-- Hoogle documentation, generated by Haddock
-- See Hoogle, http://www.haskell.org/hoogle/


-- | Generalized bananas, lenses and barbed wire
--   
--   Generalized bananas, lenses and barbed wire
@package recursion-schemes
@version 5


module Data.Functor.Foldable

-- | Base functor of <tt>[]</tt>.
data ListF a b
Nil :: ListF a b
Cons :: a -> b -> ListF a b
newtype Fix f
Fix :: (f (Fix f)) -> Fix f
unfix :: Fix f -> f (Fix f)
newtype Mu f
Mu :: (forall a. (f a -> a) -> a) -> Mu f
data Nu f
[Nu] :: (a -> f a) -> a -> Nu f
class Functor (Base t) => Recursive t where cata f = c where c = f . fmap c . project para t = p where p x = t . fmap ((,) <*> p) $ project x gpara t = gzygo embed t prepro e f = c where c = f . fmap (c . cata (embed . e)) . project gprepro k e f = extract . c where c = fmap f . k . fmap (duplicate . c . cata (embed . e)) . project
project :: Recursive t => t -> Base t t
cata :: Recursive t => (Base t a -> a) -> t -> a
para :: Recursive t => (Base t (t, a) -> a) -> t -> a
gpara :: (Recursive t, Corecursive t, Comonad w) => (forall b. Base t (w b) -> w (Base t b)) -> (Base t (EnvT t w a) -> a) -> t -> a

-- | Fokkinga's prepromorphism
prepro :: (Recursive t, Corecursive t) => (forall b. Base t b -> Base t b) -> (Base t a -> a) -> t -> a
gprepro :: (Recursive t, Corecursive t, Comonad w) => (forall b. Base t (w b) -> w (Base t b)) -> (forall c. Base t c -> Base t c) -> (Base t (w a) -> a) -> t -> a
gapo :: Corecursive t => (b -> Base t b) -> (a -> Base t (Either b a)) -> a -> t

-- | A generalized catamorphism
gcata :: (Recursive t, Comonad w) => (forall b. Base t (w b) -> w (Base t b)) -> (Base t (w a) -> a) -> t -> a
zygo :: Recursive t => (Base t b -> b) -> (Base t (b, a) -> a) -> t -> a
gzygo :: (Recursive t, Comonad w) => (Base t b -> b) -> (forall c. Base t (w c) -> w (Base t c)) -> (Base t (EnvT b w a) -> a) -> t -> a

-- | Course-of-value iteration
histo :: Recursive t => (Base t (Cofree (Base t) a) -> a) -> t -> a
ghisto :: (Recursive t, Functor h) => (forall b. Base t (h b) -> h (Base t b)) -> (Base t (Cofree h a) -> a) -> t -> a
futu :: Corecursive t => (a -> Base t (Free (Base t) a)) -> a -> t
chrono :: Functor f => (f (Cofree f b) -> b) -> (a -> f (Free f a)) -> (a -> b)
gchrono :: (Functor f, Functor w, Functor m) => (forall c. f (w c) -> w (f c)) -> (forall c. m (f c) -> f (m c)) -> (f (Cofree w b) -> b) -> (a -> f (Free m a)) -> (a -> b)
distCata :: Functor f => f (Identity a) -> Identity (f a)
distPara :: Corecursive t => Base t (t, a) -> (t, Base t a)
distParaT :: (Corecursive t, Comonad w) => (forall b. Base t (w b) -> w (Base t b)) -> Base t (EnvT t w a) -> EnvT t w (Base t a)
distZygo :: Functor f => (f b -> b) -> (f (b, a) -> (b, f a))
distZygoT :: (Functor f, Comonad w) => (f b -> b) -> (forall c. f (w c) -> w (f c)) -> f (EnvT b w a) -> EnvT b w (f a)
distHisto :: Functor f => f (Cofree f a) -> Cofree f (f a)
distGHisto :: (Functor f, Functor h) => (forall b. f (h b) -> h (f b)) -> f (Cofree h a) -> Cofree h (f a)
distFutu :: Functor f => Free f (f a) -> f (Free f a)
distGFutu :: (Functor f, Functor h) => (forall b. h (f b) -> f (h b)) -> Free h (f a) -> f (Free h a)
class Functor (Base t) => Corecursive t where ana g = a where a = embed . fmap a . g apo g = a where a = embed . (fmap (either id a)) . g postpro e g = a where a = embed . fmap (ana (e . project) . a) . g gpostpro k e g = a . return where a = embed . fmap (ana (e . project) . a . join) . k . liftM g
embed :: Corecursive t => Base t t -> t
ana :: Corecursive t => (a -> Base t a) -> a -> t
apo :: Corecursive t => (a -> Base t (Either t a)) -> a -> t

