{-# LANGUAGE UndecidableInstances #-} module Pandora.Paradigm.Primary.Transformer.Instruction where import Pandora.Core.Functor (type (:.), type (:=)) import Pandora.Core.Appliable ((!)) import Pandora.Pattern.Semigroupoid ((.)) import Pandora.Pattern.Category (($)) import Pandora.Pattern.Morphism.Straight (Straight (Straight)) import Pandora.Pattern.Functor.Covariant (Covariant ((<$>)), (<$$>)) import Pandora.Pattern.Functor.Semimonoidal (Semimonoidal (mult)) import Pandora.Pattern.Functor.Monoidal (Monoidal (unit)) import Pandora.Pattern.Functor.Traversable (Traversable ((<<-)), (-<<-<<-)) import Pandora.Pattern.Functor.Bindable (Bindable ((=<<))) import Pandora.Pattern.Functor.Monad (Monad) import Pandora.Pattern.Transformer.Liftable (Liftable (lift)) import Pandora.Pattern.Transformer.Lowerable (Lowerable (lower)) import Pandora.Pattern.Transformer.Hoistable (Hoistable ((/|\))) import Pandora.Paradigm.Primary.Algebraic.Exponential (type (-->)) import Pandora.Paradigm.Primary.Algebraic.Product ((:*:)((:*:))) import Pandora.Paradigm.Primary.Algebraic.One (One (One)) import Pandora.Paradigm.Primary.Algebraic (point) data Instruction t a = Enter a | Instruct (t :. Instruction t := a) instance Covariant (->) (->) t => Covariant (->) (->) (Instruction t) where a -> b f <$> :: (a -> b) -> Instruction t a -> Instruction t b <$> Enter a x = b -> Instruction t b forall (t :: * -> *) a. a -> Instruction t a Enter (b -> Instruction t b) -> b -> Instruction t b forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b) $ a -> b f a x a -> b f <$> Instruct (t :. Instruction t) := a xs = ((t :. Instruction t) := b) -> Instruction t b forall (t :: * -> *) a. ((t :. Instruction t) := a) -> Instruction t a Instruct (((t :. Instruction t) := b) -> Instruction t b) -> ((t :. Instruction t) := b) -> Instruction t b forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b) $ (a -> b) -> ((t :. Instruction t) := a) -> (t :. Instruction t) := b forall (source :: * -> * -> *) (between :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) (u :: * -> *) a b. (Covariant source between u, Covariant between target t) => source a b -> target (t (u a)) (t (u b)) (<$$>) @(->) @(->) a -> b f (t :. Instruction t) := a xs instance (Covariant (->) (->) t, Semimonoidal (-->) (:*:) (:*:) t) => Semimonoidal (-->) (:*:) (:*:) (Instruction t) where mult :: (Instruction t a :*: Instruction t b) --> Instruction t (a :*: b) mult = ((Instruction t a :*: Instruction t b) -> Instruction t (a :*: b)) -> (Instruction t a :*: Instruction t b) --> Instruction t (a :*: b) forall (v :: * -> * -> *) a e. v a e -> Straight v a e Straight (((Instruction t a :*: Instruction t b) -> Instruction t (a :*: b)) -> (Instruction t a :*: Instruction t b) --> Instruction t (a :*: b)) -> ((Instruction t a :*: Instruction t b) -> Instruction t (a :*: b)) -> (Instruction t a :*: Instruction t b) --> Instruction t (a :*: b) forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b) $ \case Enter a x :*: Enter b y -> (a :*: b) -> Instruction t (a :*: b) forall (t :: * -> *) a. a -> Instruction t a Enter ((a :*: b) -> Instruction t (a :*: b)) -> (a :*: b) -> Instruction t (a :*: b) forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b) $ a x a -> b -> a :*: b forall s a. s -> a -> s :*: a :*: b y Enter a x :*: Instruct (t :. Instruction t) := b y -> (a x a -> b -> a :*: b forall s a. s -> a -> s :*: a :*:) (b -> a :*: b) -> Instruction t b -> Instruction t (a :*: b) forall (source :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) a b. Covariant source target t => source a b -> target (t a) (t b) <$> ((t :. Instruction t) := b) -> Instruction t b forall (t :: * -> *) a. ((t :. Instruction t) := a) -> Instruction t a Instruct (t :. Instruction t) := b y Instruct (t :. Instruction t) := a x :*: Enter b y -> (a -> b -> a :*: b forall s a. s -> a -> s :*: a :*: b