module Pandora.Paradigm.Schemes.TUT where import Pandora.Core.Functor (type (:.), type (:=), type (~>)) import Pandora.Core.Appliable ((!)) 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 (mult)) import Pandora.Pattern.Functor.Monoidal (Monoidal (unit)) import Pandora.Pattern.Functor.Extendable (Extendable ((<<=))) import Pandora.Pattern.Functor.Distributive (Distributive ((-<<))) import Pandora.Pattern.Functor.Bindable (Bindable ((=<<))) import Pandora.Pattern.Functor.Bivariant ((<->)) import Pandora.Pattern.Functor.Adjoint (Adjoint ((-|), (|-))) import Pandora.Pattern.Transformer.Liftable (Liftable (lift)) import Pandora.Pattern.Transformer.Lowerable (Lowerable (lower)) import Pandora.Paradigm.Primary.Algebraic.Exponential (type (<--), type (-->)) import Pandora.Paradigm.Primary.Algebraic.Product ((:*:)((:*:))) import Pandora.Paradigm.Primary.Algebraic.One (One (One)) import Pandora.Paradigm.Primary.Algebraic (point, extract) import Pandora.Pattern.Morphism.Flip (Flip (Flip)) import Pandora.Pattern.Morphism.Straight (Straight (Straight)) import Pandora.Paradigm.Controlflow.Effect.Interpreted (Interpreted (Primary, run, unite, (||=))) newtype TUT ct ct' cu t t' u a = TUT (t :. u :. t' := a) infix 3 <:<.>:>, >:<.>:>, <:<.>:<, >:<.>:<, <:>.<:>, >:>.<:>, <:>.<:<, >:>.<:< type (<:<.>:>) = TUT Covariant Covariant Covariant type (>:<.>:>) = TUT Contravariant Covariant Covariant type (<:<.>:<) = TUT Covariant Covariant Contravariant type (>:<.>:<) = TUT Contravariant Covariant Contravariant type (<:>.<:>) = TUT Covariant Contravariant Covariant type (>:>.<:>) = TUT Contravariant Contravariant Covariant type (<:>.<:<) = TUT Covariant Contravariant Contravariant type (>:>.<:<) = TUT Contravariant Contravariant Contravariant instance Interpreted (->) (TUT ct ct' cu t t' u) where type Primary (TUT ct ct' cu t t' u) a = t :. u :. t' := a run :: TUT ct ct' cu t t' u a -> Primary (TUT ct ct' cu t t' u) a run ~(TUT (t :. (u :. t')) := a x) = (t :. (u :. t')) := a Primary (TUT ct ct' cu t t' u) a x unite :: Primary (TUT ct ct' cu t t' u) a -> TUT ct ct' cu t t' u a unite = Primary (TUT ct ct' cu t t' u) a -> TUT ct ct' cu t t' u a forall k k k k k k (ct :: k) (ct' :: k) (cu :: k) (t :: k -> *) (t' :: k -> k) (u :: k -> k) (a :: k). ((t :. (u :. t')) := a) -> TUT ct ct' cu t t' u a TUT instance (Covariant m m t, Covariant m m u, Covariant m m t', Interpreted m (t <:<.>:> t' := u)) => Covariant m m (t <:<.>:> t' := u) where <$> :: m a b -> m ((:=) (t <:<.>:> t') u a) ((:=) (t <:<.>:> t') u b) (<$>) m a b f = m (Primary ((t <:<.>:> t') := u) a) (Primary ((t <:<.>:> t') := u) b) -> m ((:=) (t <:<.>:> t') u a) ((:=) (t <:<.>:> 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 (t (u (t' a))) (t (u (t' b))) forall (source :: * -> * -> *) (between1 :: * -> * -> *) (between2 :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) (u :: * -> *) (v :: * -> *) a b. (Covariant source between1 v, Covariant between1 between2 u, Covariant between2 target t) => source a b -> target (t (u (v a))) (t (u (v b))) (<$$$>) @m @m @m m a b f) instance (Covariant (->) (->) t, Covariant (->) (->) t', Covariant (->) (->) u, Semimonoidal (-->) (:*:) (:*:) t, Semimonoidal (-->) (:*:) (:*:) u, Semimonoidal (-->) (:*:) (:*:) t') => Semimonoidal (-->) (:*:) (:*:) (t <:<.