{-# LANGUAGE UndecidableInstances #-} module Pandora.Paradigm.Schemes.TT 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.Kernel (constant) 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 ((:+:), bitraverse_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 TT ct ct' t t' a = TT (t :. t' >>> a) infixr 6 <::>, >::>, <::<, >::< type (<::>) = TT Covariant Covariant type (>::>) = TT Contravariant Covariant type (<::<) = TT Covariant Contravariant type (>::<) = TT Contravariant Contravariant instance Interpreted (->) (TT ct ct' t t') where type Primary (TT ct ct' t t') a = t :. t' >>> a run :: ((->) < TT ct ct' t t' a) < Primary (TT ct ct' t t') a run ~(TT (t :. t') >>> a x) = (t :. t') >>> a Primary (TT ct ct' t t') a x unite :: ((->) < Primary (TT ct ct' t t') a) < TT ct ct' t t' a unite = ((->) < Primary (TT ct ct' t t') a) < TT ct ct' t t' a forall k k k k (ct :: k) (ct' :: k) (t :: k -> *) (t' :: k -> k) (a :: k). ((t :. t') >>> a) -> TT ct ct' t t' a TT instance (Semigroupoid m, Covariant m m t, Covariant (Betwixt m m) m t, Covariant m (Betwixt m m) t', Interpreted m (t <::> t')) => Covariant m m (t <::> t') where <-|- :: m a b -> m ((<::>) t t' a) ((<::>) t t' b) (<-|-) m a b f = ((m < Primary (t <::> t') a) < Primary (t <::> t') b) -> m ((<::>) t t' a) ((<::>) t t' 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 (t' a)) (t (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 (->) (->) t, Semimonoidal (-->) (:*:) (:*:) t, Semimonoidal (-->) (:*:) (:*:) t') => Semimonoidal (-->) (:*:) (:*:) (t <::> t') where mult :: ((<::>) t t' a :*: (<::>) t t' b) --> (<::>) t t' (a :*: b) mult = (((<::>) t t' a :*: (<::>) t t' b) -> (<::>) t t' (a :*: b)) -> ((<::>) t t' a :*: (<::>) t t' b) --> (<::>) t t' (a :*: b) forall (v :: * -> * -> *) a e. v a e -> Straight v a e Straight ((((<::>) t t' a :*: (<::>) t t' b) -> (<::>) t t' (a :*: b)) -> ((<::>) t t' a :*: (<::>) t t' b) --> (<::>) t t' (a :*: b)) -> (((<::>) t t' a :*: (<::>) t t' b) -> (<::>) t t' (a :*: b)) -> ((<::>) t t' a :*: (<::>) t t' b) --> (<::>) t t' (a :*: b) forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b) <-- ((t :. t') >>> (a :*: b)) -> (<::>) t t' (a :*: b) forall k k k k (ct :: k) (ct' :: k) (t :: k -> *) (t' :: k -> k) (a :: k). ((t :. t') >>> a) -> TT ct ct' t t' a TT (((t :. t') >>> (a :*: b)) -> (<::>) t t' (a :*: b)) -> (((<::>) t t' a :*: (<::>) t t' b) -> (t :. t') >>> (a :*: b)) -> ((<::>) t t' a :*: (<::>) t t' b) -> (<::>) t t' (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 (t' a :*: t' b) -> (t :. 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 <~) (t (t' a :*: t' b) -> (t :. t') >>> (a :*: b)) -> (((<::>) t t' a :*: (<::>) t t' b) -> t (t' a :*: t' b)) -> ((<::>) t t' a :*: (<::>) t t' b) -> (t :. 