y) (a -> a :*: b) -> Instruction t a -> Instruction t (a :*: b) forall (source :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) a b. Covariant source target t => source a b -> target (t a) (t b) <$> ((t :. Instruction t) := a) -> Instruction t a forall (t :: * -> *) a. ((t :. Instruction t) := a) -> Instruction t a Instruct (t :. Instruction t) := a x Instruct (t :. Instruction t) := a x :*: Instruct (t :. Instruction t) := b y -> ((t :. Instruction t) := (a :*: b)) -> Instruction t (a :*: b) forall (t :: * -> *) a. ((t :. Instruction t) := a) -> Instruction t a Instruct (((t :. Instruction t) := (a :*: b)) -> Instruction t (a :*: b)) -> ((t :. Instruction t) := (a :*: b)) -> Instruction t (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 (source :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) a b. Semimonoidal (-->) source target t => source (t a) (t b) --> t (target a b) mult @(-->) ((Instruction t a :*: Instruction t b) --> Instruction t (a :*: b)) -> (Instruction t a :*: Instruction t b) -> Instruction t (a :*: b) forall k k k k (m :: k -> k -> *) (a :: k) (b :: k) (n :: k -> k -> *) (c :: k) (d :: k). Appliable m a b n c d => m a b -> n c d !) ((Instruction t a :*: Instruction t b) -> Instruction t (a :*: b)) -> t (Instruction t a :*: Instruction t b) -> (t :. Instruction t) := (a :*: b) forall (source :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) a b. Covariant source target t => source a b -> target (t a) (t 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 (source :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) a b. Semimonoidal (-->) source target t => source (t a) (t b) --> t (target a b) mult @(-->) ((((t :. Instruction t) := a) :*: ((t :. Instruction t) := b)) --> t (Instruction t a :*: Instruction t b)) -> (((t :. Instruction t) := a) :*: ((t :. Instruction t) := b)) -> t (Instruction t a :*: Instruction t b) forall k k k k (m :: k -> k -> *) (a :: k) (b :: k) (n :: k -> k -> *) (c :: k) (d :: k). Appliable m a b n c d => m a b -> n c d ! (t :. Instruction t) := a x ((t :. Instruction t) := a) -> ((t :. Instruction t) := b) -> ((t :. Instruction t) := a) :*: ((t :. Instruction t) := b) forall s a. s -> a -> s :*: a :*: (t :. Instruction t) := b y) instance (Covariant (->) (->) t, Semimonoidal (-->) (:*:) (:*:) t) => Monoidal (-->) (->) (:*:) (:*:) (Instruction t) where unit :: Proxy (:*:) -> (Unit (:*:) -> a) --> Instruction t a unit Proxy (:*:) _ = ((One -> a) -> Instruction t a) -> Straight (->) (One -> a) (Instruction t a) forall (v :: * -> * -> *) a e. v a e -> Straight v a e Straight (((One -> a) -> Instruction t a) -> Straight (->) (One -> a) (Instruction t a)) -> ((One -> a) -> Instruction t a) -> Straight (->) (One -> a) (Instruction t a) forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b) $ a -> Instruction t a forall (t :: * -> *) a. a -> Instruction t a Enter (a -> Instruction t a) -> ((One -> a) -> a) -> (One -> a) -> Instruction t a forall (m :: * -> * -> *) b c a. Semigroupoid m => m b c -> m a b -> m a c . ((One -> a) -> One -> a forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b) $ One One) instance Covariant (->) (->) t => Bindable (->) (Instruction t) where a -> Instruction t b f =<< :: (a -> Instruction t b) -> Instruction t a -> Instruction t b =<< Enter a x = a -> Instruction t b f a x a -> Instruction t b f =<< Instruct (t :. Instruction t) := a xs = ((t :. Instruction t) := b) -> Instruction t b forall (t :: * -> *) a. ((t :. Instruction t) := a) -> Instruction t a Instruct (((t :. Instruction t) := b) -> Instruction t b) -> ((t :. Instruction t) := b) -> Instruction t b forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b) $ (a -> Instruction t b f (a -> Instruction t b) -> Instruction t a -> Instruction t b forall (source :: * -> * -> *) (t :: * -> *) a b. Bindable source t => source a (t b) -> source (t a) (t