compdata-0.5: src/Data/Comp/Param/Algebra.hs
{-# LANGUAGE GADTs, RankNTypes, ScopedTypeVariables, TypeOperators,
FlexibleContexts, CPP #-}
--------------------------------------------------------------------------------
-- |
-- Module : Data.Comp.Param.Algebra
-- Copyright : (c) 2011 Patrick Bahr, Tom Hvitved
-- License : BSD3
-- Maintainer : Tom Hvitved <hvitved@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.Param.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,
compHomSigFun,
compSigFunHom,
hom,
compAlg,
compAlgSigFun,
-- * Monadic Term Homomorphisms
CxtFunM,
SigFunM,
HomM,
SigFunMD,
HomMD,
sigFunM,
appHomM,
appTHomM,
appHomM',
appTHomM',
homM,
homMD,
appSigFunM,
appTSigFunM,
appSigFunM',
appTSigFunM',
appSigFunMD,
appTSigFunMD,
compHomM,
compHomM',
compSigFunM,
compSigFunHomM,
compSigFunHomM',
compAlgSigFunM,
compAlgSigFunM',
compAlgM,
compAlgM',
-- * Coalgebras & Anamorphisms
Coalg,
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 Prelude hiding (sequence, mapM)
import Control.Monad hiding (sequence, mapM)
import Data.Comp.Param.Term
import Data.Comp.Param.Ops
import Data.Comp.Param.Difunctor
import Data.Comp.Param.Ditraversable
{-| This type represents an algebra over a difunctor @f@ and carrier @a@. -}
type Alg f a = f a a -> a
{-| Construct a catamorphism for contexts over @f@ with holes of type @b@, from
the given algebra. -}
free :: forall h f a b. Difunctor f
=> Alg f a -> (b -> a) -> Cxt h f a b -> a
free f g = run
where run :: Cxt h f a b -> a
run (In t) = f (difmap run t)
run (Hole x) = g x
run (Var p) = p
{-| Construct a catamorphism from the given algebra. -}
cata :: forall f a. Difunctor f => Alg f a -> Term f -> a
{-# NOINLINE [1] cata #-}
cata f (Term t) = run t
where run :: Trm f a -> a
run (In t) = f (difmap run t)
run (Var x) = x
{-| A generalisation of 'cata' from terms over @f@ to contexts over @f@, where
the holes have the type of the algebra carrier. -}
cata' :: Difunctor f => Alg f a -> Cxt h f a a -> a
{-# INLINE cata' #-}
cata' f = free f id
{-| This function applies a whole context into another context. -}
appCxt :: Difunctor f => Context f a (Cxt h f a b) -> Cxt h f a b
appCxt (In t) = In (difmap appCxt t)
appCxt (Hole x) = x
appCxt (Var p) = Var p
{-| This type represents a monadic algebra. It is similar to 'Alg' but
the return type is monadic. -}
type AlgM m f a = f a a -> m a
{-| Convert a monadic algebra into an ordinary algebra with a monadic
carrier. -}
algM :: (Ditraversable f, Monad m) => AlgM m f a -> Alg f (m a)
algM f x = disequence (dimap return id x) >>= f
{-| Construct a monadic catamorphism for contexts over @f@ with holes of type
@b@, from the given monadic algebra. -}
freeM :: forall m h f a b. (Ditraversable f, Monad m)
=> AlgM m f a -> (b -> m a) -> Cxt h f a b -> m a
freeM f g = run
where run :: Cxt h f a b -> m a
run (In t) = f =<< dimapM run t
run (Hole x) = g x
run (Var p) = return p
{-| Construct a monadic catamorphism from the given monadic algebra. -}
cataM :: forall m f a. (Ditraversable f, Monad m) => AlgM m f a -> Term f -> m a
{-# NOINLINE [1] cataM #-}
cataM algm (Term t) = run t
where run :: Trm f a -> m a
run (In t) = algm =<< dimapM run t
run (Var x) = return x
{-| 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 m h f a. (Ditraversable f, Monad m)
=> AlgM m f a -> Cxt h f a (m a) -> m a
{-# NOINLINE [1] cataM' #-}
cataM' f = freeM f id
{-| This type represents a context function. -}
type CxtFun f g = forall h a b. Cxt h f a b -> Cxt h g a b