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 @(-->) ((t (t' a) :*: t (t' b)) --> t (t' a :*: t' b)) -> (t (t' a) :*: t (t' b)) -> t (t' a :*: t' b) forall (m :: * -> * -> *) (t :: * -> *) a. Interpreted m t => (m < t a) < Primary t a <~) ((t (t' a) :*: t (t' b)) -> t (t' a :*: t' b)) -> (((<::>) t t' a :*: (<::>) t t' b) -> t (t' a) :*: t (t' b)) -> ((<::>) t t' a :*: (<::>) t t' b) -> t (t' a :*: t' b) forall (m :: * -> * -> *) b c a. Semigroupoid m => m b c -> m a b -> m a c . ((<::>) t t' a -> t (t' a) forall (m :: * -> * -> *) (t :: * -> *) a. Interpreted m t => (m < t a) < Primary t a run ((<::>) t t' a -> t (t' a)) -> ((<::>) t t' a :*: t (t' b)) -> t (t' a) :*: t (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 t' a :*: t (t' b)) -> t (t' a) :*: t (t' b)) -> (((<::>) t t' a :*: (<::>) t t' b) -> (<::>) t t' a :*: t (t' b)) -> ((<::>) t t' a :*: (<::>) t t' b) -> t (t' a) :*: t (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' b -> t (t' b)) -> ((<::>) t t' a :*: (<::>) t t' b) -> (<::>) t t' a :*: t (t' b) forall (source :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) a b. Covariant source target t => source a b -> target (t a) (t b) <-|-) instance (Covariant (->) (->) t, Covariant (->) (->) t', Semimonoidal (-->) (:*:) (:*:) t', Monoidal (-->) (-->) (:*:) (:*:) t, Monoidal (-->) (-->) (:*:) (:*:) t') => Monoidal (-->) (-->) (:*:) (:*:) (t <::> t') where unit :: Proxy (:*:) -> (Unit (:*:) --> a) --> (<::>) t t' a unit Proxy (:*:) _ = (Straight (->) One a -> (<::>) t t' a) -> Straight (->) (Straight (->) One a) ((<::>) t t' a) forall (v :: * -> * -> *) a e. v a e -> Straight v a e Straight ((Straight (->) One a -> (<::>) t t' a) -> Straight (->) (Straight (->) One a) ((<::>) t t' a)) -> (Straight (->) One a -> (<::>) t t' a) -> Straight (->) (Straight (->) One a) ((<::>) t t' a) forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b) <-- ((t :. t') >>> a) -> (<::>) t t' a forall k k k k (ct :: k) (ct' :: k) (t :: k -> *) (t' :: k -> k) (a :: k). ((t :. t') >>> a) -> TT ct ct' t t' a TT (((t :. t') >>> a) -> (<::>) t t' a) -> (Straight (->) One a -> (t :. t') >>> a) -> Straight (->) One a -> (<::>) t t' a forall (m :: * -> * -> *) b c a. Semigroupoid m => m b c -> m a b -> m a c . t' a -> (t :. t') >>> a forall (t :: * -> *) a. Pointable t => a -> t a point (t' a -> (t :. t') >>> a) -> (Straight (->) One a -> t' a) -> Straight (->) One a -> (t :. 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 (Covariant (->) (->) t, Covariant (->) (->) t', Semimonoidal (-->) (:*:) (:+:) t) => Semimonoidal (-->) (:*:) (:+:) (t <::> t') where mult :: ((<::>) t t' a :*: (<::>) t t' b) --> (<::>) t t' (a :+: b) mult = (((<::>) t t' a :*: (<::>) t t' b) -> (<::>) t t' (a :+: b)) -> ((<::>) t t' a :*: (<::>) t t' b) --> (<::>) t t' (a :+: b) forall (v :: * -> * -> *) a e. v a e -> Straight v a e Straight ((((<::>) t t' a :*: (<::>) t t' b) -> (<::>) t t' (a :+: b)) -> ((<::>) t t' a :*: (<::>) t t' b) --> (<::>) t t' (a :+: b)) -> (((<::>) t t' a :*: (<::>) t t' b) -> (<::>) t t' (a :+: b)) -> ((<::>) t t' a :*: (<::>) t t' b) --> (<::>) t t' (a :+: b) forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b) <-- \(TT (t :. t') >>> a x :*: TT (t :. t') >>> b y) -> ((t :. t') >>> (a :+: b)) -> (<::>) t t' (a :+: b) forall k k k k (ct :: k) (ct' :: k) (t :: k -> *) (t' :: k -> k) (a :: k). ((t :. t') >>> a) -> TT ct ct' t t' a TT (((t :. t') >>> (a :+: b)) -> (<::>) t t' (a :+: b)) -> ((t :. t') >>> (a :+: b)) -> (<::>) t t' (a :+: b) forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b) <----- (t' a -> t' a) -> (t' b -> t' b) -> (t' a :+: t' b) -> t' (a :+: b) forall (t :: * -> *) e e' a a'. Covariant (->) (->) t => (e -> t e') -> (a -> t a') -> (e :+: a) -> t (e' :+: a') bitraverse_sum t' a -> t' a forall (m :: * -> * -> *) a. Category m => m a a identity t' b -> t' b forall (m :: * -> * -> *) a. Category m => m a a identity ((t' a :+: t' b) -> t' (a :+: b)) -> t (t' a :+: t' b) -> (t :. 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 @(-->) @(:*:) @(:+:) ((((t :. t') >>> a) :*: ((t :. t') >>> b)) --> t (t' a :+: t' b)) -> (((t :. t') >>> a) :*: ((t :. t') >>> b)) -> t (t' a :+: t' b) forall (m :: * -> * -> *) (t :: * -> *) a. Interpreted m t => (m < t a) < Primary t a <~~~ (t :. t') >>> a x ((t :. t') >>> a) -> ((t :. t') >>> b) -> ((t :. t') >>> a) :*: ((t :. t') >>> b) forall s a. s -> a -> s :*: a :*: (t :. t') >>> b y instance (Covariant (->) (->) t, Covariant (->) (->) t', Semimonoidal (-->) (:*:) (:+:) t, Monoidal (-->) (-->) (:*:) (:+:) t) => Monoidal (-->) (-->) (:*:) (:+:) (t <::> t') where unit :: Proxy (:*:) -> (Unit (:+:) --> a) --> (<::>) t t' a unit Proxy (:*:) _ = ((Zero --> a) -> (<::>) t t' a) -> Straight (->) (Zero --> a) ((<::>) t t' a) forall (v :: * -> * -> *) a e. v a e -> Straight v a e Straight (((Zero --> a) -> (<::>) t t' a) -> Straight (->) (Zero --> a) ((<::>) t t' a)) -> ((Zero --> a) -> (<::>) t t' a) -> Straight (->) (Zero --> a) ((<::>) t t' a) forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b) <-- \Zero --> a _ -> ((t :. t') >>> a) -> (<::>) t t' a forall k k k k (ct :: k) (ct' :: k) (t :: k -> *) (t' :: k -> k) (a :: k). ((t :. t') >>> a) -> TT ct ct' t t' a TT (t :. t') >>> a forall (t :: * -> *) a. Emptiable t => t a empty instance (Covariant (->) (->) t, Semimonoidal (<--) (:*:) (:*:) t, Semimonoidal (<--) (:*:) (:*:) t') => Semimonoidal (<--) (:*:) (:*:) (t <::> t') where mult :: ((<::>) t t' a :*: (<::>) t t' b) <-- (<::>) t t' (a :*: b) mult = ((<::>) t t' (a :*: b) -> (<::>) t t' a :*: (<::>) t t' b) -> ((<::>) t t' a :*: (<::>) t t' b) <-- (<::>) t t' (a :*: b) forall (v :: * -> * -> *) a e. v e a -> Flip v a e Flip (((<::>) t t' (a :*: b) -> (<::>) t t' a :*: (<::>) t t' b) -> ((<::>) t t' a :*: (<::>) t t' b) <-- (<::>) t t' (a :*: b)) -> ((<::>) t t' (a :*: b) -> (<::>) t t' a :*: (<::>) t t' b) -> ((<::>) t t' a :*: (<::>) t t' b) <-- (<::>) t t' (a :*: b) forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b) <-- \(TT (t :. t') >>> (a :*: b) xys) -> ((t :. t') >>> a) -> (<::>) t t' a forall k k k k (ct :: k) (ct' :: k) (t :: k -> *) (t' :: k -> k) (a :: k). ((t :. t') >>> a) -> TT ct ct' t t' a TT (((t :. t') >>> a) -> (<::>) t t' a) -> (((t :. t') >>> a) :*: (<::>) t t' b) -> (<::>) t t' a :*: (<::>) t 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 :. t') >>> b) -> (<::>) t t' b forall k k k k (ct :: k) (ct' :: k) (t :: k -> *) (t' :: k -> k) (a :: k). ((t :. t') >>> a) -> TT ct ct' t t' a TT (((t :. t') >>> b) -> (<::>) t t' b) -> (((t :. t') >>> a) :*: ((t :. t') >>> b)) -> ((t :. t') >>> a) :*: (<::>) t 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 @(<--) ((((t :. t') >>> a) :*: ((t :. t') >>> b)) <-- t (t' a :*: t' b)) -> t (t' a :*: t' b) -> ((t :. t') >>> a) :*: ((t :. 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) -> ((t :. t') >>> (a :*: b)) -> t (t' a :*: t' b) forall (source :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) a b. Covariant source target t => source a b -> target (t a) (t b) <-|- (t :. t') >>> (a :*: b) xys instance (Covariant (->) (->) t, Monoidal (<--) (-->) (:*:) (:*:) t, Monoidal (<--) (-->) (:*:) (:*:) t') => Monoidal (<--) (-->) (:*:) (:*:) (t <::> t') where unit :: Proxy (:*:) -> (Unit (:*:) --> a) <-- (<::>) t t' a unit Proxy (:*:) _ = ((<::>) t t' a -> Straight (->) One a) -> Flip (->) (Straight (->) One a) ((<::>) t t' a) forall (v :: * -> * -> *) a e. v e a -> Flip v a e Flip (((<::>) t t' a -> Straight (->) One a) -> Flip (->) (Straight (->) One a) ((<::>) t t' a)) -> ((<::>) t t' a -> Straight (->) One a) -> Flip (->) (Straight (->) One a) ((<::>) t t' a) forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b) <-- \(TT (t :. t') >>> a x) -> (One -> a) -> Straight (->) One a forall (v :: * -> * -> *) a e. v a e -> Straight v a e Straight ((One -> a) -> Straight (->) One a) -> (One -> a) -> Straight (->) One a forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b) <---- a -> One -> a forall (m :: * -> * -> *) a i. Kernel m => m a (m i a) constant (a -> One -> a) -> a -> One -> a forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b) <--- 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) <-- ((t :. t') >>> a) -> t' a forall (t :: * -> *) a. Extractable t => t a -> a extract (t :. t') >>> a x instance (Traversable (->) (->) t, Traversable (->) (->) t') => Traversable (->) (->) (t <::> t') where a -> u b f <-/- :: (a -> u b) -> (<::>) t t' a -> u ((<::>) t t' b) <-/- (<::>) t t' a x = ((t :. t') >>> b) -> (<::>) t t' b forall k k k k (ct :: k) (ct' :: k) (t :: k -> *) (t' :: k -> k) (a :: k). ((t :. t') >>> a) -> TT ct ct' t t' a TT (((t :. t') >>> b) -> (<::>) t t' b) -> u ((t :. t') >>> b) -> u ((<::>) t 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 (t' a) -> u ((t :. t') >>> 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 t' a -> t (t' a) forall (m :: * -> * -> *) (t :: * -> *) a. Interpreted m t => (m < t a) < Primary t a run (<::>) t t' a x) instance (Bindable (->) t, Distributive (->) (->) t, Covariant (->) (->) t', Bindable (->) t') => Bindable (->) (t <::> t') where a -> (<::>) t t' b f =<< :: (a -> (<::>) t t' b) -> (<::>) t t' a -> (<::>) t t' b =<< TT (t :. t') >>> a x = ((t :. t') >>> b) -> (<::>) t t' b forall k k k k (ct :: k) (ct' :: k) (t :: k -> *) (t' :: k -> k) (a :: k). ((t :. t') >>> a) -> TT ct ct' t t' a TT (((t :. t') >>> b) -> (<::>) t t' b) -> ((t :. t') >>> b) -> (<::>) t t' b forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b) <--- (\t' a i -> (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) -> t (t' (t' b)) -> (t :. t') >>> b forall (source :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) a b. Covariant source target t => source a b -> target (t a) (t b) <-|- (<::>) t t' b -> (t :. t') >>> b forall (m :: * -> * -> *) (t :: * -> *) a. Interpreted m t => (m < t a) < Primary t a run ((<::>) t t' b -> (t :. t') >>> b) -> (a -> (<::>) t t' b) -> a -> (t :. t') >>> b forall (m :: * -> * -> *) b c a. Semigroupoid m => m b c -> m a b -> m a c . a -> (<::>) t t' b f (a -> (t :. t') >>> b) -> t' a -> t (t' (t' 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)) -<< t' a i) (t' a -> (t :. t') >>> b) -> ((t :. t') >>> a) -> (t :. t') >>> b forall (source :: * -> * -> *) (t :: * -> *) a b. Bindable source t => source a (t b) -> source (t a) (t b) =<< (t :. t') >>> a x instance Monoidal (-->) (-->) (:*:) (:*:) t => Liftable (->) (TT Covariant Covariant t) where lift :: Covariant (->) (->) t' => t' ~> t <::> t' lift :: t' ~> (t <::> t') lift = ((t :. t') >>> a) -> TT Covariant Covariant t t' a forall k k k k (ct :: k) (ct' :: k) (t :: k -> *) (t' :: k -> k) (a :: k). ((t :. t') >>> a) -> TT ct ct' t t' a TT (((t :. t') >>> a) -> TT Covariant Covariant t t' a) -> (t' a -> (t :. t') >>> a) -> t' a -> TT Covariant Covariant t t' a forall (m :: * -> * -> *) b c a. Semigroupoid m => m b c -> m a b -> m a c . t' a -> (t :. t') >>> a forall (t :: * -> *) a. Pointable t => a -> t a point instance Monoidal (<--) (-->) (:*:) (:*:) t => Lowerable (->) (TT Covariant Covariant t) where lower :: t <::> t' ~> t' lower :: (<::>) t t' a -> t' a lower (TT (t :. t') >>> a x) = ((t :. t') >>> a) -> t' a forall (t :: * -> *) a. Extractable t => t a -> a extract (t :. t') >>> a x instance Covariant (->) (->) t => Hoistable (->) (TT Covariant Covariant t) where (/|\) :: t' ~> v -> (t <::> t' ~> t <::> v) t' ~> v f /|\ :: (t' ~> v) -> (t <::> t') ~> (t <::> v) /|\ TT (t :. t') >>> a x = ((t :. v) >>> a) -> TT Covariant Covariant t v a forall k k k k (ct :: k) (ct' :: k) (t :: k -> *) (t' :: k -> k) (a :: k). ((t :. t') >>> a) -> TT ct ct' t t' a TT (((t :. v) >>> a) -> TT Covariant Covariant t v a) -> ((t :. v) >>> a) -> TT Covariant Covariant t v a forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b) <---- t' a -> v a t' ~> v f (t' a -> v a) -> ((t :. t') >>> a) -> (t :. v) >>> a forall (source :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) a b. Covariant source target t => source a b -> target (t a) (t b) <-|- (t :. t') >>> a x