Data.Fix

Fix-point type. It allows to define generic recursion schemes.

Fix f = f (Fix f)

Type f should be a Functor if you want to use simple recursion schemes or Traversable if you want to use monadic recursion schemes. This style allows you to express recursive functions in non-recursive manner. You can imagine that a non-recursive function holds values of the previous iteration.

Little example:

type List a = Fix (L a)

data L a b = Nil | Cons a b

instance Functor (L a) where
   fmap f x = case x of
       Nil      -> Nil
       Cons a b -> Cons a (f b)

length :: List a -> Int
length = cata $ \x -> case x of
   Nil      -> 0
   Cons _ n -> n + 1

sum :: Num a => List a -> a
sum = cata $ \x -> case x of
   Nil      -> 0
   Cons a s -> a + s

Documentation

newtype Fix f

A fix-point type.

Constructors

Fix

Fields

• unFix :: f (Fix f)

Instances

Eq (f (Fix f)) => Eq (Fix f)

Methods

(==) :: Fix f -> Fix f -> Bool

(/=) :: Fix f -> Fix f -> Bool

(Typeable (* -> *) f, Data (f (Fix f))) => Data (Fix f)

Methods

gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> Fix f -> c (Fix f)

gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (Fix f)

toConstr :: Fix f -> Constr

dataTypeOf :: Fix f -> DataType

dataCast1 :: Typeable (* -> *) t => (forall d. Data d => c (t d)) -> Maybe (c (Fix f))

dataCast2 :: Typeable (* -> * -> *) t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (Fix f))

gmapT :: (forall b. Data b => b -> b) -> Fix f -> Fix f

gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> Fix f -> r

gmapQr :: (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> Fix f -> r

gmapQ :: (forall d. Data d => d -> u) -> Fix f -> [u]

gmapQi :: Int -> (forall d. Data d => d -> u) -> Fix f -> u

gmapM :: Monad m => (forall d. Data d => d -> m d) -> Fix f -> m (Fix f)

gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> Fix f -> m (Fix f)

gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> Fix f -> m (Fix f)

Ord (f (Fix f)) => Ord (Fix f)

Methods

compare :: Fix f -> Fix f -> Ordering

(<) :: Fix f -> Fix f -> Bool

(<=) :: Fix f -> Fix f -> Bool

(>) :: Fix f -> Fix f -> Bool

(>=) :: Fix f -> Fix f -> Bool

max :: Fix f -> Fix f -> Fix f

min :: Fix f -> Fix f -> Fix f

Read (f (Fix f)) => Read (Fix f)

Methods

readsPrec :: Int -> ReadS (Fix f)

readList :: ReadS [Fix f]

readPrec :: ReadPrec (Fix f)

readListPrec :: ReadPrec [Fix f]

Show (f (Fix f)) => Show (Fix f)

Methods

showsPrec :: Int -> Fix f -> ShowS

show :: Fix f -> String

showList :: [Fix f] -> ShowS

Generic (Fix f)

Associated Types

type Rep (Fix f) :: * -> *

Methods

from :: Fix f -> Rep (Fix f) x

to :: Rep (Fix f) x -> Fix f

type Rep (Fix f)

type Rep (Fix f) = D1 (MetaData "Fix" "Data.Fix" "data-fix-0.2.0-1k1ciJC0eHJziJI3WrHqP" True) (C1 (MetaCons "Fix" PrefixI True) (S1 (MetaSel (Just Symbol "unFix") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 (f (Fix f)))))

Simple recursion

Type f should be a Functor. They transform non-recursive functions to recursive ones.

cata :: Functor f => (f a -> a) -> Fix f -> a

Catamorphism or generic function fold.

ana :: Functor f => (a -> f a) -> a -> Fix f

Anamorphism or generic function unfold.

hylo :: Functor f => (f b -> b) -> (a -> f a) -> a -> b

Hylomorphism is anamorphism followed by catamorphism.

(~>) :: Functor f => (a -> f a) -> (f b -> b) -> a -> b

Infix version of hylo.

Monadic recursion

Type f should be a Traversable.

cataM :: (Applicative m, Monad m, Traversable t) => (t a -> m a) -> Fix t -> m a

Monadic catamorphism.

anaM :: (Applicative m, Monad m, Traversable t) => (a -> m (t a)) -> a -> m (Fix t)

Monadic anamorphism.

hyloM :: (Applicative m, Monad m, Traversable t) => (t b -> m b) -> (a -> m (t a)) -> a -> m b

Monadic hylomorphism.