>:> t' := u) where mult :: ((:=) (t <:<.>:> t') u a :*: (:=) (t <:<.>:> t') u b) --> (:=) (t <:<.>:> t') u (a :*: b) mult = (((:=) (t <:<.>:> t') u a :*: (:=) (t <:<.>:> t') u b) -> (:=) (t <:<.>:> t') u (a :*: b)) -> ((:=) (t <:<.>:> t') u a :*: (:=) (t <:<.>:> t') u b) --> (:=) (t <:<.>:> t') u (a :*: b) forall (v :: * -> * -> *) a e. v a e -> Straight v a e Straight ((((:=) (t <:<.>:> t') u a :*: (:=) (t <:<.>:> t') u b) -> (:=) (t <:<.>:> t') u (a :*: b)) -> ((:=) (t <:<.>:> t') u a :*: (:=) (t <:<.>:> t') u b) --> (:=) (t <:<.>:> t') u (a :*: b)) -> (((:=) (t <:<.>:> t') u a :*: (:=) (t <:<.>:> t') u b) -> (:=) (t <:<.>:> t') u (a :*: b)) -> ((:=) (t <:<.>:> t') u a :*: (:=) (t <:<.>:> t') u b) --> (:=) (t <:<.>:> t') u (a :*: b) forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b) $ ((t :. (u :. t')) := (a :*: b)) -> (:=) (t <:<.>:> t') u (a :*: b) forall k k k k k k (ct :: k) (ct' :: k) (cu :: k) (t :: k -> *) (t' :: k -> k) (u :: k -> k) (a :: k). ((t :. (u :. t')) := a) -> TUT ct ct' cu t t' u a TUT (((t :. (u :. t')) := (a :*: b)) -> (:=) (t <:<.>:> t') u (a :*: b)) -> (((:=) (t <:<.>:> t') u a :*: (:=) (t <:<.>:> t') u b) -> (t :. (u :. t')) := (a :*: b)) -> ((:=) (t <:<.>:> t') u a :*: (:=) (t <:<.>:> t') u b) -> (:=) (t <:<.>:> t') u (a :*: b) forall (m :: * -> * -> *) b c a. Semigroupoid m => m b c -> m a b -> m a c . ((t' a :*: t' b) -> t' (a :*: b)) -> t (u (t' a :*: t' b)) -> (t :. (u :. t')) := (a :*: b) forall (source :: * -> * -> *) (between :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) (u :: * -> *) a b. (Covariant source between u, Covariant between target t) => source a b -> target (t (u a)) (t (u b)) (<$$>) @_ @(->) (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 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 (u (t' a :*: t' b)) -> (t :. (u :. t')) := (a :*: b)) -> (((:=) (t <:<.>:> t') u a :*: (:=) (t <:<.>:> t') u b) -> t (u (t' a :*: t' b))) -> ((:=) (t <:<.>:> t') u a :*: (:=) (t <:<.>:> t') u b) -> (t :. (u :. t')) := (a :*: b) forall (m :: * -> * -> *) b c a. Semigroupoid m => m b c -> m a b -> m a c . ((u (t' a) :*: u (t' b)) -> u (t' a :*: t' b)) -> t (u (t' a) :*: u (t' b)) -> t (u (t' a :*: t' 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) :*: u (t' b)) -> u (t' a :*: 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 (u (t' a) :*: u (t' b)) -> t (u (t' a :*: t' b))) -> (((:=) (t <:<.>:> t') u a :*: (:=) (t <:<.>:> t') u b) -> t (u (t' a) :*: u (t' b))) -> ((:=) (t <:<.>:> t') u a :*: (:=) (t <:<.