module Pandora.Paradigm.Schemes.TUT where

import Pandora.Core.Functor (type (:.), type (:=), type (~>))
import Pandora.Pattern.Semigroupoid ((.))
import Pandora.Pattern.Category (identity, ($))
import Pandora.Pattern.Functor.Covariant (Covariant, Covariant ((-<$>-)), (-<$$>-), (-<$$$>-))
import Pandora.Pattern.Functor.Contravariant (Contravariant)
import Pandora.Pattern.Functor.Semimonoidal (Semimonoidal (multiply_))
import Pandora.Pattern.Functor.Pointable (Pointable (point))
import Pandora.Pattern.Functor.Extractable (Extractable (extract))
import Pandora.Pattern.Functor.Extendable (Extendable ((<<=)))
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.Primary.Algebraic.Product ((:*:)((:*:)))
import Pandora.Paradigm.Controlflow.Effect.Interpreted (Interpreted (Primary, run, unite))

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

infix 3 <:<.>:>, >:<.>:>, <:<.>:<, >:<.>:<, <:>.<:>, >:>.<:>, <:>.<:<, >:>.<:<

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
	unite :: Primary (TUT ct ct' cu t t' u) a -> TUT ct ct' cu t t' u a
unite = Primary (TUT ct ct' cu t t' u) a -> TUT ct ct' cu 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

instance (Covariant t (->) (->), Covariant t' (->) (->), Covariant u (->) (->)) => Covariant (t <:<.>:> t' := u) (->) (->) where
	a -> b
f -<$>- :: (a -> b) -> (:=) (t <:<.>:> t') u a -> (:=) (t <:<.>:> t') u b
-<$>- TUT (t :. (u :. t')) := a
x = ((t :. (u :. t')) := b) -> (:=) (t <:<.>:> t') u b
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')) := b) -> (:=) (t <:<.>:> t') u b)
-> ((t :. (u :. t')) := b) -> (:=) (t <:<.>:> t') u b
forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b)
$ a -> b
f (a -> b) -> ((t :. (u :. t')) := a) -> (t :. (u :. t')) := b
forall (t :: * -> *) (u :: * -> *) (v :: * -> *)
       (category :: * -> * -> *) a b.
(Covariant t category category, Covariant u category category,
 Covariant v category category) =>
category a b -> category (t (u (v a))) (t (u (v b)))
-<$$$>- (t :. (u :. t')) := a
x

