Copyright	(C) 2008-2013 Edward Kmett
License	BSD-style (see the file LICENSE)
Maintainer	Edward Kmett <ekmett@gmail.com>
Stability	provisional
Portability	MPTCs, fundeps
Safe Haskell	Safe
Language	Haskell2010

Control.Comonad.Cofree

Contents

Lenses into cofree comonads

Description

Cofree comonads

Synopsis

data Cofree f a = a :< (f (Cofree f a))
class (Functor f, Comonad w) => ComonadCofree f w | w -> f where
- unwrap :: w a -> f (w a)
section :: Comonad f => f a -> Cofree f a
coiter :: Functor f => (a -> f a) -> a -> Cofree f a
coiterW :: (Comonad w, Functor f) => (w a -> f (w a)) -> w a -> Cofree f a
unfold :: Functor f => (b -> (a, f b)) -> b -> Cofree f a
unfoldM :: (Traversable f, Monad m) => (b -> m (a, f b)) -> b -> m (Cofree f a)
hoistCofree :: Functor f => (forall x. f x -> g x) -> Cofree f a -> Cofree g a
_extract :: Functor f => (a -> f a) -> Cofree g a -> f (Cofree g a)
_unwrap :: Functor f => (g (Cofree g a) -> f (g (Cofree g a))) -> Cofree g a -> f (Cofree g a)
telescoped :: Functor f => [(Cofree g a -> f (Cofree g a)) -> g (Cofree g a) -> f (g (Cofree g a))] -> (a -> f a) -> Cofree g a -> f (Cofree g a)
telescoped_ :: Functor f => [(Cofree g a -> f (Cofree g a)) -> g (Cofree g a) -> f (g (Cofree g a))] -> (Cofree g a -> f (Cofree g a)) -> Cofree g a -> f (Cofree g a)
shoots :: (Applicative f, Traversable g) => (a -> f a) -> Cofree g a -> f (Cofree g a)
leaves :: (Applicative f, Traversable g) => (a -> f a) -> Cofree g a -> f (Cofree g a)

Documentation

data Cofree f a Source #

The Cofree Comonad of a functor f.

Formally

A Comonad v is a cofree Comonad for f if every comonad homomorphism from another comonad w to v is equivalent to a natural transformation from w to f.

A cofree functor is right adjoint to a forgetful functor.

Cofree is a functor from the category of functors to the category of comonads that is right adjoint to the forgetful functor from the category of comonads to the category of functors that forgets how to extract and duplicate, leaving you with only a Functor.

In practice, cofree comonads are quite useful for annotating syntax trees, or talking about streams.

A number of common comonads arise directly as cofree comonads.

For instance,

Cofree Maybe forms the comonad for a non-empty list.
Cofree (Const b) is a product.
Cofree Identity forms an infinite stream.
Cofree ((->) b)' describes a Moore machine with states labeled with values of type a, and transitions on edges of type b.

Furthermore, if the functor f forms a monoid (for example, by being an instance of Alternative), the resulting Comonad is also a Monad. See Monadic Augment and Generalised Shortcut Fusion by Neil Ghani et al., Section 4.3 for more details.

In particular, if f a ≡ [a], the resulting data structure is a Rose tree. For a practical application, check Higher Dimensional Trees, Algebraically by Neil Ghani et al.

