module Pandora.Paradigm.Schemes.TUT where

import Pandora.Core.Functor (type (:.), type (:=), type (~>))
import Pandora.Pattern.Category (identity, (.), ($))
import Pandora.Pattern.Functor ((<*+>))
import Pandora.Pattern.Functor.Covariant (Covariant ((<$>), (<$$$>)), Covariant_ ((-<$>-)), (-<$$$>-))
import Pandora.Pattern.Functor.Contravariant (Contravariant)
import Pandora.Pattern.Functor.Applicative (Applicative ((<*>)))
import Pandora.Pattern.Functor.Alternative (Alternative ((<+>)))
import Pandora.Pattern.Functor.Avoidable (Avoidable (empty))
import Pandora.Pattern.Functor.Pointable (Pointable (point))
import Pandora.Pattern.Functor.Extractable (Extractable (extract))
import Pandora.Pattern.Functor.Bindable (Bindable ((>>=), ($>>=)))
import Pandora.Pattern.Functor.Extendable (Extendable ((=>>), ($=>>)))
import Pandora.Pattern.Functor.Distributive (Distributive ((>>-)))
import Pandora.Pattern.Functor.Adjoint (Adjoint ((-|), (|-)))
import Pandora.Pattern.Transformer.Liftable (Liftable (lift))
import Pandora.Pattern.Transformer.Lowerable (Lowerable (lower))
import Pandora.Paradigm.Controlflow.Effect.Interpreted (Interpreted (Primary, run, unite))

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 (Covariant t, Covariant t', Covariant u) => Covariant (t <:<.>:> t' := u) where
	a -> b
f <$> :: (a -> b) -> (:=) (t <:<.>:> t') u a -> (:=) (t <:<.>:> t') u b
<$> TUT (t :. (u :. t')) := 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)
$ a -> b
f (a -> b) -> ((t :. (u :. t')) := a) -> (t :. (u :. t')) := b
forall (t :: * -> *) (u :: * -> *) (v :: * -> *) a b.
(Covariant t, Covariant u, Covariant v) =>
(a -> b) -> ((t :. (u :. v)) := a) -> (t :. (u :. v)) := b
<$$$> (t :. (u :. t')) := a
x

instance (Covariant_ t (->) (->), Covariant_ t' (->) (->), Covariant_ u (->) (->)) => Covariant_ (t <:<.>:> t' := u) (->) (->)where
	a -> b
f -<$>- :: (a -> b) -> (:=) (t <:<.>:> t') u a -> (:=) (t <:<.>:> t') u b
-<$>- TUT (t :. (u :. t')) := 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)
$ a -> b
f (a -> b) -> ((t :. (u :. t')) := a) -> (t :. (u :. t')) := b
forall (t :: * -> *) (u :: * -> *) (v :: * -> *)
       (source :: * -> * -> *) (target :: * -> * -> *) a b.
(Covariant_ u source source, Covariant_ t source target,
 Covariant_ v target target) =>
source a b -> target (v (t (u a))) (v (t (u b)))
-<$$$>- (t :. (u :. t')) := a
x

instance (Covariant t, Covariant t', Adjoint t' t, Bindable u) => Applicative (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)
$ (u (t' (a -> b)) -> (t' (a -> b) -> u (t' b)) -> u (t' b)
forall (t :: * -> *) a b. Bindable t => t a -> (a -> t b) -> t b
>>= (t' (a -> b) -> ((a -> b) -> (t :. (u :. t')) := b) -> u (t' b)
forall (t :: * -> *) (u :: * -> *) a b.
Adjoint t u =>
t a -> (a -> u b) -> b
|- ((a -> b) -> ((t :. (u :. t')) := a) -> (t :. (u :. t')) := b
forall (t :: * -> *) (u :: * -> *) (v :: * -> *) a b.
(Covariant t, Covariant u, Covariant v) =>
(a -> b) -> ((t :. (u :. v)) := a) -> (t :. (u :. v)) := b
<$$$> (:=) (t <:<.>:> t') u a -> Primary ((t <:<.>:> t') := u) a
forall (t :: * -> *) a. Interpreted t => t a -> Primary t a
run (:=) (t <:<.>:> t') u a
x))) (u (t' (a -> b)) -> u (t' b))
-> t (u (t' (a -> b))) -> (t :. (u :. t')) := b
forall (t :: * -> *) a b. Covariant t => (a -> b) -> t a -> t b
<$> (:=) (t <:<.>:> t') u (a -> b)
-> Primary ((t <:<.>:> t') := u) (a -> b)
forall (t :: * -> *) a. Interpreted t => t a -> Primary t a
run (:=) (t <:<.>:> t') u (a -> b)
f

