*: f) g a Source #conquer :: (* :*: f) g a Source # (Divisible f, Divisible g) => Divisible (Product * f g) Source # Methodsdivide :: (a -> (b, c)) -> Product * f g b -> Product * f g c -> Product * f g a Source #conquer :: Product * f g a Source # Divisible m => Divisible (ReaderT * r m) Source # Methodsdivide :: (a -> (b, c)) -> ReaderT * r m b -> ReaderT * r m c -> ReaderT * r m a Source #conquer :: ReaderT * r m a Source # Divisible f => Divisible (M1 * i c f) Source # Methodsdivide :: (a -> (b, c)) -> M1 * i c f b -> M1 * i c f c -> M1 * i c f a Source #conquer :: M1 * i c f a Source # (Applicative f, Divisible g) => Divisible ((:.:) * * f g) Source # Methodsdivide :: (a -> (b, c)) -> (* :.: *) f g b -> (* :.: *) f g c -> (* :.: *) f g a Source #conquer :: (* :.: *) f g a Source # (Applicative f, Divisible g) => Divisible (Compose * * f g) Source # Methodsdivide :: (a -> (b, c)) -> Compose * * f g b -> Compose * * f g c -> Compose * * f g a Source #conquer :: Compose * * f g a Source # Divisible m => Divisible (RWST r w s m) Source # Methodsdivide :: (a -> (b, c)) -> RWST r w s m b -> RWST r w s m c -> RWST r w s m a Source #conquer :: RWST r w s m a Source # Divisible m => Divisible (RWST r w s m) Source # Methodsdivide :: (a -> (b, c)) -> RWST r w s m b -> RWST r w s m c -> RWST r w s m a Source #conquer :: RWST r w s m a Source #

divided :: Divisible f => f a -> f b -> f (a, b) Source #

divided = divide id


conquered :: Divisible f => f () Source #

Redundant, but provided for symmetry.

conquered = conquer


liftD :: Divisible f => (a -> b) -> f b -> f a Source #

This is the divisible analogue of liftA. It gives a viable default definition for contramap in terms of the members of Divisible.

liftD f = divide ((,) () . f) conquer


# Contravariant Alternative

class Divisible f => Decidable f where Source #

A Decidable contravariant functor is the contravariant analogue of Alternative.

Noting the superclass constraint that f must also be Divisible, a Decidable functor has the ability to "fan out" input, under the intuition that contravariant functors consume input.

In the dicussion for Divisible, an example was demonstrated with Serializers, that turn as into ByteStrings. Divisible allowed us to serialize the product of multiple values by concatenation. By making our Serializer also Decidable- we now have the ability to serialize the sum of multiple values - for example different constructors in an ADT.

Consider serializing arbitrary identifiers that can be either Strings or Ints:

data Identifier = StringId String | IntId Int


We know we have serializers for Strings and Ints, but how do we combine them into a Serializer for Identifier? Essentially, our Serializer needs to scrutinise the incoming value and choose how to serialize it:

identifier :: Serializer Identifier
identifier = Serializer $identifier -> case identifier of StringId s -> runSerializer string s IntId i -> runSerializer int i  It is exactly this notion of choice that Decidable encodes. Hence if we add an instance of Decidable for Serializer... instance Decidable Serializer where lose f = Serializer$ a -> absurd (f a)
choose split l r = Serializer \$ a ->
either (runSerializer l) (runSerializer r) (split a)


Then our identifier Serializer is

identifier :: Serializer Identifier
identifier = choose toEither string int where
toEither (StringId s) = Left s
toEither (IntId i) = Right i


Minimal complete definition

Methods

lose :: (a -> Void) -> f a Source #

Acts as identity to choose.

