Data.Functor.Foldable

Base functors

type family Base t :: * -> *

Obtain the base functor for a recursive datatype.

The core idea of this library is that instead of writing recursive functions on a recursive datatype, we prefer to write non-recursive functions on a related, non-recursive datatype we call the "base functor".

For example, [a] is a recursive type, and its corresponding base functor is ListF a:

data ListF a b = Nil | Cons a b
type instance Base [a] = ListF a

The relationship between those two types is that if we replace b with ListF a, we obtain a type which is isomorphic to [a].

Instances

type Base Natural

Defined in Data.Functor.Foldable

type Base Natural = Maybe

type Base [a]

Defined in Data.Functor.Foldable

type Base [a] = ListF a

type Base (Maybe a)

Example boring stub for non-recursive data types

Defined in Data.Functor.Foldable

type Base (Maybe a) = Const (Maybe a) :: Type -> Type

type Base (NonEmpty a)

Defined in Data.Functor.Foldable

type Base (NonEmpty a) = NonEmptyF a

type Base (Tree a)

Defined in Data.Functor.Foldable

type Base (Tree a) = TreeF a

type Base (Fix f)

Defined in Data.Functor.Foldable

type Base (Fix f) = f

type Base (Mu f)

Defined in Data.Functor.Foldable

type Base (Mu f) = f

type Base (Nu f)

Defined in Data.Functor.Foldable

type Base (Nu f) = f

type Base (Either a b)

Example boring stub for non-recursive data types

Defined in Data.Functor.Foldable

type Base (Either a b) = Const (Either a b) :: Type -> Type

type Base (Cofree f a)

Cofree comonads are Recursive/Corecursive

Defined in Data.Functor.Foldable

type Base (Cofree f a) = CofreeF f a

type Base (Free f a)

Free monads are Recursive/Corecursive

Defined in Data.Functor.Foldable

type Base (Free f a) = FreeF f a

type Base (F f a)

Church encoded free monads are Recursive/Corecursive, in the same way that Mu is.

Defined in Data.Functor.Foldable

type Base (F f a) = FreeF f a

type Base (CofreeT f w a)

Cofree tranformations of comonads are Recursive/Corecusive

Defined in Data.Functor.Foldable

type Base (CofreeT f w a) = Compose w (CofreeF f a)

type Base (FreeT f m a)

Free transformations of monads are Recursive/Corecursive

Defined in Data.Functor.Foldable

type Base (FreeT f m a) = Compose m (FreeF f a)

data ListF a b

Base functor of [].

Constructors

Nil

Cons a b

Instances

Bitraversable ListF

Defined in Data.Functor.Base

Methods

bitraverse :: Applicative f => (a -> f c) -> (b -> f d) -> ListF a b -> f (ListF c d)

Bifoldable ListF

Defined in Data.Functor.Base

Methods

bifold :: Monoid m => ListF m m -> m

bifoldMap :: Monoid m => (a -> m) -> (b -> m) -> ListF a b -> m

bifoldr :: (a -> c -> c) -> (b -> c -> c) -> c -> ListF a b -> c

bifoldl :: (c -> a -> c) -> (c -> b -> c) -> c -> ListF a b -> c

Bifunctor ListF

Defined in Data.Functor.Base

Methods

bimap :: (a -> b) -> (c -> d) -> ListF a c -> ListF b d

first :: (a -> b) -> ListF a c -> ListF b c

second :: (b -> c) -> ListF a b -> ListF a c

Eq2 ListF

Defined in Data.Functor.Base

Methods

liftEq2 :: (a -> b -> Bool) -> (c -> d -> Bool) -> ListF a c -> ListF b d -> Bool

Ord2 ListF

Defined in Data.Functor.Base

Methods

liftCompare2 :: (a -> b -> Ordering) -> (c -> d -> Ordering) -> ListF a c -> ListF b d -> Ordering

Read2 ListF

Defined in Data.Functor.Base

Methods

liftReadsPrec2 :: (Int -> ReadS a) -> ReadS [a] -> (Int -> ReadS b) -> ReadS [b] -> Int -> ReadS (ListF a b)

liftReadList2 :: (Int -> ReadS a) -> ReadS [a] -> (Int -> ReadS b) -> ReadS [b] -> ReadS [ListF a b]

liftReadPrec2 :: ReadPrec a -> ReadPrec [a] -> ReadPrec b -> ReadPrec [b] -> ReadPrec (ListF a b)

liftReadListPrec2 :: ReadPrec a -> ReadPrec [a] -> ReadPrec b -> ReadPrec [b] -> ReadPrec [ListF a b]

Show2 ListF

Defined in Data.Functor.Base

Methods

liftShowsPrec2 :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> (Int -> b -> ShowS) -> ([b] -> ShowS) -> Int -> ListF a b -> ShowS

liftShowList2 :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> (Int -> b -> ShowS) -> ([b] -> ShowS) -> [ListF a b] -> ShowS

Functor (ListF a)

Defined in Data.Functor.Base

Methods

fmap :: (a0 -> b) -> ListF a a0 -> ListF a b

(<$) :: a0 -> ListF a b -> ListF a a0

Foldable (ListF a)

Defined in Data.Functor.Base

Methods

fold :: Monoid m => ListF a m -> m

foldMap :: Monoid m => (a0 -> m) -> ListF a a0 -> m

foldMap' :: Monoid m => (a0 -> m) -> ListF a a0 -> m

foldr :: (a0 -> b -> b) -> b -> ListF a a0 -> b

foldr' :: (a0 -> b -> b) -> b -> ListF a a0 -> b

foldl :: (b -> a0 -> b) -> b -> ListF a a0 -> b

foldl' :: (b -> a0 -> b) -> b -> ListF a a0 -> b

foldr1 :: (a0 -> a0 -> a0) -> ListF a a0 -> a0

foldl1 :: (a0 -> a0 -> a0) -> ListF a a0 -> a0

toList :: ListF a a0 -> [a0]

null :: ListF a a0 -> Bool

length :: ListF a a0 -> Int

elem :: Eq a0 => a0 -> ListF a a0 -> Bool

maximum :: Ord a0 => ListF a a0 -> a0

minimum :: Ord a0 => ListF a a0 -> a0

sum :: Num a0 => ListF a a0 -> a0

product :: Num a0 => ListF a a0 -> a0

Traversable (ListF a)

Defined in Data.Functor.Base

Methods

traverse :: Applicative f => (a0 -> f b) -> ListF a a0 -> f (ListF a b)

sequenceA :: Applicative f => ListF a (f a0) -> f (ListF a a0)

mapM :: Monad m => (a0 -> m b) -> ListF a a0 -> m (ListF a b)

sequence :: Monad m => ListF a (m a0) -> m (ListF a a0)

Eq a => Eq1 (ListF a)

Defined in Data.Functor.Base

Methods

liftEq :: (a0 -> b -> Bool) -> ListF a a0 -> ListF a b -> Bool

Ord a => Ord1 (ListF a)

Defined in Data.Functor.Base

Methods

liftCompare :: (a0 -> b -> Ordering) -> ListF a a0 -> ListF a b -> Ordering

Read a => Read1 (ListF a)