Constructors

a :< (f (Cofree f a)) infixr 5

Instances

Instances details

ComonadTrans Cofree Source #	This is not a true `Comonad` transformer, but this instance is convenient.
Instance details Defined in Control.Comonad.Cofree Methods lower :: Comonad w => Cofree w a -> w a #
ComonadHoist Cofree Source #
Instance details Defined in Control.Comonad.Cofree Methods cohoist :: (Comonad w, Comonad v) => (forall x. w x -> v x) -> Cofree w a -> Cofree v a #
ComonadEnv e w => ComonadEnv e (Cofree w) Source #
Instance details Defined in Control.Comonad.Cofree Methods ask :: Cofree w a -> e #
ComonadStore s w => ComonadStore s (Cofree w) Source #
Instance details Defined in Control.Comonad.Cofree Methods pos :: Cofree w a -> s # peek :: s -> Cofree w a -> a # peeks :: (s -> s) -> Cofree w a -> a # seek :: s -> Cofree w a -> Cofree w a # seeks :: (s -> s) -> Cofree w a -> Cofree w a # experiment :: Functor f => (s -> f s) -> Cofree w a -> f a #
ComonadTraced m w => ComonadTraced m (Cofree w) Source #
Instance details Defined in Control.Comonad.Cofree Methods trace :: m -> Cofree w a -> a #
Functor f => ComonadCofree f (Cofree f) Source #
Instance details Defined in Control.Comonad.Cofree Methods unwrap :: Cofree f a -> f (Cofree f a) Source #
Alternative f => Monad (Cofree f) Source #
Instance details Defined in Control.Comonad.Cofree Methods (>>=) :: Cofree f a -> (a -> Cofree f b) -> Cofree f b # (>>) :: Cofree f a -> Cofree f b -> Cofree f b # return :: a -> Cofree f a #
Functor f => Functor (Cofree f) Source #
Instance details Defined in Control.Comonad.Cofree Methods fmap :: (a -> b) -> Cofree f a -> Cofree f b # (<$) :: a -> Cofree f b -> Cofree f a #
Alternative f => Applicative (Cofree f) Source #
Instance details Defined in Control.Comonad.Cofree Methods pure :: a -> Cofree f a # (<>) :: Cofree f (a -> b) -> Cofree f a -> Cofree f b # liftA2 :: (a -> b -> c) -> Cofree f a -> Cofree f b -> Cofree f c # (>) :: Cofree f a -> Cofree f b -> Cofree f b # (<*) :: Cofree f a -> Cofree f b -> Cofree f a #
Foldable f => Foldable (Cofree f) Source #
Instance details Defined in Control.Comonad.Cofree Methods fold :: Monoid m => Cofree f m -> m # foldMap :: Monoid m => (a -> m) -> Cofree f a -> m # foldMap' :: Monoid m => (a -> m) -> Cofree f a -> m # foldr :: (a -> b -> b) -> b -> Cofree f a -> b # foldr' :: (a -> b -> b) -> b -> Cofree f a -> b # foldl :: (b -> a -> b) -> b -> Cofree f a -> b # foldl' :: (b -> a -> b) -> b -> Cofree f a -> b # foldr1 :: (a -> a -> a) -> Cofree f a -> a # foldl1 :: (a -> a -> a) -> Cofree f a -> a # toList :: Cofree f a -> [a] # null :: Cofree f a -> Bool # length :: Cofree f a -> Int # elem :: Eq a => a -> Cofree f a -> Bool # maximum :: Ord a => Cofree f a -> a # minimum :: Ord a => Cofree f a -> a # sum :: Num a => Cofree f a -> a # product :: Num a => Cofree f a -> a #
Traversable f => Traversable (Cofree f) Source #
Instance details Defined in Control.Comonad.Cofree Methods traverse :: Applicative f0 => (a -> f0 b) -> Cofree f a -> f0 (Cofree f b) # sequenceA :: Applicative f0 => Cofree f (f0 a) -> f0 (Cofree f a) # mapM :: Monad m => (a -> m b) -> Cofree f a -> m (Cofree f b) # sequence :: Monad m => Cofree f (m a) -> m (Cofree f a) #
Eq1 f => Eq1 (Cofree f) Source #
Instance details Defined in Control.Comonad.Cofree Methods liftEq :: (a -> b -> Bool) -> Cofree f a -> Cofree f b -> Bool #
Ord1 f => Ord1 (Cofree f) Source #
Instance details Defined in Control.Comonad.Cofree Methods liftCompare :: (a -> b -> Ordering) -> Cofree f a -> Cofree f b -> Ordering #
Read1 f => Read1 (Cofree f) Source #
