{-# LANGUAGE UndecidableInstances #-}

module Pandora.Paradigm.Primary.Transformer.Reverse where

import Pandora.Pattern.Semigroupoid ((.))
import Pandora.Pattern.Category (($), (#))
import Pandora.Pattern.Functor.Covariant (Covariant ((-<$>-)))
import Pandora.Pattern.Functor.Contravariant (Contravariant ((->$<-)))
import Pandora.Pattern.Functor.Semimonoidal (Semimonoidal (multiply))
import Pandora.Pattern.Functor.Monoidal (Monoidal (unit))
import Pandora.Pattern.Functor.Traversable (Traversable ((<<-)))
import Pandora.Pattern.Functor.Distributive (Distributive ((-<<)))
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.Primary.Transformer.Backwards (Backwards (Backwards))
import Pandora.Paradigm.Primary.Algebraic.Exponential (type (<--))
import Pandora.Paradigm.Primary.Algebraic.Product ((:*:) ((:*:)))
import Pandora.Paradigm.Primary.Algebraic.One (One (One))
import Pandora.Paradigm.Primary.Algebraic (point, extract)
import Pandora.Paradigm.Primary.Transformer.Flip (Flip (Flip))
import Pandora.Paradigm.Controlflow.Effect.Interpreted (Interpreted (Primary, run, unite))

newtype Reverse t a = Reverse (t a)

instance Covariant (->) (->) t => Covariant (->) (->) (Reverse t) where
	a -> b
f -<$>- :: (a -> b) -> Reverse t a -> Reverse t b
-<$>- Reverse t a
x = t b -> Reverse t b
forall k (t :: k -> *) (a :: k). t a -> Reverse t a
Reverse (t b -> Reverse t b) -> t b -> Reverse t b
forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b)
# a -> b
f (a -> b) -> t a -> t b
forall (source :: * -> * -> *) (target :: * -> * -> *)
       (t :: * -> *) a b.
Covariant source target t =>
source a b -> target (t a) (t b)
-<$>- t a
x

instance (Semimonoidal (->) (:*:) (:*:) t, Covariant (->) (->) t) => Semimonoidal (->) (:*:) (:*:) (Reverse t) where
	multiply :: (Reverse t a :*: Reverse t b) -> Reverse t (a :*: b)
multiply (Reverse t a
x :*: Reverse t b
y) = t (a :*: b) -> Reverse t (a :*: b)
forall k (t :: k -> *) (a :: k). t a -> Reverse t a
Reverse (t (a :*: b) -> Reverse t (a :*: b))
-> t (a :*: b) -> Reverse t (a :*: b)
forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b)
# (t a :*: t b) -> t (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))
multiply (t a
x t a -> t b -> t a :*: t b
forall s a. s -> a -> s :*: a
:*: t b
y)

instance (Covariant (->) (->) t, Monoidal (->) (->) (:*:) (:*:) t) => Monoidal (->) (->) (:*:) (:*:) (Reverse t) where
	unit :: Proxy (:*:) -> (Unit (:*:) -> a) -> Reverse t a
unit Proxy (:*:)
_ Unit (:*:) -> a
f = t a -> Reverse t a
forall k (t :: k -> *) (a :: k). t a -> Reverse t a
Reverse (t a -> Reverse t a) -> (a -> t a) -> a -> Reverse t a
forall (m :: * -> * -> *) b c a.
Semigroupoid m =>
m b c -> m a b -> m a c
. a -> t a
forall (t :: * -> *) a.
Monoidal (->) (->) (:*:) (:*:) t =>
a -> t a
point (a -> Reverse t a) -> a -> Reverse t a
forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b)
$ Unit (:*:) -> a
f One
Unit (:*:)
One

instance (Semimonoidal (<--) (:*:) (:*:) t, Covariant (->) (->) t) => Semimonoidal (<--) (:*:) (:*:) (Reverse t) where
	multiply :: (Reverse t a :*: Reverse t b) <-- Reverse t (a :*: b)
multiply = (Reverse t (a :*: b) -> Reverse t a :*: Reverse t b)
-> (Reverse t a :*: Reverse t b) <-- Reverse t (a :*: b)
forall (v :: * -> * -> *) a e. v e a -> Flip v a e
Flip ((Reverse t (a :*: b) -> Reverse t a :*: Reverse t b)
 -> (Reverse t a :*: Reverse t b) <-- Reverse t (a :*: b))
-> (Reverse t (a :*: b) -> Reverse t a :*: Reverse t b)
-> (Reverse t a :*: Reverse t b) <-- Reverse t (a :*: b)
forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b)
$ \(Reverse t (a :*: b)
x) -> 
		let Flip t (a :*: b) -> t a :*: t b
f = 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)
multiply @(<--) @(:*:) @(:*:) in
		(t a -> Reverse t a
forall k (t :: k -> *) (a :: k). t a -> Reverse t a
Reverse (t a -> Reverse t a)
-> (t b -> Reverse t b)
-> (t a :*: t b)
-> Reverse t a :*: Reverse 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 b -> Reverse t b
forall k (t :: k -> *) (a :: k). t a -> Reverse t a
Reverse) ((t a :*: t b) -> Reverse t a :*: Reverse t b)
-> (t a :*: t b) -> Reverse t a :*: Reverse t b
forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b)
$ t (a :*: b) -> t a :*: t b
forall a b. t (a :*: b) -> t a :*: t b
f t (a :*: b)
x