Defined in Data.Functor.Base

Methods

liftReadsPrec :: (Int -> ReadS a0) -> ReadS [a0] -> Int -> ReadS (ListF a a0)

liftReadList :: (Int -> ReadS a0) -> ReadS [a0] -> ReadS [ListF a a0]

liftReadPrec :: ReadPrec a0 -> ReadPrec [a0] -> ReadPrec (ListF a a0)

liftReadListPrec :: ReadPrec a0 -> ReadPrec [a0] -> ReadPrec [ListF a a0]

Show a => Show1 (ListF a)

Defined in Data.Functor.Base

Methods

liftShowsPrec :: (Int -> a0 -> ShowS) -> ([a0] -> ShowS) -> Int -> ListF a a0 -> ShowS

liftShowList :: (Int -> a0 -> ShowS) -> ([a0] -> ShowS) -> [ListF a a0] -> ShowS

Generic1 (ListF a :: Type -> Type)

Defined in Data.Functor.Base

Associated Types

type Rep1 (ListF a) :: k -> Type

Methods

from1 :: forall (a0 :: k). ListF a a0 -> Rep1 (ListF a) a0

to1 :: forall (a0 :: k). Rep1 (ListF a) a0 -> ListF a a0

(Eq a, Eq b) => Eq (ListF a b)

Defined in Data.Functor.Base

Methods

(==) :: ListF a b -> ListF a b -> Bool

(/=) :: ListF a b -> ListF a b -> Bool

(Ord a, Ord b) => Ord (ListF a b)

Defined in Data.Functor.Base

Methods

compare :: ListF a b -> ListF a b -> Ordering

(<) :: ListF a b -> ListF a b -> Bool

(<=) :: ListF a b -> ListF a b -> Bool

(>) :: ListF a b -> ListF a b -> Bool

(>=) :: ListF a b -> ListF a b -> Bool

max :: ListF a b -> ListF a b -> ListF a b

min :: ListF a b -> ListF a b -> ListF a b

(Read a, Read b) => Read (ListF a b)

Defined in Data.Functor.Base

Methods

readsPrec :: Int -> ReadS (ListF a b)

readList :: ReadS [ListF a b]

readPrec :: ReadPrec (ListF a b)

readListPrec :: ReadPrec [ListF a b]

(Show a, Show b) => Show (ListF a b)

Defined in Data.Functor.Base

Methods

showsPrec :: Int -> ListF a b -> ShowS

show :: ListF a b -> String

showList :: [ListF a b] -> ShowS

Generic (ListF a b)

Defined in Data.Functor.Base

Associated Types

type Rep (ListF a b) :: Type -> Type

Methods

from :: ListF a b -> Rep (ListF a b) x

to :: Rep (ListF a b) x -> ListF a b

type Rep1 (ListF a :: Type -> Type)

Defined in Data.Functor.Base

type Rep1 (ListF a :: Type -> Type) = D1 ('MetaData "ListF" "Data.Functor.Base" "recursion-schemes-5.2.2.2-inplace" 'False) (C1 ('MetaCons "Nil" 'PrefixI 'False) (U1 :: Type -> Type) :+: C1 ('MetaCons "Cons" 'PrefixI 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 a) :*: S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) Par1))

type Rep (ListF a b)

Defined in Data.Functor.Base

type Rep (ListF a b) = D1 ('MetaData "ListF" "Data.Functor.Base" "recursion-schemes-5.2.2.2-inplace" 'False) (C1 ('MetaCons "Nil" 'PrefixI 'False) (U1 :: Type -> Type) :+: C1 ('MetaCons "Cons" 'PrefixI 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 a) :*: S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 b)))

Type classes

class Functor (Base t) => Recursive t where

A recursive datatype which can be unrolled one recursion layer at a time.

For example, a value of type [a] can be unrolled into a ListF a [a]. Ifthat unrolled value is a Cons, it contains another [a] which can be unrolled as well, and so on.

Typically, Recursive types also have a Corecursive instance, in which case project and embed are inverses.

Minimal complete definition

Nothing

Methods

project :: t -> Base t t

Unroll a single recursion layer.

>>> project [1,2,3]
Cons 1 [2,3]

default project :: (Generic t, Generic (Base t t), GCoerce (Rep t) (Rep (Base t t))) => t -> Base t t

Instances

Recursive Natural

Defined in Data.Functor.Foldable

Methods

project :: Natural -> Base Natural Natural

cata :: (Base Natural a -> a) -> Natural -> a

para :: (Base Natural (Natural, a) -> a) -> Natural -> a

gpara :: (Corecursive Natural, Comonad w) => (forall b. Base Natural (w b) -> w (Base Natural b)) -> (Base Natural (EnvT Natural w a) -> a) -> Natural -> a

prepro :: Corecursive Natural => (forall b. Base Natural b -> Base Natural b) -> (Base Natural a -> a) -> Natural -> a

gprepro :: (Corecursive Natural, Comonad w) => (forall b. Base Natural (w b) -> w (Base Natural b)) -> (forall c. Base Natural c -> Base Natural c) -> (Base Natural (w a) -> a) -> Natural -> a

Recursive [a]

Defined in Data.Functor.Foldable

Methods

project :: [a] -> Base [a] [a]

cata :: (Base [a] a0 -> a0) -> [a] -> a0

para :: (Base [a] ([a], a0) -> a0) -> [a] -> a0

gpara :: (Corecursive [a], Comonad w) => (forall b. Base [a] (w b) -> w (Base [a] b)) -> (Base [a] (EnvT [a] w a0) -> a0) -> [a] -> a0

prepro :: Corecursive [a] => (forall b. Base [a] b -> Base [a] b) -> (Base [a] a0 -> a0) -> [a] -> a0

gprepro :: (Corecursive [a], Comonad w) => (forall b. Base [a] (w b) -> w (Base [a] b)) -> (forall c. Base [a] c -> Base [a] c) -> (Base [a] (w a0) -> a0) -> [a] -> a0

Recursive (Maybe a)

Defined in Data.Functor.Foldable

Methods

project :: Maybe a -> Base (Maybe a) (Maybe a)

cata :: (Base (Maybe a) a0 -> a0) -> Maybe a -> a0

para :: (Base (Maybe a) (Maybe a, a0) -> a0) -> Maybe a -> a0

gpara :: (Corecursive (Maybe a), Comonad w) => (forall b. Base (Maybe a) (w b) -> w (Base (Maybe a) b)) -> (Base (Maybe a) (EnvT (Maybe a) w a0) -> a0) -> Maybe a -> a0

prepro :: Corecursive (Maybe a) => (forall b. Base (Maybe a) b -> Base (Maybe a) b) -> (Base (Maybe a) a0 -> a0) -> Maybe a -> a0

gprepro :: (Corecursive (Maybe a), Comonad w) => (forall b. Base (Maybe a) (w b) -> w (Base (Maybe a) b)) -> (forall c. Base (Maybe a) c -> Base (Maybe a) c) -> (Base (Maybe a) (w a0) -> a0) -> Maybe a -> a0

Recursive (NonEmpty a)