{-| This type represents a signature function. -}
type SigFun f g = forall a b. f a b -> g a b
{-| This type represents a term homomorphism. -}
type Hom f g = SigFun f (Context g)
{-| Apply a term homomorphism recursively to a term/context. -}
appHom :: forall f g. (Difunctor f, Difunctor g) => Hom f g -> CxtFun f g
{-# NOINLINE [1] appHom #-}
appHom f = run where
run :: CxtFun f g
run (In t) = appCxt (f (difmap run t))
run (Hole x) = Hole x
run (Var p) = Var p
{-| Apply a term homomorphism recursively to a term/context. -}
appHom' :: forall f g. (Difunctor g) => Hom f g -> CxtFun f g
{-# NOINLINE [1] appHom' #-}
appHom' f = run where
run :: CxtFun f g
run (In t) = appCxt (fmapCxt run (f t))
run (Hole x) = Hole x
run (Var p) = Var p
fmapCxt :: Difunctor f => (b -> b') -> Cxt h f a b -> Cxt h f a b'
fmapCxt f = run
where run (In t) = In $ difmap run t
run (Var a) = Var a
run (Hole b) = Hole $ f b
{-| Compose two term homomorphisms. -}
compHom :: (Difunctor g, Difunctor h)
=> Hom g h -> Hom f g -> Hom f h
compHom f g = appHom f . g
{-| Compose an algebra with a term homomorphism to get a new algebra. -}
compAlg :: (Difunctor f, Difunctor g) => Alg g a -> Hom f g -> Alg f a
compAlg alg talg = cata' alg . talg
compAlgSigFun :: Alg g a -> SigFun f g -> Alg f a
compAlgSigFun alg sig = alg . sig
{-| This function applies a signature function to the given context. -}
appSigFun :: forall f g. (Difunctor f) => SigFun f g -> CxtFun f g
{-# NOINLINE [1] appSigFun #-}
appSigFun f = run
where run (In t) = In $ f $ difmap run t
run (Var x) = Var x
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-bottom variant of 'appSigFun'.
appSigFun' :: forall f g. (Difunctor g) => SigFun f g -> CxtFun f g
{-# NOINLINE [1] appSigFun' #-}
appSigFun' f = run
where run (In t) = In $ difmap run $ f t
run (Var x) = Var x
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 term homomorphism and a signature function. -}
compHomSigFun :: Hom g h -> SigFun f g -> Hom f h
compHomSigFun f g = f . g
{-| This function composes a term homomorphism and a signature function. -}
compSigFunHom :: (Difunctor g) => SigFun g h -> Hom f g -> Hom f h
compSigFunHom f g = appSigFun f . g
{-| Lifts the given signature function to the canonical term homomorphism. -}
hom :: Difunctor g => SigFun f g -> Hom f g
hom f = simpCxt . f
{-| This type represents a monadic signature function. -}
type SigFunM m f g = forall a b. f a b -> m (g a b)
{-| This type represents a monadic context function. -}
type CxtFunM m f g = forall h . SigFunM m (Cxt h f) (Cxt h g)
{-| 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 b. f a (m b) -> m (g a b)
{-| 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 given signature function to a monadic term homomorphism. -}
homM :: (Difunctor 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. The monad is sequenced bottom-up.
appHomM :: forall f g m. (Ditraversable f, Difunctor g, Monad m)
=> HomM m f g -> CxtFunM m f g
{-# NOINLINE [1] appHomM #-}
appHomM f = run
where run :: CxtFunM m f g
run (In t) = liftM appCxt . f =<< dimapM run t
run (Hole x) = return (Hole x)
run (Var p) = return (Var p)
{-| A restricted form of |appHomM| which only works for terms. -}
appTHomM :: (Ditraversable f, ParamFunctor m, Monad m, Difunctor g)
=> HomM m f g -> Term f -> m (Term g)
appTHomM f (Term t) = termM (appHomM f t)
-- | Apply a monadic term homomorphism recursively to a
-- term/context. The monad is sequence top-down.