instance (Covariant t (->) (->), Covariant t' (->) (->), Covariant u (->) (->), Semimonoidal t (->) (:*:) (:*:), Semimonoidal u (->) (:*:) (:*:), Semimonoidal t' (->) (:*:) (:*:)) => Semimonoidal (t <:<.>:> t' := u) (->) (:*:) (:*:) where
	multiply_ :: ((:=) (t <:<.>:> t') u a :*: (:=) (t <:<.>:> t') u b)
-> (:=) (t <:<.>:> t') u (a :*: b)
multiply_ (TUT (t :. (u :. t')) := a
x :*: TUT (t :. (u :. t')) := b
y) = ((t :. (u :. t')) := (a :*: b)) -> (:=) (t <:<.>:> t') u (a :*: b)
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 :*: b))
 -> (:=) (t <:<.>:> t') u (a :*: b))
-> ((t :. (u :. t')) := (a :*: b))
-> (:=) (t <:<.>:> t') u (a :*: b)
forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b)
$ forall k (t :: k -> *) (p :: * -> * -> *) (source :: * -> * -> *)
       (target :: k -> k -> k) (a :: k) (b :: k).
Semimonoidal t p source target =>
p (source (t a) (t b)) (t (target a b))
forall (target :: * -> * -> *) a b.
Semimonoidal t' (->) (:*:) target =>
(t' a :*: t' b) -> t' (target a b)
multiply_ @_ @(->) @(:*:) ((t' a :*: t' b) -> t' (a :*: b))
-> ((u (t' a) :*: u (t' b)) -> u (t' a :*: t' b))
-> (u (t' a) :*: u (t' b))
-> u (t' (a :*: b))
forall (t :: * -> *) (u :: * -> *) (category :: * -> * -> *) a b.
(Covariant u category category, Covariant t category category) =>
category a b -> category (t (u a)) (t (u b))
-<$$>- forall k (t :: k -> *) (p :: * -> * -> *) (source :: * -> * -> *)
       (target :: k -> k -> k) (a :: k) (b :: k).
Semimonoidal t p source target =>
p (source (t a) (t b)) (t (target a b))
forall (target :: * -> * -> *) a b.
Semimonoidal u (->) (:*:) target =>
(u a :*: u b) -> u (target a b)
multiply_ @_ @(->) @(:*:) ((u (t' a) :*: u (t' b)) -> u (t' (a :*: b)))
-> t (u (t' a) :*: u (t' b)) -> (t :. (u :. t')) := (a :*: b)
forall (t :: * -> *) (source :: * -> * -> *)
       (target :: * -> * -> *) a b.
Covariant t source target =>
source a b -> target (t a) (t b)
-<$>- (((t :. (u :. t')) := a) :*: ((t :. (u :. t')) := b))
-> t (u (t' a) :*: u (t' b))
forall k (t :: k -> *) (p :: * -> * -> *) (source :: * -> * -> *)
       (target :: k -> k -> k) (a :: k) (b :: k).
Semimonoidal t p source target =>
p (source (t a) (t b)) (t (target a b))
multiply_ ((t :. (u :. t')) := a
x ((t :. (u :. t')) := a)
-> ((t :. (u :. t')) := b)
-> ((t :. (u :. t')) := a) :*: ((t :. (u :. t')) := b)
forall s a. s -> a -> s :*: a
:*: (t :. (u :. t')) := b
y)

instance (Covariant t (->) (->), Covariant t' (->) (->), Pointable u (->), Adjoint t' t (->) (->)) => Pointable (t <:<.>:> t' := u) (->) where
	point :: a -> (:=) (t <:<.>:> t') u a
point = ((t :. (u :. t')) := a) -> (:=) (t <:<.>:> t') u a
forall (t :: * -> *) a. Interpreted t => Primary t a -> t a
unite (((t :. (u :. t')) := a) -> (:=) (t <:<.>:> t') u a)
-> (a -> (t :. (u :. t')) := a) -> a -> (:=) (t <:<.>:> t') u a
forall (m :: * -> * -> *) b c a.
Semigroupoid m =>
m b c -> m a b -> m a c
. (forall a. Pointable u (->) => a -> u a
forall (t :: * -> *) (source :: * -> * -> *) a.
Pointable t source =>
source a (t a)
point @_ @(->) (t' a -> u (t' a)) -> a -> (t :. (u :. t')) := a
forall (t :: * -> *) (u :: * -> *) (source :: * -> * -> *)
       (target :: * -> * -> *) a b.
Adjoint t u source target =>
source (t a) b -> target a (u b)
-|)

