module Pandora.Paradigm.Schemes.TU 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.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.Primary.Algebraic.Product ((:*:)((:*:)))

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 ct cu t u a -> Primary (TU ct cu t u) a
run ~(TU (t :. u) := a
x) = (t :. u) := a
Primary (TU ct cu t u) a
x
	unite :: Primary (TU ct cu t u) a -> TU ct cu t u a
unite = Primary (TU ct cu t u) a -> TU ct cu t u a
forall k k k k (ct :: k) (cu :: k) (t :: k -> *) (u :: k -> k)
       (a :: k).
((t :. u) := a) -> TU ct cu t u a
TU

instance (Covariant t (->) (->), Covariant u (->) (->)) => Covariant (t <:.> u) (->) (->) where
	a -> b
f -<$>- :: (a -> b) -> (<:.>) t u a -> (<:.>) t u b
-<$>- (<:.>) t u a
x = ((t :. u) := b) -> (<:.>) t u b
forall k k k k (ct :: k) (cu :: k) (t :: k -> *) (u :: k -> k)
       (a :: k).
((t :. u) := a) -> TU ct cu t u a
TU (((t :. u) := b) -> (<:.>) t u b)
-> ((t :. u) := b) -> (<:.>) t u b
forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b)
$ a -> b
f (a -> b) -> t (u a) -> (t :. u) := 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))
-<$$>- (<:.>) t u a -> Primary (t <:.> u) a
forall (t :: * -> *) a. Interpreted t => t a -> Primary t a
run (<:.>) t u a
x

instance (Covariant t (->) (->), Semimonoidal t (->) (:*:) (:*:), Semimonoidal u (->) (:*:) (:*:)) => Semimonoidal (t <:.> u) (->) (:*:) (:*:) where
	multiply_ :: ((<:.>) t u a :*: (<:.>) t u b) -> (<:.>) t u (a :*: b)
multiply_ (TU (t :. u) := a
x :*: TU (t :. u) := b
y) = ((t :. u) := (a :*: b)) -> (<:.>) t u (a :*: b)
forall k k k k (ct :: k) (cu :: k) (t :: k -> *) (u :: k -> k)
       (a :: k).
((t :. u) := a) -> TU ct cu t u a
TU (((t :. u) := (a :*: b)) -> (<:.>) t u (a :*: b))
-> ((t :. u) := (a :*: b)) -> (<:.>) 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 u (->) (:*:) target =>
(u a :*: u b) -> u (target a b)
multiply_ @_ @(->) @(:*:) ((u a :*: u b) -> u (a :*: b))
-> t (u a :*: u b) -> (t :. u) := (a :*: b)
forall (t :: * -> *) (source :: * -> * -> *)
       (target :: * -> * -> *) a b.
Covariant t source target =>
source a b -> target (t a) (t b)
-<$>- (((t :. u) := a) :*: ((t :. u) := b)) -> t (u a :*: 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))
multiply_ ((t :. u) := a
x ((t :. u) := a)
-> ((t :. u) := b) -> ((t :. u) := a) :*: ((t :. u) := b)
forall s a. s -> a -> s :*: a
:*: (t :. u) := b
y)

instance (Pointable t (->), Pointable u (->)) => Pointable (t <:.> u) (->) where
	point :: a -> (<:.>) t u a
point = ((t :. u) := a) -> (<:.>) t u a
forall k k k k (ct :: k) (cu :: k) (t :: k -> *) (u :: k -> k)
       (a :: k).
((t :. u) := a) -> TU ct cu t u a
TU (((t :. u) := a) -> (<:.>) t u a)
-> (a -> (t :. u) := a) -> a -> (<:.>) t u a
forall (m :: * -> * -> *) b c a.
Semigroupoid m =>
m b c -> m a b -> m a c
. u a -> (t :. u) := a
forall (t :: * -> *) (source :: * -> * -> *) a.
Pointable t source =>
source a (t a)
point (u a -> (t :. u) := a) -> (a -> u a) -> a -> (t :. u) := a
forall (m :: * -> * -> *) b c a.
Semigroupoid m =>
m b c -> m a b -> m a c
. a -> u a
forall (t :: * -> *) (source :: * -> * -> *) a.
Pointable t source =>
source a (t a)
point

instance (Extractable t (->), Extractable u (->)) => Extractable (t <:.> u) (->) where
	extract :: (<:.>) t u a -> a
extract = u a -> a
forall (t :: * -> *) (source :: * -> * -> *) a.
Extractable t source =>
source (t a) a
extract (u a -> a) -> ((<:.>) t u a -> u a) -> (<:.>) t u a -> a
forall (m :: * -> * -> *) b c a.
Semigroupoid m =>
m b c -> m a b -> m a c
. t (u a) -> u a
forall (t :: * -> *) (source :: * -> * -> *) a.
Extractable t source =>
source (t a) a
extract (t (u a) -> u a)
-> ((<:.>) t u a -> t (u a)) -> (<:.>) t u a -> u a
forall (m :: * -> * -> *) b c a.
Semigroupoid m =>
m b c -> m a b -> m a c
. (<:.>) t u a -> t (u a)
forall (t :: * -> *) a. Interpreted t => t a -> Primary t a
run

