{-# 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