>:> t') u b) -> t (u (t' a :*: t' 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 @(-->) ((((t :. (u :. t')) := a) :*: ((t :. (u :. t')) := b)) --> t (u (t' a) :*: u (t' b))) -> (((t :. (u :. t')) := a) :*: ((t :. (u :. t')) := b)) -> t (u (t' a) :*: u (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 :. (u :. t')) := a) :*: ((t :. (u :. t')) := b)) -> t (u (t' a) :*: u (t' b))) -> (((:=) (t <:<.>:> t') u a :*: (:=) (t <:<.>:> t') u b) -> ((t :. (u :. t')) := a) :*: ((t :. (u :. t')) := b)) -> ((:=) (t <:<.>:> t') u a :*: (:=) (t <:<.>:> t') u b) -> t (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 <:<.>:> t') u a -> (t :. (u :. t')) := a) -> ((:=) (t <:<.>:> t') u b -> (t :. (u :. t')) := b) -> ((:=) (t <:<.>:> t') u a :*: (:=) (t <:<.>:> t') u b) -> ((t :. (u :. t')) := a) :*: ((t :. (u :. t')) := 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) <-> forall (t :: * -> *) a. Interpreted (->) t => t a -> Primary t a forall (m :: * -> * -> *) (t :: * -> *) a. Interpreted m t => m (t a) (Primary t a) run @(->)) instance (Covariant (->) (->) t, Semimonoidal (<--) (:*:) (:*:) t, Covariant (->) (->) u, Semimonoidal (<--) (:*:) (:*:) u, Covariant (->) (->) t', Semimonoidal (<--) (:*:) (:*:) t') => Semimonoidal (<--) (:*:) (:*:) (t <:<.>:> t' := u) where mult :: ((:=) (t <:<.>:> t') u a :*: (:=) (t <:<.>:> t') u b) <-- (:=) (t <:<.>:> t') u (a :*: b) mult = ((:=) (t <:<.>:> t') u (a :*: b) -> (:=) (t <:<.>:> t') u a :*: (:=) (t <:<.>:> t') u b) -> ((:=) (t <:<.>:> t') u a :*: (:=) (t <:<.>:> t') u b) <-- (:=) (t <:<.>:> t') u (a :*: b) forall (v :: * -> * -> *) a e. v e a -> Flip v a e Flip (((:=) (t <:<.>:> t') u (a :*: b) -> (:=) (t <:<.>:> t') u a :*: (:=) (t <:<.>:> t') u b) -> ((:=) (t <:<.>:> t') u a :*: (:=) (t <:<.>:> t') u b) <-- (:=) (t <:<.>:> t') u (a :*: b)) -> ((:=) (t <:<.>:> t') u (a :*: b) -> (:=) (t <:<.>:> t') u a :*: (:=) (t <:<.>:> t') u b) -> ((:=) (t <:<.>:> t') u a :*: (:=) (t <:<.>:> t') u b) <-- (:=) (t <:<.>:> t') u (a :*: b) forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b) $ (((t :. (u :. t')) := a) -> (:=) (t <:<.>:> t') u a forall k k k k k k (ct :: k) (ct' :: k) (cu :: k) (t :: k -> *) (t' :: k -> k) (u :: k -> k) (a :: k). ((t :. (u :. t')) := a) -> TUT ct ct' cu t t' u a TUT (((t :. (u :. t')) := a) -> (:=) (t <:<.>:> t') u a) -> (((t :. (u :. t')) := b) -> (:=) (t <:<.>:> t') u b) -> (((t :. (u :. t')) := a) :*: ((t :. (u :. t')) := b)) -> (:=) (t <:<.>:> t') u a :*: (:=) (t <:<.>:> 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 :. t')) := b) -> (:=) (t <:<.>:> t') u b forall k k k k k k (ct :: k) (ct' :: k) (cu :: k) (t :: k -> *) (t' :: k -> k) (u :: k -> k) (a :: k). ((t :. (u :. t')) := a) -> TUT ct ct' cu t t' u a TUT) ((((t :. (u :. t')) := a) :*: ((t :. (u :. t')) := b)) -> (:=) (t <:<.>:> t') u a :*: (:=) (t <:<.>:> t') u b) -> ((:=) (t <:<.>:> t') u (a :*: b) -> ((t :. (u :. t')) := a) :*: ((t :. (u :. t')) := b)) -> (:=) (t <:<.>:> t') u (a :*: b) -> (:=) (t <:<.>:> t') u a :*: (:=) (t <:<.>:> t') u 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 @(<--) ((((t :. (u :. t')) := a) :*: ((t :. (u :. t')) := b)) <-- t ((:.) u t' a :*: u (t' b))) -> t ((:.) u t' a :*: u (t' b)) -> ((t :. (u :. t')) := a) :*: ((t :. (u :. 