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

import Pandora.Core.Functor (type (:.), 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, 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.Adjoint (Adjoint ((-|), (--|), (|-), (|--)))
import Pandora.Pattern.Transformer.Liftable (Liftable (lift))
import Pandora.Pattern.Transformer.Lowerable (Lowerable (lower))
import Pandora.Paradigm.Algebraic.Exponential (type (<--), type (-->))
import Pandora.Paradigm.Algebraic.Product ((:*:) ((:*:)))
import Pandora.Paradigm.Algebraic.Sum ((:+:) (Option, Adoption))
import Pandora.Paradigm.Algebraic.One (One (One))
import Pandora.Paradigm.Algebraic (point, extract, (<-||-))
import Pandora.Pattern.Morphism.Flip (Flip (Flip))
import Pandora.Pattern.Morphism.Straight (Straight (Straight))

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 (Semigroupoid m, Covariant m m t, Covariant (Betwixt (Betwixt m m) m) m t, Covariant (Betwixt m (Betwixt m m)) (Betwixt (Betwixt m m) m) u, Covariant m (Betwixt m (Betwixt 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 :: * -> * -> *) (target :: * -> * -> *)
       (t :: * -> *) (v :: * -> *) (u :: * -> *) a b.
(Covariant source target t,
 Covariant source (Betwixt source (Betwixt source target)) v,
 Covariant
   (Betwixt source (Betwixt source target))
   (Betwixt (Betwixt source target) target)
   u,
 Covariant (Betwixt (Betwixt source target) target) target t) =>
source a b -> target (t (u (v a))) (t (u (v b)))
(<-|-|-|-) m a b
f)

instance (Adjoint (->) (->) t' t, Bindable (->) u) => 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)
<-- \(TUT (t :. (u :. t')) >>> a
x :*: TUT (t :. (u :. t')) >>> b
y) -> ((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 ((((\a
r -> (b -> a :*: b)
-> ((t :. (u :. t')) >>> b) -> (t :. (u :. t')) >>> (a :*: b)
forall (source :: * -> * -> *) (target :: * -> * -> *)
       (t :: * -> *) (v :: * -> *) (u :: * -> *) a b.
(Covariant source target t,
 Covariant source (Betwixt source (Betwixt source target)) v,
 Covariant
   (Betwixt source (Betwixt source target))
   (Betwixt (Betwixt source target) target)
   u,
 Covariant (Betwixt (Betwixt source target) target) target t) =>
source a b -> target (t (u (v a))) (t (u (v b)))
(<-|-|-|-) (a
r a -> b -> a :*: b
forall s a. s -> a -> s :*: a
:*:) (t :. (u :. t')) >>> b
y) (a -> (t :. (u :. t')) >>> (a :*: b)) -> t' a -> u (t' (a :*: 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' (a :*: b))) -> u (t' a) -> u (t' (a :*: b))
forall (source :: * -> * -> *) (t :: * -> *) a b.
Bindable source t =>
source a (t b) -> source (t a) (t b)
=<<) (u (t' a) -> u (t' (a :*: b)))
-> ((t :. (u :. t')) >>> a) -> (t :. (u :. t')) >>> (a :*: b)
forall (source :: * -> * -> *) (target :: * -> * -> *)
       (t :: * -> *) a b.
Covariant source target t =>
source a b -> target (t a) (t b)
<-|- (t :. (u :. t')) >>> a
x)

