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