Linear.hs

{-# LANGUAGE EmptyCase #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE LambdaCase #-}
{-# LANGUAGE LinearTypes #-}
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE TupleSections #-}
{-# LANGUAGE NoImplicitPrelude #-}

-- | This module provides (linear) Kleisli and CoKleisli arrows
--
-- This module is meant to be imported qualified, perhaps as below.
--
-- > import qualified Data.Profunctor.Kleisli as Linear
--
-- == What are Kleisli arrows?
--
-- The basic idea is that a Kleisli arrow is like a function arrow
-- and @Kleisli m a b@ is similar to a function from @a@ to @b@. Basically:
--
-- > type Kleisli m a b = a %1-> m b
--
-- == Why make this definition?
--
-- It let's us view @Kleisli m@ for a certain @m@ as a certain kind of
-- function arrow, give it instances, abstract over it an so on.
--
-- For instance, if @m@ is any functor, @Kleisli m@ is a @Profunctor@.
--
-- == CoKleisli
--
-- A CoKleisli arrow is just one that represents a computation from
-- a @m a@ to an @a@ via a linear arrow. (It's a Co-something because it
-- reverses the order of the function arrows in the something.)
module Data.Profunctor.Kleisli.Linear
  ( Kleisli (..),
    CoKleisli (..),
  )
where

import qualified Control.Functor.Linear as Control
import qualified Data.Functor.Linear as Data
import Data.Profunctor.Linear
import Data.Void
import Prelude.Linear (Either (..), either)
import Prelude.Linear.Internal

-- Ideally, there would only be one Kleisli arrow, parametrised by
-- a multiplicity parameter:
-- newtype Kleisli p m a b = Kleisli { runKleisli :: a # p -> m b }
--
-- Some instances would also still work, eg
-- instance Functor p f => Profunctor (Kleisli p f)

-- | Linear Kleisli arrows for the monad `m`. These arrows are still useful
-- in the case where `m` is not a monad however, and some profunctorial
-- properties still hold in this weaker setting.
newtype Kleisli m a b = Kleisli {runKleisli :: a %1 -> m b}

instance (Data.Functor f) => Profunctor (Kleisli f) where
  dimap f g (Kleisli h) = Kleisli (Data.fmap g . h . f)

instance (Control.Functor f) => Strong (,) () (Kleisli f) where
  first (Kleisli f) = Kleisli (\(a, b) -> (,b) Control.<$> f a)
  second (Kleisli g) = Kleisli (\(a, b) -> (a,) Control.<$> g b)

instance (Control.Applicative f) => Strong Either Void (Kleisli f) where
  first (Kleisli f) = Kleisli (either (Data.fmap Left . f) (Control.pure . Right))
  second (Kleisli g) = Kleisli (either (Control.pure . Left) (Data.fmap Right . g))

instance (Data.Applicative f) => Monoidal (,) () (Kleisli f) where
  Kleisli f *** Kleisli g = Kleisli $ \(x, y) -> (,) Data.<$> f x Data.<*> g y
  unit = Kleisli $ \() -> Data.pure ()

instance (Data.Functor f) => Monoidal Either Void (Kleisli f) where
  Kleisli f *** Kleisli g = Kleisli $ \case
    Left a -> Left Data.<$> f a
    Right b -> Right Data.<$> g b
  unit = Kleisli $ \case {}

instance (Control.Applicative f) => Wandering (Kleisli f) where
  wander traverse (Kleisli f) = Kleisli (traverse f)

-- | Linear co-Kleisli arrows for the comonad `w`. These arrows are still
-- useful in the case where `w` is not a comonad however, and some
-- profunctorial properties still hold in this weaker setting.
-- However stronger requirements on `f` are needed for profunctorial
-- strength, so we have fewer instances.
newtype CoKleisli w a b = CoKleisli {runCoKleisli :: w a %1 -> b}

instance (Data.Functor f) => Profunctor (CoKleisli f) where
  dimap f g (CoKleisli h) = CoKleisli (g . h . Data.fmap f)

instance Strong Either Void (CoKleisli (Data.Const x)) where
  first (CoKleisli f) = CoKleisli (\(Data.Const x) -> Left (f (Data.Const x)))