choose :: (a -> Either b c) -> f b -> f c -> f a Source #

Instances

 Source # Methodslose :: (a -> Void) -> SettableStateVar a Source #choose :: (a -> Either b c) -> SettableStateVar b -> SettableStateVar c -> SettableStateVar a Source # Source # Methodslose :: (a -> Void) -> Equivalence a Source #choose :: (a -> Either b c) -> Equivalence b -> Equivalence c -> Equivalence a Source # Source # Methodslose :: (a -> Void) -> Comparison a Source #choose :: (a -> Either b c) -> Comparison b -> Comparison c -> Comparison a Source # Source # Methodslose :: (a -> Void) -> Predicate a Source #choose :: (a -> Either b c) -> Predicate b -> Predicate c -> Predicate a Source # Source # Methodslose :: (a -> Void) -> U1 * a Source #choose :: (a -> Either b c) -> U1 * b -> U1 * c -> U1 * a Source # Source # Methodslose :: (a -> Void) -> Proxy * a Source #choose :: (a -> Either b c) -> Proxy * b -> Proxy * c -> Proxy * a Source # Divisible m => Decidable (MaybeT m) Source # Methodslose :: (a -> Void) -> MaybeT m a Source #choose :: (a -> Either b c) -> MaybeT m b -> MaybeT m c -> MaybeT m a Source # Divisible m => Decidable (ListT m) Source # Methodslose :: (a -> Void) -> ListT m a Source #choose :: (a -> Either b c) -> ListT m b -> ListT m c -> ListT m a Source # Monoid r => Decidable (Op r) Source # Methodslose :: (a -> Void) -> Op r a Source #choose :: (a -> Either b c) -> Op r b -> Op r c -> Op r a Source # Decidable f => Decidable (Rec1 * f) Source # Methodslose :: (a -> Void) -> Rec1 * f a Source #choose :: (a -> Either b c) -> Rec1 * f b -> Rec1 * f c -> Rec1 * f a Source # Decidable f => Decidable (Alt * f) Source # Methodslose :: (a -> Void) -> Alt * f a Source #choose :: (a -> Either b c) -> Alt * f b -> Alt * f c -> Alt * f a Source # Decidable f => Decidable (Reverse * f) Source # Methodslose :: (a -> Void) -> Reverse * f a Source #choose :: (a -> Either b c) -> Reverse * f b -> Reverse * f c -> Reverse * f a Source # Decidable m => Decidable (WriterT w m) Source # Methodslose :: (a -> Void) -> WriterT w m a Source #choose :: (a -> Either b c) -> WriterT w m b -> WriterT w m c -> WriterT w m a Source # Decidable m => Decidable (WriterT w m) Source # Methodslose :: (a -> Void) -> WriterT w m a Source #choose :: (a -> Either b c) -> WriterT w m b -> WriterT w m c -> WriterT w m a Source # Decidable m => Decidable (StateT s m) Source # Methodslose :: (a -> Void) -> StateT s m a Source #choose :: (a -> Either b c) -> StateT s m b -> StateT s m c -> StateT s m a Source # Decidable m => Decidable (StateT s m) Source # Methodslose :: (a -> Void) -> StateT s m a Source #choose :: (a -> Either b c) -> StateT s m b -> StateT s m c -> StateT s m a Source # Decidable f => Decidable (IdentityT * f) Source # Methodslose :: (a -> Void) -> IdentityT * f a Source #choose :: (a -> Either b c) -> IdentityT * f b -> IdentityT * f c -> IdentityT * f a Source # Decidable f => Decidable (Backwards * f) Source # Methodslose :: (a -> Void) -> Backwards * f a Source #choose :: (a -> Either b c) -> Backwards * f b -> Backwards * f c -> Backwards * f a Source # (Applicative f, Decidable g) => Decidable (ComposeFC f g) Source # Methodslose :: (a -> Void) -> ComposeFC f g a Source #choose :: (a -> Either b c) -> ComposeFC f g b -> ComposeFC f g c -> ComposeFC f g a Source # (Decidable f, Decidable g) => Decidable ((:*:) * f g) Source # Methodslose :: (a -> Void) -> (* :*: f) g a Source #choose :: (a -> Either b c) -> (* :*: f) g b -> (* :*: f) g c -> (* :*: f) g a Source # (Decidable f, Decidable g) => Decidable (Product * f g) Source # Methodslose :: (a -> Void) -> Product * f g a Source #choose :: (a -> Either b c) -> Product * f g b -> Product * f g c -> Product * f g a Source # Decidable m => Decidable (ReaderT * r m) Source # Methodslose :: (a -> Void) -> ReaderT * r m a Source #choose :: (a -> Either b c) -> ReaderT * r m b -> ReaderT * r m c -> ReaderT * r m a Source # Decidable f => Decidable (M1 * i c f) Source # Methodslose :: (a -> Void) -> M1 * i c f a Source #choose :: (a -> Either b c) -> M1 * i c f b -> M1 * i c f c -> M1 * i c f a Source # (Applicative f, Decidable g) => Decidable ((:.:) * * f g) Source # Methodslose :: (a -> Void) -> (* :.: *) f g a Source #choose :: (a -> Either b c) -> (* :.: *) f g b -> (* :.: *) f g c -> (* :.: *) f g a Source # (Applicative f, Decidable g) => Decidable (Compose * * f g) Source # Methodslose :: (a -> Void) -> Compose * * f g a Source #choose :: (a -> Either b c) -> Compose * * f g b -> Compose * * f g c -> Compose * * f g a Source # Decidable m => Decidable (RWST r w s m) Source # Methodslose :: (a -> Void) -> RWST r w s m a Source #choose :: (a -> Either b c) -> RWST r w s m b -> RWST r w s m c -> RWST r w s m a Source # Decidable m => Decidable (RWST r w s m) Source # Methodslose :: (a -> Void) -> RWST r w s m a Source #choose :: (a -> Either b c) -> RWST r w s m b -> RWST r w s m c -> RWST r w s m a Source #