-- | Fokkinga's postpromorphism
postpro :: (Corecursive t, Recursive t) => (forall b. Base t b -> Base t b) -> (a -> Base t a) -> a -> t

-- | A generalized postpromorphism
gpostpro :: (Corecursive t, Recursive t, Monad m) => (forall b. m (Base t b) -> Base t (m b)) -> (forall c. Base t c -> Base t c) -> (a -> Base t (m a)) -> a -> t

-- | A generalized anamorphism
gana :: (Corecursive t, Monad m) => (forall b. m (Base t b) -> Base t (m b)) -> (a -> Base t (m a)) -> a -> t
distAna :: Functor f => Identity (f a) -> f (Identity a)
distApo :: Recursive t => Either t (Base t a) -> Base t (Either t a)
distGApo :: Functor f => (b -> f b) -> Either b (f a) -> f (Either b a)
distGApoT :: (Functor f, Functor m) => (b -> f b) -> (forall c. m (f c) -> f (m c)) -> ExceptT b m (f a) -> f (ExceptT b m a)
hylo :: Functor f => (f b -> b) -> (a -> f a) -> a -> b

-- | A generalized hylomorphism
ghylo :: (Comonad w, Functor f, Monad m) => (forall c. f (w c) -> w (f c)) -> (forall d. m (f d) -> f (m d)) -> (f (w b) -> b) -> (a -> f (m a)) -> a -> b
refix :: (Recursive s, Corecursive t, Base s ~ Base t) => s -> t
fold :: Recursive t => (Base t a -> a) -> t -> a

-- | A generalized catamorphism
gfold :: (Recursive t, Comonad w) => (forall b. Base t (w b) -> w (Base t b)) -> (Base t (w a) -> a) -> t -> a
unfold :: Corecursive t => (a -> Base t a) -> a -> t

-- | A generalized anamorphism
gunfold :: (Corecursive t, Monad m) => (forall b. m (Base t b) -> Base t (m b)) -> (a -> Base t (m a)) -> a -> t
refold :: Functor f => (f b -> b) -> (a -> f a) -> a -> b

-- | A generalized hylomorphism
grefold :: (Comonad w, Functor f, Monad m) => (forall c. f (w c) -> w (f c)) -> (forall d. m (f d) -> f (m d)) -> (f (w b) -> b) -> (a -> f (m a)) -> a -> b

-- | Mendler-style iteration
mcata :: (forall y. (y -> c) -> f y -> c) -> Fix f -> c

-- | Mendler-style course-of-value iteration
mhisto :: (forall y. (y -> c) -> (y -> f y) -> f y -> c) -> Fix f -> c

-- | Elgot algebras
elgot :: Functor f => (f a -> a) -> (b -> Either a (f b)) -> b -> a

-- | Elgot coalgebras:
--   <a>http://comonad.com/reader/2008/elgot-coalgebras/</a>
coelgot :: Functor f => ((a, f b) -> b) -> (a -> f a) -> a -> b

