{-# LANGUAGE UndecidableInstances #-}

module Pandora.Paradigm.Controlflow.Effect.Transformer.Monadic (Monadic (..), (:>) (..)) where

import Pandora.Core.Appliable ((!))
import Pandora.Core.Functor (type (~>))
import Pandora.Pattern.Morphism.Straight (Straight (Straight))
import Pandora.Pattern.Semigroupoid ((.))
import Pandora.Pattern.Category (($))
import Pandora.Pattern.Functor.Covariant (Covariant ((<$>)))
import Pandora.Pattern.Functor.Semimonoidal (Semimonoidal (mult))
import Pandora.Pattern.Functor.Monoidal (Monoidal (unit))
import Pandora.Pattern.Functor.Distributive (Distributive ((-<<)))
import Pandora.Pattern.Functor.Traversable (Traversable ((<<-)))
import Pandora.Pattern.Functor.Bindable (Bindable ((=<<)))
import Pandora.Pattern.Functor.Extendable (Extendable ((<<=)))
import Pandora.Pattern.Functor.Monad (Monad)
import Pandora.Pattern.Transformer.Liftable (Liftable (lift))
import Pandora.Pattern.Transformer.Hoistable (Hoistable ((/|\)))
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 (Pointable, point)
import Pandora.Paradigm.Controlflow.Effect.Interpreted (Schematic, Interpreted (Primary, run, unite))

class Interpreted (->) t => Monadic t where
	{-# MINIMAL wrap #-}
	wrap :: Pointable u => t ~> t :> u

infixr 3 :>
newtype (:>) t u a = TM { (:>) t u a -> Schematic Monad t u a
tm :: Schematic Monad t u a }

instance Covariant (->) (->) (Schematic Monad t u) => Covariant (->) (->) (t :> u) where
	a -> b
f <$> :: (a -> b) -> (:>) t u a -> (:>) t u b
<$> TM Schematic Monad t u a
x = Schematic Monad t u b -> (:>) t u b
forall (t :: * -> *) (u :: * -> *) a.
Schematic Monad t u a -> (:>) t u a
TM (Schematic Monad t u b -> (:>) t u b)
-> Schematic Monad t u b -> (:>) t u b
forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b)
$ a -> b
f (a -> b) -> Schematic Monad t u a -> Schematic Monad t u b
forall (source :: * -> * -> *) (target :: * -> * -> *)
       (t :: * -> *) a b.
Covariant source target t =>
source a b -> target (t a) (t b)
<$> Schematic Monad t u a
x

instance Semimonoidal (-->) (:*:) (:*:) (Schematic Monad t u) => Semimonoidal (-->) (:*:) (:*:) (t :> u) where
	mult :: ((:>) t u a :*: (:>) t u b) --> (:>) t u (a :*: b)
mult = (((:>) t u a :*: (:>) t u b) -> (:>) t u (a :*: b))
-> ((:>) t u a :*: (:>) t u b) --> (:>) t u (a :*: b)
forall (v :: * -> * -> *) a e. v a e -> Straight v a e
Straight ((((:>) t u a :*: (:>) t u b) -> (:>) t u (a :*: b))
 -> ((:>) t u a :*: (:>) t u b) --> (:>) t u (a :*: b))
-> (((:>) t u a :*: (:>) t u b) -> (:>) t u (a :*: b))
-> ((:>) t u a :*: (:>) t u b) --> (:>) t u (a :*: b)
forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b)
$ \(TM Schematic Monad t u a
f :*: TM Schematic Monad t u b
x) -> Schematic Monad t u (a :*: b) -> (:>) t u (a :*: b)
forall (t :: * -> *) (u :: * -> *) a.
Schematic Monad t u a -> (:>) t u a
TM (Schematic Monad t u (a :*: b) -> (:>) t u (a :*: b))
-> Schematic Monad t u (a :*: b) -> (:>) t u (a :*: b)
forall (m :: * -> * -> *) a b. Category m => m (m a b) (m 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))
forall (t :: * -> *) a b.
Semimonoidal (-->) (:*:) (:*:) t =>
(t a :*: t b) --> t (a :*: b)
mult @(-->) @(:*:) @(:*:) ((Schematic Monad t u a :*: Schematic Monad t u b)
 --> Schematic Monad t u (a :*: b))
-> (Schematic Monad t u a :*: Schematic Monad t u b)
-> Schematic Monad t u (a :*: b)
forall k k k k (m :: k -> k -> *) (a :: k) (b :: k)
       (n :: k -> k -> *) (c :: k) (d :: k).
Appliable m a b n c d =>
m a b -> n c d
! Schematic Monad t u a
f Schematic Monad t u a
-> Schematic Monad t u b
-> Schematic Monad t u a :*: Schematic Monad t u b
forall s a. s -> a -> s :*: a
:*: Schematic Monad t u b
x

