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.Monoidal (Monoidal (unit)) import Pandora.Pattern.Functor.Traversable (Traversable ((<<-)), (-<<-<<-)) import Pandora.Pattern.Functor.Distributive (Distributive ((-<<))) import Pandora.Pattern.Functor.Bindable (Bindable ((=<<))) import Pandora.Pattern.Functor.Bivariant ((<->)) 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.Exponential (type (<--)) import Pandora.Paradigm.Primary.Algebraic.Product ((:*:) ((:*:))) import Pandora.Paradigm.Primary.Algebraic.Sum ((:+:) (Option, Adoption), sum) import Pandora.Paradigm.Primary.Algebraic.One (One (One)) import Pandora.Paradigm.Primary.Algebraic (empty, point, extract) import Pandora.Paradigm.Primary.Transformer.Flip (Flip (Flip)) 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 category category u, Covariant category category t) => 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 (p :: * -> * -> *) (source :: * -> * -> *) (target :: k -> k -> k) (t :: k -> *) (a :: k) (b :: k). Semimonoidal p source target t => p (source (t a) (t b)) (t (target a b)) forall (target :: * -> * -> *) (t :: * -> *) a b. Semimonoidal (->) (:*:) target t => (t a :*: t b) -> t (target a b) multiply @(->) @(:*:) ((u a :*: u b) -> u (a :*: b)) -> t (u a :*: u b) -> (t :. u) := (a :*: b) forall (source :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) a b. Covariant source target t => source a b -> target (t a) (t b) -<$>- (((t :. u) := a) :*: ((t :. u) := b)) -> t (u a :*: u b) forall k (p :: * -> * -> *) (source :: * -> * -> *) (target :: k -> k -> k) (t :: k -> *) (a :: k) (b :: k). Semimonoidal p source target t => 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 (Covariant (->) (->) t, Covariant (->) (->) u, Semimonoidal (->) (:*:) (:*:) u, Monoidal (->) (->) (:*:) (:*:) t, Monoidal (->) (->) (:*:) (:*:) u) => Monoidal (->) (->) (:*:) (:*:) (t <:.> u) where unit :: Proxy (:*:) -> (Unit (:*:) -> a) -> (<:.>) t u a unit Proxy (:*:) _ Unit (:*:) -> a f = ((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 :: * -> *) a. Monoidal (->) (->) (:*:) (:*:) t => 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 :: * -> *) a. Monoidal (->) (->) (:*:) (:*:) t => a -> t a point (a -> (<:.>) t u a) -> a -> (<:.>) t u a forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b) $ Unit (:*:) -> a f One Unit (:*:) One instance (Covariant (->) (->) t, Covariant (->) (->) u, Semimonoidal (->) (:*:) (:+:) t) => 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) $ (u a -> u (a :+: b)) -> (u b -> u (a :+: b)) -> (u a :+: u b) -> u (a :+: b) forall e r a. (e -> r) -> (a -> r) -> (e :+: a) -> r sum (a -> a :+: b forall s a. s -> s :+: a Option (a -> a :+: b) -> u a -> u (a :+: b) forall (source :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) a b. Covariant source target t => source a b -> target (t a) (t b) -<$>-) (b -> a :+: b forall s a. a -> s :+: a Adoption (b -> a :+: b) -> u b -> u (a :+: b) forall (source :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) a b. Covariant source target t => source a b -> target (t a) (t b) -<$>-) ((u a :+: u b) -> u (a :+: b)) -> t (u a :+: u b) -> (t :. u) := (a :+: b) forall (source :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) a b. Covariant source target t => source a b -> target (t a) (t b) -<$>- (((t :. u) := a) :*: ((t :. u) := b)) -> t (u a :+: u b) forall k (p :: * -> * -> *) (source :: * -> * -> *) (target :: k -> k -> k) (t :: k -> *) (a :: k) (b :: k). Semimonoidal p source target t => 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 (Covariant (->) (->) t, Covariant (->) (->) u, Monoidal (->) (->) (:*:) (:+:) t) => Monoidal (->) (->) (:*:) (:+:) (t <:.> u) where unit :: Proxy (:*:) -> (Unit (:+:) -> a) -> (<:.>) t u a unit Proxy (:*:) _ Unit (:+:) -> a _ = ((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 forall (t :: * -> *) a. Monoidal (->) (->) (:*:) (:+:) t => t a empty instance (Covariant (->) (->) t, Semimonoidal (<--) (:*:) (:*:) t, Semimonoidal (<--) (:*:) (:*:) u) => Semimonoidal (<--) (:*:) (:*:) (t <:.> u) where multiply :: ((<:.>) t u a :*: (<:.>) t u b) <-- (<:.>) t u (a :*: b) multiply = ((<:.>) t u (a :*: b) -> (<:.>) t u a :*: (<:.>) t u b) -> ((<:.>) t u a :*: (<:.>) t u b) <-- (<:.