{-# LANGUAGE GADTs, Rank2Types, ScopedTypeVariables, TypeOperators,
  FlexibleContexts, CPP #-}
--------------------------------------------------------------------------------
-- |
-- Module      :  Data.Comp.Algebra
-- Copyright   :  (c) 2010-2011 Patrick Bahr, Tom Hvitved
-- License     :  BSD3
-- Maintainer  :  Patrick Bahr <paba@diku.dk>
-- Stability   :  experimental
-- Portability :  non-portable (GHC Extensions)
--
-- This module defines the notion of algebras and catamorphisms, and their
-- generalizations to e.g. monadic versions and other (co)recursion schemes.
--
--------------------------------------------------------------------------------

module Data.Comp.Algebra (
      -- * Algebras & Catamorphisms
      Alg,
      free,
      cata,
      cata',
      appCxt,
      
      -- * Monadic Algebras & Catamorphisms
      AlgM,
      algM,
      freeM,
      cataM,
      cataM',

      -- * Term Homomorphisms
      CxtFun,
      SigFun,
      Hom,
      appHom,
      appHom',
      compHom,
      appSigFun,
      appSigFun',
      compSigFun,
      compSigFunHom,
      compHomSigFun,
      compAlgSigFun,
      hom,
      compAlg,
      compCoalg,
      compCVCoalg,

      -- * Monadic Term Homomorphisms
      CxtFunM,
      SigFunM,
      HomM,
      SigFunMD,
      HomMD,
      sigFunM,
      hom',
      appHomM,
      appHomM',
      homM,
      homMD,
      appSigFunM,
      appSigFunM',
      appSigFunMD,
      compHomM,
      compSigFunM,
      compSigFunHomM,
      compHomSigFunM,
      compAlgSigFunM,
      compAlgM,
      compAlgM',

      -- * Coalgebras & Anamorphisms
      Coalg,
      ana,
      ana',
      CoalgM,
      anaM,

      -- * R-Algebras & Paramorphisms
      RAlg,
      para,
      RAlgM,
      paraM,

      -- * R-Coalgebras & Apomorphisms
      RCoalg,
      apo,
      RCoalgM,
      apoM,

      -- * CV-Algebras & Histomorphisms
      CVAlg,
      histo,
      CVAlgM,
      histoM,

      -- * CV-Coalgebras & Futumorphisms
      CVCoalg,
      futu,
      CVCoalg',
      futu',
      CVCoalgM,
      futuM
    ) where

import Data.Comp.Term
import Data.Comp.Ops
import Data.Traversable
import Control.Monad hiding (sequence, mapM)

import Prelude hiding (sequence, mapM)



{-| This type represents an algebra over a functor @f@ and carrier
@a@. -}

type Alg f a = f a -> a

{-| Construct a catamorphism for contexts over @f@ with holes of type @a@, from
  the given algebra. -}
free :: forall f h a b . (Functor f) => Alg f b -> (a -> b) -> Cxt h f a -> b
free f g = run
    where run :: Cxt h f a -> b
          run (Hole x) = g x
          run (Term t) = f (fmap run t)

{-| Construct a catamorphism from the given algebra. -}
cata :: forall f a . (Functor f) => Alg f a -> Term f -> a 
{-# NOINLINE [1] cata #-}
-- cata f = free f undefined
-- the above definition is safe since terms do not contain holes
--
-- a direct implementation:
cata f = run 
    where run :: Term f -> a
          run  = f . fmap run . unTerm


{-| A generalisation of 'cata' from terms over @f@ to contexts over @f@, where
  the holes have the type of the algebra carrier. -}
cata' :: (Functor f) => Alg f a -> Cxt h f a -> a
{-# INLINE cata' #-}
cata' f = free f id


{-| This function applies a whole context into another context. -}

appCxt :: (Functor f) => Context f (Cxt h f a) -> Cxt h f a
-- appCxt = cata' Term
appCxt (Hole x) = x
appCxt (Term t) = Term (fmap appCxt t)



{-| This type represents a monadic algebra. It is similar to 'Alg' but
the return type is monadic.  -}

type AlgM m f a = f a -> m a 

{-| Convert a monadic algebra into an ordinary algebra with a monadic
  carrier. -}
algM :: (Traversable f, Monad m) => AlgM m f a -> Alg f (m a)
algM f x = sequence x >>= f

{-| Construct a monadic catamorphism for contexts over @f@ with holes of type
  @a@, from the given monadic algebra. -}
freeM :: forall h f a m b. (Traversable f, Monad m) =>
               AlgM m f b -> (a -> m b) -> Cxt h f a -> m b
-- freeM alg var = free (algM alg) var
freeM algm var = run
    where run :: Cxt h f a -> m b
          run (Hole x) = var x
          run (Term t) = algm =<< mapM run t

{-| Construct a monadic catamorphism from the given monadic algebra. -}
cataM :: forall f m a. (Traversable f, Monad m) => AlgM m f a -> Term f -> m a 
{-# NOINLINE [1] cataM #-}
-- cataM = cata . algM
cataM algm = run
    where run :: Term f -> m a
          run = algm <=< mapM run . unTerm