instance Monoidal (-->) (->) (:*:) (:*:) (Schematic Monad t u) => Monoidal (-->) (->) (:*:) (:*:) (t :> u) where
	unit :: Proxy (:*:) -> (Unit (:*:) -> a) --> (:>) t u a
unit Proxy (:*:)
_ = ((One -> a) -> (:>) t u a) -> Straight (->) (One -> a) ((:>) t u a)
forall (v :: * -> * -> *) a e. v a e -> Straight v a e
Straight (((One -> a) -> (:>) t u a)
 -> Straight (->) (One -> a) ((:>) t u a))
-> ((One -> a) -> (:>) t u a)
-> Straight (->) (One -> a) ((:>) t u a)
forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b)
$ Schematic Monad t u a -> (:>) t u a
forall (t :: * -> *) (u :: * -> *) a.
Schematic Monad t u a -> (:>) t u a
TM (Schematic Monad t u a -> (:>) t u a)
-> ((One -> a) -> Schematic Monad t u a)
-> (One -> a)
-> (:>) t u a
forall (m :: * -> * -> *) b c a.
Semigroupoid m =>
m b c -> m a b -> m a c
. a -> Schematic Monad t u a
forall (t :: * -> *) a. Pointable t => a -> t a
point (a -> Schematic Monad t u a)
-> ((One -> a) -> a) -> (One -> a) -> Schematic Monad t u a
forall (m :: * -> * -> *) b c a.
Semigroupoid m =>
m b c -> m a b -> m a c
. ((One -> a) -> One -> a
forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b)
$ One
One)

instance Traversable (->) (->) (Schematic Monad t u) => Traversable (->) (->) (t :> u) where
	a -> u b
f <<- :: (a -> u b) -> (:>) t u a -> u ((:>) t u b)
<<- TM Schematic Monad t u a
x = Schematic Monad t u b -> (:>) t u b
forall (t :: * -> *) (u :: * -> *) a.
Schematic Monad t u a -> (:>) t u a
TM (Schematic Monad t u b -> (:>) t u b)
-> u (Schematic Monad t u b) -> u ((:>) t u 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) -> Schematic Monad t u a -> u (Schematic Monad t u b)
forall (source :: * -> * -> *) (target :: * -> * -> *)
       (t :: * -> *) (u :: * -> *) a b.
(Traversable source target t, Covariant source target u,
 Monoidal (Straight source) target (:*:) (:*:) u) =>
source a (u b) -> target (t a) (u (t b))
<<- Schematic Monad t u a
x

instance Distributive (->) (->) (Schematic Monad t u) => Distributive (->) (->) (t :> u) where
	a -> (:>) t u b
f -<< :: (a -> (:>) t u b) -> u a -> (:>) t u (u b)
-<< u a
x = Schematic Monad t u (u b) -> (:>) t u (u b)
forall (t :: * -> *) (u :: * -> *) a.
Schematic Monad t u a -> (:>) t u a
TM (Schematic Monad t u (u b) -> (:>) t u (u b))
-> Schematic Monad t u (u b) -> (:>) t u (u b)
forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b)
$ (:>) t u b -> Schematic Monad t u b
forall (t :: * -> *) (u :: * -> *) a.
(:>) t u a -> Schematic Monad t u a
tm ((:>) t u b -> Schematic Monad t u b)
-> (a -> (:>) t u b) -> a -> Schematic Monad t u b
forall (m :: * -> * -> *) b c a.
Semigroupoid m =>
m b c -> m a b -> m a c
. a -> (:>) t u b
f (a -> Schematic Monad t u b) -> u a -> Schematic Monad t u (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 Bindable (->) (Schematic Monad t u) => Bindable (->) (t :> u) where
	a -> (:>) t u b
f =<< :: (a -> (:>) t u b) -> (:>) t u a -> (:>) t u b
=<< TM Schematic Monad t u a
x = Schematic Monad t u b -> (:>) t u b
forall (t :: * -> *) (u :: * -> *) a.
Schematic Monad t u a -> (:>) t u a
TM (Schematic Monad t u b -> (:>) t u b)
-> Schematic Monad t u b -> (:>) t u b
forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b)
$ (:>) t u b -> Schematic Monad t u b
forall (t :: * -> *) (u :: * -> *) a.
(:>) t u a -> Schematic Monad t u a
tm ((:>) t u b -> Schematic Monad t u b)
-> (a -> (:>) t u b) -> a -> Schematic Monad t u b
forall (m :: * -> * -> *) b c a.
Semigroupoid m =>
m b c -> m a b -> m a c
. a -> (:>) t u b
f (a -> Schematic Monad t u b)
-> Schematic Monad t u a -> Schematic Monad t u b
forall (source :: * -> * -> *) (t :: * -> *) a b.
Bindable source t =>
source a (t b) -> source (t a) (t b)
=<< Schematic Monad t u a
x