chosen :: Decidable f => f b -> f c -> f (Either b c) Source #

chosen = choose id


lost :: Decidable f => f Void Source #

lost = lose id


# Mathematical definitions

## Divisible

In denser jargon, a Divisible contravariant functor is a monoid object in the category of presheaves from Hask to Hask, equipped with Day convolution mapping the Cartesian product of the source to the Cartesian product of the target.

By way of contrast, an Applicative functor can be viewed as a monoid object in the category of copresheaves from Hask to Hask, equipped with Day convolution mapping the Cartesian product of the source to the Cartesian product of the target.

Given the canonical diagonal morphism:

delta a = (a,a)


divide delta should be associative with conquer as a unit

divide delta m conquer = m
divide delta conquer m = m
divide delta (divide delta m n) o = divide delta m (divide delta n o)


With more general arguments you'll need to reassociate and project using the monoidal structure of the source category. (Here fst and snd are used in lieu of the more restricted lambda and rho, but this construction works with just a monoidal category.)

divide f m conquer = contramap (fst . f) m
divide f conquer m = contramap (snd . f) m
divide f (divide g m n) o = divide f' m (divide id n o) where
f' a = case f a of (bc,d) -> case g bc of (b,c) -> (a,(b,c))


### A note on conquer

The underlying theory would suggest that this should be:

conquer :: (a -> ()) -> f a


However, as we are working over a Cartesian category (Hask) and the Cartesian product, such an input morphism is uniquely determined to be const mempty, so we elide it.

## Decidable

A Divisible contravariant functor is a monoid object in the category of presheaves from Hask to Hask, equipped with Day convolution mapping the cartesian product of the source to the Cartesian product of the target.

choose Left m (lose f)  = m
choose Right (lose f) m = m
choose f (choose g m n) o = divide f' m (divide id n o) where
f' bcd = either (either id (Right . Left) . g) (Right . Right) . f


In addition, we expect the same kind of distributive law as is satisfied by the usual covariant Alternative, w.r.t Applicative, which should be fully formulated and added here at some point!