Instance details Defined in Control.Comonad.Cofree Methods liftReadsPrec :: (Int -> ReadS a) -> ReadS [a] -> Int -> ReadS (Cofree f a) # liftReadList :: (Int -> ReadS a) -> ReadS [a] -> ReadS [Cofree f a] # liftReadPrec :: ReadPrec a -> ReadPrec [a] -> ReadPrec (Cofree f a) # liftReadListPrec :: ReadPrec a -> ReadPrec [a] -> ReadPrec [Cofree f a] #
Show1 f => Show1 (Cofree f) Source #
Instance details Defined in Control.Comonad.Cofree Methods liftShowsPrec :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> Int -> Cofree f a -> ShowS # liftShowList :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> [Cofree f a] -> ShowS #
(Alternative f, MonadZip f) => MonadZip (Cofree f) Source #
Instance details Defined in Control.Comonad.Cofree Methods mzip :: Cofree f a -> Cofree f b -> Cofree f (a, b) # mzipWith :: (a -> b -> c) -> Cofree f a -> Cofree f b -> Cofree f c # munzip :: Cofree f (a, b) -> (Cofree f a, Cofree f b) #
Functor f => Comonad (Cofree f) Source #
Instance details Defined in Control.Comonad.Cofree Methods extract :: Cofree f a -> a # duplicate :: Cofree f a -> Cofree f (Cofree f a) # extend :: (Cofree f a -> b) -> Cofree f a -> Cofree f b #
ComonadApply f => ComonadApply (Cofree f) Source #
Instance details Defined in Control.Comonad.Cofree Methods (<@>) :: Cofree f (a -> b) -> Cofree f a -> Cofree f b # (@>) :: Cofree f a -> Cofree f b -> Cofree f b # (<@) :: Cofree f a -> Cofree f b -> Cofree f a #
Distributive f => Distributive (Cofree f) Source #
Instance details Defined in Control.Comonad.Cofree Methods distribute :: Functor f0 => f0 (Cofree f a) -> Cofree f (f0 a) # collect :: Functor f0 => (a -> Cofree f b) -> f0 a -> Cofree f (f0 b) # distributeM :: Monad m => m (Cofree f a) -> Cofree f (m a) # collectM :: Monad m => (a -> Cofree f b) -> m a -> Cofree f (m b) #
Traversable1 f => Traversable1 (Cofree f) Source #
Instance details Defined in Control.Comonad.Cofree Methods traverse1 :: Apply f0 => (a -> f0 b) -> Cofree f a -> f0 (Cofree f b) # sequence1 :: Apply f0 => Cofree f (f0 b) -> f0 (Cofree f b) #
Apply f => Apply (Cofree f) Source #
Instance details Defined in Control.Comonad.Cofree Methods (<.>) :: Cofree f (a -> b) -> Cofree f a -> Cofree f b # (.>) :: Cofree f a -> Cofree f b -> Cofree f b # (<.) :: Cofree f a -> Cofree f b -> Cofree f a # liftF2 :: (a -> b -> c) -> Cofree f a -> Cofree f b -> Cofree f c #
Foldable1 f => Foldable1 (Cofree f) Source #
Instance details Defined in Control.Comonad.Cofree Methods fold1 :: Semigroup m => Cofree f m -> m # foldMap1 :: Semigroup m => (a -> m) -> Cofree f a -> m # toNonEmpty :: Cofree f a -> NonEmpty a #
Functor f => Extend (Cofree f) Source #
Instance details Defined in Control.Comonad.Cofree Methods duplicated :: Cofree f a -> Cofree f (Cofree f a) # extended :: (Cofree f a -> b) -> Cofree f a -> Cofree f b #
Functor f => Generic1 (Cofree f :: Type -> Type) Source #
Instance details Defined in Control.Comonad.Cofree Associated Types type Rep1 (Cofree f) :: k -> Type # Methods from1 :: forall (a :: k). Cofree f a -> Rep1 (Cofree f) a # to1 :: forall (a :: k). Rep1 (Cofree f) a -> Cofree f a #
FunctorWithIndex i f => FunctorWithIndex [i] (Cofree f) Source #
Instance details Defined in Control.Comonad.Cofree Methods imap :: ([i] -> a -> b) -> Cofree f a -> Cofree f b #
FoldableWithIndex i f => FoldableWithIndex [i] (Cofree f) Source #
Instance details Defined in Control.Comonad.Cofree Methods ifoldMap :: Monoid m => ([i] -> a -> m) -> Cofree f a -> m # ifoldMap' :: Monoid m => ([i] -> a -> m) -> Cofree f a -> m # ifoldr :: ([i] -> a -> b -> b) -> b -> Cofree f a -> b # ifoldl :: ([i] -> b -> a -> b) -> b -> Cofree f a -> b # ifoldr' :: ([i] -> a -> b -> b) -> b -> Cofree f a -> b # ifoldl' :: ([i] -> b -> a -> b) -> b -> Cofree f a -> b #