instance Extendable (->) (Schematic Monad t u) => Extendable (->) (t :> u) where
	(:>) t u a -> b
f <<= :: ((:>) t u a -> b) -> (:>) t u a -> (:>) t u b
<<= TM Schematic Monad t u a
x = Schematic Monad t u b -> (:>) t u b
forall (t :: * -> *) (u :: * -> *) a.
Schematic Monad t u a -> (:>) t u a
TM (Schematic Monad t u b -> (:>) t u b)
-> Schematic Monad t u b -> (:>) t u b
forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b)
$ (:>) t u a -> b
f ((:>) t u a -> b)
-> (Schematic Monad t u a -> (:>) t u a)
-> Schematic Monad t u a
-> b
forall (m :: * -> * -> *) b c a.
Semigroupoid m =>
m b c -> m a b -> m a c
. Schematic Monad t u a -> (:>) t u a
forall (t :: * -> *) (u :: * -> *) a.
Schematic Monad t u a -> (:>) t u a
TM (Schematic Monad t u a -> b)
-> Schematic Monad t u a -> Schematic Monad t u b
forall (source :: * -> * -> *) (t :: * -> *) a b.
Extendable source t =>
source (t a) b -> source (t a) (t b)
<<= Schematic Monad t u a
x

instance (Covariant (->) (->) (Schematic Monad t u), Monoidal (-->) (->) (:*:) (:*:) (Schematic Monad t u), Bindable (->) (t :> u)) => Monad (->) (t :> u) where

instance Liftable (->) (Schematic Monad t) => Liftable (->) ((:>) t) where
	lift :: u a -> (:>) t u a
lift = Schematic Monad t u a -> (:>) t u a
forall (t :: * -> *) (u :: * -> *) a.
Schematic Monad t u a -> (:>) t u a
TM (Schematic Monad t u a -> (:>) t u a)
-> (u a -> Schematic Monad t u a) -> u a -> (:>) t u a
forall (m :: * -> * -> *) b c a.
Semigroupoid m =>
m b c -> m a b -> m a c
. u a -> Schematic Monad t u a
forall (cat :: * -> * -> *) (t :: (* -> *) -> * -> *) (u :: * -> *)
       a.
(Liftable cat t, Covariant cat cat u) =>
cat (u a) (t u a)
lift

instance Hoistable (Schematic Monad t) => Hoistable ((:>) t) where
	u ~> v
f /|\ :: (u ~> v) -> (t :> u) ~> (t :> v)
/|\ TM Schematic Monad t u a
x = Schematic Monad t v a -> (:>) t v a
forall (t :: * -> *) (u :: * -> *) a.
Schematic Monad t u a -> (:>) t u a
TM (Schematic Monad t v a -> (:>) t v a)
-> Schematic Monad t v a -> (:>) t v a
forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b)
$ u ~> v
f (u ~> v) -> Schematic Monad t u a -> Schematic Monad t v a
forall k (t :: (* -> *) -> k -> *) (u :: * -> *) (v :: * -> *).
(Hoistable t, Covariant (->) (->) u) =>
(u ~> v) -> t u ~> t v
/|\ Schematic Monad t u a
x

instance (Interpreted (->) (Schematic Monad t u)) => Interpreted (->) (t :> u) where
	type Primary (t :> u) a = Primary (Schematic Monad t u) a
	run :: (:>) t u a -> Primary (t :> u) a
run ~(TM Schematic Monad t u a
x) = Schematic Monad t u a -> Primary (Schematic Monad t u) a
forall (m :: * -> * -> *) (t :: * -> *) a.
Interpreted m t =>
m (t a) (Primary t a)
run Schematic Monad t u a
x
	unite :: Primary (t :> u) a -> (:>) t u a
unite = Schematic Monad t u a -> (:>) t u a
forall (t :: * -> *) (u :: * -> *) a.
Schematic Monad t u a -> (:>) t u a
TM (Schematic Monad t u a -> (:>) t u a)
-> (Primary (Schematic Monad t u) a -> Schematic Monad t u a)
-> Primary (Schematic Monad t u) a
-> (:>) t u a
forall (m :: * -> * -> *) b c a.
Semigroupoid m =>
m b c -> m a b -> m a c
. Primary (Schematic Monad t u) a -> Schematic Monad t u a
forall (m :: * -> * -> *) (t :: * -> *) a.
Interpreted m t =>
m (Primary t a) (t a)
unite