Defined in Data.Functor.Foldable

Methods

project :: NonEmpty a -> Base (NonEmpty a) (NonEmpty a)

cata :: (Base (NonEmpty a) a0 -> a0) -> NonEmpty a -> a0

para :: (Base (NonEmpty a) (NonEmpty a, a0) -> a0) -> NonEmpty a -> a0

gpara :: (Corecursive (NonEmpty a), Comonad w) => (forall b. Base (NonEmpty a) (w b) -> w (Base (NonEmpty a) b)) -> (Base (NonEmpty a) (EnvT (NonEmpty a) w a0) -> a0) -> NonEmpty a -> a0

prepro :: Corecursive (NonEmpty a) => (forall b. Base (NonEmpty a) b -> Base (NonEmpty a) b) -> (Base (NonEmpty a) a0 -> a0) -> NonEmpty a -> a0

gprepro :: (Corecursive (NonEmpty a), Comonad w) => (forall b. Base (NonEmpty a) (w b) -> w (Base (NonEmpty a) b)) -> (forall c. Base (NonEmpty a) c -> Base (NonEmpty a) c) -> (Base (NonEmpty a) (w a0) -> a0) -> NonEmpty a -> a0

Recursive (Tree a)

Defined in Data.Functor.Foldable

Methods

project :: Tree a -> Base (Tree a) (Tree a)

cata :: (Base (Tree a) a0 -> a0) -> Tree a -> a0

para :: (Base (Tree a) (Tree a, a0) -> a0) -> Tree a -> a0

gpara :: (Corecursive (Tree a), Comonad w) => (forall b. Base (Tree a) (w b) -> w (Base (Tree a) b)) -> (Base (Tree a) (EnvT (Tree a) w a0) -> a0) -> Tree a -> a0

prepro :: Corecursive (Tree a) => (forall b. Base (Tree a) b -> Base (Tree a) b) -> (Base (Tree a) a0 -> a0) -> Tree a -> a0

gprepro :: (Corecursive (Tree a), Comonad w) => (forall b. Base (Tree a) (w b) -> w (Base (Tree a) b)) -> (forall c. Base (Tree a) c -> Base (Tree a) c) -> (Base (Tree a) (w a0) -> a0) -> Tree a -> a0

Functor f => Recursive (Fix f)

Defined in Data.Functor.Foldable

Methods

project :: Fix f -> Base (Fix f) (Fix f)

cata :: (Base (Fix f) a -> a) -> Fix f -> a

para :: (Base (Fix f) (Fix f, a) -> a) -> Fix f -> a

gpara :: (Corecursive (Fix f), Comonad w) => (forall b. Base (Fix f) (w b) -> w (Base (Fix f) b)) -> (Base (Fix f) (EnvT (Fix f) w a) -> a) -> Fix f -> a

prepro :: Corecursive (Fix f) => (forall b. Base (Fix f) b -> Base (Fix f) b) -> (Base (Fix f) a -> a) -> Fix f -> a

gprepro :: (Corecursive (Fix f), Comonad w) => (forall b. Base (Fix f) (w b) -> w (Base (Fix f) b)) -> (forall c. Base (Fix f) c -> Base (Fix f) c) -> (Base (Fix f) (w a) -> a) -> Fix f -> a

Functor f => Recursive (Mu f)

Defined in Data.Functor.Foldable

Methods

project :: Mu f -> Base (Mu f) (Mu f)

cata :: (Base (Mu f) a -> a) -> Mu f -> a

para :: (Base (Mu f) (Mu f, a) -> a) -> Mu f -> a

gpara :: (Corecursive (Mu f), Comonad w) => (forall b. Base (Mu f) (w b) -> w (Base (Mu f) b)) -> (Base (Mu f) (EnvT (Mu f) w a) -> a) -> Mu f -> a

prepro :: Corecursive (Mu f) => (forall b. Base (Mu f) b -> Base (Mu f) b) -> (Base (Mu f) a -> a) -> Mu f -> a

gprepro :: (Corecursive (Mu f), Comonad w) => (forall b. Base (Mu f) (w b) -> w (Base (Mu f) b)) -> (forall c. Base (Mu f) c -> Base (Mu f) c) -> (Base (Mu f) (w a) -> a) -> Mu f -> a

Functor f => Recursive (Nu f)

Defined in Data.Functor.Foldable

Methods

project :: Nu f -> Base (Nu f) (Nu f)

cata :: (Base (Nu f) a -> a) -> Nu f -> a

para :: (Base (Nu f) (Nu f, a) -> a) -> Nu f -> a

gpara :: (Corecursive (Nu f), Comonad w) => (forall b. Base (Nu f) (w b) -> w (Base (Nu f) b)) -> (Base (Nu f) (EnvT (Nu f) w a) -> a) -> Nu f -> a

prepro :: Corecursive (Nu f) => (forall b. Base (Nu f) b -> Base (Nu f) b) -> (Base (Nu f) a -> a) -> Nu f -> a

gprepro :: (Corecursive (Nu f), Comonad w) => (forall b. Base (Nu f) (w b) -> w (Base (Nu f) b)) -> (forall c. Base (Nu f) c -> Base (Nu f) c) -> (Base (Nu f) (w a) -> a) -> Nu f -> a

Recursive (Either a b)

Defined in Data.Functor.Foldable

Methods

project :: Either a b -> Base (Either a b) (Either a b)

cata :: (Base (Either a b) a0 -> a0) -> Either a b -> a0

para :: (Base (Either a b) (Either a b, a0) -> a0) -> Either a b -> a0

gpara :: (Corecursive (Either a b), Comonad w) => (forall b0. Base (Either a b) (w b0) -> w (Base (Either a b) b0)) -> (Base (Either a b) (EnvT (Either a b) w a0) -> a0) -> Either a b -> a0

prepro :: Corecursive (Either a b) => (forall b0. Base (Either a b) b0 -> Base (Either a b) b0) -> (Base (Either a b) a0 -> a0) -> Either a b -> a0

gprepro :: (Corecursive (Either a b), Comonad w) => (forall b0. Base (Either a b) (w b0) -> w (Base (Either a b) b0)) -> (forall c. Base (Either a b) c -> Base (Either a b) c) -> (Base (Either a b) (w a0) -> a0) -> Either a b -> a0

Functor f => Recursive (Cofree f a)

Defined in Data.Functor.Foldable

Methods

project :: Cofree f a -> Base (Cofree f a) (Cofree f a)

cata :: (Base (Cofree f a) a0 -> a0) -> Cofree f a -> a0

para :: (Base (Cofree f a) (Cofree f a, a0) -> a0) -> Cofree f a -> a0

gpara :: (Corecursive (Cofree f a), Comonad w) => (forall b. Base (Cofree f a) (w b) -> w (Base (Cofree f a) b)) -> (Base (Cofree f a) (EnvT (Cofree f a) w a0) -> a0) -> Cofree f a -> a0

prepro :: Corecursive (Cofree f a) => (forall b. Base (Cofree f a) b -> Base (Cofree f a) b) -> (Base (Cofree f a) a0 -> a0) -> Cofree f a -> a0

gprepro :: (Corecursive (Cofree f a), Comonad w) => (forall b. Base (Cofree f a) (w b) -> w (Base (Cofree f a) b)) -> (forall c. Base (Cofree f a) c -> Base (Cofree f a) c) -> (Base (Cofree f a) (w a0) -> a0) -> Cofree f a -> a0