appHomM' :: forall f g m. (Ditraversable g, Monad m)
=> HomM m f g -> CxtFunM m f g
appHomM' f = run
where run :: CxtFunM m f g
run (In t) = liftM appCxt . dimapMCxt run =<< f t
run (Var p) = return (Var p)
run (Hole x) = return (Hole x)
dimapMCxt :: (Ditraversable f, Monad m)
=> (b -> m b') -> Cxt h f a b -> m (Cxt h f a b')
dimapMCxt f = run
where run (In t) = liftM In $ dimapM run t
run (Var a) = return $ Var a
run (Hole b) = liftM Hole (f b)
{-| A restricted form of |appHomM'| which only works for terms. -}
appTHomM' :: (Ditraversable g, ParamFunctor m, Monad m, Difunctor g)
=> HomM m f g -> Term f -> m (Term g)
appTHomM' f (Term t) = termM (appHomM' f t)
{-| This function constructs the unique monadic homomorphism from the
initial term algebra to the given term algebra. -}
homMD :: forall f g m. (Difunctor f, Difunctor g, Monad m)
=> HomMD m f g -> CxtFunM m f g
homMD f = run
where run :: CxtFunM m f g
run (In t) = liftM appCxt (f (difmap run t))
run (Hole x) = return (Hole x)
run (Var p) = return (Var p)
{-| This function applies a monadic signature function to the given context. -}
appSigFunM :: forall m f g. (Ditraversable f, Monad m)
=> SigFunM m f g -> CxtFunM m f g
appSigFunM f = run
where run :: CxtFunM m f g
run (In t) = liftM In . f =<< dimapM run t
run (Var x) = return $ Var x
run (Hole x) = return $ Hole x
-- implementation via term homomorphisms
-- appSigFunM f = appHomM $ hom' f
{-| A restricted form of |appSigFunM| which only works for terms. -}
appTSigFunM :: (Ditraversable f, ParamFunctor m, Monad m, Difunctor g)
=> SigFunM m f g -> Term f -> m (Term g)
appTSigFunM f (Term t) = termM (appSigFunM f t)
-- | This function applies a monadic signature function to the given
-- context. This is a 'top-down variant of 'appSigFunM'.
appSigFunM' :: forall m f g. (Ditraversable g, Monad m)
=> SigFunM m f g -> CxtFunM m f g
appSigFunM' f = run
where run :: CxtFunM m f g
run (In t) = liftM In . dimapM run =<< f t
run (Var x) = return $ Var x
run (Hole x) = return $ Hole x
{-| A restricted form of |appSigFunM'| which only works for terms. -}
appTSigFunM' :: (Ditraversable g, ParamFunctor m, Monad m, Difunctor g)
=> SigFunM m f g -> Term f -> m (Term g)
appTSigFunM' f (Term t) = termM (appSigFunM' f t)
{-| This function applies a signature function to the given context. -}
appSigFunMD :: forall f g m. (Ditraversable f, Difunctor g, Monad m)
=> SigFunMD m f g -> CxtFunM m f g
appSigFunMD f = run
where run :: CxtFunM m f g
run (In t) = liftM In (f (difmap run t))
run (Hole x) = return (Hole x)
run (Var p) = return (Var p)
{-| A restricted form of |appSigFunMD| which only works for terms. -}
appTSigFunMD :: (Ditraversable f, ParamFunctor m, Monad m, Difunctor g)
=> SigFunMD m f g -> Term f -> m (Term g)
appTSigFunMD f (Term t) = termM (appSigFunMD f t)
{-| Compose two monadic term homomorphisms. -}
compHomM :: (Ditraversable g, Difunctor 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' :: (Ditraversable 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_ :: (Difunctor h, Difunctor g, Monad m)