instance (Applicative t, Covariant t', Alternative u) => Alternative (t <:<.>:> t' := u) where
	(:=) (t <:<.>:> t') u a
x <+> :: (:=) (t <:<.>:> t') u a
-> (:=) (t <:<.>:> t') u a -> (:=) (t <:<.>:> t') u a
<+> (:=) (t <:<.>:> t') u a
y = ((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 a
forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b)
$ (:=) (t <:<.>:> t') u a -> Primary ((t <:<.>:> t') := u) a
forall (t :: * -> *) a. Interpreted t => t a -> Primary t a
run (:=) (t <:<.>:> t') u a
x ((t :. (u :. t')) := a)
-> ((t :. (u :. t')) := a) -> (t :. (u :. t')) := a
forall (t :: * -> *) (u :: * -> *) a.
(Applicative t, Alternative u) =>
((t :. u) := a) -> ((t :. u) := a) -> (t :. u) := a
<*+> (:=) (t <:<.>:> t') u a -> Primary ((t <:<.>:> t') := u) a
forall (t :: * -> *) a. Interpreted t => t a -> Primary t a
run (:=) (t <:<.>:> t') u a
y

instance (Pointable t (->), Applicative t, Covariant t', Avoidable u) => Avoidable (t <:<.>:> t' := u) where
	empty :: (:=) (t <:<.>:> t') u a
empty = ((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 a
forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b)
$ (:.) u t' a -> (t :. (u :. t')) := a
forall (t :: * -> *) (source :: * -> * -> *) a.
Pointable t source =>
source a (t a)
point (:.) u t' a
forall (t :: * -> *) a. Avoidable t => t a
empty

instance (Covariant_ t (->) (->), Covariant_ t' (->) (->), Pointable u (->), Adjoint t' t) => Pointable (t <:<.>:> t' := u) (->) where
	point :: a -> (:=) (t <:<.>:> t') u a
point = ((t :. (u :. t')) := a) -> (:=) (t <:<.>:> t') u a
forall (t :: * -> *) a. Interpreted t => Primary t a -> t a
unite (((t :. (u :. t')) := a) -> (:=) (t <:<.>:> t') u a)
-> (a -> (t :. (u :. t')) := a) -> a -> (:=) (t <:<.>:> t') u a
forall (m :: * -> * -> *) b c a.
Category m =>
m b c -> m a b -> m a c
. (a -> (t' a -> u (t' a)) -> (t :. (u :. t')) := a
forall (t :: * -> *) (u :: * -> *) a b.
Adjoint t u =>
a -> (t a -> b) -> u b
-| t' a -> u (t' a)
forall (t :: * -> *) (source :: * -> * -> *) a.
Pointable t source =>
source a (t a)
point)

instance (Covariant t, Covariant t', Adjoint t' t, Bindable u) => Bindable (t <:<.>:> t' := u) where
	(:=) (t <:<.>:> t') u a
x >>= :: (:=) (t <:<.>:> t') u a
-> (a -> (:=) (t <:<.>:> t') u b) -> (:=) (t <:<.>:> t') u b
>>= a -> (:=) (t <:<.>:> t') u b
f = ((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 -> Primary ((t <:<.>:> t') := u) a
forall (t :: * -> *) a. Interpreted t => t a -> Primary t a
run (:=) (t <:<.>:> t') u a
x ((t :. (u :. t')) := a)
-> (t' a -> u (t' b)) -> (t :. (u :. t')) := b
forall (t :: * -> *) (u :: * -> *) a b.
(Bindable t, Covariant u) =>
((u :. t) := a) -> (a -> t b) -> (u :. t) := b
$>>= (t' a -> (a -> (t :. (u :. t')) := b) -> u (t' b)
forall (t :: * -> *) (u :: * -> *) a b.
Adjoint t u =>
t a -> (a -> u b) -> b
|- (:=) (t <:<.>:> t') u b -> (t :. (u :. t')) := b
forall (t :: * -> *) a. Interpreted t => 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.
Category m =>
m b c -> m a b -> m a c
. a -> (:=) (t <:<.>:> t') u b
f)