{-| A generalisation of 'cataM' from terms over @f@ to contexts over @f@, where
  the holes have the type of the monadic algebra carrier. -}
cataM' :: forall h f a m . (Traversable f, Monad m)
            => AlgM m f a -> Cxt h f a -> m a
{-# NOINLINE [1] cataM' #-}
-- cataM' f = free (\x -> sequence x >>= f) return
cataM' f = run
    where run :: Cxt h f a -> m a
          run (Hole x) = return x
          run (Term t) = f =<< mapM run t


{-| This type represents a context function. -}
type CxtFun f g = forall a h. Cxt h f a -> Cxt h g a

{-| This type represents a signature function.-}
type SigFun f g = forall a. f a -> g a

{-| This type represents a term homomorphism. -}
type Hom f g = SigFun f (Context g)

{-| This function applies the given term homomorphism to a
term/context. -}
appHom :: forall f g . (Functor f, Functor g) => Hom f g -> CxtFun f g
{-# NOINLINE [1] appHom #-}
-- Note: The rank 2 type polymorphism is not necessary. Alternatively, also the type
-- (Functor f, Functor g) => (f (Cxt h g b) -> Context g (Cxt h g b)) -> Cxt h f b -> Cxt h g b
-- would achieve the same. The given type is chosen for clarity.
appHom f = run where
    run :: CxtFun f g
    run (Hole x) = Hole x
    run (Term t) = appCxt (f (fmap run t))

-- | Apply a term homomorphism recursively to a term/context. This is
-- a top-down variant of 'appHom'.
appHom' :: forall f g . (Functor g) => Hom f g -> CxtFun f g
{-# NOINLINE [1] appHom' #-}
-- Note: The rank 2 type polymorphism is not necessary. Alternatively, also the type
-- (Functor f, Functor g) => (f (Cxt h g b) -> Context g (Cxt h g b)) -> Cxt h f b -> Cxt h g b
-- would achieve the same. The given type is chosen for clarity.
appHom' f = run where
    run :: CxtFun f g
    run (Hole x) = Hole x
    run (Term t) = appCxt  (fmap run (f t))

{-| Compose two term homomorphisms. -}
compHom :: (Functor g, Functor h) => Hom g h -> Hom f g -> Hom f h
-- Note: The rank 2 type polymorphism is not necessary. Alternatively, also the type
-- (Functor f, Functor g) => (f (Cxt h g b) -> Context g (Cxt h g b))
-- -> (a -> Cxt h f b) -> a -> Cxt h g b
-- would achieve the same. The given type is chosen for clarity.
compHom f g = appHom f . g

{-| Compose an algebra with a term homomorphism to get a new algebra. -}
compAlg :: (Functor g) => Alg g a -> Hom f g -> Alg f a
compAlg alg talg = cata' alg . talg

{-| Compose a term homomorphism with a coalgebra to get a cv-coalgebra. -}
compCoalg :: Hom f g -> Coalg f a -> CVCoalg' g a
compCoalg hom coa = hom . coa

{-| Compose a term homomorphism with a cv-coalgebra to get a new cv-coalgebra.
 -}
compCVCoalg :: (Functor f, Functor g)
  => Hom f g -> CVCoalg' f a -> CVCoalg' g a
compCVCoalg hom coa = appHom hom . coa


{-| This function applies a signature function to the given context. -}
appSigFun :: (Functor f) => SigFun f g -> CxtFun f g
{-# NOINLINE [1] appSigFun #-}
appSigFun f = run
    where run (Term t) = Term $ f $ fmap run t
          run (Hole x) = Hole x
-- implementation via term homomorphisms:
--  appSigFun f = appHom_ $ hom f

-- | This function applies a signature function to the given
-- context. This is a top-down variant of 'appSigFun'.
appSigFun' :: (Functor g) => SigFun f g -> CxtFun f g
{-# NOINLINE [1] appSigFun' #-}
appSigFun' f = run
    where run (Term t) = Term $ fmap run  $ f t
          run (Hole x) = Hole x


{-| This function composes two signature functions. -}
compSigFun :: SigFun g h -> SigFun f g -> SigFun f h
compSigFun f g = f . g

-- | This function composes a signature function with a term
-- homomorphism.
compSigFunHom :: (Functor g) => SigFun g h -> Hom f g -> Hom f h
compSigFunHom f g = appSigFun f . g

-- | This function composes a term homomorphism with a signature function.
compHomSigFun :: Hom g h -> SigFun f g -> Hom f h
compHomSigFun f g = f . g

-- | This function composes an algebra with a signature function.
compAlgSigFun :: Alg g a -> SigFun f g -> Alg f a
compAlgSigFun f g = f . g


-- | Lifts the given signature function to the canonical term
-- homomorphism.
hom :: (Functor g) => SigFun f g -> Hom f g
hom f = simpCxt . f

{-|
  This type represents a monadic context function.
-}
type CxtFunM m f g = forall a h. Cxt h f a -> m (Cxt h g a)

{-| This type represents a monadic signature function. -}

type SigFunM m f g = forall a. f a -> m (g a)

{-| This type represents a monadic signature function.  It is similar
to 'SigFunM' but has monadic values also in the domain. -}
type SigFunMD m f g = forall a. f (m a) -> m (g a)