=> Hom g h -> HomM m f g -> HomM m f h
compHomM_ f g = liftM (appHom f) . g
{-| Compose two monadic term homomorphisms. -}
compHomSigFunM :: Monad m => HomM m g h -> SigFunM m f g -> HomM m f h
compHomSigFunM f g = f <=< g-}
{-| Compose two monadic term homomorphisms. -}
compSigFunHomM :: (Ditraversable g, 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' :: (Ditraversable h, Monad m)
=> SigFunM m g h -> HomM m f g -> HomM m f h
compSigFunHomM' f g = appSigFunM' f <=< g
{-| Compose a monadic algebra with a monadic term homomorphism to get a new
monadic algebra. -}
compAlgM :: (Ditraversable g, Monad m) => AlgM m g a -> HomM m f g -> AlgM m f a
compAlgM alg talg = freeM alg return <=< talg
{-| Compose a monadic algebra with a term homomorphism to get a new monadic
algebra. -}
compAlgM' :: (Ditraversable g, Monad m) => AlgM m g a -> Hom f g -> AlgM m f a
compAlgM' alg talg = freeM alg return . talg
{-| Compose a monadic algebra with a monadic signature function to get a new
monadic algebra. -}
compAlgSigFunM :: Monad m => AlgM m g a -> SigFunM m f g -> AlgM m f a
compAlgSigFunM alg talg = alg <=< talg
{-| Compose a monadic algebra with a signature function to get a new monadic
algebra. -}
compAlgSigFunM' :: AlgM m g a -> SigFun f g -> AlgM m f a
compAlgSigFunM' alg talg = 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
----------------
-- Coalgebras --
----------------
{-| This type represents a coalgebra over a difunctor @f@ and carrier @a@. The
list of @(a,b)@s represent the parameters that may occur in the constructed
value. The first component represents the seed of the parameter,
and the second component is the (polymorphic) parameter itself. If @f@ is
itself a binder, then the parameters bound by @f@ can be passed to the
covariant argument, thereby making them available to sub terms. -}
type Coalg f a = forall b. a -> [(a,b)] -> Either b (f b (a,[(a,b)]))
{-| Construct an anamorphism from the given coalgebra. -}
ana :: Difunctor f => Coalg f a -> a -> Term f
ana f x = Term $ anaAux f x
where anaAux :: Difunctor f => Coalg f a -> a -> (forall a. Trm f a)
anaAux f x = run (x,[])
where run (a,bs) = case f a bs of
Left p -> Var p
Right t -> In $ difmap run t
{-| This type represents a monadic coalgebra over a difunctor @f@ and carrier
@a@. -}
type CoalgM m f a = forall b. a -> [(a,b)] -> m (Either b (f b (a,[(a,b)])))
{-| Construct a monadic anamorphism from the given monadic coalgebra. -}
anaM :: forall a m f. (Ditraversable f, Monad m)
=> CoalgM m f a -> a -> forall a. m (Trm f a)
anaM f x = run (x,[])
where run (a,bs) = do c <- f a bs
case c of
Left p -> return $ Var p
Right t -> liftM In $ dimapM run t
--------------------------------
-- R-Algebras & Paramorphisms --
--------------------------------
{-| This type represents an r-algebra over a difunctor @f@ and carrier @a@. -}
type RAlg f a = f a (Trm f a, a) -> a
{-| Construct a paramorphism from the given r-algebra. -}
para :: forall f a. Difunctor f => RAlg f a -> Term f -> a
para f (Term t) = run t
where run :: Trm f a -> a