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 ((:.) u t' a :*: u (t' b)) -> ((t :. (u :. t')) := a) :*: ((t :. (u :. t')) := b)) -> ((:=) (t <:<.>:> t') u (a :*: b) -> t ((:.) u t' a :*: u (t' b))) -> (:=) (t <:<.>:> t') u (a :*: b) -> ((t :. (u :. t')) := a) :*: ((t :. (u :. t')) := b) forall (m :: * -> * -> *) b c a. Semigroupoid m => m b c -> m a b -> m a c . (u (t' a :*: t' b) -> (:.) u t' a :*: u (t' b)) -> t (u (t' a :*: t' b)) -> t ((:.) u t' a :*: u (t' 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 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 (u (t' a :*: t' b)) -> t ((:.) u t' a :*: u (t' b))) -> ((:=) (t <:<.>:> t') u (a :*: b) -> t (u (t' a :*: t' b))) -> (:=) (t <:<.>:> t') u (a :*: b) -> t ((:.) u t' a :*: u (t' b)) forall (m :: * -> * -> *) b c a. Semigroupoid m => m b c -> m a b -> m a c . (t' (a :*: b) -> t' a :*: t' b) -> t (u (t' (a :*: b))) -> t (u (t' a :*: 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)) (<$$>) @_ @(->) (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 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 (u (t' (a :*: b))) -> t (u (t' a :*: t' b))) -> ((:=) (t <:<.>:> t') u (a :*: b) -> t (u (t' (a :*: b)))) -> (:=) (t <:<.>:> t') u (a :*: b) -> t (u (t' a :*: t' b)) forall (m :: * -> * -> *) b c a. Semigroupoid m => m b c -> m a b -> m a c . (:=) (t <:<.>:> t') u (a :*: b) -> t (u (t' (a :*: b))) forall (m :: * -> * -> *) (t :: * -> *) a. Interpreted m t => m (t a) (Primary t a) run instance (Covariant (->) (->) t, Covariant (->) (->) u, Semimonoidal (<--) (:*:) (:*:) t, Semimonoidal (<--) (:*:) (:*:) t', Monoidal (<--) (->) (:*:) (:*:) u, Adjoint (->) (->) t t') => Monoidal (<--) (->) (:*:) (:*:) (t <:<.>:> t' := u) where unit :: Proxy (:*:) -> (Unit (:*:) -> a) <-- (:=) (t <:<.>:> t') u a unit Proxy (:*:) _ = ((:=) (t <:<.>:> t') u a -> One -> a) -> Flip (->) (One -> a) ((:=) (t <:<.>:> t') u a) forall (v :: * -> * -> *) a e. v e a -> Flip v a e Flip (((:=) (t <:<.>:> t') u a -> One -> a) -> Flip (->) (One -> a) ((:=) (t <:<.>:> t') u a)) -> ((:=) (t <:<.>:> t') u a -> One -> a) -> Flip (->) (One -> a) ((:=) (t <:<.>:> t') u a) forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b) $ \(TUT (t :. (u :. t')) := a xys) -> (\One _ -> (u (t' a) -> t' a forall (t :: * -> *) a. Extractable t => t a -> a extract (u (t' a) -> t' a) -> ((t :. (u :. t')) := a) -> a forall (source :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) (u :: * -> *) a b. Adjoint source target t u => target a (u b) -> source (t a) b |-) (t :. (u :. t')) := a xys) instance (Covariant (->) (->) t, Covariant (->) (->) t', Adjoint (->) (->) t' t, Bindable (->) u) => Bindable (->) (t <:<.>:> t' := u) where a -> (:=) (t <:<.>:> t') u b f =<< :: (a -> (:=) (t <:<.>:> t') u b) -> (:=) (t <:<.>:> t') u a -> (:=) (t <:<.>:> t') u b =<< (:=) (t <:<.>:> t') u a x = ((t :. (u :. t')) := b) -> (:=) (t <:<.