Functor f => Recursive (Free f a)

Defined in Data.Functor.Foldable

Methods

project :: Free f a -> Base (Free f a) (Free f a)

cata :: (Base (Free f a) a0 -> a0) -> Free f a -> a0

para :: (Base (Free f a) (Free f a, a0) -> a0) -> Free f a -> a0

gpara :: (Corecursive (Free f a), Comonad w) => (forall b. Base (Free f a) (w b) -> w (Base (Free f a) b)) -> (Base (Free f a) (EnvT (Free f a) w a0) -> a0) -> Free f a -> a0

prepro :: Corecursive (Free f a) => (forall b. Base (Free f a) b -> Base (Free f a) b) -> (Base (Free f a) a0 -> a0) -> Free f a -> a0

gprepro :: (Corecursive (Free f a), Comonad w) => (forall b. Base (Free f a) (w b) -> w (Base (Free f a) b)) -> (forall c. Base (Free f a) c -> Base (Free f a) c) -> (Base (Free f a) (w a0) -> a0) -> Free f a -> a0

Functor f => Recursive (F f a)

Defined in Data.Functor.Foldable

Methods

project :: F f a -> Base (F f a) (F f a)

cata :: (Base (F f a) a0 -> a0) -> F f a -> a0

para :: (Base (F f a) (F f a, a0) -> a0) -> F f a -> a0

gpara :: (Corecursive (F f a), Comonad w) => (forall b. Base (F f a) (w b) -> w (Base (F f a) b)) -> (Base (F f a) (EnvT (F f a) w a0) -> a0) -> F f a -> a0

prepro :: Corecursive (F f a) => (forall b. Base (F f a) b -> Base (F f a) b) -> (Base (F f a) a0 -> a0) -> F f a -> a0

gprepro :: (Corecursive (F f a), Comonad w) => (forall b. Base (F f a) (w b) -> w (Base (F f a) b)) -> (forall c. Base (F f a) c -> Base (F f a) c) -> (Base (F f a) (w a0) -> a0) -> F f a -> a0

(Functor w, Functor f) => Recursive (CofreeT f w a)

Defined in Data.Functor.Foldable

Methods

project :: CofreeT f w a -> Base (CofreeT f w a) (CofreeT f w a)

cata :: (Base (CofreeT f w a) a0 -> a0) -> CofreeT f w a -> a0

para :: (Base (CofreeT f w a) (CofreeT f w a, a0) -> a0) -> CofreeT f w a -> a0

gpara :: (Corecursive (CofreeT f w a), Comonad w0) => (forall b. Base (CofreeT f w a) (w0 b) -> w0 (Base (CofreeT f w a) b)) -> (Base (CofreeT f w a) (EnvT (CofreeT f w a) w0 a0) -> a0) -> CofreeT f w a -> a0

prepro :: Corecursive (CofreeT f w a) => (forall b. Base (CofreeT f w a) b -> Base (CofreeT f w a) b) -> (Base (CofreeT f w a) a0 -> a0) -> CofreeT f w a -> a0

gprepro :: (Corecursive (CofreeT f w a), Comonad w0) => (forall b. Base (CofreeT f w a) (w0 b) -> w0 (Base (CofreeT f w a) b)) -> (forall c. Base (CofreeT f w a) c -> Base (CofreeT f w a) c) -> (Base (CofreeT f w a) (w0 a0) -> a0) -> CofreeT f w a -> a0

(Functor m, Functor f) => Recursive (FreeT f m a)

Defined in Data.Functor.Foldable

Methods

project :: FreeT f m a -> Base (FreeT f m a) (FreeT f m a)

cata :: (Base (FreeT f m a) a0 -> a0) -> FreeT f m a -> a0

para :: (Base (FreeT f m a) (FreeT f m a, a0) -> a0) -> FreeT f m a -> a0

gpara :: (Corecursive (FreeT f m a), Comonad w) => (forall b. Base (FreeT f m a) (w b) -> w (Base (FreeT f m a) b)) -> (Base (FreeT f m a) (EnvT (FreeT f m a) w a0) -> a0) -> FreeT f m a -> a0

prepro :: Corecursive (FreeT f m a) => (forall b. Base (FreeT f m a) b -> Base (FreeT f m a) b) -> (Base (FreeT f m a) a0 -> a0) -> FreeT f m a -> a0

gprepro :: (Corecursive (FreeT f m a), Comonad w) => (forall b. Base (FreeT f m a) (w b) -> w (Base (FreeT f m a) b)) -> (forall c. Base (FreeT f m a) c -> Base (FreeT f m a) c) -> (Base (FreeT f m a) (w a0) -> a0) -> FreeT f m a -> a0

class Functor (Base t) => Corecursive t where

A recursive datatype which can be rolled up one recursion layer at a time.

For example, a value of type ListF a [a] can be rolled up into a [a]. This [a] can then be used in a Cons to construct another ListF a [a], which can be rolled up as well, and so on.

Typically, Corecursive types also have a Recursive instance, in which case embed and project are inverses.

Minimal complete definition

Nothing

Methods

embed :: Base t t -> t

Roll up a single recursion layer.

>>> embed (Cons 1 [2,3])
[1,2,3]

default embed :: (Generic t, Generic (Base t t), GCoerce (Rep (Base t t)) (Rep t)) => Base t t -> t

Instances

Corecursive Natural

Defined in Data.Functor.Foldable

Methods

embed :: Base Natural Natural -> Natural

ana :: (a -> Base Natural a) -> a -> Natural

apo :: (a -> Base Natural (Either Natural a)) -> a -> Natural

postpro :: Recursive Natural => (forall b. Base Natural b -> Base Natural b) -> (a -> Base Natural a) -> a -> Natural

gpostpro :: (Recursive Natural, Monad m) => (forall b. m (Base Natural b) -> Base Natural (m b)) -> (forall c. Base Natural c -> Base Natural c) -> (a -> Base Natural (m a)) -> a -> Natural

Corecursive [a]

Defined in Data.Functor.Foldable

Methods

embed :: Base [a] [a] -> [a]

ana :: (a0 -> Base [a] a0) -> a0 -> [a]

apo :: (a0 -> Base [a] (Either [a] a0)) -> a0 -> [a]

postpro :: Recursive [a] => (forall b. Base [a] b -> Base [a] b) -> (a0 -> Base [a] a0) -> a0 -> [a]

gpostpro :: (Recursive [a], Monad m) => (forall b. m (Base [a] b) -> Base [a] (m b)) -> (forall c. Base [a] c -> Base [a] c) -> (a0 -> Base [a] (m a0)) -> a0 -> [a]

Corecursive (Maybe a)