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')) >>> a) :*: (>>>>>>>>) (t <:<.>:> t') u b)
-> (>>>>>>>>) (t <:<.>:> t') u a :*: (>>>>>>>>) (t <:<.>:> 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 :. t')) >>> a) :*: (>>>>>>>>) (t <:<.>:> t') u b)
 -> (>>>>>>>>) (t <:<.>:> t') u a :*: (>>>>>>>>) (t <:<.>:> t') u b)
-> ((>>>>>>>>) (t <:<.>:> t') u (a :*: b)
    -> ((t :. (u :. t')) >>> a) :*: (>>>>>>>>) (t <:<.>:> t') u 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
. (((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')) >>> a) :*: ((t :. (u :. t')) >>> b))
-> ((t :. (u :. t')) >>> a) :*: (>>>>>>>>) (t <:<.>:> t') u b
forall (source :: * -> * -> *) (target :: * -> * -> *)
       (t :: * -> *) a b.
Covariant source target t =>
source a b -> target (t a) (t b)
<-|-) ((((t :. (u :. t')) >>> a) :*: ((t :. (u :. t')) >>> b))
 -> ((t :. (u :. t')) >>> a) :*: (>>>>>>>>) (t <:<.>:> t') u b)
-> ((>>>>>>>>) (t <:<.>:> t') u (a :*: b)
    -> ((t :. (u :. t')) >>> a) :*: ((t :. (u :. t')) >>> b))
-> (>>>>>>>>) (t <:<.>:> t') u (a :*: b)
-> ((t :. (u :. t')) >>> 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 (m :: * -> * -> *) (t :: * -> *) a.
Interpreted m t =>
(m < t a) < Primary t a
<~) (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 (m :: * -> * -> *) (t :: * -> *) a.
Interpreted m t =>
(m < t a) < Primary t a
<~) (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 :: * -> * -> *) (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))
(<-|-|-) @_ @(->) (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 (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 -> Straight (->) One a)
-> Flip (->) (Straight (->) One a) ((>>>>>>>>) (t <:<.>:> t') u a)
forall (v :: * -> * -> *) a e. v e a -> Flip v a e
Flip (((>>>>>>>>) (t <:<.>:> t') u a -> Straight (->) One a)
 -> Flip (->) (Straight (->) One a) ((>>>>>>>>) (t <:<.>:> t') u a))
-> ((>>>>>>>>) (t <:<.>:> t') u a -> Straight (->) One a)
-> Flip (->) (Straight (->) 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 -> a) -> Straight (->) One a
forall (v :: * -> * -> *) a e. v a e -> Straight v a e
Straight (\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)

-- TODO: generalize on (->) and (:*:)
instance {-# OVERLAPS #-} (Covariant (->) (->) u, Semimonoidal (-->) (:*:) (:+:) u) => Semimonoidal (-->) (:*:) (:+:) ((->) s <:<.>:> (:*:) s >>>>>>>> u) where
 mult :: ((>>>>>>>>) ((->) s <:<.>:> (:*:) s) u a
 :*: (>>>>>>>>) ((->) s <:<.>:> (:*:) s) u b)
--> (>>>>>>>>) ((->) s <:<.>:> (:*:) s) u (a :+: b)
mult = (((>>>>>>>>) ((->) s <:<.>:> (:*:) s) u a
  :*: (>>>>>>>>) ((->) s <:<.>:> (:*:) s) u b)
 -> (>>>>>>>>) ((->) s <:<.>:> (:*:) s) u (a :+: b))
-> ((>>>>>>>>) ((->) s <:<.>:> (:*:) s) u a
    :*: (>>>>>>>>) ((->) s <:<.>:> (:*:) s) u b)
   --> (>>>>>>>>) ((->) s <:<.>:> (:*:) s) u (a :+: b)
forall (v :: * -> * -> *) a e. v a e -> Straight v a e
Straight ((((>>>>>>>>) ((->) s <:<.>:> (:*:) s) u a
   :*: (>>>>>>>>) ((->) s <:<.>:> (:*:) s) u b)
  -> (>>>>>>>>) ((->) s <:<.>:> (:*:) s) u (a :+: b))
 -> ((>>>>>>>>) ((->) s <:<.>:> (:*:) s) u a
     :*: (>>>>>>>>) ((->) s <:<.>:> (:*:) s) u b)
    --> (>>>>>>>>) ((->) s <:<.>:> (:*:) s) u (a :+: b))
-> (((>>>>>>>>) ((->) s <:<.>:> (:*:) s) u a
     :*: (>>>>>>>>) ((->) s <:<.>:> (:*:) s) u b)
    -> (>>>>>>>>) ((->) s <:<.>:> (:*:) s) u (a :+: b))
-> ((>>>>>>>>) ((->) s <:<.>:> (:*:) s) u a
    :*: (>>>>>>>>) ((->) s <:<.>:> (:*:) s) u b)
   --> (>>>>>>>>) ((->) s <:<.>:> (:*:) s) u (a :+: b)
forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b)
<-- \(TUT ((->) s :. (u :. (:*:) s)) >>> a
x :*: TUT ((->) s :. (u :. (:*:) s)) >>> b
y) -> (((->) s :. (u :. (:*:) s)) >>> (a :+: b))
-> (>>>>>>>>) ((->) s <:<.>:> (:*:) s) 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
	((((->) s :. (u :. (:*:) s)) >>> (a :+: b))
 -> (>>>>>>>>) ((->) s <:<.>:> (:*:) s) u (a :+: b))
-> (((->) s :. (u :. (:*:) s)) >>> (a :+: b))
-> (>>>>>>>>) ((->) s <:<.>:> (:*:) s) u (a :+: b)
forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b)
<------ ((s :*: a) :+: (s :*: b)) -> s :*: (a :+: b)
forall s a b. ((s :*: a) :+: (s :*: b)) -> s :*: (a :+: b)
product_over_sum
		(((s :*: a) :+: (s :*: b)) -> s :*: (a :+: b))
-> (s -> u ((s :*: a) :+: (s :*: b)))
-> ((->) s :. (u :. (:*:) s)) >>> (a :+: 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))
<-|-|- 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 (s :*: a) :*: u (s :*: b)) --> u ((s :*: a) :+: (s :*: b)))
-> (s -> u (s :*: a) :*: u (s :*: b))
-> s
-> u ((s :*: a) :+: (s :*: 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 @(-->) @(:*:) @(:*:)
				(((((->) s :. (u :. (:*:) s)) >>> a)
  :*: (((->) s :. (u :. (:*:) s)) >>> b))
 --> (s -> u (s :*: a) :*: u (s :*: b)))
-> ((((->) s :. (u :. (:*:) s)) >>> a)
    :*: (((->) s :. (u :. (:*:) s)) >>> b))
-> s
-> u (s :*: a) :*: u (s :*: b)
forall (m :: * -> * -> *) (t :: * -> *) a.
Interpreted m t =>
(m < t a) < Primary t a
<~~~ ((->) s :. (u :. (:*:) s)) >>> a
x (((->) s :. (u :. (:*:) s)) >>> a)
-> (((->) s :. (u :. (:*:) s)) >>> b)
-> (((->) s :. (u :. (:*:) s)) >>> a)
   :*: (((->) s :. (u :. (:*:) s)) >>> b)
forall s a. s -> a -> s :*: a
:*: ((->) s :. (u :. (:*:) s)) >>> b
y

