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