module Pandora.Paradigm.Schemes.UT where

import Pandora.Core.Functor (type (:.), type (:=), type (~>))
import Pandora.Pattern.Category ((.), ($))
import Pandora.Pattern.Functor.Covariant (Covariant ((<$>), (<$$>)))
import Pandora.Pattern.Functor.Contravariant (Contravariant)
import Pandora.Pattern.Functor.Applicative (Applicative ((<*>), (<**>)))
import Pandora.Pattern.Functor.Pointable (Pointable (point))
import Pandora.Pattern.Functor.Bindable (Bindable ((>>=), join))
import Pandora.Pattern.Functor.Extractable (Extractable (extract))
import Pandora.Pattern.Functor.Traversable (Traversable ((->>)))
import Pandora.Pattern.Functor.Monad (Monad)
import Pandora.Pattern.Transformer.Liftable (Liftable (lift))
import Pandora.Pattern.Transformer.Lowerable (Lowerable (lower))
import Pandora.Paradigm.Controlflow.Effect.Interpreted (Interpreted (Primary, run, unite))

newtype UT ct cu t u a = UT (u :. t := a)

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

instance Interpreted (UT ct cu t u) where
	type Primary (UT ct cu t u) a = u :. t := a
	run :: UT ct cu t u a -> Primary (UT ct cu t u) a
run ~(UT (u :. t) := a
x) = (u :. t) := a
Primary (UT ct cu t u) a
x
	unite :: Primary (UT ct cu t u) a -> UT ct cu t u a
unite = Primary (UT ct cu t u) a -> UT ct cu t u a
forall k k k k (ct :: k) (cu :: k) (t :: k -> k) (u :: k -> *)
       (a :: k).
((u :. t) := a) -> UT ct cu t u a
UT

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

instance (Applicative t, Applicative u) => Applicative (t <.:> u) where
	UT (u :. t) := (a -> b)
f <*> :: (<.:>) t u (a -> b) -> (<.:>) t u a -> (<.:>) t u b
<*> UT (u :. t) := a
x = ((u :. t) := b) -> (<.:>) t u b
forall k k k k (ct :: k) (cu :: k) (t :: k -> k) (u :: k -> *)
       (a :: k).
((u :. t) := a) -> UT ct cu t u a
UT (((u :. t) := b) -> (<.:>) t u b)
-> ((u :. t) := b) -> (<.:>) t u b
forall (m :: * -> * -> *) a b. Category m => m a b -> m a b
$ (u :. t) := (a -> b)
f ((u :. t) := (a -> b)) -> ((u :. t) := a) -> (u :. t) := b
forall (t :: * -> *) (u :: * -> *) a b.
(Applicative t, Applicative u) =>
((t :. u) := (a -> b)) -> ((t :. u) := a) -> (t :. u) := b
<**> (u :. t) := a
x

instance (Pointable t, Pointable u) => Pointable (t <.:> u) where
	point :: a |-> (t <.:> u)
point = ((u :. t) := a) -> UT Covariant Covariant t u a
forall k k k k (ct :: k) (cu :: k) (t :: k -> k) (u :: k -> *)
       (a :: k).
((u :. t) := a) -> UT ct cu t u a
UT (((u :. t) := a) -> UT Covariant Covariant t u a)
-> (a -> (u :. t) := a) -> a |-> (t <.:> u)
forall (m :: * -> * -> *) b c a.
Category m =>
m b c -> m a b -> m a c
. t a |-> u
forall (t :: * -> *) a. Pointable t => a |-> t
point (t a |-> u) -> (a -> t a) -> a -> (u :. t) := a
forall (m :: * -> * -> *) b c a.
Category m =>
m b c -> m a b -> m a c
. a -> t a
forall (t :: * -> *) a. Pointable t => a |-> t
point

instance (Traversable t, Bindable t, Applicative u, Monad u) => Bindable (t <.:> u) where
	UT (u :. t) := a
x >>= :: (<.:>) t u a -> (a -> (<.:>) t u b) -> (<.:>) t u b
>>= a -> (<.:>) t u b
f = ((u :. t) := b) -> (<.:>) t u b
forall k k k k (ct :: k) (cu :: k) (t :: k -> k) (u :: k -> *)
       (a :: k).
((u :. t) := a) -> UT ct cu t u a
UT (((u :. t) := b) -> (<.:>) t u b)
-> ((u :. t) := b) -> (<.:>) t u b
forall (m :: * -> * -> *) a b. Category m => m a b -> m a b
$ (u :. t) := a
x ((u :. t) := a) -> (t a -> (u :. t) := b) -> (u :. t) := b
forall (t :: * -> *) a b. Bindable t => t a -> (a -> t b) -> t b
>>= \t a
i -> ((t :. t) := b) -> t b
forall (t :: * -> *) a. Bindable t => ((t :. t) := a) -> t a
join (((t :. t) := b) -> t b) -> u ((t :. t) := b) -> (u :. t) := b
forall (t :: * -> *) a b. Covariant t => (a -> b) -> t a -> t b
<$> (t a
i t a -> (a -> (u :. t) := b) -> u ((t :. t) := b)
forall (t :: * -> *) (u :: * -> *) a b.
(Traversable t, Pointable u, Applicative u) =>
t a -> (a -> u b) -> (u :. t) := b
->> (<.:>) t u b -> (u :. t) := b
forall (t :: * -> *) a. Interpreted t => t a -> Primary t a
run ((<.:>) t u b -> (u :. t) := b)
-> (a -> (<.:>) t u b) -> a -> (u :. t) := b
forall (m :: * -> * -> *) b c a.
Category m =>
m b c -> m a b -> m a c
. a -> (<.:>) t u b
f)

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

instance Pointable t => Liftable (UT Covariant Covariant t) where
	lift :: Covariant u => u ~> t <.:> u
	lift :: u ~> (t <.:> u)
lift u a
x = ((u :. t) := a) -> UT Covariant Covariant t u a
forall k k k k (ct :: k) (cu :: k) (t :: k -> k) (u :: k -> *)
       (a :: k).
((u :. t) := a) -> UT ct cu t u a
UT (((u :. t) := a) -> UT Covariant Covariant t u a)
-> ((u :. t) := a) -> UT Covariant Covariant t u a
forall (m :: * -> * -> *) a b. Category m => m a b -> m a b
$ a |-> t
forall (t :: * -> *) a. Pointable t => a |-> t
point (a |-> t) -> u a -> (u :. t) := a
forall (t :: * -> *) a b. Covariant t => (a -> b) -> t a -> t b
<$> u a
x

instance Extractable t => Lowerable (UT Covariant Covariant t) where
	lower :: Covariant u => t <.:> u ~> u
	lower :: (t <.:> u) ~> u
lower (UT (u :. t) := a
x) = a <-| t
forall (t :: * -> *) a. Extractable t => a <-| t
extract (a <-| t) -> ((u :. t) := a) -> u a
forall (t :: * -> *) a b. Covariant t => (a -> b) -> t a -> t b
<$> (u :. t) := a
x