run (In t) = f $ difmap (\x -> (x, run x)) t
run (Var x) = x
{-| This type represents a monadic r-algebra over a difunctor @f@ and carrier
@a@. -}
type RAlgM m f a = f a (Trm f a, a) -> m a
{-| Construct a monadic paramorphism from the given monadic r-algebra. -}
paraM :: forall m f a. (Ditraversable f, Monad m) => RAlgM m f a -> Term f -> m a
paraM f (Term t) = run t
where run :: Trm f a -> m a
run (In t) = f =<< dimapM (\x -> run x >>= \y -> return (x, y)) t
run (Var x) = return x
--------------------------------
-- R-Coalgebras & Apomorphisms --
--------------------------------
{-| This type represents an r-coalgebra over a difunctor @f@ and carrier @a@. -}
type RCoalg f a = forall b. a -> [(a,b)] -> Either b (f b (Either (Trm f b) (a,[(a,b)])))
{-| Construct an apomorphism from the given r-coalgebra. -}
apo :: Difunctor f => RCoalg f a -> a -> Term f
apo f x = Term (apoAux f x)
where apoAux :: Difunctor f => RCoalg f a -> a -> (forall a. Trm f a)
apoAux coa x = run (x,[])
where -- run :: (a,[(a,b)]) -> Trm f b
run (a,bs) = case coa a bs of
Left x -> Var x
Right t -> In $ difmap run' t
-- run' :: Either (Trm f b) (a,[(a,b)]) -> Trm f b
run' (Left t) = t
run' (Right x) = run x
{-| This type represents a monadic r-coalgebra over a functor @f@ and carrier
@a@. -}
type RCoalgM m f a = forall b. a -> [(a,b)] -> m (Either b (f b (Either (Trm f b) (a,[(a,b)]))))
{-| Construct a monadic apomorphism from the given monadic r-coalgebra. -}
apoM :: forall f m a. (Ditraversable f, Monad m)
=> RCoalgM m f a -> a -> forall a. m (Trm f a)
apoM coa x = run (x,[])
where run (a,bs) = do
res <- coa a bs
case res of
Left x -> return $ Var x
Right t -> liftM In $ dimapM run' t
run' (Left t) = return t
run' (Right x) = run x
----------------------------------
-- CV-Algebras & Histomorphisms --
----------------------------------
{-| This type represents a cv-algebra over a difunctor @f@ and carrier @a@. -}
type CVAlg f a f' = f a (Trm f' a) -> a
-- | This function applies 'projectA' at the tip of the term.
projectTip :: DistAnn f a f' => Trm f' a -> a
projectTip (In v) = snd $ projectA v
projectTip (Var p) = p
{-| Construct a histomorphism from the given cv-algebra. -}
histo :: forall f f' a. (Difunctor f, DistAnn f a f')
=> CVAlg f a f' -> Term f -> a
histo alg = projectTip . cata run
where run :: Alg f (Trm f' a)
run v = In $ injectA (alg v') v'
where v' = dimap Var id v
{-| This type represents a monadic cv-algebra over a functor @f@ and carrier
@a@. -}
type CVAlgM m f a f' = f a (Trm f' a) -> m a
{-| Construct a monadic histomorphism from the given monadic cv-algebra. -}
histoM :: forall f f' m a. (Ditraversable f, Monad m, DistAnn f a f')
=> CVAlgM m f a f' -> Term f -> m a
histoM alg (Term t) = liftM projectTip (run t)
where run :: Trm f a -> m (Trm f' a)
run (In t) = do t' <- dimapM run t
r <- alg t'
return $ In $ injectA r t'
run (Var p) = return $ Var p
-----------------------------------
-- CV-Coalgebras & Futumorphisms --
-----------------------------------
{-| This type represents a cv-coalgebra over a difunctor @f@ and carrier @a@.
