{-# LANGUAGE UndecidableInstances #-}
module Pandora.Paradigm.Schemes.TT where

import Pandora.Core.Functor (type (:.), type (:=), type (~>))
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.Functor.Bivariant ((<->))
import Pandora.Pattern.Transformer.Liftable (Liftable (lift))
import Pandora.Pattern.Transformer.Lowerable (Lowerable (lower))
import Pandora.Pattern.Transformer.Hoistable (Hoistable ((/|\)))
import Pandora.Paradigm.Controlflow.Effect.Interpreted (Interpreted (Primary, run, unite, (!), (||=)))
import Pandora.Paradigm.Primary.Algebraic.Exponential (type (<--), type (-->))
import Pandora.Paradigm.Primary.Algebraic.Product ((:*:) ((:*:)))
import Pandora.Paradigm.Primary.Algebraic.Sum ((:+:), bitraverse_sum)
import Pandora.Paradigm.Primary.Algebraic.One (One (One))
import Pandora.Paradigm.Primary.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 3 <::>, >::>, <::<, >::<

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 :: * -> * -> *) (t :: * -> *) a.
Interpreted m t =>
m (t a) (Primary t a)
! ((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
. (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' a -> t (t' a))
-> ((<::>) t t' b -> t (t' b))
-> ((<::>) t t' a :*: (<::>) t t' b)
-> t (t' a) :*: t (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, Covariant (->) (->) t', Semimonoidal (-->) (:*:) (:*:) t', Monoidal (-->) (-->) (:*:) (:*:) t, Monoidal (-->) (-->) (:*:) (:*:) t') => Monoidal (-->) (-->) (:*:) (:*:) (t <::> t') where
	unit :: Proxy (:*:) -> (Unit (:*:) --> a) --> (<::>) t t' a
unit Proxy (:*:)
_ = ((One --> a) -> (<::>) t t' a)
-> Straight (->) (One --> a) ((<::>) t t' a)
forall (v :: * -> * -> *) a e. v a e -> Straight v a e
Straight (((One --> a) -> (<::>) t t' a)
 -> Straight (->) (One --> a) ((<::>) t t' a))
-> ((One --> a) -> (<::>) t t' a)
-> Straight (->) (One --> a) ((<::>) t t' a)
forall (m :: * -> * -> *) (t :: * -> *) a.
Interpreted m t =>
m (t a) (Primary t 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) -> (<::>) t t' a)
-> ((One --> a) -> (t :. t') := a) -> (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)
-> ((One --> a) -> t' a) -> (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) -> ((One --> a) -> a) -> (One --> a) -> t' a
forall (m :: * -> * -> *) b c a.
Semigroupoid m =>
m b c -> m a b -> m a c
. ((One -> a) -> One -> a
forall (m :: * -> * -> *) (t :: * -> *) a.
Interpreted m t =>
m (t a) (Primary t a)
! One
One) ((One -> a) -> a) -> ((One --> a) -> One -> a) -> (One --> a) -> a
forall (m :: * -> * -> *) b c a.
Semigroupoid m =>
m b c -> m a b -> m a c
. (One --> a) -> One -> a
forall (m :: * -> * -> *) (t :: * -> *) a.
Interpreted m t =>
m (t a) (Primary t a)
run

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 :: * -> * -> *) (t :: * -> *) a.
Interpreted m t =>
m (t a) (Primary t a)
! \(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 :: * -> * -> *) (t :: * -> *) a.
Interpreted m t =>
m (t a) (Primary t a)
! (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 :: * -> * -> *) (t :: * -> *) a.
Interpreted m t =>
m (t a) (Primary t a)
! \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 :: * -> * -> *) (t :: * -> *) a.
Interpreted m t =>
m (t a) (Primary t a)
! \(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') := b) -> (<::>) t t' b)
-> (((t :. t') := a) :*: ((t :. t') := b))
-> (<::>) t t' a :*: (<::>) t 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)
<-> ((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') := a) :*: ((t :. t') := b))
 -> (<::>) t t' a :*: (<::>) t t' b)
-> (t (t' a :*: t' b) -> ((t :. t') := a) :*: ((t :. t') := b))
-> t (t' a :*: 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 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)
!) (t (t' a :*: t' b) -> (<::>) t t' a :*: (<::>) t 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 -> One --> a)
-> Flip (->) (One --> a) ((<::>) t t' a)
forall (v :: * -> * -> *) a e. v e a -> Flip v a e
Flip (((<::>) t t' a -> One --> a)
 -> Flip (->) (One --> a) ((<::>) t t' a))
-> ((<::>) t t' a -> One --> a)
-> Flip (->) (One --> a) ((<::>) t t' a)
forall (m :: * -> * -> *) (t :: * -> *) a.
Interpreted m t =>
m (t a) (Primary t a)
! \(TT (t :. t') := a
x) -> (One -> a) -> 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 :: * -> * -> *) (t :: * -> *) a.
Interpreted m t =>
m (t a) (Primary t a)
! ((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 :: * -> * -> *) (t :: * -> *) a.
Interpreted m t =>
m (t a) (Primary t a)
! (\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 :: * -> * -> *) (t :: * -> *) a.
Interpreted m t =>
m (t a) (Primary t a)
! 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