instance (Traversable t (->) (->), Traversable u (->) (->)) => Traversable (t <:.> u) (->) (->) where
	a -> u b
f <<- :: (a -> u b) -> (<:.>) t u a -> u ((<:.>) t u b)
<<- (<:.>) t u a
x = ((t :. u) := b) -> (<:.>) t u b
forall k k k k (ct :: k) (cu :: k) (t :: k -> *) (u :: k -> k)
       (a :: k).
((t :. u) := a) -> TU ct cu t u a
TU (((t :. u) := b) -> (<:.>) t u b)
-> u ((t :. u) := b) -> u ((<:.>) t u b)
forall (t :: * -> *) (source :: * -> * -> *)
       (target :: * -> * -> *) a b.
Covariant t source target =>
source a b -> target (t a) (t b)
-<$>- a -> u b
f (a -> u b) -> t (u a) -> u ((t :. u) := b)
forall (t :: * -> *) (u :: * -> *) (v :: * -> *)
       (category :: * -> * -> *) a b.
(Traversable t category category, Covariant u category category,
 Pointable u category, Semimonoidal u category (:*:) (:*:),
 Traversable v category category) =>
category a (u b) -> category (v (t a)) (u (v (t b)))
-<<-<<- (<:.>) t u a -> Primary (t <:.> u) a
forall (t :: * -> *) a. Interpreted t => t a -> Primary t a
run (<:.>) t u a
x

instance (Bindable t (->), Distributive t (->) (->), Covariant u (->) (->), Bindable u (->)) => Bindable (t <:.> u) (->) where
	a -> (<:.>) t u b
f =<< :: (a -> (<:.>) t u b) -> (<:.>) t u a -> (<:.>) t u b
=<< TU (t :. u) := a
x = ((t :. u) := b) -> (<:.>) t u b
forall k k k k (ct :: k) (cu :: k) (t :: k -> *) (u :: k -> k)
       (a :: k).
((t :. u) := a) -> TU ct cu t u a
TU (((t :. u) := b) -> (<:.>) t u b)
-> ((t :. u) := b) -> (<:.>) t u b
forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b)
$ (\u a
i -> (u b -> u b
forall (m :: * -> * -> *) a. Category m => m a a
identity (u b -> u b) -> u (u b) -> u b
forall (t :: * -> *) (source :: * -> * -> *) a b.
Bindable t source =>
source a (t b) -> source (t a) (t b)
=<<) (u (u b) -> u b) -> t (u (u b)) -> (t :. u) := b
forall (t :: * -> *) (source :: * -> * -> *)
       (target :: * -> * -> *) a b.
Covariant t source target =>
source a b -> target (t a) (t b)
-<$>- (<:.>) t u b -> (t :. u) := b
forall (t :: * -> *) a. Interpreted t => t a -> Primary t a
run ((<:.>) t u b -> (t :. u) := b)
-> (a -> (<:.>) t u b) -> a -> (t :. u) := b
forall (m :: * -> * -> *) b c a.
Semigroupoid m =>
m b c -> m a b -> m a c
. a -> (<:.>) t u b
f (a -> (t :. u) := b) -> u a -> t (u (u b))
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
i) (u a -> (t :. u) := b) -> ((t :. u) := a) -> (t :. u) := b
forall (t :: * -> *) (source :: * -> * -> *) a b.
Bindable t source =>
source a (t b) -> source (t a) (t b)
=<< (t :. u) := a
x

instance Pointable t (->) => Liftable (TU Covariant Covariant t) where
	lift :: Covariant u (->) (->) => u ~> t <:.> u
	lift :: u ~> (t <:.> u)
lift = ((t :. u) := a) -> TU Covariant Covariant t u a
forall k k k k (ct :: k) (cu :: k) (t :: k -> *) (u :: k -> k)
       (a :: k).
((t :. u) := a) -> TU ct cu t u a
TU (((t :. u) := a) -> TU Covariant Covariant t u a)
-> (u a -> (t :. u) := a) -> u a -> TU Covariant Covariant t u a
forall (m :: * -> * -> *) b c a.
Semigroupoid m =>
m b c -> m a b -> m a c
. u a -> (t :. u) := a
forall (t :: * -> *) (source :: * -> * -> *) a.
Pointable t source =>
source a (t a)
point

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

instance Covariant t (->) (->) => Hoistable (TU Covariant Covariant t) where
	(/|\) :: u ~> v -> (t <:.> u ~> t <:.> v)
	u ~> v
f /|\ :: (u ~> v) -> (t <:.> u) ~> (t <:.> v)
/|\ TU (t :. u) := a
x = ((t :. v) := a) -> TU Covariant Covariant t v a
forall k k k k (ct :: k) (cu :: k) (t :: k -> *) (u :: k -> k)
       (a :: k).
((t :. u) := a) -> TU ct cu t u a
TU (((t :. v) := a) -> TU Covariant Covariant t v a)
-> ((t :. v) := a) -> TU Covariant Covariant t v a
forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b)
$ u a -> v a
u ~> v
f (u a -> v a) -> ((t :. u) := a) -> (t :. v) := a
forall (t :: * -> *) (source :: * -> * -> *)
       (target :: * -> * -> *) a b.
Covariant t source target =>
source a b -> target (t a) (t b)
-<$>- (t :. u) := a
x