{-| This type represents a monadic term homomorphism.  -}
type HomM m f g = SigFunM m f (Context g)

{-| This type represents a monadic term homomorphism. It is similar to
'HomM' but has monadic values also in the domain. -}
type HomMD m f g = SigFunMD m f (Context g)


{-| Lift the given signature function to a monadic signature function. Note that
  term homomorphisms are instances of signature functions. Hence this function
  also applies to term homomorphisms. -}
sigFunM :: (Monad m) => SigFun f g -> SigFunM m f g
sigFunM f = return . f

{-| Lift the give monadic signature function to a monadic term homomorphism. -}
hom' :: (Functor f, Functor g, Monad m) => SigFunM m f g -> HomM m f g
hom' f = liftM  (Term . fmap Hole) . f


{-| Lift the given signature function to a monadic term homomorphism. -}
homM :: (Functor g, Monad m) => SigFunM m f g -> HomM m f g
homM f = liftM simpCxt . f


{-| Apply a monadic term homomorphism recursively to a term/context. -}
appHomM :: forall f g m . (Traversable f, Functor g, Monad m)
         => HomM m f g -> CxtFunM m f g
{-# NOINLINE [1] appHomM #-}
appHomM f = run
    where run :: Cxt h f a -> m (Cxt h g a)
          run (Hole x) = return (Hole x)
          run (Term t) = liftM appCxt . f =<< mapM run t

-- | Apply a monadic term homomorphism recursively to a
-- term/context. This a top-down variant of 'appHomM'.
appHomM' :: forall f g m . (Traversable g, Monad m)
         => HomM m f g -> CxtFunM m f g
{-# NOINLINE [1] appHomM' #-}
appHomM' f = run
    where run :: Cxt h f a -> m (Cxt h g a)
          run (Hole x) = return (Hole x)
          run (Term t) = liftM appCxt . mapM run =<< f t

{-| This function constructs the unique monadic homomorphism from the
initial term algebra to the given term algebra. -}
homMD :: forall f g m . (Traversable f, Functor g, Monad m)
          => HomMD m f g -> CxtFunM m f g
homMD f = run 
    where run :: Cxt h f a -> m (Cxt h g a)
          run (Hole x) = return (Hole x)
          run (Term t) = liftM appCxt (f (fmap run t))


{-| This function applies a monadic signature function to the given context. -}
appSigFunM :: (Traversable f, Monad m) => SigFunM m f g -> CxtFunM m f g
{-# NOINLINE [1] appSigFunM #-}
appSigFunM f = run
    where run (Term t) = liftM Term . f =<< mapM run t
          run (Hole x) = return (Hole x)
-- implementation via term homomorphisms
-- appSigFunM f = appHomM $ hom' f



-- | This function applies a monadic signature function to the given
-- context. This is a top-down variant of 'appSigFunM'.
appSigFunM' :: (Traversable g, Monad m) => SigFunM m f g -> CxtFunM m f g
{-# NOINLINE [1] appSigFunM' #-}
appSigFunM' f = run
    where run (Term t) = liftM Term . mapM run =<< f t
          run (Hole x) = return (Hole x)

{-| This function applies a signature function to the given context. -}
appSigFunMD :: forall f g m . (Traversable f, Functor g, Monad m)
              => SigFunMD m f g -> CxtFunM m f g
appSigFunMD f = run 
    where run :: Cxt h f a -> m (Cxt h g a)
          run (Hole x) = return (Hole x)
          run (Term t) = liftM Term (f (fmap run t))

{-| Compose two monadic term homomorphisms. -}
compHomM :: (Traversable g, Functor h, Monad m)
             => HomM m g h -> HomM m f g -> HomM m f h
compHomM f g = appHomM f <=< g

{-| Compose two monadic term homomorphisms. -}
compHomM' :: (Traversable h, Monad m)
                => HomM m g h -> HomM m f g -> HomM m f h
compHomM' f g = appHomM' f <=< g

{-| Compose two monadic term homomorphisms. -}
compHomM_ :: (Functor h, Functor g, Monad m)
                => Hom g h -> HomM m f g -> HomM m f h
compHomM_ f g = liftM (appHom f) . g

{-| Compose a monadic algebra with a monadic term homomorphism to get a new
  monadic algebra. -}
compAlgM :: (Traversable g, Monad m) => AlgM m g a -> HomM m f g -> AlgM m f a
compAlgM alg talg = cataM' alg <=< talg


{-| Compose a monadic algebra with a term homomorphism to get a new monadic
  algebra. -}
compAlgM' :: (Traversable g, Monad m) => AlgM m g a -> Hom f g -> AlgM m f a
compAlgM' alg talg = cataM' alg . talg


{-| This function composes two monadic signature functions.  -}
compSigFunM :: (Monad m) => SigFunM m g h -> SigFunM m f g -> SigFunM m f h
compSigFunM f g = f <=< g

compSigFunHomM :: (Traversable g, Functor h, Monad m)
                   => SigFunM m g h -> HomM m f g -> HomM m f h
compSigFunHomM f g = appSigFunM f <=< g