instance (Covariant t', Covariant t, Adjoint t' t, Extendable u) => Extendable (t' <:<.>:> t := u) where
	(:=) (t' <:<.>:> t) u a
x =>> :: (:=) (t' <:<.>:> t) u a
-> ((:=) (t' <:<.>:> t) u a -> b) -> (:=) (t' <:<.>:> t) u b
=>> (:=) (t' <:<.>:> t) u a -> b
f = ((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 -> Primary ((t' <:<.>:> t) := u) a
forall (t :: * -> *) a. Interpreted t => t a -> Primary t a
run (:=) (t' <:<.>:> t) u a
x ((t' :. (u :. t)) := a)
-> (u (t a) -> t b) -> (t' :. (u :. t)) := b
forall (t :: * -> *) (u :: * -> *) a b.
(Extendable t, Covariant u) =>
((u :. t) := a) -> (t a -> b) -> (u :. t) := b
$=>> (u (t a) -> (((t' :. (u :. t)) := a) -> b) -> t b
forall (t :: * -> *) (u :: * -> *) a b.
Adjoint t u =>
a -> (t a -> b) -> u 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.
Category m =>
m b c -> m a b -> m a c
. ((t' :. (u :. t)) := a) -> (:=) (t' <:<.>:> t) u a
forall (t :: * -> *) a. Interpreted t => Primary t a -> t a
unite)

instance (Covariant_ t (->) (->), Covariant_ t' (->) (->), Adjoint t t', Extractable u (->)) => Extractable (t <:<.>:> t' := u) (->) where
	extract :: (:=) (t <:<.>:> t') u a -> a
extract = (t (u (t' a)) -> (u (t' a) -> t' a) -> a
forall (t :: * -> *) (u :: * -> *) a b.
Adjoint t u =>
t a -> (a -> u b) -> b
|- u (t' a) -> t' a
forall (t :: * -> *) (source :: * -> * -> *) a.
Extractable t source =>
source (t a) a
extract) (t (u (t' a)) -> a)
-> ((:=) (t <:<.>:> t') u a -> t (u (t' a)))
-> (:=) (t <:<.>:> t') u a
-> a
forall (m :: * -> * -> *) b c a.
Category m =>
m b c -> m a b -> m a c
. (:=) (t <:<.>:> t') u a -> t (u (t' a))
forall (t :: * -> *) a. Interpreted t => t a -> Primary t a
run

instance (forall u . Covariant u, 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)
$ u a
x u a -> (a -> t (t' a)) -> (t :. (u :. t')) := a
forall (t :: * -> *) (u :: * -> *) a b.
(Distributive t, Covariant u) =>
u a -> (a -> t b) -> (t :. u) := b
>>- (a -> (t' a -> t' a) -> t (t' a)
forall (t :: * -> *) (u :: * -> *) a b.
Adjoint t u =>
a -> (t a -> b) -> u b
-| t' a -> t' a
forall (m :: * -> * -> *) a. Category m => m a a
identity)

instance (forall u . Covariant u, 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) = (t :. (u :. t')) := a
x ((t :. (u :. t')) := a) -> ((:.) u t' a -> t' (u a)) -> u a
forall (t :: * -> *) (u :: * -> *) a b.
Adjoint t u =>
t a -> (a -> u b) -> b
|- ((:.) u t' a -> (t' a -> t' a) -> t' (u a)
forall (t :: * -> *) (u :: * -> *) a b.
(Distributive t, Covariant u) =>
u a -> (a -> t b) -> (t :. u) := b
>>- t' a -> t' a
forall (m :: * -> * -> *) a. Category m => m a a
identity)