Parameterized.hs

{-# OPTIONS -fglasgow-exts #-}
-----------------------------------------------------------------------------
-- |
-- Module      :  Control.Comonad.Parameterized
-- Copyright   :  (C) 2008 Edward Kmett
-- License     :  BSD-style (see the file LICENSE)
--
-- Maintainer  :  Edward Kmett <ekmett@gmail.com>
-- Stability   :  experimental
-- Portability :  portable
--
----------------------------------------------------------------------------
module Control.Comonad.Parameterized 
	( module Control.Bifunctor.Biff
	, module Control.Functor.Pointed.Parameterized
	, module Control.Comonad
	, PComonad(..)
	, pcoaugment
	) where

import Control.Arrow ((&&&))
import Control.Comonad
import Control.Bifunctor.Fix
import Control.Bifunctor.Biff
import Control.Functor.Pointed.Parameterized
import Control.Morphism.Ana

class PCopointed f => PComonad f where
	pextend :: (f b c -> a) -> f b c -> f a c

{- Parameterized comonad laws:

> pextend pextract = id
> pextract . pextend g = g
> pextend (g . pextend j) = pextend g . pextend j
> pextract . second g = pextract 
> second g . pextend (j . second g) = pextend j . second g 

-}

{-# RULES
"pextend pextract" 		pextend pextract = id
"pextract . pextend g" 		forall g. pextract . pextend g = g
"pextract . bimap id g" 	forall g. pextract . bimap id g = pextract
"bimap _ _ . pextract" 		forall j g. bimap id g . pextend (j . bimap id g) = pextend j . bimap id g
 #-}

pcoaugment :: PComonad f => ((FixB f a -> f b (FixB f a)) -> FixB f b) -> (FixB f a -> b) -> FixB f b
pcoaugment g k = g (pextend (k . InB) . outB)

instance PCopointed f => Copointed (FixB f) where
        extract = pextract . outB
	
instance PComonad f => Comonad (FixB f) where
        extend k w = pcoaugment (flip biana w) k

instance Functor f => PCopointed (BiffB (,) Identity f) where
        pextract = runIdentity . fst . runBiffB
	
instance Functor f => PComonad (BiffB (,) Identity f) where
        pextend f = BiffB . (Identity . f &&& snd . runBiffB)