Defined in Data.Functor.Foldable

Methods

embed :: Base (Maybe a) (Maybe a) -> Maybe a

ana :: (a0 -> Base (Maybe a) a0) -> a0 -> Maybe a

apo :: (a0 -> Base (Maybe a) (Either (Maybe a) a0)) -> a0 -> Maybe a

postpro :: Recursive (Maybe a) => (forall b. Base (Maybe a) b -> Base (Maybe a) b) -> (a0 -> Base (Maybe a) a0) -> a0 -> Maybe a

gpostpro :: (Recursive (Maybe a), Monad m) => (forall b. m (Base (Maybe a) b) -> Base (Maybe a) (m b)) -> (forall c. Base (Maybe a) c -> Base (Maybe a) c) -> (a0 -> Base (Maybe a) (m a0)) -> a0 -> Maybe a

Corecursive (NonEmpty a)

Defined in Data.Functor.Foldable

Methods

embed :: Base (NonEmpty a) (NonEmpty a) -> NonEmpty a

ana :: (a0 -> Base (NonEmpty a) a0) -> a0 -> NonEmpty a

apo :: (a0 -> Base (NonEmpty a) (Either (NonEmpty a) a0)) -> a0 -> NonEmpty a

postpro :: Recursive (NonEmpty a) => (forall b. Base (NonEmpty a) b -> Base (NonEmpty a) b) -> (a0 -> Base (NonEmpty a) a0) -> a0 -> NonEmpty a

gpostpro :: (Recursive (NonEmpty a), Monad m) => (forall b. m (Base (NonEmpty a) b) -> Base (NonEmpty a) (m b)) -> (forall c. Base (NonEmpty a) c -> Base (NonEmpty a) c) -> (a0 -> Base (NonEmpty a) (m a0)) -> a0 -> NonEmpty a

Corecursive (Tree a)

Defined in Data.Functor.Foldable

Methods

embed :: Base (Tree a) (Tree a) -> Tree a

ana :: (a0 -> Base (Tree a) a0) -> a0 -> Tree a

apo :: (a0 -> Base (Tree a) (Either (Tree a) a0)) -> a0 -> Tree a

postpro :: Recursive (Tree a) => (forall b. Base (Tree a) b -> Base (Tree a) b) -> (a0 -> Base (Tree a) a0) -> a0 -> Tree a

gpostpro :: (Recursive (Tree a), Monad m) => (forall b. m (Base (Tree a) b) -> Base (Tree a) (m b)) -> (forall c. Base (Tree a) c -> Base (Tree a) c) -> (a0 -> Base (Tree a) (m a0)) -> a0 -> Tree a

Functor f => Corecursive (Fix f)

Defined in Data.Functor.Foldable

Methods

embed :: Base (Fix f) (Fix f) -> Fix f

ana :: (a -> Base (Fix f) a) -> a -> Fix f

apo :: (a -> Base (Fix f) (Either (Fix f) a)) -> a -> Fix f

postpro :: Recursive (Fix f) => (forall b. Base (Fix f) b -> Base (Fix f) b) -> (a -> Base (Fix f) a) -> a -> Fix f

gpostpro :: (Recursive (Fix f), Monad m) => (forall b. m (Base (Fix f) b) -> Base (Fix f) (m b)) -> (forall c. Base (Fix f) c -> Base (Fix f) c) -> (a -> Base (Fix f) (m a)) -> a -> Fix f

Functor f => Corecursive (Mu f)

Defined in Data.Functor.Foldable

Methods

embed :: Base (Mu f) (Mu f) -> Mu f

ana :: (a -> Base (Mu f) a) -> a -> Mu f

apo :: (a -> Base (Mu f) (Either (Mu f) a)) -> a -> Mu f

postpro :: Recursive (Mu f) => (forall b. Base (Mu f) b -> Base (Mu f) b) -> (a -> Base (Mu f) a) -> a -> Mu f

gpostpro :: (Recursive (Mu f), Monad m) => (forall b. m (Base (Mu f) b) -> Base (Mu f) (m b)) -> (forall c. Base (Mu f) c -> Base (Mu f) c) -> (a -> Base (Mu f) (m a)) -> a -> Mu f

Functor f => Corecursive (Nu f)

Defined in Data.Functor.Foldable

Methods

embed :: Base (Nu f) (Nu f) -> Nu f

ana :: (a -> Base (Nu f) a) -> a -> Nu f

apo :: (a -> Base (Nu f) (Either (Nu f) a)) -> a -> Nu f

postpro :: Recursive (Nu f) => (forall b. Base (Nu f) b -> Base (Nu f) b) -> (a -> Base (Nu f) a) -> a -> Nu f

gpostpro :: (Recursive (Nu f), Monad m) => (forall b. m (Base (Nu f) b) -> Base (Nu f) (m b)) -> (forall c. Base (Nu f) c -> Base (Nu f) c) -> (a -> Base (Nu f) (m a)) -> a -> Nu f

Corecursive (Either a b)

Defined in Data.Functor.Foldable

Methods

embed :: Base (Either a b) (Either a b) -> Either a b

ana :: (a0 -> Base (Either a b) a0) -> a0 -> Either a b

apo :: (a0 -> Base (Either a b) (Either (Either a b) a0)) -> a0 -> Either a b

postpro :: Recursive (Either a b) => (forall b0. Base (Either a b) b0 -> Base (Either a b) b0) -> (a0 -> Base (Either a b) a0) -> a0 -> Either a b

gpostpro :: (Recursive (Either a b), Monad m) => (forall b0. m (Base (Either a b) b0) -> Base (Either a b) (m b0)) -> (forall c. Base (Either a b) c -> Base (Either a b) c) -> (a0 -> Base (Either a b) (m a0)) -> a0 -> Either a b

Functor f => Corecursive (Cofree f a)

Defined in Data.Functor.Foldable

Methods

embed :: Base (Cofree f a) (Cofree f a) -> Cofree f a

ana :: (a0 -> Base (Cofree f a) a0) -> a0 -> Cofree f a

apo :: (a0 -> Base (Cofree f a) (Either (Cofree f a) a0)) -> a0 -> Cofree f a

postpro :: Recursive (Cofree f a) => (forall b. Base (Cofree f a) b -> Base (Cofree f a) b) -> (a0 -> Base (Cofree f a) a0) -> a0 -> Cofree f a

gpostpro :: (Recursive (Cofree f a), Monad m) => (forall b. m (Base (Cofree f a) b) -> Base (Cofree f a) (m b)) -> (forall c. Base (Cofree f a) c -> Base (Cofree f a) c) -> (a0 -> Base (Cofree f a) (m a0)) -> a0 -> Cofree f a

Functor f => Corecursive (Free f a)

It may be better to work with the instance for F directly.

