{-# Language FlexibleInstances, MultiParamTypeClasses, ScopedTypeVariables,
             TypeFamilies, TypeOperators, UndecidableInstances #-}

-- | A /natural transformation/ is a concept from category theory for a mapping between two functors and their objects
-- that preserves a naturality condition. In Haskell the naturality condition boils down to parametricity, so a
-- natural transformation between two functors @f@ and @g@ is represented as
--
-- > type NaturalTransformation f g = ∀a. f a → g a
--
-- This type appears in several Haskell libraries, most obviously in
-- [natural-transformations](https://hackage.haskell.org/package/natural-transformation). There are times, however,
-- when we crave more control. Sometimes what we want to do depends on which type @a@ is hiding in that @f a@ we're
-- given. Sometimes, in other words, we need an /unnatural/ transformation.
--
-- This means we have to abandon parametricity for ad-hoc polymorphism, and that means type classes. There are two
-- steps to defining a transformation:
--
-- * an instance of the base class 'Transformation' declares the two functors being mapped, much like a function type
--   signature,
-- * while the actual mapping of values is performed by an arbitrary number of instances of the method '$', a bit like
--   multiple equation clauses that make up a single function definition.
--
-- The module is meant to be imported qualified.

module Transformation where

import Data.Functor.Product (Product(Pair))
import Data.Functor.Sum (Sum(InL, InR))
import Data.Kind (Type)
import qualified Rank2

import Prelude hiding (($))

-- | A 'Transformation', natural or not, maps one functor to another.
class Transformation t where
   type Domain t :: Type -> Type
   type Codomain t :: Type -> Type

-- | An unnatural 'Transformation' can behave differently at different points.
class Transformation t => At t x where
   -- | Apply the transformation @t@ at type @x@ to map 'Domain' to the 'Codomain' functor.
   ($) :: t -> Domain t x -> Codomain t x
   infixr 0 $

-- | Alphabetical synonym for '$'
apply :: t `At` x => t -> Domain t x -> Codomain t x
apply = ($)

-- | Composition of two transformations
data Compose t u = Compose t u

instance (Transformation t, Transformation u, Domain t ~ Codomain u) => Transformation (Compose t u) where
   type Domain (Compose t u) = Domain u
   type Codomain (Compose t u) = Codomain t

instance (t `At` x, u `At` x, Domain t ~ Codomain u) => Compose t u `At` x where
   Compose t u $ x =  t $ (u $ x)

instance Transformation (Rank2.Arrow p q x) where
   type Domain (Rank2.Arrow p q x) = p
   type Codomain (Rank2.Arrow p q x) = q

instance Rank2.Arrow p q x `At` x where
   ($) = Rank2.apply

instance (Transformation t1, Transformation t2, Domain t1 ~ Domain t2) => Transformation (t1, t2) where
   type Domain (t1, t2) = Domain t1
   type Codomain (t1, t2) = Product (Codomain t1) (Codomain t2)

instance (t `At` x, u `At` x, Domain t ~ Domain u) => (t, u) `At` x where
   (t, u) $ x = Pair (t $ x) (u $ x)

instance (Transformation t1, Transformation t2, Domain t1 ~ Domain t2) => Transformation (Either t1 t2) where
   type Domain (Either t1 t2) = Domain t1
   type Codomain (Either t1 t2) = Sum (Codomain t1) (Codomain t2)

instance (t `At` x, u `At` x, Domain t ~ Domain u) => Either t u `At` x where
   Left t $ x = InL (t $ x)
   Right t $ x = InR (t $ x)