{-# LANGUAGE TypeFamilies, TypeOperators, GADTs, FlexibleContexts #-}
-----------------------------------------------------------------------------
-- |
-- Module      :  Data.Category.Product
-- Copyright   :  (c) Sjoerd Visscher 2010
-- License     :  BSD-style (see the file LICENSE)
--
-- Maintainer  :  sjoerd@w3future.com
-- Stability   :  experimental
-- Portability :  non-portable
-----------------------------------------------------------------------------
module Data.Category.Product where

import Prelude hiding ((.), id, Functor)

import Data.Category
import Data.Category.Functor


data (:*:) :: (* -> * -> *) -> (* -> * -> *) -> * -> * -> * where
  (:**:) :: c1 a1 b1 -> c2 a2 b2 -> (:*:) c1 c2 (a1, a2) (b1, b2)

-- | The product category of category @c1@ and @c2@.
instance (Category c1, Category c2) => Category (c1 :*: c2) where
  
  data Obj (c1 :*: c2) a where
    ProdO :: Obj c1 a1 -> Obj c2 a2 -> Obj (c1 :*: c2) (a1, a2)
    
  src (a1 :**: a2)            = ProdO (src a1) (src a2)
  tgt (a1 :**: a2)            = ProdO (tgt a1) (tgt a2)
  
  id (ProdO x1 x2)            = id x1 :**: id x2
  
  (a1 :**: a2) . (b1 :**: b2) = (a1 . b1) :**: (a2 . b2)


  
  
    
data Proj1 (c1 :: * -> * -> *) (c2 :: * -> * -> *) = Proj1
type instance Dom (Proj1 c1 c2) = c1 :*: c2
type instance Cod (Proj1 c1 c2) = c1
type instance Proj1 c1 c2 :% (a1, a2) = a1
instance Functor (Proj1 c1 c2) where 
  Proj1 %% ProdO x1 _ = x1
  Proj1 % (f1 :**: _) = f1

data Proj2 (c1 :: * -> * -> *) (c2 :: * -> * -> *) = Proj2
type instance Dom (Proj2 c1 c2) = c1 :*: c2
type instance Cod (Proj2 c1 c2) = c2
type instance Proj2 c1 c2 :% (a1, a2) = a2
instance Functor (Proj2 c1 c2) where 
  Proj2 %% ProdO _ x2 = x2
  Proj2 % (_ :**: f2) = f2

data f1 :***: f2 where 
  (:***:) :: (Functor f1, Functor f2, Category (Cod f1), Category (Cod f2)) => f1 -> f2 -> f1 :***: f2
type instance Dom (f1 :***: f2) = Dom f1 :*: Dom f2
type instance Cod (f1 :***: f2) = Cod f1 :*: Cod f2
type instance (f1 :***: f2) :% (a1, a2) = (f1 :% a1, f2 :% a2)
instance Functor (f1 :***: f2) where 
  (g1 :***: g2) %% ProdO x1 x2 = ProdO (g1 %% x1) (g2 %% x2)
  (g1 :***: g2) % (f1 :**: f2) = (g1 % f1) :**: (g2 % f2)
  
data Hom (~>) where
  Hom :: Category (~>) => Hom (~>)
type instance Dom (Hom (~>)) = Op (~>) :*: (~>)
type instance Cod (Hom (~>)) = (->)
type instance Hom (~>) :% (a, b) = a ~> b
instance Functor (Hom (~>)) where 
  (%%) = undefined
  Hom % (Op g1 :**: g2) = \f -> g2 . f . g1
  
data DiagProd :: (* -> * -> *) -> * where 
  DiagProd :: Category (~>) => DiagProd (~>)
type instance Dom (DiagProd (~>)) = (~>)
type instance Cod (DiagProd (~>)) = (~>) :*: (~>)
type instance DiagProd (~>) :% a = (a, a)
instance Functor (DiagProd (~>)) where 
  DiagProd %% x = ProdO x x
  DiagProd % f = f :**: f