Defined in Data.Functor.Foldable

Methods

embed :: Base (Free f a) (Free f a) -> Free f a

ana :: (a0 -> Base (Free f a) a0) -> a0 -> Free f a

apo :: (a0 -> Base (Free f a) (Either (Free f a) a0)) -> a0 -> Free f a

postpro :: Recursive (Free f a) => (forall b. Base (Free f a) b -> Base (Free f a) b) -> (a0 -> Base (Free f a) a0) -> a0 -> Free f a

gpostpro :: (Recursive (Free f a), Monad m) => (forall b. m (Base (Free f a) b) -> Base (Free f a) (m b)) -> (forall c. Base (Free f a) c -> Base (Free f a) c) -> (a0 -> Base (Free f a) (m a0)) -> a0 -> Free f a

Functor f => Corecursive (F f a)

Defined in Data.Functor.Foldable

Methods

embed :: Base (F f a) (F f a) -> F f a

ana :: (a0 -> Base (F f a) a0) -> a0 -> F f a

apo :: (a0 -> Base (F f a) (Either (F f a) a0)) -> a0 -> F f a

postpro :: Recursive (F f a) => (forall b. Base (F f a) b -> Base (F f a) b) -> (a0 -> Base (F f a) a0) -> a0 -> F f a

gpostpro :: (Recursive (F f a), Monad m) => (forall b. m (Base (F f a) b) -> Base (F f a) (m b)) -> (forall c. Base (F f a) c -> Base (F f a) c) -> (a0 -> Base (F f a) (m a0)) -> a0 -> F f a

(Functor w, Functor f) => Corecursive (CofreeT f w a)

Defined in Data.Functor.Foldable

Methods

embed :: Base (CofreeT f w a) (CofreeT f w a) -> CofreeT f w a

ana :: (a0 -> Base (CofreeT f w a) a0) -> a0 -> CofreeT f w a

apo :: (a0 -> Base (CofreeT f w a) (Either (CofreeT f w a) a0)) -> a0 -> CofreeT f w a

postpro :: Recursive (CofreeT f w a) => (forall b. Base (CofreeT f w a) b -> Base (CofreeT f w a) b) -> (a0 -> Base (CofreeT f w a) a0) -> a0 -> CofreeT f w a

gpostpro :: (Recursive (CofreeT f w a), Monad m) => (forall b. m (Base (CofreeT f w a) b) -> Base (CofreeT f w a) (m b)) -> (forall c. Base (CofreeT f w a) c -> Base (CofreeT f w a) c) -> (a0 -> Base (CofreeT f w a) (m a0)) -> a0 -> CofreeT f w a

(Functor m, Functor f) => Corecursive (FreeT f m a)

Defined in Data.Functor.Foldable

Methods

embed :: Base (FreeT f m a) (FreeT f m a) -> FreeT f m a

ana :: (a0 -> Base (FreeT f m a) a0) -> a0 -> FreeT f m a

apo :: (a0 -> Base (FreeT f m a) (Either (FreeT f m a) a0)) -> a0 -> FreeT f m a

postpro :: Recursive (FreeT f m a) => (forall b. Base (FreeT f m a) b -> Base (FreeT f m a) b) -> (a0 -> Base (FreeT f m a) a0) -> a0 -> FreeT f m a

gpostpro :: (Recursive (FreeT f m a), Monad m0) => (forall b. m0 (Base (FreeT f m a) b) -> Base (FreeT f m a) (m0 b)) -> (forall c. Base (FreeT f m a) c -> Base (FreeT f m a) c) -> (a0 -> Base (FreeT f m a) (m0 a0)) -> a0 -> FreeT f m a

Folding functions

Folding functions allow you to reduce a recursive structure down to a value. The value can be a simple type such as Int or String, or it can also be a recursive structure. Each of the functions below will be accompanied by an example which folds the following Tree Int down to some String.

>>> putStr $ drawTree $ fmap show myTree
0
|
+- 1
|
+- 2
|
`- 3
   |
   `- 31
      |
      `- 311
         |
         +- 3111
         |
         `- 3112

fold :: Recursive t => (Base t a -> a) -> t -> a

Folds a recursive type down to a value, one layer at a time.

>>> :{
let mySum :: [Int] -> Int
    mySum = fold $ \case
      Nil -> 0
      Cons x sumXs -> x + sumXs
:}

>>> mySum [10,11,12]
33

In our running example, one layer consists of an Int and a list of recursive positions. In Tree Int, those recursive positions contain sub-trees of type Tree Int. Since we are working one layer at a time, the Base t a -> a function is not given a Tree Int, but a TreeF Int String. That is, each recursive position contains the String resulting from recursively folding the corresponding sub-tree.

>>> :{
let pprint1 :: Tree Int -> String
    pprint1 = fold $ \case
      NodeF i [] -> show i
      NodeF i ss -> show i ++ ": [" ++ intercalate ", " ss ++ "]"
:}

>>> putStrLn $ pprint1 myTree
0: [1, 2, 3: [31: [311: [3111, 3112]]]]

More generally, the t argument is the recursive value, the a is the final result, and the Base t a -> a function explains how to reduce a single layer full of recursive results down to a result.

cata :: Recursive t => (Base t a -> a) -> t -> a

An alias for fold.

fold is by far the most common recursion-scheme, because working one layer at a time is the most common strategy for writing a recursive function. But there are also other, rarer strategies. Researchers have given names to the most common strategies, and their name for fold is "catamorphism". They also give its Base t a -> a argument a special name, "(Base t)-algebra". More generally, a function of the form f a -> a is called an "f-algebra".

The names might seem intimidating at first, but using the standard nomenclature has benefits. If you program with others, it can be useful to have a shared vocabulary to refer to those recursion patterns. For example, you can discuss which type of recursion is the most appropriate for the problem at hand. Names can also help to structure your thoughts while writing recursive functions.

The rest of this module lists a few of the other recursion-schemes which are common enough to have a name. In this section, we restrict our attention to those which fold a recursive structure down to a value. In the examples all functions will be of type Tree Int -> String.

cataA :: Recursive t => (Base t (f a) -> f a) -> t -> f a

A specialization of cata for effectful folds.

cataA is the same as cata, but with a more specialized type. The only reason it exists is to make it easier to discover how to use this library with effects.

For our running example, let's improve the output format of our pretty-printer by using indentation. To do so, we will need to keep track of the current indentation level. We will do so using a Reader Int effect. Our recursive positions will thus contain Reader Int String actions, not Strings. This means we need to run those actions in order to get the results.

>>> :{
let pprint2 :: Tree Int -> String
    pprint2 = flip runReader 0 . cataA go
      where
        go :: TreeF Int (Reader Int String)
           -> Reader Int String
        go (NodeF i rss) = do
          -- rss :: [Reader Int String]
          -- ss  :: [String]
          ss <- local (+ 2) $ sequence rss
          indent <- ask
          let s = replicate indent ' ' ++ "* " ++ show i
          pure $ intercalate "\n" (s : ss)
:}

>>> putStrLn $ pprint2 myTree
* 0
  * 1
  * 2
  * 3
    * 31
      * 311
        * 3111
        * 3112

The fact that the recursive positions contain Reader actions instead of Strings gives us some flexibility. Here, we are able to increase the indentation by running those actions inside a local block. More generally, we can control the order of their side-effects, interleave them with other effects, etc.

A similar technique is to specialize cata so that the result is a function. This makes it possible for data to flow down in addition to up. In this modified version of our running example, the indentation level flows down from the root to the leaves, while the resulting strings flow up from the leaves to the root.

>>> :{
let pprint3 :: Tree Int -> String
    pprint3 t = cataA go t 0
      where
        go :: TreeF Int (Int -> String)
           -> Int -> String
        go (NodeF i fs) indent
            -- fs :: [Int -> String]
          = let indent' = indent + 2
                ss = map (\f -> f indent') fs
                s = replicate indent ' ' ++ "* " ++ show i
            in intercalate "\n" (s : ss)
:}

>>> putStrLn $ pprint3 myTree
* 0
  * 1
  * 2
  * 3
    * 31
      * 311
        * 3111
        * 3112

para :: Recursive t => (Base t (t, a) -> a) -> t -> a

A variant of cata in which recursive positions also include the original sub-tree, in addition to the result of folding that sub-tree.