{-| Compose two monadic term homomorphisms. -}
compSigFunHomM' :: (Traversable h, Monad m)
                    => SigFunM m g h -> HomM m f g -> HomM m f h
compSigFunHomM' f g = appSigFunM' f <=< g

{-| This function composes two monadic signature functions.  -}
compHomSigFunM :: (Monad m) => HomM m g h -> SigFunM m f g -> HomM m f h
compHomSigFunM f g = f <=< g


{-| This function composes two monadic signature functions.  -}
compAlgSigFunM :: (Monad m) => AlgM m g a -> SigFunM m f g -> AlgM m f a
compAlgSigFunM f g = f <=< g

----------------
-- Coalgebras --
----------------

{-| This type represents a coalgebra over a functor @f@ and carrier @a@. -}
type Coalg f a = a -> f a

{-| Construct an anamorphism from the given coalgebra. -}
ana :: forall a f . Functor f => Coalg f a -> a -> Term f
ana f = run
    where run :: a -> Term f
          run t = Term $ fmap run (f t)

-- | Shortcut fusion variant of 'ana'.
ana' :: forall a f . Functor f => Coalg f a -> a -> Term f
ana' f t = build $ run t
    where run :: forall b . a -> Alg f b -> b
          run t con = run' t where
              run' :: a ->  b
              run' t = con $ fmap run' (f t)

build :: (forall a. Alg f a -> a) -> Term f
{-# INLINE [1] build #-}
build g = g Term

{-| This type represents a monadic coalgebra over a functor @f@ and carrier
  @a@. -}
type CoalgM m f a = a -> m (f a)

{-| Construct a monadic anamorphism from the given monadic coalgebra. -}
anaM :: forall a m f. (Traversable f, Monad m)
          => CoalgM m f a -> a -> m (Term f)
anaM f = run 
    where run :: a -> m (Term f)
          run t = liftM Term $ f t >>= mapM run


--------------------------------
-- R-Algebras & Paramorphisms --
--------------------------------

{-| This type represents an r-algebra over a functor @f@ and carrier @a@. -}
type RAlg f a = f (Term f, a) -> a

{-| Construct a paramorphism from the given r-algebra. -}
para :: (Functor f) => RAlg f a -> Term f -> a
para f = snd . cata run
    where run t = (Term $ fmap fst t, f t)

{-| This type represents a monadic r-algebra over a functor @f@ and carrier
  @a@. -}
type RAlgM m f a = f (Term f, a) -> m a

{-| Construct a monadic paramorphism from the given monadic r-algebra. -}
paraM :: (Traversable f, Monad m) => 
         RAlgM m f a -> Term f -> m a
paraM f = liftM snd . cataM run
    where run t = do
            a <- f t
            return (Term $ fmap fst t, a)

--------------------------------
-- R-Coalgebras & Apomorphisms --
--------------------------------

{-| This type represents an r-coalgebra over a functor @f@ and carrier @a@. -}
type RCoalg f a = a -> f (Either (Term f) a)

{-| Construct an apomorphism from the given r-coalgebra. -}
apo :: (Functor f) => RCoalg f a -> a -> Term f
apo f = run 
    where run = Term . fmap run' . f
          run' (Left t) = t
          run' (Right a) = run a
-- can also be defined in terms of anamorphisms (but less
-- efficiently):
-- apo f = ana run . Right
--     where run (Left (Term t)) = fmap Left t
--           run (Right a) = f a

{-| This type represents a monadic r-coalgebra over a functor @f@ and carrier
  @a@. -}
type RCoalgM m f a = a -> m (f (Either (Term f) a))

{-| Construct a monadic apomorphism from the given monadic r-coalgebra. -}
apoM :: (Traversable f, Monad m) =>
        RCoalgM m f a -> a -> m (Term f)
apoM f = run 
    where run a = do
            t <- f a
            t' <- mapM run' t
            return $ Term t'
          run' (Left t) = return t
          run' (Right a) = run a

-- can also be defined in terms of anamorphisms (but less
-- efficiently):
-- apoM f = anaM run . Right
--     where run (Left (Term t)) = return $ fmap Left t
--           run (Right a) = f a


----------------------------------
-- CV-Algebras & Histomorphisms --
----------------------------------

{-| This type represents a cv-algebra over a functor @f@ and carrier @a@. -}
type CVAlg f a f' = f (Term f') -> a


-- | This function applies 'projectA' at the tip of the term.
projectTip  :: (DistAnn f a f') => Term f' -> (f (Term f'), a)
projectTip (Term v) = projectA v

{-| Construct a histomorphism from the given cv-algebra. -}
histo :: (Functor f,DistAnn f a f') => CVAlg f a f' -> Term f -> a
histo alg  = snd . projectTip . cata run
    where run v = Term $ injectA (alg v) v

{-| This type represents a monadic cv-algebra over a functor @f@ and carrier
  @a@. -}
type CVAlgM m f a f' = f (Term f') -> m a

{-| Construct a monadic histomorphism from the given monadic cv-algebra. -}
histoM :: (Traversable f, Monad m, DistAnn f a f') =>
          CVAlgM m f a f' -> Term f -> m a
histoM alg  = liftM (snd . projectTip) . cataM run
    where run v = do r <- alg v
                     return $ Term $ injectA r v

