{-# LANGUAGE UndecidableInstances #-} module Pandora.Paradigm.Controlflow.Effect.Transformer.Monadic (Monadic (..), (:>) (..)) where import Pandora.Pattern.Morphism.Straight (Straight (Straight)) import Pandora.Pattern.Semigroupoid ((.)) import Pandora.Pattern.Functor.Covariant (Covariant ((<-|-))) import Pandora.Pattern.Functor.Semimonoidal (Semimonoidal (mult)) import Pandora.Pattern.Functor.Monoidal (Monoidal (unit)) import Pandora.Pattern.Functor.Distributive (Distributive ((-<<), (--<<))) import Pandora.Pattern.Functor.Traversable (Traversable ((<<-))) import Pandora.Pattern.Functor.Bindable (Bindable ((=<<), (==<<))) import Pandora.Pattern.Functor.Extendable (Extendable ((<<=), (<<==))) import Pandora.Pattern.Functor.Monad (Monad) import Pandora.Pattern.Transformer.Liftable (Liftable (lift)) import Pandora.Pattern.Transformer.Hoistable (Hoistable ((/|\))) import Pandora.Paradigm.Primary.Algebraic.Exponential (type (-->)) import Pandora.Paradigm.Primary.Algebraic.Product ((:*:)((:*:))) import Pandora.Paradigm.Primary.Algebraic.Sum ((:+:)) import Pandora.Paradigm.Primary.Algebraic.One (One (One)) import Pandora.Paradigm.Primary.Algebraic (Pointable, point) import Pandora.Paradigm.Controlflow.Effect.Interpreted (Schematic, Interpreted (Primary, run, unite, (!))) class Interpreted m t => Monadic m t where {-# MINIMAL wrap #-} wrap :: Pointable u => m (t a) ((t :> u) a) infixr 3 :> newtype (:>) t u a = TM { (:>) t u a -> Schematic Monad t u a tm :: Schematic Monad t u a } instance Covariant (->) (->) (Schematic Monad t u) => Covariant (->) (->) (t :> u) where a -> b f <-|- :: (a -> b) -> (:>) t u a -> (:>) t u b <-|- TM Schematic Monad t u a x = Schematic Monad t u b -> (:>) t u b forall (t :: * -> *) (u :: * -> *) a. Schematic Monad t u a -> (:>) t u a TM (Schematic Monad t u b -> (:>) t u b) -> Schematic Monad t u b -> (:>) t u b forall (m :: * -> * -> *) (t :: * -> *) a. Interpreted m t => m (t a) (Primary t a) ! a -> b f (a -> b) -> Schematic Monad t u a -> Schematic Monad t u b forall (source :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) a b. Covariant source target t => source a b -> target (t a) (t b) <-|- Schematic Monad t u a x instance Semimonoidal (-->) (:*:) (:*:) (Schematic Monad t u) => Semimonoidal (-->) (:*:) (:*:) (t :> u) where mult :: ((:>) t u a :*: (:>) t u b) --> (:>) t u (a :*: b) mult = (((:>) t u a :*: (:>) t u b) -> (:>) t u (a :*: b)) -> ((:>) t u a :*: (:>) t u b) --> (:>) t u (a :*: b) forall (v :: * -> * -> *) a e. v a e -> Straight v a e Straight ((((:>) t u a :*: (:>) t u b) -> (:>) t u (a :*: b)) -> ((:>) t u a :*: (:>) t u b) --> (:>) t u (a :*: b)) -> (((:>) t u a :*: (:>) t u b) -> (:>) t u (a :*: b)) -> ((:>) t u a :*: (:>) t u b) --> (:>) t u (a :*: b) forall (m :: * -> * -> *) (t :: * -> *) a. Interpreted m t => m (t a) (Primary t a) ! \(TM Schematic Monad t u a f :*: TM Schematic Monad t u b x) -> Schematic Monad t u (a :*: b) -> (:>) t u (a :*: b) forall (t :: * -> *) (u :: * -> *) a. Schematic Monad t u a -> (:>) t u a TM (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) mult @(-->) @(:*:) @(:*:) ((Schematic Monad t u a :*: Schematic Monad t u b) --> Schematic Monad t u (a :*: b)) -> (Schematic Monad t u a :*: Schematic Monad t u b) -> Schematic Monad t u (a :*: b) forall (m :: * -> * -> *) (t :: * -> *) a. Interpreted m t => m (t a) (Primary t a) ! Schematic Monad t u a f Schematic Monad t u a -> Schematic Monad t u b -> Schematic Monad t u a :*: Schematic Monad t u b forall s a. s -> a -> s :*: a :*: Schematic Monad t u b x) instance Monoidal (-->) (-->) (:*:) (:*:) (Schematic Monad t u) => Monoidal (-->) (-->) (:*:) (:*:) (t :> u) where unit :: Proxy (:*:) -> (Unit (:*:) --> a) --> (:>) t u a unit Proxy (:*:) _ = ((One --> a) -> (:>) t u a) -> Straight (->) (One --> a) ((:>) t u a) forall (v :: * -> * -> *) a e. v a e -> Straight v a e Straight (((One --> a) -> (:>) t u a) -> Straight (->) (One --> a) ((:>) t u a)) -> ((One --> a) -> (:>) t u a) -> Straight (->) (One --> a) ((:>) t u a) forall (m :: * -> * -> *) (t :: * -> *) a. Interpreted m t => m (t a) (Primary t a) ! Schematic Monad t u a -> (:>) t u a forall (t :: * -> *) (u :: * -> *) a. Schematic Monad t u a -> (:>) t u a TM (Schematic Monad t u a -> (:>) t u a) -> ((One --> a) -> Schematic Monad t u a) -> (One --> a) -> (:>) t u a forall (m :: * -> * -> *) b c a. Semigroupoid m => m b c -> m a b -> m a c . a -> Schematic Monad t u a forall (t :: * -> *) a. Pointable t => a -> t a point (a -> Schematic Monad t u a) -> ((One --> a) -> a) -> (One --> a) -> Schematic Monad t u a forall (m :: * -> * -> *) b c a. Semigroupoid m => m b c -> m a b -> m a c . ((One -> a) -> One -> a forall (m :: * -> * -> *) (t :: * -> *) a. Interpreted m t => m (t a) (Primary t a) ! One One) ((One -> a) -> a) -> ((One --> a) -> One -> a) -> (One --> a) -> a forall (m :: * -> * -> *) b c a. Semigroupoid m => m b c -> m a b -> m a c . (One --> a) -> One -> a forall (m :: * -> * -> *) (t :: * -> *) a. Interpreted m t => m (t a) (Primary t a) run instance Semimonoidal (-->) (:*:) (:+:) (Schematic Monad t u) => Semimonoidal (-->) (:*:) (:+:) (t :> u) where mult :: ((:>) t u a :*: (:>) t u b) --> (:>) t u (a :+: b) mult = (((:>) t u a :*: (:>) t u b) -> (:>) t u (a :+: b)) -> ((:>) t u a :*: (:>) t u b) --> (:>) t u (a :+: b) forall (v :: * -> * -> *) a e. v a e -> Straight v a e Straight ((((:>) t u a :*: (:>) t u b) -> (:>) t u (a :+: b)) -> ((:>) t u a :*: (:>) t u b) --> (:>) t u (a :+: b)) -> (((:>) t u a :*: (:>) t u b) -> (:>) t u (a :+: b)) -> ((:>) t u a :*: (:>) t u b) --> (:>) t u (a :+: b) forall (m :: * -> * -> *) (t :: * -> *) a. Interpreted m t => m (t a) (Primary t a) ! \(TM Schematic Monad t u a f :*: TM Schematic Monad t u b x) -> Schematic Monad t u (a :+: b) -> (:>) t u (a :+: b) forall (t :: * -> *) (u :: * -> *) a. Schematic Monad t u a -> (:>) t u a TM (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) mult @(-->) @(:*:) @(:+:) ((Schematic Monad t u a :*: Schematic Monad t u b) --> Schematic Monad t u (a :+: b)) -> (Schematic Monad t u a :*: Schematic Monad t u b) -> Schematic Monad t u (a :+: b) forall (m :: * -> * -> *) (t :: * -> *) a. Interpreted m t => m (t a) (Primary t a) ! Schematic Monad t u a f Schematic Monad t u a -> Schematic Monad t u b -> Schematic Monad t u a :*: Schematic Monad t u b forall s a. s -> a -> s :*: a :*: Schematic Monad t u b x) instance Traversable (->) (->) (Schematic Monad t u) => Traversable (->) (->) (t :> u) where a -> u b f <<- :: (a -> u b) -> (:>) t u a -> u ((:>) t u b) <<- TM Schematic Monad t u a x = Schematic Monad t u b -> (:>) t u b forall (t :: * -> *) (u :: * -> *) a. Schematic Monad t u a -> (:>) t u a TM (Schematic Monad t u b -> (:>) t u b) -> u (Schematic Monad 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) -> Schematic Monad t u a -> u (Schematic Monad t u b) forall (source :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) (u :: * -> *) a b. (Traversable source target t, Covariant source target u, Monoidal (Straight source) (Straight target) (:*:) (:*:) u) => source a (u b) -> target (t a) (u (t b)) <<- Schematic Monad t u a x instance Distributive (->) (->) (Schematic Monad t u) => Distributive (->) (->) (t :> u) where a -> (:>) t u b f -<< :: (a -> (:>) t u b) -> u a -> (:>) t u (u b) -<< u a x = Schematic Monad t u (u b) -> (:>) t u (u b) forall (t :: * -> *) (u :: * -> *) a. Schematic Monad t u a -> (:>) t u a TM (Schematic Monad t u (u b) -> (:>) t u (u b)) -> Schematic Monad t u (u b) -> (:>) t u (u b) forall (m :: * -> * -> *) (t :: * -> *) a. Interpreted m t => m (t a) (Primary t a) ! (:>) t u b -> Schematic Monad t u b forall (t :: * -> *) (u :: * -> *) a. (:>) t u a -> Schematic Monad t u a tm ((:>) t u b -> Schematic Monad t u b) -> (a -> (:>) t u b) -> a -> Schematic Monad 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 -> Schematic Monad t u b) -> u a -> Schematic Monad 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 x instance Bindable (->) (Schematic Monad t u) => Bindable (->) (t :> u) where a -> (:>) t u b f =<< :: (a -> (:>) t u b) -> (:>) t u a -> (:>) t u b =<< TM Schematic Monad t u a x = Schematic Monad t u b -> (:>) t u b forall (t :: * -> *) (u :: * -> *) a. Schematic Monad t u a -> (:>) t u a TM (Schematic Monad t u b -> (:>) t u b) -> Schematic Monad t u b -> (:>) t u b forall (m :: * -> * -> *) (t :: * -> *) a. Interpreted m t => m (t a) (Primary t a) ! (:>) t u b -> Schematic Monad t u b forall (t :: * -> *) (u :: * -> *) a. (:>) t u a -> Schematic Monad t u a tm ((:>) t u b -> Schematic Monad t u b) -> (a -> (:>) t u b) -> a -> Schematic Monad 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 -> Schematic Monad t u b) -> Schematic Monad t u a -> Schematic Monad t u b forall (source :: * -> * -> *) (t :: * -> *) a b. Bindable source t => source a (t b) -> source (t a) (t b) ==<< Schematic Monad t u a x instance Extendable (->) (Schematic Monad t u) => Extendable (->) (t :> u) where (:>) t u a -> b f <<= :: ((:>) t u a -> b) -> (:>) t u a -> (:>) t u b <<= TM Schematic Monad t u a x = Schematic Monad t u b -> (:>) t u b forall (t :: * -> *) (u :: * -> *) a. Schematic Monad t u a -> (:>) t u a TM (Schematic Monad t u b -> (:>) t u b) -> Schematic Monad t u b -> (:>) t u b forall (m :: * -> * -> *) (t :: * -> *) a. Interpreted m t => m (t a) (Primary t a) ! (:>) t u a -> b f ((:>) t u a -> b) -> (Schematic Monad t u a -> (:>) t u a) -> Schematic Monad t u a -> b forall (m :: * -> * -> *) b c a. Semigroupoid m => m b c -> m a b -> m a c . Schematic Monad t u a -> (:>) t u a forall (t :: * -> *) (u :: * -> *) a. Schematic Monad t u a -> (:>) t u a TM (Schematic Monad t u a -> b) -> Schematic Monad t u a -> Schematic Monad t u b forall (source :: * -> * -> *) (t :: * -> *) a b. Extendable source t => source (t a) b -> source (t a) (t b) <<== Schematic Monad t u a x instance (Covariant (->) (->) (Schematic Monad t u), Monoidal (-->) (-->) (:*:) (:*:) (Schematic Monad t u), Bindable (->) (t :> u)) => Monad (->) (t :> u) where instance Liftable (->) (Schematic Monad t) => Liftable (->) ((:>) t) where lift :: u a -> (:>) t u a lift = Schematic Monad t u a -> (:>) t u a forall (t :: * -> *) (u :: * -> *) a. Schematic Monad t u a -> (:>) t u a TM (Schematic Monad t u a -> (:>) t u a) -> (u a -> Schematic Monad t u a) -> u a -> (:>) t u a forall (m :: * -> * -> *) b c a. Semigroupoid m => m b c -> m a b -> m a c . u a -> Schematic Monad t u a forall (cat :: * -> * -> *) (t :: (* -> *) -> * -> *) (u :: * -> *) a. (Liftable cat t, Covariant cat cat u) => cat (u a) (t u a) lift instance Hoistable (->) (Schematic Monad t) => Hoistable (->) ((:>) t) where forall a. u a -> v a f /|\ :: (forall a. u a -> v a) -> forall a. (:>) t u a -> (:>) t v a /|\ TM Schematic Monad t u a x = Schematic Monad t v a -> (:>) t v a forall (t :: * -> *) (u :: * -> *) a. Schematic Monad t u a -> (:>) t u a TM (Schematic Monad t v a -> (:>) t v a) -> Schematic Monad t v a -> (:>) t v a forall (m :: * -> * -> *) (t :: * -> *) a. Interpreted m t => m (t a) (Primary t a) ! forall a. u a -> v a f (forall a. u a -> v a) -> Schematic Monad t u a -> Schematic Monad t v a forall k (m :: * -> * -> *) (t :: (* -> *) -> k -> *) (u :: * -> *) (v :: * -> *). (Hoistable m t, Covariant m m u) => (forall a. m (u a) (v a)) -> forall (a :: k). m (t u a) (t v a) /|\ Schematic Monad t u a x instance (Interpreted (->) (Schematic Monad t u)) => Interpreted (->) (t :> u) where type Primary (t :> u) a = Primary (Schematic Monad t u) a run :: (:>) t u a -> Primary (t :> u) a run ~(TM Schematic Monad t u a x) = Schematic Monad t u a -> Primary (Schematic Monad t u) a forall (m :: * -> * -> *) (t :: * -> *) a. Interpreted m t => m (t a) (Primary t a) run Schematic Monad t u a x unite :: Primary (t :> u) a -> (:>) t u a unite = Schematic Monad t u a -> (:>) t u a forall (t :: * -> *) (u :: * -> *) a. Schematic Monad t u a -> (:>) t u a TM (Schematic Monad t u a -> (:>) t u a) -> (Primary (Schematic Monad t u) a -> Schematic Monad t u a) -> Primary (Schematic Monad t u) a -> (:>) t u a forall (m :: * -> * -> *) b c a. Semigroupoid m => m b c -> m a b -> m a c . Primary (Schematic Monad t u) a -> Schematic Monad t u a forall (m :: * -> * -> *) (t :: * -> *) a. Interpreted m t => m (Primary t a) (t a) unite