b) =<<) (Instruction t a -> Instruction t b) -> ((t :. Instruction t) := a) -> (t :. Instruction t) := b forall (source :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) a b. Covariant source target t => source a b -> target (t a) (t b) <$> (t :. Instruction t) := a xs instance Monad (->) t => Monad (->) (Instruction t) where instance Traversable (->) (->) t => Traversable (->) (->) (Instruction t) where a -> u b f <<- :: (a -> u b) -> Instruction t a -> u (Instruction t b) <<- Enter a x = b -> Instruction t b forall (t :: * -> *) a. a -> Instruction t a Enter (b -> Instruction t b) -> u b -> u (Instruction t b) forall (source :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) a b. Covariant source target t => source a b -> target (t a) (t b) <$> a -> u b f a x a -> u b f <<- Instruct (t :. Instruction t) := a xs = ((t :. Instruction t) := b) -> Instruction t b forall (t :: * -> *) a. ((t :. Instruction t) := a) -> Instruction t a Instruct (((t :. Instruction t) := b) -> Instruction t b) -> u ((t :. Instruction t) := b) -> u (Instruction t 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 :. Instruction t) := a) -> u ((t :. Instruction t) := b) forall (t :: * -> *) (u :: * -> *) (v :: * -> *) (category :: * -> * -> *) a b. (Traversable category category t, Covariant category category u, Monoidal (Straight category) category (:*:) (:*:) u, Traversable category category v) => category a (u b) -> category (v (t a)) (u (v (t b))) -<<-<<- (t :. Instruction t) := a xs instance Liftable (->) Instruction where lift :: u a -> Instruction u a lift u a x = ((u :. Instruction u) := a) -> Instruction u a forall (t :: * -> *) a. ((t :. Instruction t) := a) -> Instruction t a Instruct (((u :. Instruction u) := a) -> Instruction u a) -> ((u :. Instruction u) := a) -> Instruction u a forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b) $ a -> Instruction u a forall (t :: * -> *) a. a -> Instruction t a Enter (a -> Instruction u a) -> u a -> (u :. Instruction u) := a forall (source :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) a b. Covariant source target t => source a b -> target (t a) (t b) <$> u a x instance (forall t . Bindable (->) t, forall t . Monoidal (-->) (->) (:*:) (:*:) t) => Lowerable (->) Instruction where lower :: Instruction u a -> u a lower (Enter a x) = a -> u a forall (t :: * -> *) a. Pointable t => a -> t a point a x lower (Instruct (u :. Instruction u) := a xs) = Instruction u a -> u a forall (cat :: * -> * -> *) (t :: (* -> *) -> * -> *) (u :: * -> *) a. (Lowerable cat t, Covariant cat cat u) => cat (t u a) (u a) lower (Instruction u a -> u a) -> ((u :. Instruction u) := a) -> u a forall (source :: * -> * -> *) (t :: * -> *) a b. Bindable source t => source a (t b) -> source (t a) (t b) =<< (u :. Instruction u) := a xs instance (forall v . Covariant (->) (->) v) => Hoistable Instruction where u ~> v _ /|\ :: (u ~> v) -> Instruction u ~> Instruction v /|\ Enter a x = a -> Instruction v a forall (t :: * -> *) a. a -> Instruction t a Enter a x u ~> v f /|\ Instruct (u :. Instruction u) := a xs = ((v :. Instruction v) := a) -> Instruction v a forall (t :: * -> *) a. ((t :. Instruction t) := a) -> Instruction t a Instruct (((v :. Instruction v) := a) -> Instruction v a) -> ((v :. Instruction v) := a) -> Instruction v a forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b) $ (u ~> v f (u ~> v) -> Instruction u a -> Instruction v a forall k (t :: (* -> *) -> k -> *) (u :: * -> *) (v :: * -> *). (Hoistable t, Covariant (->) (->) u) => (u ~> v) -> t u ~> t v /|\) (Instruction u a -> Instruction v a) -> v (Instruction u a) -> (v :. Instruction v) := a forall (source :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) a b. Covariant source target t => source a b -> target (t a) (t b) <$> ((u :. Instruction u) := a) -> v (Instruction u a) u ~> v f (u :. Instruction u) := a xs