-----------------------------------
-- CV-Coalgebras & Futumorphisms --
-----------------------------------

{-| This type represents a cv-coalgebra over a functor @f@ and carrier @a@. -}
type CVCoalg f a = a -> f (Context f a)

{-| Construct a futumorphism from the given cv-coalgebra. -}
futu :: forall f a . Functor f => CVCoalg f a -> a -> Term f
futu coa = ana run . Hole
    where run :: Coalg f (Context f a)
          run (Hole x) = coa x
          run (Term t) = t

{-| This type represents a monadic cv-coalgebra over a functor @f@ and carrier
  @a@. -}
type CVCoalgM m f a = a -> m (f (Context f a))

{-| Construct a monadic futumorphism from the given monadic cv-coalgebra. -}
futuM :: forall f a m . (Traversable f, Monad m) =>
         CVCoalgM m f a -> a -> m (Term f)
futuM coa = anaM run . Hole
    where run :: CoalgM m f (Context f a)
          run (Hole x) = coa x
          run (Term t) = return t

{-| This type represents a generalised cv-coalgebra over a functor @f@ and
  carrier @a@. -}
type CVCoalg' f a = a -> Context f a

{-| Construct a futumorphism from the given generalised cv-coalgebra. -}
futu' :: forall f a . Functor f => CVCoalg' f a -> a -> Term f
futu' coa = run
    where run :: a -> Term f
          run x = appCxt $ fmap run (coa x)


-------------------------------------------
-- functions only used for rewrite rules --
-------------------------------------------


appAlgHom :: forall f g d . (Functor g) => Alg g d -> Hom f g -> Term f -> d
appAlgHom alg hom = run where
    run :: Term f -> d
    run (Term t) = run' $ hom t
    run' :: Context g (Term f) -> d
    run' (Term t) = alg $ fmap run' t
    run' (Hole x) = run x


-- | This function applies a signature function after a term homomorphism.
appSigFunHom :: forall f g h. (Functor g)
                 => SigFun g h -> Hom f g -> CxtFun f h
{-# NOINLINE [1] appSigFunHom #-}
appSigFunHom f g = run where
    run :: CxtFun f h
    run (Term t) = run' $ g t
    run (Hole h) = Hole h
    run' :: Context g (Cxt h' f b) -> Cxt h' h b
    run' (Term t) = Term $ f $ fmap run' t
    run' (Hole h) = run h

-- | This function applies the given algebra bottom-up while applying
-- the given term homomorphism top-down. Thereby we have no
-- requirements on the source signature @f@.
appAlgHomM :: forall m f g a. (Traversable g, Monad m)
               => AlgM m g a -> HomM m f g -> Term f -> m a
appAlgHomM alg hom = run
    where run :: Term f -> m a
          run (Term t) = hom t >>= mapM run >>= run'
          run' :: Context g a -> m a
          run' (Term t) = mapM run' t >>= alg
          run' (Hole x) = return x


appHomHomM :: forall m f g h . (Monad m, Traversable g, Functor h)
                   => HomM m g h -> HomM m f g -> CxtFunM m f h
appHomHomM f g = run where
    run :: CxtFunM m f h
    run (Term t) = run' =<< g t
    run (Hole h) = return $ Hole h
    run' :: Context g (Cxt h' f b) -> m (Cxt h' h b)
    run' (Term t) = liftM appCxt $ f =<< mapM run' t
    run' (Hole h) = run h


appSigFunHomM :: forall m f g h . (Traversable g, Monad m)
                   => SigFunM m g h -> HomM m f g -> CxtFunM m f h
appSigFunHomM f g = run where
    run :: CxtFunM m f h
    run (Term t) = run' =<< g t
    run (Hole h) = return $ Hole h
    run' :: Context g (Cxt h' f b) -> m (Cxt h' h b)
    run' (Term t) = liftM Term $ f =<< mapM run' t
    run' (Hole h) = run h


-------------------
-- rewrite rules --
-------------------

#ifndef NO_RULES
{-# RULES
  "cata/appHom" forall (a :: Alg g d) (h :: Hom f g) x.
    cata a (appHom h x) = cata (compAlg a h) x;

  "cata/appHom'" forall (a :: Alg g d) (h :: Hom f g) x.
    cata a (appHom' h x) = appAlgHom a h x;

  "cata/appSigFun" forall (a :: Alg g d) (h :: SigFun f g) x.
    cata a (appSigFun h x) = cata (compAlgSigFun a h) x;

  "cata/appSigFun'" forall (a :: Alg g d) (h :: SigFun f g) x.
    cata a (appSigFun' h x) = appAlgHom a (hom h) x;

  "cata/appSigFunHom" forall (f :: Alg f3 d) (g :: SigFun f2 f3)
                                      (h :: Hom f1 f2) x.
    cata f (appSigFunHom g h x) = appAlgHom (compAlgSigFun f g) h x;

  "appAlgHom/appHom" forall (a :: Alg h d) (f :: Hom f g) (h :: Hom g h) x.
    appAlgHom a h (appHom f x) = cata (compAlg a (compHom h f)) x;

