TU.hs

{-# LANGUAGE UndecidableInstances #-}
module Pandora.Paradigm.Schemes.TU where

import Pandora.Core.Functor (type (:.), type (>), type (~>))
import Pandora.Pattern.Betwixt (Betwixt)
import Pandora.Pattern.Semigroupoid (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.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.Algebraic.Exponential (type (<--), type (-->))
import Pandora.Paradigm.Algebraic.Product ((:*:) ((:*:)))
import Pandora.Paradigm.Algebraic.Sum ((:+:))
import Pandora.Paradigm.Algebraic.One (One (One))
import Pandora.Paradigm.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 (Semigroupoid m, Covariant m m t, Covariant (Betwixt m m) m t, Covariant m (Betwixt m m) u, Interpreted m (t <:.> u)) => Covariant m m (t <:.> u) where
	(<-|-) f = (=#-) ((<-|-|-) 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 (-->) (:*:) (:+:) u) => Semimonoidal (-->) (:*:) (:+:) (t <:.> u) where
	mult = Straight <-- \(TU x :*: TU y) -> TU
		<----- mult @(-->) @(:*:) @(:+:)
			<-|-- mult @(-->) @(:*:) @(:*:)
				<~~~ x :*: y

instance (Covariant (->) (->) t, Covariant (->) (->) u, Semimonoidal (-->) (:*:) (:*:) t, Semimonoidal (-->) (:*:) (:+:) 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) -> Straight (\_ -> 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