module Pandora.Paradigm.Schemes.TUT where

import Pandora.Core.Functor (type (:.), type (:=), type (~>))
import Pandora.Pattern.Category (identity, ($))
import Pandora.Pattern.Functor.Covariant (Covariant)
import Pandora.Pattern.Functor.Contravariant (Contravariant)
import Pandora.Pattern.Functor.Distributive (Distributive ((>>-)))
import Pandora.Pattern.Functor.Adjoint (Adjoint ((-|), (|-)))
import Pandora.Pattern.Transformer.Liftable (Liftable (lift))
import Pandora.Pattern.Transformer.Lowerable (Lowerable (lower))
import Pandora.Paradigm.Controlflow.Effect.Interpreted (Interpreted (Primary, run))

newtype TUT ct ct' cu t t' u a = TUT (t :. u :. t' := a)

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

instance Interpreted (TUT ct ct' cu t t' u) where
	type Primary (TUT ct ct' cu t t' u) a = t :. u :. t' := a
	run :: TUT ct ct' cu t t' u a -> Primary (TUT ct ct' cu t t' u) a
run ~(TUT (t :. (u :. t')) := a
x) = (t :. (u :. t')) := a
Primary (TUT ct ct' cu t t' u) a
x

instance (Adjoint t' t, Distributive t) => Liftable (t <:<.>:> t') where
	lift :: Covariant u => u ~> t <:<.>:> t' := u
	lift :: u ~> ((t <:<.>:> t') := u)
lift u a
x = ((t :. (u :. t')) := a)
-> TUT Covariant Covariant Covariant t t' u a
forall k k k k k k (ct :: k) (ct' :: k) (cu :: k) (t :: k -> *)
       (t' :: k -> k) (u :: k -> k) (a :: k).
((t :. (u :. t')) := a) -> TUT ct ct' cu t t' u a
TUT (((t :. (u :. t')) := a)
 -> TUT Covariant Covariant Covariant t t' u a)
-> ((t :. (u :. t')) := a)
-> TUT Covariant Covariant Covariant t t' u a
forall (m :: * -> * -> *) a b. Category m => m a b -> m a b
$ u a
x u a -> (a -> t (t' a)) -> (t :. (u :. t')) := a
forall (t :: * -> *) (u :: * -> *) a b.
(Distributive t, Covariant u) =>
u a -> (a -> t b) -> (t :. u) := b
>>- (a -> (t' a -> t' a) -> t (t' a)
forall (t :: * -> *) (u :: * -> *) a b.
Adjoint t u =>
a -> (t a -> b) -> u b
-| t' a -> t' a
forall (m :: * -> * -> *) a. Category m => m a a
identity)

instance (Adjoint t t', Distributive t') => Lowerable (t <:<.>:> t') where
	lower :: Covariant u => (t <:<.>:> t' := u) ~> u
	lower :: ((t <:<.>:> t') := u) ~> u
lower (TUT (t :. (u :. t')) := a
x) = (t :. (u :. t')) := a
x ((t :. (u :. t')) := a) -> ((:.) u t' a -> t' (u a)) -> u a
forall (t :: * -> *) (u :: * -> *) a b.
Adjoint t u =>
t a -> (a -> u b) -> b
|- ((:.) u t' a -> (t' a -> t' a) -> t' (u a)
forall (t :: * -> *) (u :: * -> *) a b.
(Distributive t, Covariant u) =>
u a -> (a -> t b) -> (t :. u) := b
>>- t' a -> t' a
forall (m :: * -> * -> *) a. Category m => m a a
identity)