  "appAlgHom/appHom'" forall (a :: Alg h d) (f :: Hom f g) (h :: Hom g h) x.
    appAlgHom a h (appHom' f x) = appAlgHom a (compHom h f) x;

  "appAlgHom/appSigFun" forall (a :: Alg h d) (f :: SigFun f g) (h :: Hom g h) x.
    appAlgHom a h (appSigFun f x) = cata (compAlg a (compHomSigFun h f)) x;

  "appAlgHom/appSigFun'" forall (a :: Alg h d) (f :: SigFun f g) (h :: Hom g h) x.
    appAlgHom a h (appSigFun' f x) = appAlgHom a (compHomSigFun h f) x;

  "appAlgHom/appSigFunHom" forall (a :: Alg i d) (f :: Hom f g) (g :: SigFun g h)
                                          (h :: Hom h i) x.
    appAlgHom a h (appSigFunHom g f x)
      = appAlgHom a (compHom (compHomSigFun h g) f) x;

  "appHom/appHom" forall (a :: Hom g h) (h :: Hom f g) x.
    appHom a (appHom h x) = appHom (compHom a h) x;

  "appHom'/appHom'" forall (a :: Hom g h) (h :: Hom f g) x.
    appHom' a (appHom' h x) = appHom' (compHom a h) x;

  "appHom'/appHom" forall (a :: Hom g h) (h :: Hom f g) x.
    appHom' a (appHom h x) = appHom (compHom a h) x;

  "appHom/appHom'" forall (a :: Hom g h) (h :: Hom f g) x.
    appHom a (appHom' h x) = appHom' (compHom a h) x;
    
  "appSigFun/appSigFun" forall (f :: SigFun g h) (g :: SigFun f g) x.
    appSigFun f (appSigFun g x) = appSigFun (compSigFun f g) x;

  "appSigFun'/appSigFun'" forall (f :: SigFun g h) (g :: SigFun f g) x.
    appSigFun' f (appSigFun' g x) = appSigFun' (compSigFun f g) x;

  "appSigFun/appSigFun'" forall (f :: SigFun g h) (g :: SigFun f g) x.
    appSigFun f (appSigFun' g x) = appSigFunHom f (hom g) x;

  "appSigFun'/appSigFun" forall (f :: SigFun g h) (g :: SigFun f g) x.
    appSigFun' f (appSigFun g x) = appSigFun (compSigFun f g) x;

  "appHom/appSigFun" forall (f :: Hom g h) (g :: SigFun f g) x.
    appHom f (appSigFun g x) = appHom (compHomSigFun f g) x;

  "appHom/appSigFun'" forall (f :: Hom g h) (g :: SigFun f g) x.
    appHom f (appSigFun' g x) =  appHom' (compHomSigFun f g) x;

  "appHom'/appSigFun'" forall (f :: Hom g h) (g :: SigFun f g) x.
    appHom' f (appSigFun' g x) =  appHom' (compHomSigFun f g) x;

  "appHom'/appSigFun" forall (f :: Hom g h) (g :: SigFun f g) x.
    appHom' f (appSigFun g x) = appHom (compHomSigFun f g) x;
    
  "appSigFun/appHom" forall (f :: SigFun g h) (g :: Hom f g) x.
    appSigFun f (appHom g x) = appSigFunHom f g x;

  "appSigFun'/appHom'" forall (f :: SigFun g h) (g :: Hom f g) x.
    appSigFun' f (appHom' g x) = appHom' (compSigFunHom f g) x;

  "appSigFun/appHom'" forall (f :: SigFun g h) (g :: Hom f g) x.
    appSigFun f (appHom' g x) = appSigFunHom f g x;

  "appSigFun'/appHom" forall (f :: SigFun g h) (g :: Hom f g) x.
    appSigFun' f (appHom g x) = appHom (compSigFunHom f g) x;
    
  "appSigFunHom/appSigFun" forall (f :: SigFun f3 f4) (g :: Hom f2 f3)
                                      (h :: SigFun f1 f2) x.
    appSigFunHom f g (appSigFun h x)
    = appSigFunHom f (compHomSigFun g h) x;

  "appSigFunHom/appSigFun'" forall (f :: SigFun f3 f4) (g :: Hom f2 f3)
                                      (h :: SigFun f1 f2) x.
    appSigFunHom f g (appSigFun' h x)
    = appSigFunHom f (compHomSigFun g h) x;

  "appSigFunHom/appHom" forall (f :: SigFun f3 f4) (g :: Hom f2 f3)
                                      (h :: Hom f1 f2) x.
    appSigFunHom f g (appHom h x)
    = appSigFunHom f (compHom g h) x;

  "appSigFunHom/appHom'" forall (f :: SigFun f3 f4) (g :: Hom f2 f3)
                                      (h :: Hom f1 f2) x.
    appSigFunHom f g (appHom' h x)
    = appSigFunHom f (compHom g h) x;

  "appSigFun/appSigFunHom" forall (f :: SigFun f3 f4) (g :: SigFun f2 f3)
                                      (h :: Hom f1 f2) x.
    appSigFun f (appSigFunHom g h x) = appSigFunHom (compSigFun f g) h x;

