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.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 (<--))
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.Paradigm.Primary.Transformer.Flip (Flip (Flip))

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 category category u, Covariant category category t) =>
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 (p :: * -> * -> *) (source :: * -> * -> *)
       (target :: k -> k -> k) (t :: k -> *) (a :: k) (b :: k).
Semimonoidal p source target t =>
p (source (t a) (t b)) (t (target a b))
forall (target :: * -> * -> *) (t :: * -> *) a b.
Semimonoidal (->) (:*:) target t =>
(t a :*: t b) -> t (target a b)
multiply @(->) @(:*:) ((u a :*: u b) -> u (a :*: b))
-> t (u a :*: u b) -> (t :. u) := (a :*: b)
forall (source :: * -> * -> *) (target :: * -> * -> *)
       (t :: * -> *) a b.
Covariant source target t =>
source a b -> target (t a) (t b)
-<$>- (((t :. u) := a) :*: ((t :. u) := b)) -> t (u a :*: u b)
forall k (p :: * -> * -> *) (source :: * -> * -> *)
       (target :: k -> k -> k) (t :: k -> *) (a :: k) (b :: k).
Semimonoidal p source target t =>
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 (Covariant (->) (->) t, Covariant (->) (->) u, Semimonoidal (->) (:*:) (:*:) u, Monoidal (->) (->) (:*:) (:*:) t, Monoidal (->) (->) (:*:) (:*:) u) => Monoidal (->) (->) (:*:) (:*:) (t <:.> u) where
	unit :: Proxy (:*:) -> (Unit (:*:) -> a) -> (<:.>) t u a
unit Proxy (:*:)
_ Unit (:*:) -> a
f = ((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 :: * -> *) a.
Monoidal (->) (->) (:*:) (:*:) t =>
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 :: * -> *) a.
Monoidal (->) (->) (:*:) (:*:) t =>
a -> t a
point (a -> (<:.>) t u a) -> a -> (<:.>) t u a
forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b)
$ Unit (:*:) -> a
f One
Unit (:*:)
One

instance (Covariant (->) (->) t, Covariant (->) (->) u, Semimonoidal (->) (:*:) (:+:) t) => 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)
$ (u a -> u (a :+: b))
-> (u b -> u (a :+: b)) -> (u a :+: u b) -> u (a :+: b)
forall e r a. (e -> r) -> (a -> r) -> (e :+: a) -> r
sum (a -> a :+: b
forall s a. s -> s :+: a
Option (a -> a :+: b) -> u a -> u (a :+: b)
forall (source :: * -> * -> *) (target :: * -> * -> *)
       (t :: * -> *) a b.
Covariant source target t =>
source a b -> target (t a) (t b)
-<$>-) (b -> a :+: b
forall s a. a -> s :+: a
Adoption (b -> a :+: b) -> u b -> u (a :+: b)
forall (source :: * -> * -> *) (target :: * -> * -> *)
       (t :: * -> *) a b.
Covariant source target t =>
source a b -> target (t a) (t b)
-<$>-) ((u a :+: u b) -> u (a :+: b))
-> t (u a :+: u b) -> (t :. u) := (a :+: b)
forall (source :: * -> * -> *) (target :: * -> * -> *)
       (t :: * -> *) a b.
Covariant source target t =>
source a b -> target (t a) (t b)
-<$>- (((t :. u) := a) :*: ((t :. u) := b)) -> t (u a :+: u b)
forall k (p :: * -> * -> *) (source :: * -> * -> *)
       (target :: k -> k -> k) (t :: k -> *) (a :: k) (b :: k).
Semimonoidal p source target t =>
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 (Covariant (->) (->) t, Covariant (->) (->) u, Monoidal (->) (->) (:*:) (:+:) t) => Monoidal (->) (->) (:*:) (:+:) (t <:.> u) where
	unit :: Proxy (:*:) -> (Unit (:+:) -> a) -> (<:.>) t u a
unit Proxy (:*:)
_ Unit (:+:) -> a
_ = ((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
forall (t :: * -> *) a. Monoidal (->) (->) (:*:) (:+:) t => t a
empty

