module Pandora.Paradigm.Schemes.UT where import Pandora.Core.Functor (type (:.), type (:=), type (~>)) import Pandora.Pattern.Category ((.), ($), identity) import Pandora.Pattern.Functor.Covariant (Covariant ((<$>), (<$$>)), Covariant_ ((-<$>-)), (-<$$>-)) import Pandora.Pattern.Functor.Contravariant (Contravariant) import Pandora.Pattern.Functor.Applicative (Applicative ((<*>), (<**>)), Semimonoidal) import Pandora.Pattern.Functor.Pointable (Pointable (point)) import Pandora.Pattern.Functor.Bindable (Bindable ((=<<))) import Pandora.Pattern.Functor.Extractable (Extractable (extract)) import Pandora.Pattern.Functor.Traversable (Traversable ((<<-))) import Pandora.Pattern.Transformer.Liftable (Liftable (lift)) import Pandora.Pattern.Transformer.Lowerable (Lowerable (lower)) import Pandora.Paradigm.Controlflow.Effect.Interpreted (Interpreted (Primary, run, unite)) import Pandora.Paradigm.Primary.Algebraic.Product ((:*:)) newtype UT ct cu t u a = UT (u :. t := a) infixr 3 <.:>, >.:>, <.:<, >.:< 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 (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 <$$> (u :. t) := a x instance (Covariant_ t (->) (->), Covariant_ u (->) (->)) => Covariant_ (t <.:> u) (->) (->) where a -> b f -<$>- :: (a -> b) -> (<.:>) t u a -> (<.:>) t u b -<$>- UT (u :. t) := a x = ((u :. t) := b) -> (<.:>) t u b forall k k k k (ct :: k) (cu :: k) (t :: k -> k) (u :: k -> *) (a :: k). ((u :. t) := a) -> UT ct cu t u a UT (((u :. t) := b) -> (<.:>) t u b) -> ((u :. t) := b) -> (<.:>) t u b forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b) $ a -> b f (a -> b) -> ((u :. t) := a) -> (u :. t) := b forall (t :: * -> *) (u :: * -> *) (category :: * -> * -> *) a b. (Covariant_ u category category, Covariant_ t category category) => category a b -> category (t (u a)) (t (u b)) -<$$>- (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 (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 a point = ((u :. t) := a) -> (<.:>) 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) -> (<.:>) t u a) -> (a -> (u :. t) := a) -> a -> (<.:>) t u a forall (m :: * -> * -> *) b c a. Category m => m b c -> m a b -> m a c . t a -> (u :. t) := a forall (t :: * -> *) (source :: * -> * -> *) a. Pointable t source => source a (t a) point (t a -> (u :. t) := a) -> (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 :: * -> *) (source :: * -> * -> *) a. Pointable t source => source a (t a) point instance (Traversable t (->) (->), Bindable t (->), Semimonoidal u (:*:) (->) (->), Pointable u (->), Bindable u (->)) => Bindable (t <.:> u) (->) where a -> (<.:>) t u b f =<< :: (a -> (<.:>) t u 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 (m a b) (m a b) $ ((t b -> t b forall (m :: * -> * -> *) a. Category m => m a a identity (t b -> t b) -> t (t b) -> t b forall (t :: * -> *) (source :: * -> * -> *) a b. Bindable t source => source a (t b) -> source (t a) (t b) =<<) (t (t b) -> t b) -> u (t (t b)) -> (u :. t) := b forall (t :: * -> *) (source :: * -> * -> *) (target :: * -> * -> *) a b. Covariant_ t source target => source a b -> target (t a) (t b) -<$>-) (u (t (t b)) -> (u :. t) := b) -> (t a -> u (t (t b))) -> t a -> (u :. t) := b forall (m :: * -> * -> *) b c a. Category m => m b c -> m a b -> m a c . ((<.:>) 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 (a -> (u :. t) := b) -> t a -> u (t (t b)) forall (t :: * -> *) (source :: * -> * -> *) (target :: * -> * -> *) (u :: * -> *) a b. (Traversable t source target, Covariant_ u source target, Pointable u target, Semimonoidal u (:*:) source target) => source a (u b) -> target (t a) (u (t b)) <<-) (t a -> (u :. t) := b) -> ((u :. t) := a) -> (u :. t) := b forall (t :: * -> *) (source :: * -> * -> *) a b. Bindable t source => source a (t b) -> source (t a) (t b) =<< (u :. t) := a x instance (Extractable t (->), Extractable u (->)) => Extractable (t <.:> u) (->) where extract :: (<.:>) t u a -> a extract = t a -> a forall (t :: * -> *) (source :: * -> * -> *) a. Extractable t source => source (t a) a extract (t a -> a) -> ((<.:>) t u a -> t a) -> (<.:>) t u a -> a forall (m :: * -> * -> *) b c a. Category m => m b c -> m a b -> m a c . u (t a) -> t a forall (t :: * -> *) (source :: * -> * -> *) a. Extractable t source => source (t a) a extract (u (t a) -> t a) -> ((<.:>) t u a -> u (t a)) -> (<.:>) t u a -> t a forall (m :: * -> * -> *) b c a. Category m => m b c -> m a b -> m a c . (<.:>) 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 (m a b) (m a b) $ forall a. Pointable t (->) => a -> t a forall (t :: * -> *) (source :: * -> * -> *) a. Pointable t source => source a (t a) point @_ @(->) (a -> t a) -> u a -> (u :. t) := a forall (t :: * -> *) (source :: * -> * -> *) (target :: * -> * -> *) a b. Covariant_ t source target => source a b -> target (t a) (t b) -<$>- u a x instance Extractable t (->) => Lowerable (UT Covariant Covariant t) where lower :: Covariant_ u (->) (->) => t <.:> u ~> u lower :: (t <.:> u) ~> u lower (UT (u :. t) := a x) = forall a. Extractable t (->) => t a -> a forall (t :: * -> *) (source :: * -> * -> *) a. Extractable t source => source (t a) a extract @_ @(->) (t a -> a) -> ((u :. t) := a) -> u a forall (t :: * -> *) (source :: * -> * -> *) (target :: * -> * -> *) a b. Covariant_ t source target => source a b -> target (t a) (t b) -<$>- (u :. t) := a x