TraversableWithIndex i f => TraversableWithIndex [i] (Cofree f) Source #
Instance details Defined in Control.Comonad.Cofree Methods itraverse :: Applicative f0 => ([i] -> a -> f0 b) -> Cofree f a -> f0 (Cofree f b) #
(Eq1 f, Eq a) => Eq (Cofree f a) Source #
Instance details Defined in Control.Comonad.Cofree Methods (==) :: Cofree f a -> Cofree f a -> Bool # (/=) :: Cofree f a -> Cofree f a -> Bool #
(Typeable f, Data (f (Cofree f a)), Data a) => Data (Cofree f a) Source #
Instance details Defined in Control.Comonad.Cofree Methods gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> Cofree f a -> c (Cofree f a) # gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (Cofree f a) # toConstr :: Cofree f a -> Constr # dataTypeOf :: Cofree f a -> DataType # dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c (Cofree f a)) # dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (Cofree f a)) # gmapT :: (forall b. Data b => b -> b) -> Cofree f a -> Cofree f a # gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> Cofree f a -> r # gmapQr :: forall r r'. (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> Cofree f a -> r # gmapQ :: (forall d. Data d => d -> u) -> Cofree f a -> [u] # gmapQi :: Int -> (forall d. Data d => d -> u) -> Cofree f a -> u # gmapM :: Monad m => (forall d. Data d => d -> m d) -> Cofree f a -> m (Cofree f a) # gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> Cofree f a -> m (Cofree f a) # gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> Cofree f a -> m (Cofree f a) #
(Ord1 f, Ord a) => Ord (Cofree f a) Source #
Instance details Defined in Control.Comonad.Cofree Methods compare :: Cofree f a -> Cofree f a -> Ordering # (<) :: Cofree f a -> Cofree f a -> Bool # (<=) :: Cofree f a -> Cofree f a -> Bool # (>) :: Cofree f a -> Cofree f a -> Bool # (>=) :: Cofree f a -> Cofree f a -> Bool # max :: Cofree f a -> Cofree f a -> Cofree f a # min :: Cofree f a -> Cofree f a -> Cofree f a #
(Read1 f, Read a) => Read (Cofree f a) Source #
Instance details Defined in Control.Comonad.Cofree Methods readsPrec :: Int -> ReadS (Cofree f a) # readList :: ReadS [Cofree f a] # readPrec :: ReadPrec (Cofree f a) # readListPrec :: ReadPrec [Cofree f a] #
(Show1 f, Show a) => Show (Cofree f a) Source #
Instance details Defined in Control.Comonad.Cofree Methods showsPrec :: Int -> Cofree f a -> ShowS # show :: Cofree f a -> String # showList :: [Cofree f a] -> ShowS #
Generic (Cofree f a) Source #
Instance details Defined in Control.Comonad.Cofree Associated Types type Rep (Cofree f a) :: Type -> Type # Methods from :: Cofree f a -> Rep (Cofree f a) x # to :: Rep (Cofree f a) x -> Cofree f a #
type Rep1 (Cofree f :: Type -> Type) Source #
Instance details Defined in Control.Comonad.Cofree type Rep1 (Cofree f :: Type -> Type) = D1 ('MetaData "Cofree" "Control.Comonad.Cofree" "free-5.1.9-6mZiXNlacuxASVeUzXvCgj" 'False) (C1 ('MetaCons ":<" ('InfixI 'RightAssociative 5) 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) Par1 :*: S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (f :.: Rec1 (Cofree f))))
type Rep (Cofree f a) Source #
Instance details Defined in Control.Comonad.Cofree type Rep (Cofree f a) = D1 ('MetaData "Cofree" "Control.Comonad.Cofree" "free-5.1.9-6mZiXNlacuxASVeUzXvCgj" 'False) (C1 ('MetaCons ":<" ('InfixI 'RightAssociative 5) 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 a) :*: S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 (f (Cofree f a)))))

