{-# LANGUAGE UndecidableInstances #-} module Pandora.Paradigm.Schemes.UT where import Pandora.Core.Functor (type (:.), type (>>>), type (~>)) import Pandora.Core.Interpreted (Interpreted (Primary, run, unite, (<~), (<~~~), (<~~~~), (=#-))) import Pandora.Pattern.Betwixt (Betwixt) import Pandora.Pattern.Semigroupoid (Semigroupoid ((.))) import Pandora.Pattern.Category (identity, (<--), (<---), (<----), (<-----)) import Pandora.Pattern.Morphism.Straight (Straight (Straight)) import Pandora.Pattern.Functor.Covariant (Covariant ((<-|-), (<-|--), (<-|---), (<-|-|-))) import Pandora.Pattern.Functor.Contravariant (Contravariant) import Pandora.Pattern.Functor.Semimonoidal (Semimonoidal (mult)) import Pandora.Pattern.Functor.Monoidal (Monoidal (unit)) import Pandora.Pattern.Functor.Bindable (Bindable ((=<<), (==<<))) import Pandora.Pattern.Functor.Traversable (Traversable ((<-/--))) import Pandora.Pattern.Transformer.Liftable (Liftable (lift)) import Pandora.Pattern.Transformer.Lowerable (Lowerable (lower)) import Pandora.Paradigm.Algebraic.Exponential (type (<--), type (-->)) import Pandora.Paradigm.Algebraic.Product ((:*:) ((:*:))) import Pandora.Paradigm.Algebraic.Sum ((:+:)) import Pandora.Paradigm.Algebraic.One (One (One)) import Pandora.Paradigm.Algebraic (point, extract, (<-||-), (<-||---)) import Pandora.Pattern.Morphism.Flip (Flip (Flip)) newtype UT ct cu t u a = UT (u :. t >>> a) infixr 3 <.:>, >.:>, <.:<, >.:< type (<.:>) = UT Covariant Covariant type (>.:>) = UT Contravariant Covariant type (<.:<) = UT Covariant Contravariant type (>.:<) = UT Contravariant Contravariant instance Interpreted (->) (UT ct cu t u) where type Primary (UT ct cu t u) a = u :. t >>> a run :: ((->) < UT ct cu t u a) < Primary (UT ct cu t u) a run ~(UT (u :. t) >>> a x) = (u :. t) >>> a Primary (UT ct cu t u) a x unite :: ((->) < Primary (UT ct cu t u) a) < UT ct cu t u a unite = ((->) < Primary (UT ct cu t u) a) < UT ct cu t u a forall k k k k (ct :: k) (cu :: k) (t :: k -> k) (u :: k -> *) (a :: k). ((u :. t) >>> a) -> UT ct cu t u a UT instance (Semigroupoid m, Covariant m m u, Covariant m m t, Covariant m (Betwixt m m) t, Covariant (Betwixt m m) m u, Interpreted m (t <.:> u)) => Covariant m m (t <.:> u) where <-|- :: m a b -> m ((<.:>) t u a) ((<.:>) t u b) (<-|-) m a b f = ((m < Primary (t <.:> u) a) < Primary (t <.:> u) b) -> m ((<.:>) t u a) ((<.:>) t u 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 (=#-) (m a b -> m (u (t a)) (u (t b)) forall (source :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) (u :: * -> *) a b. (Covariant source target t, Covariant source (Betwixt source target) u, Covariant (Betwixt source target) target t) => source a b -> target (t (u a)) (t (u b)) (<-|-|-) m a b f) instance (Covariant (->) (->) u, Semimonoidal (-->) (:*:) (:*:) t, Semimonoidal (-->) (:*:) (:*:) 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 :: * -> * -> *) a b. Category m => m (m a b) (m a b) <-- ((u :. t) >>> (a :*: b)) -> (<.:>) t u (a :*: b) forall k k k k (ct :: k) (cu :: k) (t :: k -> k) (u :: k -> *) (a :: k). ((u :. t) >>> a) -> UT ct cu t u a UT (((u :. t) >>> (a :*: b)) -> (<.:>) t u (a :*: b)) -> (((<.:>) t u a :*: (<.:>) t u b) -> (u :. t) >>> (a :*: b)) -> ((<.:>) t u a :*: (<.:>) t u b) -> (<.:>) t u (a :*: b) forall (m :: * -> * -> *) b c a. Semigroupoid m => m b c -> m a b -> m a c . ((t a :*: t b) -> t (a :*: b)) -> u (t a :*: t b) -> (u :. 