The list of @(a,b)@s represent the parameters that may occur in the
constructed value. The first component represents the seed of the parameter,
and the second component is the (polymorphic) parameter itself. If @f@ is
itself a binder, then the parameters bound by @f@ can be passed to the
covariant argument, thereby making them available to sub terms. -}
type CVCoalg f a = forall b. a -> [(a,b)]
-> Either b (f b (Context f b (a,[(a,b)])))
{-| Construct a futumorphism from the given cv-coalgebra. -}
futu :: Difunctor f => CVCoalg f a -> a -> Term f
futu f x = Term (futuAux f x)
where futuAux :: Difunctor f => CVCoalg f a -> a -> (forall a. Trm f a)
futuAux coa x = run (x,[])
where run (a,bs) = case coa a bs of
Left p -> Var p
Right t -> In $ difmap run' t
run' (In t) = In $ difmap run' t
run' (Hole x) = run x
run' (Var p) = Var p
{-| This type represents a monadic cv-coalgebra over a difunctor @f@ and carrier
@a@. -}
type CVCoalgM m f a = forall b. a -> [(a,b)]
-> m (Either b (f b (Context f b (a,[(a,b)]))))
{-| Construct a monadic futumorphism from the given monadic cv-coalgebra. -}
futuM :: forall f a m. (Ditraversable f, Monad m) =>
CVCoalgM m f a -> a -> forall a. m (Trm f a)
futuM coa x = run (x,[])
where run (a,bs) = do c <- coa a bs
case c of
Left p -> return $ Var p
Right t -> liftM In $ dimapM run' t
run' (In t) = liftM In $ dimapM run' t
run' (Hole x) = run x
run' (Var p) = return $ Var p
{-| This type represents a generalised cv-coalgebra over a difunctor @f@ and
carrier @a@. -}
type CVCoalg' f a = forall b. a -> [(a,b)] -> Context f b (a,[(a,b)])
{-| Construct a futumorphism from the given generalised cv-coalgebra. -}
futu' :: Difunctor f => CVCoalg' f a -> a -> Term f
futu' f x = Term (futuAux' f x)
where futuAux' :: Difunctor f => CVCoalg' f a -> a -> (forall a. Trm f a)
futuAux' coa x = run (x,[])
where run (a,bs) = run' $ coa a bs
run' (In t) = In $ difmap run' t
run' (Hole x) = run x
run' (Var p) = Var p
{--------------------------------------------
-- functions only used for rewrite rules --
-------------------------------------------
appAlgHom :: forall f g d. Difunctor g => Alg g d -> Hom f g -> Term f -> d
{-# NOINLINE [1] appAlgHom #-}
appAlgHom alg hom (Term t) = run t where
run :: Trm f d -> d
run (In t) = run' $ hom t
run (Var a) = a
run' :: Context g d (Trm f d) -> d
run' (In t) = alg $ fmap run' t
run' (Var a) = a
run' (Hole x) = run x
-- | This function applies a signature function after a term homomorphism.
appSigFunHom :: forall f g h. (Difunctor g)
=> SigFun g h -> Hom f g -> CxtFun f h
{-# NOINLINE [1] appSigFunHom #-}
appSigFunHom f g = run where
run :: CxtFun f h
run (In t) = run' $ g t
run (Var a) = Var a
run (Hole h) = Hole h
run' :: Context g a (Cxt h' f a b) -> Cxt h' h a b
run' (In t) = In $ f $ fmap run' t
run' (Var a) = Var a
run' (Hole h) = run h
appAlgHomM :: forall m g f d. Ditraversable g
=> AlgM m g d -> HomM m f g -> Term f -> m d
appAlgHomM alg hom (Term t) = run t where
run :: Trm f d -> m d
run (In t) = run' =<< hom t
run (Var a) = return a
run' :: Context g d (Trm f d) -> m d
run' (In t) = alg =<< dimapM run' t
run' (Var a) = return a
run' (Hole x) = run x
appHomHomM :: forall m f g h. (Ditraversable g, Difunctor h)
=> HomM m g h -> HomM m f g -> CxtFunM m f h
appHomHomM f g = run where
-- run :: CxtFunM m f h
run (In t) = run' =<< g t
run (Var a) = return $ Var a
run (Hole h) = return $ Hole h
-- run' :: Context g Any (Cxt h' f Any b) -> m (Cxt h' h Any b)
run' (In t) = liftM appCxt $ f =<< dimapM run' t
run' (Var a) = return $ Var a
run' (Hole h) = run h
appSigFunHomM :: forall m f g h. Ditraversable g
=> SigFunM m g h -> HomM m f g -> CxtFunM m f h
appSigFunHomM f g = run where
-- run :: CxtFunM m f h
run (In t) = run' =<< g t
run (Var a) = return $ Var a
run (Hole h) = return $ Hole h
-- run' :: Context g Any (Cxt h' f Any b) -> m (Cxt h' h Any b)
run' (In t) = liftM In $ f =<< dimapM run' t
run' (Var a) = return $ Var a
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;
#-}
#endif
-}