module Pandora.Paradigm.Schemes.T_U where import Pandora.Core.Functor (type (:=)) import Pandora.Pattern.Functor.Covariant (Covariant, Covariant ((<$>))) import Pandora.Pattern.Functor.Contravariant (Contravariant, Contravariant ((>$<))) import Pandora.Pattern.Functor.Bivariant (Bivariant ((<->))) import Pandora.Pattern.Functor.Divariant (Divariant ((>->))) import Pandora.Paradigm.Controlflow.Effect.Interpreted (Interpreted (Primary, run, unite, (||=))) newtype T_U ct cu p t u a = T_U (p (t a) (u a)) infixr 2 <:.:>, >:.:>, <:.:<, >:.:< type (<:.:>) t u p = T_U Covariant Covariant p t u type (>:.:>) t u p = T_U Contravariant Covariant p t u type (<:.:<) t u p = T_U Covariant Contravariant p t u type (>:.:<) t u p = T_U Contravariant Contravariant p t u instance Interpreted (->) (T_U ct cu p t u) where type Primary (T_U ct cu p t u) a = p (t a) (u a) run :: T_U ct cu p t u a -> Primary (T_U ct cu p t u) a run ~(T_U p (t a) (u a) x) = p (t a) (u a) Primary (T_U ct cu p t u) a x unite :: Primary (T_U ct cu p t u) a -> T_U ct cu p t u a unite = Primary (T_U ct cu p t u) a -> T_U ct cu p t u a forall k k k k k (ct :: k) (cu :: k) (p :: k -> k -> *) (t :: k -> k) (u :: k -> k) (a :: k). p (t a) (u a) -> T_U ct cu p t u a T_U instance (forall i . Covariant (->) (->) (p i), Bivariant (->) (->) (->) p, Covariant (->) (->) t, Covariant (->) (->) u) => Covariant (->) (->) (t <:.:> u := p) where a -> b f <$> :: (a -> b) -> (:=) (t <:.:> u) p a -> (:=) (t <:.:> u) p b <$> (:=) (t <:.:> u) p a x = (a -> b f (a -> b) -> t a -> t b forall (source :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) a b. Covariant source target t => source a b -> target (t a) (t b) <$>) (t a -> t b) -> (u a -> u b) -> p (t a) (u a) -> p (t b) (u b) forall (left :: * -> * -> *) (right :: * -> * -> *) (target :: * -> * -> *) (v :: * -> * -> *) a b c d. Bivariant left right target v => left a b -> right c d -> target (v a c) (v b d) <-> (a -> b f (a -> b) -> u a -> u b forall (source :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) a b. Covariant source target t => source a b -> target (t a) (t b) <$>) (Primary ((t <:.:> u) := p) a -> Primary ((t <:.:> u) := p) b) -> (:=) (t <:.:> u) p a -> (:=) (t <:.:> u) p b forall (m :: * -> * -> *) (t :: * -> *) (u :: * -> *) a b. (Interpreted m t, Semigroupoid m, Interpreted m u) => m (Primary t a) (Primary u b) -> m (t a) (u b) ||= (:=) (t <:.:> u) p a x instance (Divariant (->) (->) (->) p, Contravariant (->) (->) t, Covariant (->) (->) u) => Covariant (->) (->) (t >:.:> u := p) where a -> b f <$> :: (a -> b) -> (:=) (t >:.:> u) p a -> (:=) (t >:.:> u) p b <$> (:=) (t >:.:> u) p a x = (a -> b f (a -> b) -> t b -> t a forall (source :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) a b. Contravariant source target t => source a b -> target (t b) (t a) >$<) (t b -> t a) -> (u a -> u b) -> p (t a) (u a) -> p (t b) (u b) forall (left :: * -> * -> *) (right :: * -> * -> *) (target :: * -> * -> *) (v :: * -> * -> *) a b c d. Divariant left right target v => left a b -> right c d -> target (v b c) (v a d) >-> (a -> b f (a -> b) -> u a -> u b forall (source :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) a b. Covariant source target t => source a b -> target (t a) (t b) <$>) (Primary ((t >:.:> u) := p) a -> Primary ((t >:.:> u) := p) b) -> (:=) (t >:.:> u) p a -> (:=) (t >:.:> u) p b forall (m :: * -> * -> *) (t :: * -> *) (u :: * -> *) a b. (Interpreted m t, Semigroupoid m, Interpreted m u) => m (Primary t a) (Primary u b) -> m (t a) (u b) ||= (:=) (t >:.:> u) p a x instance (forall i . Covariant (->) (->) (p i), Bivariant (->) (->) (->) p, Contravariant (->) (->) t, Contravariant (->) (->) u) => Contravariant (->) (->) (t >:.:< u := p) where a -> b f >$< :: (a -> b) -> (:=) (t >:.:< u) p b -> (:=) (t >:.:< u) p a >$< (:=) (t >:.:< u) p b x = (a -> b f (a -> b) -> t b -> t a forall (source :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) a b. Contravariant source target t => source a b -> target (t b) (t a) >$<) (t b -> t a) -> (u b -> u a) -> p (t b) (u b) -> p (t a) (u a) forall (left :: * -> * -> *) (right :: * -> * -> *) (target :: * -> * -> *) (v :: * -> * -> *) a b c d. Bivariant left right target v => left a b -> right c d -> target (v a c) (v b d) <-> (a -> b f (a -> b) -> u b -> u a forall (source :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) a b. Contravariant source target t => source a b -> target (t b) (t a) >$<) (Primary ((t >:.:< u) := p) b -> Primary ((t >:.:< u) := p) a) -> (:=) (t >:.:< u) p b -> (:=) (t >:.:< u) p a forall (m :: * -> * -> *) (t :: * -> *) (u :: * -> *) a b. (Interpreted m t, Semigroupoid m, Interpreted m u) => m (Primary t a) (Primary u b) -> m (t a) (u b) ||= (:=) (t >:.:< u) p b x