module Pandora.Paradigm.Primary.Transformer.Backwards 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.Algebraic ((-<*>-))
import Pandora.Paradigm.Primary.Algebraic.Product ((:*:) ((:*:)))
import Pandora.Paradigm.Primary.Algebraic.Exponential (type (<--), (%))
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 Backwards t a = Backwards (t a)

instance Covariant (->) (->) t => Covariant (->) (->) (Backwards t) where
	a -> b
f -<$>- :: (a -> b) -> Backwards t a -> Backwards t b
-<$>- Backwards t a
x = t b -> Backwards t b
forall k (t :: k -> *) (a :: k). t a -> Backwards t a
Backwards (t b -> Backwards t b) -> t b -> Backwards 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

-- TODO: check that effects evaluation goes in opposite order
instance (Semimonoidal (->) (:*:) (:*:) t, Covariant (->) (->) t) => Semimonoidal (->) (:*:) (:*:) (Backwards t) where
	multiply :: (Backwards t a :*: Backwards t b) -> Backwards t (a :*: b)
multiply (Backwards t a
x :*: Backwards t b
y) = t (a :*: b) -> Backwards t (a :*: b)
forall k (t :: k -> *) (a :: k). t a -> Backwards t a
Backwards (t (a :*: b) -> Backwards t (a :*: b))
-> t (a :*: b) -> Backwards t (a :*: b)
forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b)
#
		(a -> b -> a :*: b
forall s a. s -> a -> s :*: a
(:*:) (a -> b -> a :*: b) -> b -> a -> a :*: b
forall a b c. (a -> b -> c) -> b -> a -> c
%) (b -> a -> a :*: b) -> t b -> t (a -> a :*: b)
forall (source :: * -> * -> *) (target :: * -> * -> *)
       (t :: * -> *) a b.
Covariant source target t =>
source a b -> target (t a) (t b)
-<$>- t b
y t (a -> a :*: b) -> t a -> t (a :*: b)
forall (t :: * -> *) a b.
(Covariant (->) (->) t, Semimonoidal (->) (:*:) (:*:) t) =>
t (a -> b) -> t a -> t b
-<*>- t a
x

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

instance Traversable (->) (->) t => Traversable (->) (->) (Backwards t) where
	a -> u b
f <<- :: (a -> u b) -> Backwards t a -> u (Backwards t b)
<<- Backwards t a
x = t b -> Backwards t b
forall k (t :: k -> *) (a :: k). t a -> Backwards t a
Backwards (t b -> Backwards t b) -> u (t b) -> u (Backwards 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 a -> 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 (->) (->) (Backwards t) where
	a -> Backwards t b
f -<< :: (a -> Backwards t b) -> u a -> Backwards t (u b)
-<< u a
x = t (u b) -> Backwards t (u b)
forall k (t :: k -> *) (a :: k). t a -> Backwards t a
Backwards (t (u b) -> Backwards t (u b)) -> t (u b) -> Backwards t (u b)
forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b)
$ Backwards t b -> t b
forall (t :: * -> *) a. Interpreted t => t a -> Primary t a
run (Backwards t b -> t b) -> (a -> Backwards t b) -> a -> t b
forall (m :: * -> * -> *) b c a.
Semigroupoid m =>
m b c -> m a b -> m a c
. a -> Backwards 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 (->) (->) (Backwards t) where
	a -> b
f ->$<- :: (a -> b) -> Backwards t b -> Backwards t a
->$<- Backwards t b
x = t a -> Backwards t a
forall k (t :: k -> *) (a :: k). t a -> Backwards t a
Backwards (t a -> Backwards t a) -> t a -> Backwards 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 (Backwards t) where
	type Primary (Backwards t) a = t a
	run :: Backwards t a -> Primary (Backwards t) a
run ~(Backwards t a
x) = t a
Primary (Backwards t) a
x
	unite :: Primary (Backwards t) a -> Backwards t a
unite = Primary (Backwards t) a -> Backwards t a
forall k (t :: k -> *) (a :: k). t a -> Backwards t a
Backwards

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

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

instance Hoistable Backwards where
	u ~> v
f /|\ :: (u ~> v) -> Backwards u ~> Backwards v
/|\ Backwards u a
x = v a -> Backwards v a
forall k (t :: k -> *) (a :: k). t a -> Backwards t a
Backwards (v a -> Backwards v a) -> v a -> Backwards 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