instance (Covariant (->) (->) t, Semimonoidal (<--) (:*:) (:*:) t, Semimonoidal (<--) (:*:) (:*:) u) => Semimonoidal (<--) (:*:) (:*:) (t <:.> u) where
	multiply :: ((<:.>) t u a :*: (<:.>) t u b) <-- (<:.>) t u (a :*: b)
multiply = ((<:.>) t u (a :*: b) -> (<:.>) t u a :*: (<:.>) t u b)
-> ((<:.>) t u a :*: (<:.>) t u b) <-- (<:.>) t u (a :*: b)
forall (v :: * -> * -> *) a e. v e a -> Flip v a e
Flip (((<:.>) t u (a :*: b) -> (<:.>) t u a :*: (<:.>) t u b)
 -> ((<:.>) t u a :*: (<:.>) t u b) <-- (<:.>) t u (a :*: b))
-> ((<:.>) t u (a :*: b) -> (<:.>) t u a :*: (<:.>) t u b)
-> ((<:.>) t u a :*: (<:.>) t u b) <-- (<:.>) t u (a :*: b)
forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b)
$ \(TU (t :. u) := (a :*: b)
xys) ->
		let Flip u (a :*: b) -> u a :*: u b
f = forall k (p :: * -> * -> *) (source :: * -> * -> *)
       (target :: k -> k -> k) (t :: k -> *) (a :: k) (b :: k).
Semimonoidal p source target t =>
p (source (t a) (t b)) (t (target a b))
forall (t :: * -> *) a b.
Semimonoidal (<--) (:*:) (:*:) t =>
(t a :*: t b) <-- t (a :*: b)
multiply @(<--) @(:*:) @(:*:) in
		let Flip t (a :*: b) -> t a :*: t b
g = forall k (p :: * -> * -> *) (source :: * -> * -> *)
       (target :: k -> k -> k) (t :: k -> *) (a :: k) (b :: k).
Semimonoidal p source target t =>
p (source (t a) (t b)) (t (target a b))
forall (t :: * -> *) a b.
Semimonoidal (<--) (:*:) (:*:) t =>
(t a :*: t b) <-- t (a :*: b)
multiply @(<--) @(:*:) @(:*:) in
		(((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)
-> (t (u b) -> (<:.>) t u b)
-> (((t :. u) := a) :*: t (u b))
-> (<:.>) t u a :*: (<:.>) t u b
forall (left :: * -> * -> *) (right :: * -> * -> *)
       (target :: * -> * -> *) (v :: * -> * -> *) a b c d.
Bivariant left right target v =>
left a b -> right c d -> target (v a c) (v b d)
<-> 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) := a) :*: t (u b)) -> (<:.>) t u a :*: (<:.>) t u b)
-> (((t :. u) := a) :*: t (u b)) -> (<:.>) t u a :*: (<:.>) t u b
forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b)
$ t (u a :*: u b) -> ((t :. u) := a) :*: t (u b)
forall a b. t (a :*: b) -> t a :*: t b
g (u (a :*: b) -> u a :*: u b
forall a b. u (a :*: b) -> u a :*: u b
f (u (a :*: b) -> u a :*: u b)
-> ((t :. u) := (a :*: b)) -> t (u a :*: u b)
forall (source :: * -> * -> *) (target :: * -> * -> *)
       (t :: * -> *) a b.
Covariant source target t =>
source a b -> target (t a) (t b)
-<$>- (t :. u) := (a :*: b)
xys) where

instance (Covariant (->) (->) t, Monoidal (<--) (->) (:*:) (:*:) t, Monoidal (<--) (->) (:*:) (:*:) u) => Monoidal (<--) (->) (:*:) (:*:) (t <:.> u) where
	unit :: Proxy (:*:) -> (Unit (:*:) -> a) <-- (<:.>) t u a
unit Proxy (:*:)
_ = ((<:.>) t u a -> One -> a) -> Flip (->) (One -> a) ((<:.>) t u a)
forall (v :: * -> * -> *) a e. v e a -> Flip v a e
Flip (((<:.>) t u a -> One -> a) -> Flip (->) (One -> a) ((<:.>) t u a))
-> ((<:.>) t u a -> One -> a)
-> Flip (->) (One -> a) ((<:.>) t u a)
forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b)
$ \(TU (t :. u) := a
x) -> (\One
_ -> u a -> a
forall (t :: * -> *) a. Extractable_ t => t a -> a
extract (u a -> a) -> u a -> a
forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b)
$ ((t :. u) := a) -> u a
forall (t :: * -> *) a. Extractable_ t => t a -> a
extract (t :. u) := a
x)

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 (source :: * -> * -> *) (target :: * -> * -> *)
       (t :: * -> *) a b.
Covariant source target t =>
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 category category t, Covariant category category u,
 Monoidal category category (:*:) (:*:) u,
 Traversable category category v) =>
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 (source :: * -> * -> *) (t :: * -> *) a b.
Bindable source t =>
source a (t b) -> source (t a) (t b)
=<<) (u (u b) -> u b) -> t (u (u b)) -> (t :. u) := b
forall (source :: * -> * -> *) (target :: * -> * -> *)
       (t :: * -> *) a b.
Covariant source target t =>
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 (source :: * -> * -> *) (target :: * -> * -> *)
       (t :: * -> *) (u :: * -> *) a b.
(Distributive source target t, Covariant source target u) =>
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 (source :: * -> * -> *) (t :: * -> *) a b.
Bindable source t =>
source a (t b) -> source (t a) (t b)
=<< (t :. u) := a
x

instance Monoidal (->) (->) (:*:) (:*:) 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 :: * -> *) a.
Monoidal (->) (->) (:*:) (:*:) t =>
a -> t a
point

instance Monoidal (<--) (->) (:*:) (:*:) 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 :: * -> *) a. Extractable_ t => 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 (source :: * -> * -> *) (target :: * -> * -> *)
       (t :: * -> *) a b.
Covariant source target t =>
source a b -> target (t a) (t b)
-<$>- (t :. u) := a
x