instance (Covariant (->) (->) t, Monoidal (<--) (->) (:*:) (:*:) t) => Monoidal (<--) (->) (:*:) (:*:) (Reverse t) where
	unit :: Proxy (:*:) -> (Unit (:*:) -> a) <-- Reverse t a
unit Proxy (:*:)
_ = (Reverse t a -> One -> a) -> Flip (->) (One -> a) (Reverse t a)
forall (v :: * -> * -> *) a e. v e a -> Flip v a e
Flip ((Reverse t a -> One -> a) -> Flip (->) (One -> a) (Reverse t a))
-> (Reverse t a -> One -> a) -> Flip (->) (One -> a) (Reverse t a)
forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b)
$ \(Reverse t a
x) -> (\One
_ -> t a -> a
forall (t :: * -> *) a. Extractable_ t => t a -> a
extract t a
x)

instance Traversable (->) (->) t => Traversable (->) (->) (Reverse t) where
	a -> u b
f <<- :: (a -> u b) -> Reverse t a -> u (Reverse t b)
<<- Reverse t a
x = t b -> Reverse t b
forall k (t :: k -> *) (a :: k). t a -> Reverse t a
Reverse (t b -> Reverse t b) -> u (t b) -> u (Reverse t b)
forall (source :: * -> * -> *) (target :: * -> * -> *)
       (t :: * -> *) a b.
Covariant source target t =>
source a b -> target (t a) (t b)
-<$>- Backwards u (t b) -> Primary (Backwards u) (t b)
forall (t :: * -> *) a. Interpreted t => t a -> Primary t a
run (u b -> Backwards u b
forall k (t :: k -> *) (a :: k). t a -> Backwards t a
Backwards (u b -> Backwards u b) -> (a -> u b) -> a -> Backwards u b
forall (m :: * -> * -> *) b c a.
Semigroupoid m =>
m b c -> m a b -> m a c
. a -> u b
f (a -> Backwards u b) -> t a -> Backwards u (t b)
forall (source :: * -> * -> *) (target :: * -> * -> *)
       (t :: * -> *) (u :: * -> *) a b.
(Traversable source target t, Covariant source target u,
 Monoidal source target (:*:) (:*:) u) =>
source a (u b) -> target (t a) (u (t b))
<<- t a
x)

instance Distributive (->) (->) t => Distributive (->) (->) (Reverse t) where
	a -> Reverse t b
f -<< :: (a -> Reverse t b) -> u a -> Reverse t (u b)
-<< u a
x = t (u b) -> Reverse t (u b)
forall k (t :: k -> *) (a :: k). t a -> Reverse t a
Reverse (t (u b) -> Reverse t (u b)) -> t (u b) -> Reverse t (u b)
forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b)
$ Reverse t b -> t b
forall (t :: * -> *) a. Interpreted t => t a -> Primary t a
run (Reverse t b -> t b) -> (a -> Reverse t b) -> a -> t b
forall (m :: * -> * -> *) b c a.
Semigroupoid m =>
m b c -> m a b -> m a c
. a -> Reverse t b
f (a -> t b) -> u a -> t (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
x

instance Contravariant (->) (->) t => Contravariant (->) (->) (Reverse t) where
	a -> b
f ->$<- :: (a -> b) -> Reverse t b -> Reverse t a
->$<- Reverse t b
x = t a -> Reverse t a
forall k (t :: k -> *) (a :: k). t a -> Reverse t a
Reverse (t a -> Reverse t a) -> t a -> Reverse t a
forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b)
# a -> b
f (a -> b) -> t b -> t a
forall (source :: * -> * -> *) (target :: * -> * -> *)
       (t :: * -> *) a b.
Contravariant source target t =>
source a b -> target (t b) (t a)
->$<- t b
x

instance Interpreted (Reverse t) where
	type Primary (Reverse t) a = t a
	run :: Reverse t a -> Primary (Reverse t) a
run ~(Reverse t a
x) = t a
Primary (Reverse t) a
x
	unite :: Primary (Reverse t) a -> Reverse t a
unite = Primary (Reverse t) a -> Reverse t a
forall k (t :: k -> *) (a :: k). t a -> Reverse t a
Reverse

instance Liftable (->) Reverse where
	lift :: u a -> Reverse u a
lift = u a -> Reverse u a
forall k (t :: k -> *) (a :: k). t a -> Reverse t a
Reverse

instance Lowerable (->) Reverse where
	lower :: Reverse u a -> u a
lower = Reverse u a -> u a
forall (t :: * -> *) a. Interpreted t => t a -> Primary t a
run

instance Hoistable Reverse where
	u ~> v
f /|\ :: (u ~> v) -> Reverse u ~> Reverse v
/|\ Reverse u a
x = v a -> Reverse v a
forall k (t :: k -> *) (a :: k). t a -> Reverse t a
Reverse (v a -> Reverse v a) -> v a -> Reverse v a
forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b)
# u a -> v a
u ~> v
f u a
x