class (Functor f, Comonad w) => ComonadCofree f w | w -> f where Source #

Allows you to peel a layer off a cofree comonad.

Methods

unwrap :: w a -> f (w a) Source #

Remove a layer.

Instances

Instances details

ComonadCofree [] Tree Source #
Instance details Defined in Control.Comonad.Cofree.Class Methods unwrap :: Tree a -> [Tree a] Source #
ComonadCofree Maybe NonEmpty Source #
Instance details Defined in Control.Comonad.Cofree.Class Methods unwrap :: NonEmpty a -> Maybe (NonEmpty a) Source #
Functor f => ComonadCofree f (Cofree f) Source #
Instance details Defined in Control.Comonad.Cofree Methods unwrap :: Cofree f a -> f (Cofree f a) Source #
Comonad w => ComonadCofree Identity (CoiterT w) Source #
Instance details Defined in Control.Comonad.Trans.Coiter Methods unwrap :: CoiterT w a -> Identity (CoiterT w a) Source #
(ComonadCofree f w, Monoid m) => ComonadCofree f (TracedT m w) Source #
Instance details Defined in Control.Comonad.Cofree.Class Methods unwrap :: TracedT m w a -> f (TracedT m w a) Source #
ComonadCofree f w => ComonadCofree f (StoreT s w) Source #
Instance details Defined in Control.Comonad.Cofree.Class Methods unwrap :: StoreT s w a -> f (StoreT s w a) Source #
ComonadCofree f w => ComonadCofree f (EnvT e w) Source #
Instance details Defined in Control.Comonad.Cofree.Class Methods unwrap :: EnvT e w a -> f (EnvT e w a) Source #
ComonadCofree f w => ComonadCofree f (IdentityT w) Source #
Instance details Defined in Control.Comonad.Cofree.Class Methods unwrap :: IdentityT w a -> f (IdentityT w a) Source #
(Functor f, Comonad w) => ComonadCofree f (CofreeT f w) Source #
Instance details Defined in Control.Comonad.Trans.Cofree Methods unwrap :: CofreeT f w a -> f (CofreeT f w a) Source #
ComonadCofree (Const b :: Type -> Type) ((,) b) Source #
Instance details Defined in Control.Comonad.Cofree.Class Methods unwrap :: (b, a) -> Const b (b, a) Source #

section :: Comonad f => f a -> Cofree f a Source #

lower . section = id

coiter :: Functor f => (a -> f a) -> a -> Cofree f a Source #

Use coiteration to generate a cofree comonad from a seed.

coiter f = unfold (id &&& f)

coiterW :: (Comonad w, Functor f) => (w a -> f (w a)) -> w a -> Cofree f a Source #

Like coiter for comonadic values.

unfold :: Functor f => (b -> (a, f b)) -> b -> Cofree f a Source #

Unfold a cofree comonad from a seed.

unfoldM :: (Traversable f, Monad m) => (b -> m (a, f b)) -> b -> m (Cofree f a) Source #

Unfold a cofree comonad from a seed, monadically.

hoistCofree :: Functor f => (forall x. f x -> g x) -> Cofree f a -> Cofree g a Source #

Lenses into cofree comonads

_extract :: Functor f => (a -> f a) -> Cofree g a -> f (Cofree g a) Source #

This is a lens that can be used to read or write from the target of extract.