For our running example, let's add a number to each node indicating how many children are below it. To do so, we will need to count those nodes from the original sub-tree.

>>> :{
let pprint4 :: Tree Int -> String
    pprint4 = flip runReader 0 . para go
      where
        go :: TreeF Int (Tree Int, Reader Int String)
           -> Reader Int String
        go (NodeF i trss) = do
          -- trss :: [(Tree Int, Reader Int String)]
          -- ts   :: [Tree Int]
          -- rss  :: [Reader Int String]
          -- ss   :: [String]
          let (ts, rss) = unzip trss
          let count = sum $ fmap length ts
          ss <- local (+ 2) $ sequence rss
          indent <- ask
          let s = replicate indent ' '
               ++ "* " ++ show i
               ++ " (" ++ show count ++ ")"
          pure $ intercalate "\n" (s : ss)
:}

>>> putStrLn $ pprint4 myTree
* 0 (7)
  * 1 (0)
  * 2 (0)
  * 3 (4)
    * 31 (3)
      * 311 (2)
        * 3111 (0)
        * 3112 (0)

One common use for para is to construct a new tree which reuses most of the sub-trees from the original. In the following example, we insert a new node under the leftmost leaf. This requires allocating new nodes along a path from the root to that leaf, while keeping every other sub-tree untouched.

>>> :{
let insertLeftmost :: Int -> Tree Int -> Tree Int
    insertLeftmost new = para go
      where
        go :: TreeF Int (Tree Int, Tree Int)
           -> Tree Int
        go (NodeF i []) = Node i [Node new []]
        go (NodeF i ((_orig, recur) : tts))
            -- tts :: [(Tree Int, Tree Int)]
          = let (origs, _recurs) = unzip tts
            in Node i (recur : origs)
:}

>>> putStrLn $ pprint4 $ insertLeftmost 999 myTree
* 0 (8)
  * 1 (1)
    * 999 (0)
  * 2 (0)
  * 3 (4)
    * 31 (3)
      * 311 (2)
        * 3111 (0)
        * 3112 (0)

histo :: Recursive t => (Base t (Cofree (Base t) a) -> a) -> t -> a

A variant of cata which includes the results of all the descendents, not just the direct children.

Like para, a sub-tree is provided for each recursive position. Each node in that sub-tree is annotated with the result for that descendent. The Cofree type is used to add those annotations.

For our running example, let's recreate GitHub's directory compression algorithm. Notice that in the repository for this package, GitHub displays src/Data/Functor, not src:

GitHub does this because src only contains one entry: Data. Similarly, Data only contains one entry: Functor. Functor contains several entries, so the compression stops there. This helps users get to the interesting folders more quickly.

Before we use histo, we need to define a helper function rollup. It collects nodes until it reaches a node which doesn't have exactly one child. It also returns the labels of that node's children.

>>> :{
let rollup :: [Cofree (TreeF node) label]
           -> ([node], [label])
    rollup [_ :< NodeF node cofrees] =
      let (nodes, label) = rollup cofrees
      in (node : nodes, label)
    rollup cofrees =
      ([], fmap extract cofrees)
:}

>>> let foobar xs = 1 :< NodeF "foo" [2 :< NodeF "bar" xs]
>>> rollup [foobar []]
(["foo","bar"],[])
>>> rollup [foobar [3 :< NodeF "baz" [], 4 :< NodeF "quux" []]]
(["foo","bar"],[3,4])

The value foobar [] can be interpreted as the tree NodeF "foo" [NodeF "bar" []], plus two annotations. The "foo" node is annotated with 1, while the "bar" node is annotated with 2. When we call histo below, those annotations are recursive results of type Int -> String.

>>> :{
let pprint5 :: Tree Int -> String
    pprint5 t = histo go t 0
      where
        go :: TreeF Int (Cofree (TreeF Int) (Int -> String))
           -> Int -> String
        go (NodeF node cofrees) indent
            -- cofrees :: [Cofree (TreeF Int) (Int -> String)]
            -- fs :: [Int -> String]
          = let indent' = indent + 2
                (nodes, fs) = rollup cofrees
                ss = map (\f -> f indent') fs
                s = replicate indent ' '
                 ++ "* " ++ intercalate " / " (fmap show (node : nodes))
            in intercalate "\n" (s : ss)
:}

>>> putStrLn $ pprint5 myTree
* 0
  * 1
  * 2
  * 3 / 31 / 311
    * 3111
    * 3112

One common use for histo is to cache the value computed for smaller sub-trees. In the Fibonacci example below, the recursive type is Natural, which is isomorphic to [()]. Our annotated sub-tree is thus isomorphic to a list of annotations. In our case, each annotation is the result which was computed for a smaller number. We thus have access to a list which caches all the Fibonacci numbers we have computed so far.

>>> :{
let fib :: Natural -> Integer
    fib = histo go
      where
        go :: Maybe (Cofree Maybe Integer) -> Integer
        go Nothing = 1
        go (Just (_ :< Nothing)) = 1
        go (Just (fibNMinus1 :< Just (fibNMinus2 :< _)))
          = fibNMinus1 + fibNMinus2
:}

>>> fmap fib [0..10]
[1,1,2,3,5,8,13,21,34,55,89]

In general, Cofree f a can be thought of as a cache that has the same shape as the recursive structure which was given as input.

zygo :: Recursive t => (Base t b -> b) -> (Base t (b, a) -> a) -> t -> a

Unfolding functions

unfold :: Corecursive t => (a -> Base t a) -> a -> t

A generalization of unfoldr. The starting seed is expanded into a base functor whose recursive positions contain more seeds, which are themselves expanded, and so on.

>>> :{
>>> let ourEnumFromTo :: Int -> Int -> [Int]
>>> ourEnumFromTo lo hi = ana go lo where
>>> go i = if i > hi then Nil else Cons i (i + 1)
>>> :}

>>> ourEnumFromTo 1 4
[1,2,3,4]

ana

Arguments

:: Corecursive t

=> (a -> Base t a)

a (Base t)-coalgebra

-> a

seed

-> t

resulting fixed point

An alias for unfold.

apo :: Corecursive t => (a -> Base t (Either t a)) -> a -> t

futu :: Corecursive t => (a -> Base t (Free (Base t) a)) -> a -> t

Combining unfolds and folds

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

An optimized version of fold f . unfold g.

Useful when your recursion structure is shaped like a particular recursive datatype, but you're neither consuming nor producing that recursive datatype. For example, the recursion structure of quick sort is a binary tree, but its input and output is a list, not a binary tree.