>:> t') u b forall k k k k k k (ct :: k) (ct' :: k) (cu :: k) (t :: k -> *) (t' :: k -> k) (u :: k -> k) (a :: k). ((t :. (u :. t')) := a) -> TUT ct ct' cu t t' u a TUT (((t :. (u :. t')) := b) -> (:=) (t <:<.>:> t') u b) -> ((t :. (u :. t')) := b) -> (:=) (t <:<.>:> t') u b forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b) $ (((:=) (t <:<.>:> t') u b -> (t :. (u :. t')) := b forall (m :: * -> * -> *) (t :: * -> *) a. Interpreted m t => m (t a) (Primary t a) run ((:=) (t <:<.>:> t') u b -> (t :. (u :. t')) := b) -> (a -> (:=) (t <:<.>:> t') u b) -> a -> (t :. (u :. t')) := b forall (m :: * -> * -> *) b c a. Semigroupoid m => m b c -> m a b -> m a c . a -> (:=) (t <:<.>:> t') u b f (a -> (t :. (u :. t')) := b) -> t' a -> u (t' b) forall (source :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) (u :: * -> *) a b. Adjoint source target t u => target a (u b) -> source (t a) 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) -> u (t' b)) -> t (u (t' a)) -> (t :. (u :. t')) := b forall (source :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) a b. Covariant source target t => source a b -> target (t a) (t b) <$> (:=) (t <:<.>:> t') u a -> t (u (t' a)) forall (m :: * -> * -> *) (t :: * -> *) a. Interpreted m t => m (t a) (Primary t a) run (:=) (t <:<.>:> t') u a x instance (Covariant (->) (->) t, Covariant (->) (->) u, Covariant (->) (->) t', Semimonoidal (-->) (:*:) (:*:) t, Semimonoidal (-->) (:*:) (:*:) t', Monoidal (-->) (->) (:*:) (:*:) u, Adjoint (->) (->) t' t) => Monoidal (-->) (->) (:*:) (:*:) (t <:<.>:> t' := u) where unit :: Proxy (:*:) -> (Unit (:*:) -> a) --> (:=) (t <:<.>:> t') u a unit Proxy (:*:) _ = ((One -> a) -> (:=) (t <:<.>:> t') u a) -> Straight (->) (One -> a) ((:=) (t <:<.>:> t') u a) forall (v :: * -> * -> *) a e. v a e -> Straight v a e Straight (((One -> a) -> (:=) (t <:<.>:> t') u a) -> Straight (->) (One -> a) ((:=) (t <:<.>:> t') u a)) -> ((One -> a) -> (:=) (t <:<.>:> t') u a) -> Straight (->) (One -> a) ((:=) (t <:<.>:> t') u a) forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b) $ ((t :. (u :. t')) := a) -> (:=) (t <:<.>:> t') u a forall (m :: * -> * -> *) (t :: * -> *) a. Interpreted m t => m (Primary t a) (t a) unite (((t :. (u :. t')) := a) -> (:=) (t <:<.>:> t') u a) -> ((One -> a) -> (t :. (u :. t')) := a) -> (One -> a) -> (:=) (t <:<.>:> 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)) -> a -> (t :. (u :. t')) := a forall (source :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) (u :: * -> *) a b. Adjoint source target t u => source (t a) b -> target a (u b) -|) (a -> (t :. (u :. t')) := a) -> ((One -> a) -> a) -> (One -> a) -> (t :. (u :. 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 (Adjoint (->) (->) t' t, Extendable (->) u) => Extendable (->) (t' <:<.>:> t := u) where (:=) (t' <:<.>:> t) u a -> b f <<= :: ((:=) (t' <:<.>:> t) u a -> b) -> (:=) (t' <:<.>:> t) u a -> (:=) (t' <:<.>:> t) u b <<= (:=) (t' <:<.>:> t) u a x = ((t' :. (u :. t)) := b) -> (:=) (t' <:<.