>) t u (a :*: b) forall (v :: * -> * -> *) a e. v e a -> Flip v a e Flip (((<:.>) t u (a :*: b) -> (<:.>) t u a :*: (<:.>) t u b) -> ((<:.>) t u a :*: (<:.>) t u b) <-- (<:.>) t u (a :*: b)) -> ((<:.>) t u (a :*: b) -> (<:.>) t u a :*: (<:.>) t u b) -> ((<:.>) t u a :*: (<:.>) t u b) <-- (<:.>) t u (a :*: b) forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b) $ \(TU (t :. u) := (a :*: b) xys) -> let Flip u (a :*: b) -> u a :*: u b f = forall k (p :: * -> * -> *) (source :: * -> * -> *) (target :: k -> k -> k) (t :: k -> *) (a :: k) (b :: k). Semimonoidal p source target t => p (source (t a) (t b)) (t (target a b)) forall (t :: * -> *) a b. Semimonoidal (<--) (:*:) (:*:) t => (t a :*: t b) <-- t (a :*: b) multiply @(<--) @(:*:) @(:*:) in let Flip t (a :*: b) -> t a :*: t b g = forall k (p :: * -> * -> *) (source :: * -> * -> *) (target :: k -> k -> k) (t :: k -> *) (a :: k) (b :: k). Semimonoidal p source target t => p (source (t a) (t b)) (t (target a b)) forall (t :: * -> *) a b. Semimonoidal (<--) (:*:) (:*:) t => (t a :*: t b) <-- t (a :*: b) multiply @(<--) @(:*:) @(:*:) in (((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) -> (t (u b) -> (<:.>) t u b) -> (((t :. u) := a) :*: t (u b)) -> (<:.>) t u a :*: (<:.>) t 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) <-> 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) := a) :*: t (u b)) -> (<:.>) t u a :*: (<:.>) t u b) -> (((t :. u) := a) :*: t (u b)) -> (<:.>) t u a :*: (<:.>) t u b forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b) $ t (u a :*: u b) -> ((t :. u) := a) :*: t (u b) forall a b. t (a :*: b) -> t a :*: t b g (u (a :*: b) -> u a :*: u b forall a b. u (a :*: b) -> u a :*: u b f (u (a :*: b) -> u a :*: u b) -> ((t :. u) := (a :*: b)) -> t (u a :*: u b) forall (source :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) a b. Covariant source target t => source a b -> target (t a) (t b) -<$>- (t :. u) := (a :*: b) xys) where instance (Covariant (->) (->) t, Monoidal (<--) (->) (:*:) (:*:) t, Monoidal (<--) (->) (:*:) (:*:) u) => Monoidal (<--) (->) (:*:) (:*:) (t <:.> u) where unit :: Proxy (:*:) -> (Unit (:*:) -> a) <-- (<:.>) t u a unit Proxy (:*:) _ = ((<:.>) t u a -> One -> a) -> Flip (->) (One -> a) ((<:.>) t u a) forall (v :: * -> * -> *) a e. v e a -> Flip v a e Flip (((<:.>) t u a -> One -> a) -> Flip (->) (One -> a) ((<:.>) t u a)) -> ((<:.>) t u a -> One -> a) -> Flip (->) (One -> a) ((<:.>) t u a) forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b) $ \(TU (t :. u) := a x) -> (\One _ -> u a -> a forall (t :: * -> *) a. Extractable_ t => t a -> a extract (u a -> a) -> u a -> a forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b) $ ((t :. u) := a) -> u a forall (t :: * -> *) a. Extractable_ t => t a -> a extract (t :. u) := a x) 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 (source :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) a b. Covariant source target t => 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 category category t, Covariant category category u, Monoidal category category (:*:) (:*:) u, Traversable category category v) => 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 (source :: * -> * -> *) (t :: * -> *) a b. Bindable source t => source a (t b) -> source (t a) (t b) =<<) (u (u b) -> u b) -> t (u (u b)) -> (t :. u) := b forall (source :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) a b. Covariant source target t => 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 (source :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) (u :: * -> *) a b. (Distributive source target t, Covariant source target u) => 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 (source :: * -> * -> *) (t :: * -> *) a b. Bindable source t => source a (t b) -> source (t a) (t b) =<< (t :. u) := a x instance Monoidal (->) (->) (:*:) (:*:) 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 :: * -> *) a. Monoidal (->) (->) (:*:) (:*:) t => a -> t a point instance Monoidal (<--) (->) (:*:) (:*:) 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 :: * -> *) a. Extractable_ t => 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 (source :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) a b. Covariant source target t => source a b -> target (t a) (t b) -<$>- (t :. u) := a x