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 a :*: t b) --> t (a :*: b)) -> (t a :*: t b) -> t (a :*: b) forall (m :: * -> * -> *) (t :: * -> *) a. Interpreted m t => (m < t a) < Primary t a <~) (u (t a :*: t b) -> (u :. t) >>> (a :*: b)) -> (((<.:>) t u a :*: (<.:>) t u b) -> u (t a :*: t b)) -> ((<.:>) t u a :*: (<.:>) t u b) -> (u :. t) >>> (a :*: b) forall (m :: * -> * -> *) b c a. Semigroupoid m => m b c -> m a b -> m a c . (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 @(-->) ((u (t a) :*: u (t b)) --> u (t a :*: t b)) -> (u (t a) :*: u (t b)) -> u (t a :*: t b) forall (m :: * -> * -> *) (t :: * -> *) a. Interpreted m t => (m < t a) < Primary t a <~) ((u (t a) :*: u (t b)) -> u (t a :*: t b)) -> (((<.:>) t u a :*: (<.:>) t u b) -> u (t a) :*: u (t b)) -> ((<.:>) t u a :*: (<.:>) t u b) -> u (t a :*: t b) forall (m :: * -> * -> *) b c a. Semigroupoid m => m b c -> m a b -> m a c . ((<.:>) t u a -> u (t a) forall (m :: * -> * -> *) (t :: * -> *) a. Interpreted m t => (m < t a) < Primary t a run ((<.:>) t u a -> u (t a)) -> ((<.:>) t u a :*: u (t b)) -> u (t a) :*: u (t b) forall (m :: * -> * -> *) (p :: * -> * -> *) a b c. (Covariant m m (Flip p c), Interpreted m (Flip p c)) => m a b -> m (p a c) (p b c) <-||-) (((<.:>) t u a :*: u (t b)) -> u (t a) :*: u (t b)) -> (((<.:>) t u a :*: (<.:>) t u b) -> (<.:>) t u a :*: u (t b)) -> ((<.:>) t u a :*: (<.:>) t u b) -> u (t a) :*: u (t b) forall (m :: * -> * -> *) b c a. Semigroupoid m => m b c -> m a b -> m a c . (forall (t :: * -> *) a. Interpreted (->) t => ((->) < t a) < Primary t a forall (m :: * -> * -> *) (t :: * -> *) a. Interpreted m t => (m < t a) < Primary t a run @(->) ((<.:>) t u b -> u (t b)) -> ((<.:>) t u a :*: (<.:>) t u b) -> (<.:>) t u a :*: u (t b) forall (source :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) a b. Covariant source target t => source a b -> target (t a) (t b) <-|-) instance (Covariant (->) (->) u, Covariant (->) (->) t, Semimonoidal (-->) (:*:) (:*:) u, Semimonoidal (-->) (:*:) (:+:) t) => 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 :: * -> * -> *) a b. Category m => m (m a b) (m a b) <-- \(UT (u :. t) >>> a x :*: UT (u :. t) >>> b y) -> ((u :. t) >>> (a :+: b)) -> (<.:>) t u (a :+: b) forall k k k k (ct :: k) (cu :: k) (t :: k -> k) (u :: k -> *) (a :: k). ((u :. t) >>> a) -> UT ct cu t u a UT (((u :. t) >>> (a :+: b)) -> (<.:>) t u (a :+: b)) -> ((u :. t) >>> (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 (t :: * -> *) a b. Semimonoidal (-->) (:*:) (:+:) t => (t a :*: t b) --> t (a :+: b) mult @(-->) @(:*:) @(:+:) ((t a :*: t b) --> t (a :+: b)) -> u (t a :*: t b) -> (u :. 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 (t :: * -> *) a b. Semimonoidal (-->) (:*:) (:*:) t => (t a :*: t b) --> t (a :*: b) mult @(-->) @(:*:) @(:*:) ((((u :. t) >>> a) :*: ((u :. t) >>> b)) --> u (t a :*: t b)) -> (((u :. t) >>> a) :*: ((u :. t) >>> b)) -> u (t a :*: t b) forall (m :: * -> * -> *) (t :: * -> *) a. Interpreted m t => (m < t a) < Primary t a <~~~ (u :. t) >>> a x ((u :. t) >>> a) -> ((u :. t) >>> b) -> ((u :. t) >>> a) :*: ((u :. t) >>> b) forall s a. s -> a -> s :*: a :*: (u :. t) >>> 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 (:*:) _ = (Straight (->) One a -> (<.:>) t u a) -> Straight (->) (Straight (->) One a) ((<.:>) t u a) forall (v :: * -> * -> *) a e. v a e -> Straight v a e Straight ((Straight (->) One a -> (<.:>) t u a) -> Straight (->) (Straight (->) One a) ((<.:>) t u a)) -> (Straight (->) One a -> (<.:>) t u a) -> Straight (->) (Straight (->) One a) ((<.:>) t u a) forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b) <-- ((u :. t) >>> a) -> (<.:>) t u a forall k k k k (ct :: k) (cu :: k) (t :: k -> k) (u :: k -> *) (a :: k). ((u :. t) >>> a) -> UT ct cu t u a UT (((u :. t) >>> a) -> (<.:>) t u a) -> (Straight (->) One a -> (u :. t) >>> a) -> Straight (->) One a -> (<.:>) t u a forall (m :: * -> * -> *) b c a. Semigroupoid m => m b c -> m a b -> m a c . t a -> (u :. t) >>> a forall (t :: * -> *) a. Pointable t => a -> t a point (t a -> (u :. t) >>> a) -> (Straight (->) One a -> t a) -> Straight (->) One a -> (u :. t) >>> a forall (m :: * -> * -> *) b c a. Semigroupoid m => m b c -> m a b -> m a c . a -> t a forall (t :: * -> *) a. Pointable t => a -> t a point (a -> t a) -> (Straight (->) One a -> a) -> Straight (->) One a -> t a forall (m :: * -> * -> *) b c a. Semigroupoid m => m b c -> m a b -> m a c . (Straight (->) One a -> One -> a forall (m :: * -> * -> *) (t :: * -> *) a. Interpreted m t => (m < t a) < Primary t a <~ One One) instance (Traversable (->) (->) t, Bindable (->) t, Semimonoidal (-->) (:*:) (:*:) u, Monoidal (-->) (-->) (:*:) (:*:) u, Bindable (->) u) => Bindable (->) (t <.:> u) where a -> (<.:>) t u b f =<< :: (a -> (<.:>) t u b) -> (<.:>) t u a -> (<.:>) t u b =<< UT (u :. t) >>> a x = ((u :. t) >>> b) -> (<.:>) t u b forall k k k k (ct :: k) (cu :: k) (t :: k -> k) (u :: k -> *) (a :: k). ((u :. t) >>> a) -> UT ct cu t u a UT (((u :. t) >>> b) -> (<.:>) t u b) -> ((u :. t) >>> b) -> (<.:>) t u b forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b) <---- ((t b -> t b forall (m :: * -> * -> *) a. Category m => m a a identity (t b -> t b) -> t (t b) -> t b forall (source :: * -> * -> *) (t :: * -> *) a b. Bindable source t => source a (t b) -> source (t a) (t b) =<<) (t (t b) -> t b) -> u (t (t b)) -> (u :. t) >>> b forall (source :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) a b. Covariant source target t => source a b -> target (t a) (t b) <-|-) (u (t (t b)) -> (u :. t) >>> b) -> (t a -> u (t (t b))) -> t a -> (u :. t) >>> b forall (m :: * -> * -> *) b c a. Semigroupoid m => m b c -> m a b -> m a c . ((<.:>) t u b -> (u :. t) >>> b forall (m :: * -> * -> *) (t :: * -> *) a. Interpreted m t => (m < t a) < Primary t a run ((<.:>) t u b -> (u :. t) >>> b) -> (a -> (<.:>) t u b) -> a -> (u :. t) >>> b forall (m :: * -> * -> *) b c a. Semigroupoid m => m b c -> m a b -> m a c . a -> (<.:>) t u b f (a -> (u :. t) >>> b) -> t a -> u (t (t 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)) <-/--) (t a -> (u :. t) >>> b) -> ((u :. t) >>> a) -> (u :. t) >>> b forall (source :: * -> * -> *) (t :: * -> *) a b. Bindable source t => source a (t b) -> source (t a) (t b) ==<< (u :. t) >>> a x instance (Covariant (->) (->) u, Semimonoidal (<--) (:*:) (:*:) t, Semimonoidal (<--) (:*:) (:*:) u) => Semimonoidal (<--) (:*:) (:*:) (t <.:> u) where mult :: ((<.:>) t u a :*: (<.:>) t u b) <-- (<.:>) t u (a :*: b) mult = ((<.:>) 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) <-- \(UT (u :. t) >>> (a :*: b) xys) -> ((u :. t) >>> a) -> (<.:>) t u a forall k k k k (ct :: k) (cu :: k) (t :: k -> k) (u :: k -> *) (a :: k). ((u :. t) >>> a) -> UT ct cu t u a UT (((u :. t) >>> a) -> (<.:>) t u a) -> (((u :. t) >>> a) :*: (<.:>) t u b) -> (<.:>) t u a :*: (<.:>) t u b forall (m :: * -> * -> *) (p :: * -> * -> *) a b c. (Covariant m m (Flip p c), Interpreted m (Flip p c)) => m a b -> m (p a c) (p b c) <-||--- ((u :. t) >>> b) -> (<.:>) t u b forall k k k k (ct :: k) (cu :: k) (t :: k -> k) (u :: k -> *) (a :: k). ((u :. t) >>> a) -> UT ct cu t u a UT (((u :. t) >>> b) -> (<.:>) t u b) -> (((u :. t) >>> a) :*: ((u :. t) >>> b)) -> ((u :. t) >>> a) :*: (<.:>) t u 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 @(<--) ((((u :. t) >>> a) :*: ((u :. t) >>> b)) <-- u (t a :*: t b)) -> u (t a :*: t b) -> ((u :. t) >>> a) :*: ((u :. t) >>> b) forall (m :: * -> * -> *) (t :: * -> *) a. Interpreted m t => (m < t a) < Primary t a <~~~~ (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 a :*: t b) <-- t (a :*: b)) -> t (a :*: b) -> t a :*: t b forall (m :: * -> * -> *) (t :: * -> *) a. Interpreted m t => (m < t a) < Primary t a <~) (t (a :*: b) -> t a :*: t b) -> ((u :. t) >>> (a :*: b)) -> u (t a :*: t b) forall (source :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) a b. Covariant source target t => source a b -> target (t a) (t b) <-|- (u :. t) >>> (a :*: b) xys instance (Covariant (->) (->) u, Monoidal (<--) (-->) (:*:) (:*:) t, Monoidal (<--) (-->) (:*:) (:*:) u) => Monoidal (<--) (-->) (:*:) (:*:) (t <.:> u) where unit :: Proxy (:*:) -> (Unit (:*:) --> a) <-- (<.:>) t u a unit Proxy (:*:) _ = ((<.:>) t u a -> Straight (->) One a) -> Flip (->) (Straight (->) One a) ((<.:>) t u a) forall (v :: * -> * -> *) a e. v e a -> Flip v a e Flip (((<.:>) t u a -> Straight (->) One a) -> Flip (->) (Straight (->) One a) ((<.:>) t u a)) -> ((<.:>) t u a -> Straight (->) One a) -> Flip (->) (Straight (->) One a) ((<.:>) t u a) forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b) <-- \(UT (u :. t) >>> a x) -> (One -> a) -> Straight (->) One a forall (v :: * -> * -> *) a e. v a e -> Straight v a e Straight (\One _ -> t a -> a forall (t :: * -> *) a. Extractable t => t a -> a extract (t a -> a) -> t a -> a forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b) <-- ((u :. t) >>> a) -> t a forall (t :: * -> *) a. Extractable t => t a -> a extract (u :. t) >>> a x) instance Monoidal (-->) (-->) (:*:) (:*:) t => Liftable (->) (UT Covariant Covariant t) where lift :: Covariant (->) (->) u => u ~> t <.:> u lift :: u ~> (t <.:> u) lift u a x = ((u :. t) >>> a) -> UT Covariant Covariant t u a forall k k k k (ct :: k) (cu :: k) (t :: k -> k) (u :: k -> *) (a :: k). ((u :. t) >>> a) -> UT ct cu t u a UT (((u :. t) >>> a) -> UT Covariant Covariant t u a) -> ((u :. t) >>> a) -> UT Covariant Covariant t u a forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b) <---- a -> t a forall (t :: * -> *) a. Pointable t => a -> t a point (a -> t a) -> u a -> (u :. t) >>> a forall (source :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) a b. Covariant source target t => source a b -> target (t a) (t b) <-|- u a x instance Monoidal (<--) (-->) (:*:) (:*:) t => Lowerable (->) (UT Covariant Covariant t) where lower :: Covariant (->) (->) u => t <.:> u ~> u lower :: (t <.:> u) ~> u lower (UT (u :. t) >>> a x) = t a -> a forall (t :: * -> *) a. Extractable t => t a -> a extract (t a -> a) -> ((u :. t) >>> a) -> u a forall (source :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) a b. Covariant source target t => source a b -> target (t a) (t b) <-|- (u :. t) >>> a x