>>> data BinTreeF a b = Tip | Branch b a b deriving (Functor)

>>> :{
>>> let quicksort :: Ord a => [a] -> [a]
>>> quicksort = refold merge split where
>>> split []     = Tip
>>> split (x:xs) = let (l, r) = partition (<x) xs in Branch l x r
>>> 
>>> merge Tip            = []
>>> merge (Branch l x r) = l ++ [x] ++ r
>>> :}

>>> quicksort [1,5,2,8,4,9,8]
[1,2,4,5,8,8,9]

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

An alias for refold.

chrono :: Functor f => (f (Cofree f b) -> b) -> (a -> f (Free f a)) -> a -> b

Changing representation

refix :: (Recursive s, Corecursive t, Base s ~ Base t) => s -> t

Convert from one recursive representation to another.

>>> refix ["foo", "bar"] :: Fix (ListF String)
Fix (Cons "foo" (Fix (Cons "bar" (Fix Nil))))

hoist :: (Recursive s, Corecursive t) => (forall a. Base s a -> Base t a) -> s -> t

Convert from one recursive type to another.

>>> showTree $ hoist (\(NonEmptyF h t) -> NodeF [h] (maybeToList t)) ( 'a' :| "bcd")
(a (b (c d)))

transverse :: (Recursive s, Corecursive t, Functor f) => (forall a. Base s (f a) -> f (Base t a)) -> s -> f t

An effectful version of hoist.

Properties:

transverse sequenceA = pure

Examples:

The weird type of first argument allows user to decide an order of sequencing:

>>> transverse (\x -> print (void x) *> sequence x) "foo" :: IO String
Cons 'f' ()
Cons 'o' ()
Cons 'o' ()
Nil
"foo"

>>> transverse (\x -> sequence x <* print (void x)) "foo" :: IO String
Nil
Cons 'o' ()
Cons 'o' ()
Cons 'f' ()
"foo"

cotransverse :: (Recursive s, Corecursive t, Functor f) => (forall a. f (Base s a) -> Base t (f a)) -> f s -> t

A coeffectful version of hoist.

Properties:

cotransverse distAna = runIdentity

Examples:

Stateful transformations:

>>> :{
cotransverse
  (\(u, b) -> case b of
    Nil -> Nil
    Cons x a -> Cons (if u then toUpper x else x) (not u, a))
  (True, "foobar") :: String
:}
"FoObAr"

We can implement a variant of zipWith

>>> data Pair a = Pair a a deriving Functor

>>> :{
let zipWith' :: forall a b. (a -> a -> b) -> [a] -> [a] -> [b]
    zipWith' f xs ys = cotransverse g (Pair xs ys) where
      g :: Pair (ListF a c) -> ListF b (Pair c)
      g (Pair Nil        _)          = Nil
      g (Pair _          Nil)        = Nil
      g (Pair (Cons x a) (Cons y b)) = Cons (f x y) (Pair a b)
    :}

>>> zipWith' (*) [1,2,3] [4,5,6]
[4,10,18]

>>> zipWith' (*) [1,2,3] [4,5,6,8]
[4,10,18]

>>> zipWith' (*) [1,2,3,3] [4,5,6]
[4,10,18]

Advanced usage

Mendler-style recursion-schemes

mcata :: (forall y. (y -> c) -> f y -> c) -> Fix f -> c

Mendler-style iteration

mpara :: (forall y. (y -> c) -> (y -> Fix f) -> f y -> c) -> Fix f -> c

Mendler-style recursion

Since: 5.2.2

mhisto :: (forall y. (y -> c) -> (y -> f y) -> f y -> c) -> Fix f -> c

Mendler-style course-of-value iteration

mzygo :: (forall y. (y -> b) -> f y -> b) -> (forall y. (y -> c) -> (y -> b) -> f y -> c) -> Fix f -> c

Mendler-style semi-mutual recursion

Since: 5.2.2

mana :: (forall y. (x -> y) -> x -> f y) -> x -> Fix f

Mendler-style coiteration

Since: 5.2.2

mapo :: (forall y. (Fix f -> y) -> (x -> y) -> x -> f y) -> x -> Fix f

Mendler-style corecursion

Since: 5.2.2

mfutu :: (forall y. (f y -> y) -> (x -> y) -> x -> f y) -> x -> Fix f

Mendler-style course-of-values coiteration

Since: 5.2.2

Fokkinga's recursion-schemes

prepro :: (Recursive t, Corecursive t) => (forall b. Base t b -> Base t b) -> (Base t a -> a) -> t -> a

Fokkinga's prepromorphism

postpro :: (Corecursive t, Recursive t) => (forall b. Base t b -> Base t b) -> (a -> Base t a) -> a -> t

Fokkinga's postpromorphism

Elgot (co)algebras

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

Elgot algebras

coelgot :: Functor f => ((a, f b) -> b) -> (a -> f a) -> a -> b

Elgot coalgebras: http://comonad.com/reader/2008/elgot-coalgebras/

Generalized recursion-schemes

gfold

Arguments

:: (Recursive t, Comonad w)

=> (forall b. Base t (w b) -> w (Base t b))

a distributive law

-> (Base t (w a) -> a)

a (Base t)-w-algebra

-> t

fixed point

-> a

A generalized catamorphism

gcata

Arguments

:: (Recursive t, Comonad w)

=> (forall b. Base t (w b) -> w (Base t b))

a distributive law

-> (Base t (w a) -> a)

a (Base t)-w-algebra

-> t

fixed point

-> a

A generalized catamorphism

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

ghisto :: (Recursive t, Comonad w) => (forall b. Base t (w b) -> w (Base t b)) -> (Base t (CofreeT (Base t) w 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

gunfold

Arguments

:: (Corecursive t, Monad m)

=> (forall b. m (Base t b) -> Base t (m b))

a distributive law

-> (a -> Base t (m a))

a (Base t)-m-coalgebra

-> a

seed

-> t

A generalized anamorphism

gana

Arguments

:: (Corecursive t, Monad m)

=> (forall b. m (Base t b) -> Base t (m b))

a distributive law

-> (a -> Base t (m a))

a (Base t)-m-coalgebra

-> a

seed

-> t

A generalized anamorphism

gapo :: Corecursive t => (b -> Base t b) -> (a -> Base t (Either b a)) -> a -> t

gfutu :: (Corecursive t, Functor m, Monad m) => (forall b. m (Base t b) -> Base t (m b)) -> (a -> Base t (FreeT (Base t) m a)) -> a -> t

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

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

A generalized hylomorphism

gchrono :: (Functor f, Functor w, Functor m, Comonad w, Monad m) => (forall c. f (w c) -> w (f c)) -> (forall c. m (f c) -> f (m c)) -> (f (CofreeT f w b) -> b) -> (a -> f (FreeT f m a)) -> a -> b

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

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 postpromorphism

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)

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 (CofreeT f h a) -> CofreeT f h (f a)

distZygo

Arguments

:: Functor f

=> (f b -> b)

-> f (b, a) -> (b, f a)

A distributive for semi-mutual recursion

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)

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)

distFutu :: Functor f => Free f (f a) -> f (Free f a)

distGFutu :: (Functor f, Functor h) => (forall b. h (f b) -> f (h b)) -> FreeT f h (f a) -> f (FreeT f h a)

Zygohistomorphic prepromorphisms

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

Zygohistomorphic prepromorphisms:

A corrected and modernized version of http://www.haskell.org/haskellwiki/Zygohistomorphic_prepromorphisms