  "appSigFun'/appSigFunHom" forall (f :: SigFun f3 f4) (g :: SigFun f2 f3)
                                      (h :: Hom f1 f2) x.
    appSigFun' f (appSigFunHom g h x) = appSigFunHom (compSigFun f g) h x;

  "appHom/appSigFunHom" forall (f :: Hom f3 f4) (g :: SigFun f2 f3)
                                      (h :: Hom f1 f2) x.
    appHom f (appSigFunHom g h x) = appHom' (compHom (compHomSigFun f g) h) x;

  "appHom'/appSigFunHom" forall (f :: Hom f3 f4) (g :: SigFun f2 f3)
                                      (h :: Hom f1 f2) x.
    appHom' f (appSigFunHom g h x) = appHom' (compHom (compHomSigFun f g) h) x;

  "appSigFunHom/appSigFunHom" forall (f1 :: SigFun f4 f5) (f2 :: Hom f3 f4)
                                             (f3 :: SigFun f2 f3) (f4 :: Hom f1 f2) x.
    appSigFunHom f1 f2 (appSigFunHom f3 f4 x)
      = appSigFunHom f1 (compHom (compHomSigFun f2 f3) f4) x; #-}

{-# RULES 
  "cataM/appHomM" forall (a :: AlgM Maybe g d) (h :: HomM Maybe f g) x.
     appHomM h x >>= cataM a =  appAlgHomM a h x;

  "cataM/appHomM'" forall (a :: AlgM Maybe g d) (h :: HomM Maybe f g) x.
     appHomM' h x >>= cataM a = appAlgHomM a h x;

  "cataM/appSigFunM" forall (a :: AlgM Maybe g d) (h :: SigFunM Maybe f g) x.
     appSigFunM h x >>= cataM a =  appAlgHomM a (homM h) x;

  "cataM/appSigFunM'" forall (a :: AlgM Maybe g d) (h :: SigFunM Maybe f g) x.
     appSigFunM' h x >>= cataM a = appAlgHomM a (homM h) x;

  "cataM/appHom" forall (a :: AlgM m g d) (h :: Hom f g) x.
     cataM a (appHom h x) = appAlgHomM a (sigFunM h) x;

  "cataM/appHom'" forall (a :: AlgM m g d) (h :: Hom f g) x.
     cataM a (appHom' h x) = appAlgHomM a (sigFunM h) x;

  "cataM/appSigFun" forall (a :: AlgM m g d) (h :: SigFun f g) x.
     cataM a (appSigFun h x) = appAlgHomM a (sigFunM $ hom h) x;

  "cataM/appSigFun'" forall (a :: AlgM m g d) (h :: SigFun f g) x.
     cataM a (appSigFun' h x) = appAlgHomM a (sigFunM $ hom h) x;

  "cataM/appSigFun" forall (a :: AlgM m g d) (h :: SigFun f g) x.
     cataM a (appSigFun h x) = appAlgHomM a (sigFunM $ hom h) x;

  "cataM/appSigFunHom" forall (a :: AlgM m h d) (g :: SigFun g h) (f :: Hom f g) x.
     cataM a (appSigFunHom g f x) = appAlgHomM a (sigFunM $ compSigFunHom g f) x;

  "appHomM/appHomM" forall (a :: HomM Maybe g h) (h :: HomM Maybe f g) x.
     appHomM h x >>= appHomM a = appHomM (compHomM a h) x;

  "appHomM/appSigFunM" forall (a :: HomM Maybe g h) (h :: SigFunM Maybe f g) x.
     appSigFunM h x >>= appHomM a = appHomM (compHomSigFunM a h) x;

  "appHomM/appHomM'" forall (a :: HomM Maybe g h) (h :: HomM Maybe f g) x.
     appHomM' h x >>= appHomM a = appHomHomM a h x;

  "appHomM/appSigFunM'" forall (a :: HomM Maybe g h) (h :: SigFunM Maybe f g) x.
     appSigFunM' h x >>= appHomM a = appHomHomM a (homM h) x;

  "appHomM'/appHomM" forall (a :: HomM Maybe g h) (h :: HomM Maybe f g) x.
     appHomM h x >>= appHomM' a = appHomM' (compHomM' a h) x;

  "appHomM'/appSigFunM" forall (a :: HomM Maybe g h) (h :: SigFunM Maybe f g) x.
     appSigFunM h x >>= appHomM' a = appHomM' (compHomSigFunM a h) x;

  "appHomM'/appHomM'" forall (a :: HomM Maybe g h) (h :: HomM Maybe f g) x.
     appHomM' h x >>= appHomM' a = appHomM' (compHomM' a h) x;

  "appHomM'/appSigFunM'" forall (a :: HomM Maybe g h) (h :: SigFunM Maybe f g) x.
     appSigFunM' h x >>= appHomM' a = appHomM' (compHomSigFunM a h) x;

  "appHomM/appHom" forall (a :: HomM m g h) (h :: Hom f g) x.
     appHomM a (appHom h x) = appHomHomM a (sigFunM h) x;

  "appHomM/appSigFun" forall (a :: HomM m g h) (h :: SigFun f g) x.
     appHomM a (appSigFun h x) = appHomHomM a (sigFunM $ hom h) x;