>:> t) u b forall k k k k k k (ct :: k) (ct' :: k) (cu :: k) (t :: k -> *) (t' :: k -> k) (u :: k -> k) (a :: k). ((t :. (u :. t')) := a) -> TUT ct ct' cu t t' u a TUT (((t' :. (u :. t)) := b) -> (:=) (t' <:<.>:> t) u b) -> ((t' :. (u :. t)) := b) -> (:=) (t' <:<.>:> t) u b forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b) $ (((:=) (t' <:<.>:> t) u a -> b f ((:=) (t' <:<.>:> t) u a -> b) -> (t' (u (t a)) -> (:=) (t' <:<.>:> t) u a) -> t' (u (t a)) -> b forall (m :: * -> * -> *) b c a. Semigroupoid m => m b c -> m a b -> m a c . t' (u (t a)) -> (:=) (t' <:<.>:> t) u a forall (m :: * -> * -> *) (t :: * -> *) a. Interpreted m t => m (Primary t a) (t a) unite (t' (u (t a)) -> b) -> u (t a) -> t b forall (source :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) (u :: * -> *) a b. Adjoint source target t u => source (t a) b -> target a (u b) -|) (u (t a) -> t b) -> u (t a) -> u (t b) forall (source :: * -> * -> *) (t :: * -> *) a b. Extendable source t => source (t a) b -> source (t a) (t b) <<=) (u (t a) -> u (t b)) -> t' (u (t a)) -> (t' :. (u :. t)) := b forall (source :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) a b. Covariant source target t => source a b -> target (t a) (t b) <$> (:=) (t' <:<.>:> t) u a -> t' (u (t a)) forall (m :: * -> * -> *) (t :: * -> *) a. Interpreted m t => m (t a) (Primary t a) run (:=) (t' <:<.>:> t) u a x instance (Adjoint (->) (->) t' t, Distributive (->) (->) t) => Liftable (->) (t <:<.>:> t') where lift :: Covariant (->) (->) u => u ~> t <:<.>:> t' := u lift :: u ~> ((t <:<.>:> t') := u) lift u a x = ((t :. (u :. t')) := a) -> TUT Covariant Covariant Covariant t t' u a forall k k k k k k (ct :: k) (ct' :: k) (cu :: k) (t :: k -> *) (t' :: k -> k) (u :: k -> k) (a :: k). ((t :. (u :. t')) := a) -> TUT ct ct' cu t t' u a TUT (((t :. (u :. t')) := a) -> TUT Covariant Covariant Covariant t t' u a) -> ((t :. (u :. t')) := a) -> TUT Covariant Covariant Covariant t t' u a forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b) $ (forall a. Category (->) => a -> a forall (m :: * -> * -> *) a. Category m => m a a identity @(->) (t' a -> t' a) -> a -> t (t' a) forall (source :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) (u :: * -> *) a b. Adjoint source target t u => source (t a) b -> target a (u b) -|) (a -> t (t' a)) -> u a -> (t :. (u :. t')) := a 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 (Adjoint (->) (->) t t', Distributive (->) (->) t') => Lowerable (->) (t <:<.>:> t') where lower :: Covariant (->) (->) u => (t <:<.>:> t' := u) ~> u lower :: ((t <:<.>:> t') := u) ~> u lower (TUT (t :. (u :. t')) := a x) = (forall a. Category (->) => a -> a forall (m :: * -> * -> *) a. Category m => m a a identity @(->) (t' a -> t' a) -> u (t' a) -> t' (u a) 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 (t' a) -> t' (u a)) -> ((t :. (u :. t')) := a) -> u a forall (source :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) (u :: * -> *) a b. Adjoint source target t u => target a (u b) -> source (t a) b |- (t :. (u :. t')) := a x