instance (Adjoint t' t (->) (->), Extendable u (->)) => Extendable (t' <:<.>:> t := u) (->) where
	(:=) (t' <:<.>:> t) u a -> b
f <<= :: ((:=) (t' <:<.>:> t) u a -> b)
-> (:=) (t' <:<.>:> t) u a -> (:=) (t' <:<.>:> t) u b
<<= (:=) (t' <:<.>:> t) u a
x = ((t' :. (u :. t)) := b) -> (:=) (t' <:<.>:> t) u b
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)) := b) -> (:=) (t' <:<.>:> t) u b)
-> ((t' :. (u :. t)) := b) -> (:=) (t' <:<.>:> t) u b
forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b)
$ (((:=) (t' <:<.>:> t) u a -> b
f ((:=) (t' <:<.>:> t) u a -> b)
-> (t' (u (t a)) -> (:=) (t' <:<.>:> t) u a) -> t' (u (t a)) -> b
forall (m :: * -> * -> *) b c a.
Semigroupoid m =>
m b c -> m a b -> m a c
. t' (u (t a)) -> (:=) (t' <:<.>:> t) u a
forall (t :: * -> *) a. Interpreted t => Primary t a -> t a
unite (t' (u (t a)) -> b) -> u (t a) -> t b
forall (t :: * -> *) (u :: * -> *) (source :: * -> * -> *)
       (target :: * -> * -> *) a b.
Adjoint t u source target =>
source (t a) b -> target a (u b)
-|) (u (t a) -> t b) -> u (t a) -> u (t b)
forall (t :: * -> *) (source :: * -> * -> *) a b.
Extendable t source =>
source (t a) b -> source (t a) (t b)
<<=) (u (t a) -> u (t b)) -> t' (u (t a)) -> (t' :. (u :. t)) := b
forall (t :: * -> *) (source :: * -> * -> *)
       (target :: * -> * -> *) a b.
Covariant t source target =>
source a b -> target (t a) (t b)
-<$>- (:=) (t' <:<.>:> t) u a -> Primary ((t' <:<.>:> t) := u) a
forall (t :: * -> *) a. Interpreted t => t a -> Primary t a
run (:=) (t' <:<.>:> t) u a
x

instance (Covariant t (->) (->), Covariant t' (->) (->), Adjoint t t' (->) (->), Extractable u (->)) => Extractable (t <:<.>:> t' := u) (->) where
	extract :: (:=) (t <:<.>:> t') u a -> a
extract = (forall a. Extractable u (->) => u a -> a
forall (t :: * -> *) (source :: * -> * -> *) a.
Extractable t source =>
source (t a) a
extract @_ @(->) (u (t' a) -> t' a) -> t (u (t' a)) -> a
forall (t :: * -> *) (u :: * -> *) (source :: * -> * -> *)
       (target :: * -> * -> *) a b.
Adjoint t u source target =>
target a (u b) -> source (t a) b
|-) (t (u (t' a)) -> a)
-> ((:=) (t <:<.>:> t') u a -> t (u (t' a)))
-> (:=) (t <:<.>:> t') u a
-> a
forall (m :: * -> * -> *) b c a.
Semigroupoid m =>
m b c -> m a b -> m a c
. (:=) (t <:<.>:> t') u a -> t (u (t' a))
forall (t :: * -> *) a. Interpreted t => t a -> Primary t a
run

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 (m a b) (m a b)
$ (forall a. Category (->) => a -> a
forall (m :: * -> * -> *) a. Category m => m a a
identity @(->) (t' a -> t' a) -> a -> t (t' a)
forall (t :: * -> *) (u :: * -> *) (source :: * -> * -> *)
       (target :: * -> * -> *) a b.
Adjoint t u source target =>
source (t a) b -> target a (u b)
-|) (a -> t (t' a)) -> u a -> (t :. (u :. t')) := a
forall (t :: * -> *) (source :: * -> * -> *)
       (target :: * -> * -> *) (u :: * -> *) a b.
(Distributive t source target, Covariant u source target) =>
source a (t b) -> target (u a) (t (u b))
-<< u a
x

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) = (forall a. Category (->) => a -> a
forall (m :: * -> * -> *) a. Category m => m a a
identity @(->) (t' a -> t' a) -> u (t' a) -> t' (u a)
forall (t :: * -> *) (source :: * -> * -> *)
       (target :: * -> * -> *) (u :: * -> *) a b.
(Distributive t source target, Covariant u source target) =>
source a (t b) -> target (u a) (t (u b))
-<<) (u (t' a) -> t' (u a)) -> ((t :. (u :. t')) := a) -> u a
forall (t :: * -> *) (u :: * -> *) (source :: * -> * -> *)
       (target :: * -> * -> *) a b.
Adjoint t u source target =>
target a (u b) -> source (t a) b
|- (t :. (u :. t')) := a
x