-- | Zygohistomorphic prepromorphisms:
--   
--   A corrected and modernized version of
--   <a>http://www.haskell.org/haskellwiki/Zygohistomorphic_prepromorphisms</a>
zygoHistoPrepro :: (Corecursive t, Recursive t) => (Base t b -> b) -> (forall c. Base t c -> Base t c) -> (Base t (EnvT b (Cofree (Base t)) a) -> a) -> t -> a
instance GHC.Generics.Generic1 (Data.Functor.Foldable.ListF a)
instance GHC.Generics.Generic (Data.Functor.Foldable.ListF a b)
instance (GHC.Read.Read a, GHC.Read.Read b) => GHC.Read.Read (Data.Functor.Foldable.ListF a b)
instance (GHC.Show.Show a, GHC.Show.Show b) => GHC.Show.Show (Data.Functor.Foldable.ListF a b)
instance (GHC.Classes.Ord a, GHC.Classes.Ord b) => GHC.Classes.Ord (Data.Functor.Foldable.ListF a b)
instance (GHC.Classes.Eq a, GHC.Classes.Eq b) => GHC.Classes.Eq (Data.Functor.Foldable.ListF a b)
instance (Data.Typeable.Internal.Typeable f, Data.Data.Data (f (Data.Functor.Foldable.Fix f))) => Data.Data.Data (Data.Functor.Foldable.Fix f)
instance Data.Functor.Classes.Eq2 Data.Functor.Foldable.ListF
instance GHC.Classes.Eq a => Data.Functor.Classes.Eq1 (Data.Functor.Foldable.ListF a)
instance Data.Functor.Classes.Ord2 Data.Functor.Foldable.ListF
instance GHC.Classes.Ord a => Data.Functor.Classes.Ord1 (Data.Functor.Foldable.ListF a)
instance GHC.Show.Show a => Data.Functor.Classes.Show1 (Data.Functor.Foldable.ListF a)
instance Data.Functor.Classes.Show2 Data.Functor.Foldable.ListF
instance Data.Functor.Classes.Read2 Data.Functor.Foldable.ListF
instance GHC.Read.Read a => Data.Functor.Classes.Read1 (Data.Functor.Foldable.ListF a)
instance GHC.Base.Functor (Data.Functor.Foldable.ListF a)
instance Data.Foldable.Foldable (Data.Functor.Foldable.ListF a)
instance Data.Traversable.Traversable (Data.Functor.Foldable.ListF a)
instance Data.Bifunctor.Bifunctor Data.Functor.Foldable.ListF
instance Data.Bifoldable.Bifoldable Data.Functor.Foldable.ListF
instance Data.Bitraversable.Bitraversable Data.Functor.Foldable.ListF
instance Data.Functor.Foldable.Recursive [a]
instance Data.Functor.Foldable.Corecursive [a]
instance Data.Functor.Foldable.Recursive (GHC.Base.Maybe a)
instance Data.Functor.Foldable.Corecursive (GHC.Base.Maybe a)
instance Data.Functor.Foldable.Recursive (Data.Either.Either a b)
instance Data.Functor.Foldable.Corecursive (Data.Either.Either a b)
instance Data.Functor.Classes.Eq1 f => GHC.Classes.Eq (Data.Functor.Foldable.Fix f)
instance Data.Functor.Classes.Ord1 f => GHC.Classes.Ord (Data.Functor.Foldable.Fix f)
instance Data.Functor.Classes.Show1 f => GHC.Show.Show (Data.Functor.Foldable.Fix f)
instance Data.Functor.Classes.Read1 f => GHC.Read.Read (Data.Functor.Foldable.Fix f)
instance GHC.Base.Functor f => Data.Functor.Foldable.Recursive (Data.Functor.Foldable.Fix f)
instance GHC.Base.Functor f => Data.Functor.Foldable.Corecursive (Data.Functor.Foldable.Fix f)
instance GHC.Base.Functor f => Data.Functor.Foldable.Recursive (Data.Functor.Foldable.Mu f)
instance GHC.Base.Functor f => Data.Functor.Foldable.Corecursive (Data.Functor.Foldable.Mu f)
instance (GHC.Base.Functor f, Data.Functor.Classes.Eq1 f) => GHC.Classes.Eq (Data.Functor.Foldable.Mu f)
instance (GHC.Base.Functor f, Data.Functor.Classes.Ord1 f) => GHC.Classes.Ord (Data.Functor.Foldable.Mu f)
instance (GHC.Base.Functor f, Data.Functor.Classes.Show1 f) => GHC.Show.Show (Data.Functor.Foldable.Mu f)
instance (GHC.Base.Functor f, Data.Functor.Classes.Read1 f) => GHC.Read.Read (Data.Functor.Foldable.Mu f)
instance GHC.Base.Functor f => Data.Functor.Foldable.Corecursive (Data.Functor.Foldable.Nu f)
instance GHC.Base.Functor f => Data.Functor.Foldable.Recursive (Data.Functor.Foldable.Nu f)
instance (GHC.Base.Functor f, Data.Functor.Classes.Eq1 f) => GHC.Classes.Eq (Data.Functor.Foldable.Nu f)
instance (GHC.Base.Functor f, Data.Functor.Classes.Ord1 f) => GHC.Classes.Ord (Data.Functor.Foldable.Nu f)
instance (GHC.Base.Functor f, Data.Functor.Classes.Show1 f) => GHC.Show.Show (Data.Functor.Foldable.Nu f)
instance (GHC.Base.Functor f, Data.Functor.Classes.Read1 f) => GHC.Read.Read (Data.Functor.Foldable.Nu f)