  "appHomM'/appHom" forall (a :: HomM m g h) (h :: Hom f g) x.
     appHomM' a (appHom h x) = appHomM' (compHomM' a (sigFunM h)) x;

  "appHomM'/appSigFun" forall (a :: HomM m g h) (h :: SigFun f g) x.
     appHomM' a (appSigFun h x) = appHomM' (compHomSigFunM a (sigFunM h)) x;

  "appHomM/appHom'" forall (a :: HomM m g h) (h :: Hom f g) x.
     appHomM a (appHom' h x) = appHomHomM a (sigFunM h) x;

  "appHomM/appSigFun'" forall (a :: HomM m g h) (h :: SigFun f g) x.
     appHomM a (appSigFun' h x) = appHomHomM a (sigFunM $ hom h) x;

  "appHomM'/appHom'" forall (a :: HomM m g h) (h :: Hom f g) x.
     appHomM' a (appHom' h x) = appHomM' (compHomM' a (sigFunM h)) x;

  "appHomM'/appSigFun'" forall (a :: HomM m g h) (h :: SigFun f g) x.
     appHomM' a (appSigFun' h x) = appHomM' (compHomSigFunM a (sigFunM h)) x;

  "appSigFunM/appHomM" forall (a :: SigFunM Maybe g h) (h :: HomM Maybe f g) x.
     appHomM h x >>= appSigFunM a = appSigFunHomM a h x;

  "appSigFunHomM/appSigFunM" forall (a :: SigFunM Maybe g h) (h :: SigFunM Maybe f g) x.
     appSigFunM h x >>= appSigFunM a = appSigFunM (compSigFunM a h) x;

  "appSigFunM/appHomM'" forall (a :: SigFunM Maybe g h) (h :: HomM Maybe f g) x.
     appHomM' h x >>= appSigFunM a = appSigFunHomM a h x;

  "appSigFunM/appSigFunM'" forall (a :: SigFunM Maybe g h) (h :: SigFunM Maybe f g) x.
     appSigFunM' h x >>= appSigFunM a = appSigFunHomM a (homM h) x;

  "appSigFunM'/appHomM" forall (a :: SigFunM Maybe g h) (h :: HomM Maybe f g) x.
     appHomM h x >>= appSigFunM' a = appHomM' (compSigFunHomM' a h) x;

  "appSigFunM'/appSigFunM" forall (a :: SigFunM Maybe g h) (h :: SigFunM Maybe f g) x.
     appSigFunM h x >>= appSigFunM' a = appSigFunM' (compSigFunM a h) x;

  "appSigFunM'/appHomM'" forall (a :: SigFunM Maybe g h) (h :: HomM Maybe f g) x.
     appHomM' h x >>= appSigFunM' a = appHomM' (compSigFunHomM' a h) x;

  "appSigFunM'/appSigFunM'" forall (a :: SigFunM Maybe g h) (h :: SigFunM Maybe f g) x.
     appSigFunM' h x >>= appSigFunM' a = appSigFunM' (compSigFunM a h) x;

  "appSigFunM/appHom" forall (a :: SigFunM m g h) (h :: Hom f g) x.
     appSigFunM a (appHom h x) = appSigFunHomM a (sigFunM h) x;

  "appSigFunM/appSigFun" forall (a :: SigFunM m g h) (h :: SigFun f g) x.
     appSigFunM a (appSigFun h x) = appSigFunHomM a (sigFunM $ hom h) x;

  "appSigFunM'/appHom" forall (a :: SigFunM m g h) (h :: Hom f g) x.
     appSigFunM' a (appHom h x) = appHomM' (compSigFunHomM' a (sigFunM h)) x;

  "appSigFunM'/appSigFun" forall (a :: SigFunM m g h) (h :: SigFun f g) x.
     appSigFunM' a (appSigFun h x) = appSigFunM' (compSigFunM a (sigFunM h)) x;

  "appSigFunM/appHom'" forall (a :: SigFunM m g h) (h :: Hom f g) x.
     appSigFunM a (appHom' h x) = appSigFunHomM a (sigFunM h) x;

  "appSigFunM/appSigFun'" forall (a :: SigFunM m g h) (h :: SigFun f g) x.
     appSigFunM a (appSigFun' h x) = appSigFunHomM a (sigFunM $ hom h) x;

  "appSigFunM'/appHom'" forall (a :: SigFunM m g h) (h :: Hom f g) x.
     appSigFunM' a (appHom' h x) = appHomM' (compSigFunHomM' a (sigFunM h)) x;

  "appSigFunM'/appSigFun'" forall (a :: SigFunM m g h) (h :: SigFun f g) x.
     appSigFunM' a (appSigFun' h x) = appSigFunM' (compSigFunM a (sigFunM h)) x;


  "appHom/appHomM" forall (a :: Hom g h) (h :: HomM m f g) x.
     appHomM h x >>= (return . appHom a) = appHomM (compHomM_ a h) x; #-}

{-# RULES
  "cata/build"  forall alg (g :: forall a . Alg f a -> a) .
                cata alg (build g) = g alg  #-}
#endif