{-# LANGUAGE UndecidableInstances #-} module Pandora.Paradigm.Schemes.TU where import Pandora.Core.Functor (type (:.), 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.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.Traversable (Traversable ((<-/-)), (<-/-/-)) import Pandora.Pattern.Functor.Distributive (Distributive ((-<<))) import Pandora.Pattern.Functor.Bindable (Bindable ((=<<))) import Pandora.Pattern.Transformer.Liftable (Liftable (lift)) import Pandora.Pattern.Transformer.Lowerable (Lowerable (lower)) import Pandora.Pattern.Transformer.Hoistable (Hoistable ((/|\))) 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 (empty, point, extract, (<-||-), (<-||---)) import Pandora.Pattern.Morphism.Flip (Flip (Flip)) import Pandora.Pattern.Morphism.Straight (Straight (Straight)) newtype TU ct cu t u a = TU (t :. u >>> a) infixr 6 <:.>, >:.>, <:.<, >:.< 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 (Semigroupoid m, Covariant m m t, Covariant (Betwixt m m) m t, Covariant m (Betwixt 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 (t (u a)) (t (u 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 (->) (->) t, 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) <-- ((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 :*: (<:.>) t u b) -> (t :. u) >>> (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 . ((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) (<-|-) (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 a :*: u b) --> u (a :*: b)) -> (u a :*: u b) -> u (a :*: b) forall (m :: * -> * -> *) (t :: * -> *) a. Interpreted m t => (m < t a) < Primary t a <~) (t (u a :*: u b) -> (t :. u) >>> (a :*: b)) -> (((<:.>) t u a :*: (<:.>) t u b) -> t (u a :*: u 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 . (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 a) :*: t (u b)) --> t (u a :*: u b)) -> (t (u a) :*: t (u b)) -> t (u a :*: u b) forall (m :: * -> * -> *) (t :: * -> *) a. Interpreted m t => (m < t a) < Primary t a <~) ((t (u a) :*: t (u b)) -> t (u a :*: u b)) -> (((<:.>) t u a :*: (<:.>) t u b) -> t (u a) :*: t (u b)) -> ((<:.>) t u a :*: (<:.>) t u b) -> t (u a :*: u b) forall (m :: * -> * -> *) b c a. Semigroupoid m => m b c -> m a b -> m a c . ((<:.>) t u a -> t (u a) forall (m :: * -> * -> *) (t :: * -> *) a. Interpreted m t => (m < t a) < Primary t a run ((<:.>) t u a -> t (u a)) -> ((<:.>) t u 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) <-||-) (((<:.>) t u a :*: t (u b)) -> t (u a) :*: t (u b)) -> (((<:.>) 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 :: * -> * -> *) 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 -> t (u b)) -> ((<:.>) t u a :*: (<:.>) t u b) -> (<:.>) t u a :*: t (u b) forall (source :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) a b. Covariant source target t => source a b -> target (t a) (t b) <-|-) 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) <-- ((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) -> (Straight (->) One a -> (t :. u) >>> a) -> Straight (->) One 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. Pointable t => a -> t a point (u a -> (t :. u) >>> a) -> (Straight (->) One a -> u a) -> Straight (->) One 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. Pointable t => a -> t a point (a -> u a) -> (Straight (->) One a -> a) -> Straight (->) One a -> u 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 (Covariant (->) (->) t, 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) <-- \(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 (t :: * -> *) a b. Semimonoidal (-->) (:*:) (:+:) t => (t a :*: t b) --> t (a :+: b) mult @(-->) @(:*:) @(:+:) ((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) <-|-- 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 :. u) >>> a) :*: ((t :. u) >>> b)) --> t (u a :*: u b)) -> (((t :. u) >>> a) :*: ((t :. u) >>> b)) -> t (u a :*: u b) forall (m :: * -> * -> *) (t :: * -> *) a. Interpreted m t => (m < t a) < Primary t a <~~~ (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 (-->) (:*:) (:*:) t, Semimonoidal (-->) (:*:) (:+:) u, Monoidal (-->) (-->) (:*:) (:+:) t) => Monoidal (-->) (-->) (:*:) (:+:) (t <:.> u) where unit :: Proxy (:*:) -> (Unit (:+:) --> a) --> (<:.>) t u a unit Proxy (:*:) _ = ((Zero --> a) -> (<:.>) t u a) -> Straight (->) (Zero --> a) ((<:.>) t u a) forall (v :: * -> * -> *) a e. v a e -> Straight v a e Straight (((Zero --> a) -> (<:.>) t u a) -> Straight (->) (Zero --> a) ((<:.>) t u a)) -> ((Zero --> a) -> (<:.>) t u a) -> Straight (->) (Zero --> a) ((<:.>) t u a) forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b) <-- \Zero --> 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. Emptiable t => t a empty instance (Covariant (->) (->) t, 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) <-- \(TU (t :. u) >>> (a :*: b) xys) -> ((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) >>> 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) <-||--- ((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) >>> a) :*: ((t :. u) >>> b)) -> ((t :. u) >>> 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 @(<--) ((((t :. u) >>> a) :*: ((t :. u) >>> b)) <-- t (u a :*: u b)) -> t (u a :*: u b) -> ((t :. u) >>> a) :*: ((t :. u) >>> 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 @(<--) ((u a :*: u b) <-- u (a :*: b)) -> u (a :*: b) -> u a :*: u b forall (m :: * -> * -> *) (t :: * -> *) a. Interpreted m t => (m < t a) < Primary t a <~) (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 instance (Covariant (->) (->) t, 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) <-- \(TU (t :. u) >>> a x) -> (One -> a) -> Straight (->) One a forall (v :: * -> * -> *) a e. v a e -> Straight v a e Straight (\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 (Straight category) (Straight category) (:*:) (:*:) u, Traversable category category v) => category a (u b) -> category (v (t a)) (u (v (t b))) <-/-/- (<:.>) t u a -> t (u a) forall (m :: * -> * -> *) (t :: * -> *) a. Interpreted m t => (m < 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 (m :: * -> * -> *) (t :: * -> *) a. Interpreted m t => (m < 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. Pointable 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