TU.hs

module Pandora.Paradigm.Schemes.TU where

import Pandora.Core.Functor (type (:.), type (:=), type (~>))
import Pandora.Core.Appliable ((!))
import Pandora.Pattern.Semigroupoid ((.))
import Pandora.Pattern.Category (($), identity)
import Pandora.Pattern.Functor.Covariant (Covariant ((<$>)), (<$$>))
import Pandora.Pattern.Functor.Contravariant (Contravariant)
import Pandora.Pattern.Functor.Semimonoidal (Semimonoidal (mult))
import Pandora.Pattern.Functor.Monoidal (Monoidal (unit))
import Pandora.Pattern.Functor.Traversable (Traversable ((<<-)), (-<<-<<-))
import Pandora.Pattern.Functor.Distributive (Distributive ((-<<)))
import Pandora.Pattern.Functor.Bindable (Bindable ((=<<)))
import Pandora.Pattern.Functor.Bivariant ((<->))
import Pandora.Pattern.Transformer.Liftable (Liftable (lift))
import Pandora.Pattern.Transformer.Lowerable (Lowerable (lower))
import Pandora.Pattern.Transformer.Hoistable (Hoistable ((/|\)))
import Pandora.Paradigm.Controlflow.Effect.Interpreted (Interpreted (Primary, run, unite, (||=)))
import Pandora.Paradigm.Primary.Algebraic.Exponential (type (<--), type (-->))
import Pandora.Paradigm.Primary.Algebraic.Product ((:*:) ((:*:)))
import Pandora.Paradigm.Primary.Algebraic.Sum ((:+:) (Option, Adoption), sum)
import Pandora.Paradigm.Primary.Algebraic.One (One (One))
import Pandora.Paradigm.Primary.Algebraic (empty, point, extract)
import Pandora.Pattern.Morphism.Flip (Flip (Flip))
import Pandora.Pattern.Morphism.Straight (Straight (Straight))

newtype TU ct cu t u a = TU (t :. u := a)

infixr 3 <:.>, >:.>, <:.<, >:.<

type (<:.>) = TU Covariant Covariant
type (>:.>) = TU Contravariant Covariant
type (<:.<) = TU Covariant Contravariant
type (>:.<) = TU Contravariant Contravariant

instance Interpreted (->) (TU ct cu t u) where
	type Primary (TU ct cu t u) a = t :. u := a
	run ~(TU x) = x
	unite = TU

instance (Covariant m m t, Covariant m m u, Interpreted m (t <:.> u)) => Covariant m m (t <:.> u) where
	(<$>) f = (||=) ((<$$>) @m @m f)

instance (Covariant (->) (->) t, Semimonoidal (-->) (:*:) (:*:) t, Semimonoidal (-->) (:*:) (:*:) u) => Semimonoidal (-->) (:*:) (:*:) (t <:.> u) where
	mult = Straight $ TU . (<$>) (mult @(-->) !) . (mult @(-->) !) . (run @(->) <-> run @(->))

instance (Covariant (->) (->) t, Covariant (->) (->) u, Semimonoidal (-->) (:*:) (:*:) u, Monoidal (-->) (->) (:*:) (:*:) t, Monoidal (-->) (->) (:*:) (:*:) u) => Monoidal (-->) (->) (:*:) (:*:) (t <:.> u) where
	unit _ = Straight $ TU . point . point . ($ One)

instance (Covariant (->) (->) t, Covariant (->) (->) u, Semimonoidal (-->) (:*:) (:+:) t) => Semimonoidal (-->) (:*:) (:+:) (t <:.> u) where
	mult = Straight $ \(TU x :*: TU y) -> TU $ sum (Option <$>) (Adoption <$>) <$> (mult @(-->) ! x :*: y)

instance (Covariant (->) (->) t, Covariant (->) (->) u, Monoidal (-->) (->) (:*:) (:+:) t) => Monoidal (-->) (->) (:*:) (:+:) (t <:.> u) where
	unit _ = Straight $ \_ -> TU empty

instance (Covariant (->) (->) t, Semimonoidal (<--) (:*:) (:*:) t, Semimonoidal (<--) (:*:) (:*:) u) => Semimonoidal (<--) (:*:) (:*:) (t <:.> u) where
	mult = Flip $ \(TU xys) -> (TU <-> TU) . (mult @(<--) !) $ (mult @(<--) !) <$> xys

instance (Covariant (->) (->) t, Monoidal (<--) (->) (:*:) (:*:) t, Monoidal (<--) (->) (:*:) (:*:) u) => Monoidal (<--) (->) (:*:) (:*:) (t <:.> u) where
	unit _ = Flip $ \(TU x) -> (\_ -> extract $ extract x)

instance (Traversable (->) (->) t, Traversable (->) (->) u) => Traversable (->) (->) (t <:.> u) where
	f <<- x = TU <$> f -<<-<<- run x

instance (Bindable (->) t, Distributive (->) (->) t, Covariant (->) (->) u, Bindable (->) u) => Bindable (->) (t <:.> u) where
	f =<< TU x = TU $ (\i -> (identity =<<) <$> run . f -<< i) =<< x

instance Monoidal (-->) (->) (:*:) (:*:) t => Liftable (->) (TU Covariant Covariant t) where
	lift :: Covariant (->) (->) u => u ~> t <:.> u
	lift = TU . point

instance Monoidal (<--) (->) (:*:) (:*:) t => Lowerable (->) (TU Covariant Covariant t) where
	lower :: t <:.> u ~> u
	lower (TU x) = extract x

instance Covariant (->) (->) t => Hoistable (TU Covariant Covariant t) where
	(/|\) :: u ~> v -> (t <:.> u ~> t <:.> v)
	f /|\ TU x = TU $ f <$> x