Using (^.) from the lens package:

foo ^. _extract == extract foo

For more on lenses see the lens package on hackage

_extract :: Lens' (Cofree g a) a

_unwrap :: Functor f => (g (Cofree g a) -> f (g (Cofree g a))) -> Cofree g a -> f (Cofree g a) Source #

This is a lens that can be used to read or write to the tails of a Cofree Comonad.

Using (^.) from the lens package:

foo ^. _unwrap == unwrap foo

For more on lenses see the lens package on hackage

_unwrap :: Lens' (Cofree g a) (g (Cofree g a))

telescoped :: Functor f => [(Cofree g a -> f (Cofree g a)) -> g (Cofree g a) -> f (g (Cofree g a))] -> (a -> f a) -> Cofree g a -> f (Cofree g a) Source #

Construct an Lens into a Cofree g given a list of lenses into the base functor. When the input list is empty, this is equivalent to _extract. When the input list is non-empty, this composes the input lenses with _unwrap to walk through the Cofree g before using _extract to get the element at the final location.

For more on lenses see the lens package on hackage.

telescoped :: [Lens' (g (Cofree g a)) (Cofree g a)]      -> Lens' (Cofree g a) a

telescoped :: [Traversal' (g (Cofree g a)) (Cofree g a)] -> Traversal' (Cofree g a) a

telescoped :: [Getter (g (Cofree g a)) (Cofree g a)]     -> Getter (Cofree g a) a

telescoped :: [Fold (g (Cofree g a)) (Cofree g a)]       -> Fold (Cofree g a) a

telescoped :: [Setter' (g (Cofree g a)) (Cofree g a)]    -> Setter' (Cofree g a) a

telescoped_ :: Functor f => [(Cofree g a -> f (Cofree g a)) -> g (Cofree g a) -> f (g (Cofree g a))] -> (Cofree g a -> f (Cofree g a)) -> Cofree g a -> f (Cofree g a) Source #

Construct an Lens into a Cofree g given a list of lenses into the base functor. The only difference between this and telescoped is that telescoped focuses on a single value, but this focuses on the entire remaining subtree. When the input list is empty, this is equivalent to id. When the input list is non-empty, this composes the input lenses with _unwrap to walk through the Cofree g.

For more on lenses see the lens package on hackage.

telescoped :: [Lens' (g (Cofree g a)) (Cofree g a)]      -> Lens' (Cofree g a) (Cofree g a)

telescoped :: [Traversal' (g (Cofree g a)) (Cofree g a)] -> Traversal' (Cofree g a) (Cofree g a)

telescoped :: [Getter (g (Cofree g a)) (Cofree g a)]     -> Getter (Cofree g a) (Cofree g a)

telescoped :: [Fold (g (Cofree g a)) (Cofree g a)]       -> Fold (Cofree g a) (Cofree g a)

telescoped :: [Setter' (g (Cofree g a)) (Cofree g a)]    -> Setter' (Cofree g a) (Cofree g a)

shoots :: (Applicative f, Traversable g) => (a -> f a) -> Cofree g a -> f (Cofree g a) Source #

A Traversal' that gives access to all non-leaf a elements of a Cofree g a, where non-leaf is defined as x from (x :< xs) where null xs is False.

Because this doesn't give access to all values in the Cofree g, it cannot be used to change types.

shoots :: Traversable g => Traversal' (Cofree g a) a

N.B. On GHC < 7.9, this is slightly less flexible, as it has to use null (toList xs) instead.

leaves :: (Applicative f, Traversable g) => (a -> f a) -> Cofree g a -> f (Cofree g a) Source #

A Traversal' that gives access to all leaf a elements of a Cofree g a, where leaf is defined as x from (x :< xs) where null xs is True.

Because this doesn't give access to all values in the Cofree g, it cannot be used to change types.

shoots :: Traversable g => Traversal' (Cofree g a) a

N.B. On GHC < 7.9, this is slightly less flexible, as it has to use null (toList xs) instead.