product_over_sum :: (s :*: a) :+: (s :*: b) -> s :*: (a :+: b)
product_over_sum :: ((s :*: a) :+: (s :*: b)) -> s :*: (a :+: b)
product_over_sum (Option (s
s :*: a
x)) = s
s s -> (a :+: b) -> s :*: (a :+: b)
forall s a. s -> a -> s :*: a
:*: a -> a :+: b
forall o a. o -> o :+: a
Option a
x
product_over_sum (Adoption (s
s :*: b
y)) = s
s s -> (a :+: b) -> s :*: (a :+: b)
forall s a. s -> a -> s :*: a
:*: b -> a :+: b
forall o a. a -> o :+: a
Adoption b
y

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 (Bindable (->) u, Monoidal (-->) (-->) (:*:) (:*:) u, Adjoint (->) (->) t' t) => Monoidal (-->) (-->) (:*:) (:*:) (t <:<.>:> t' >>>>>>>> u) where
	unit :: Proxy (:*:) -> (Unit (:*:) --> a) --> (>>>>>>>>) (t <:<.>:> t') u a
unit Proxy (:*:)
_ = (Straight (->) One a -> (>>>>>>>>) (t <:<.>:> t') u a)
-> Straight
     (->) (Straight (->) One a) ((>>>>>>>>) (t <:<.>:> t') u a)
forall (v :: * -> * -> *) a e. v a e -> Straight v a e
Straight ((Straight (->) One a -> (>>>>>>>>) (t <:<.>:> t') u a)
 -> Straight
      (->) (Straight (->) One a) ((>>>>>>>>) (t <:<.>:> t') u a))
-> (Straight (->) One a -> (>>>>>>>>) (t <:<.>:> t') u a)
-> Straight
     (->) (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)
-> (Straight (->) One a -> (t :. (u :. t')) >>> a)
-> Straight (->) 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)
-> (Straight (->) One a -> a)
-> Straight (->) One a
-> (t :. (u :. 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 (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