free-5.2: Monads for free

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 :: Type -> Type) a = a :< (f (Cofree f a))
- class (Functor f, Comonad w) => ComonadCofree (f :: Type -> Type) (w :: Type -> Type) | 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 :: forall f a (g :: Type -> Type). 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 :: forall f (g :: Type -> Type) a. (Applicative f, Traversable g) => (a -> f a) -> Cofree g a -> f (Cofree g a)
- leaves :: forall f (g :: Type -> Type) a. (Applicative f, Traversable g) => (a -> f a) -> Cofree g a -> f (Cofree g a)

# Documentation

data Cofree (f :: Type -> Type) a #

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

ComonadHoist Cofree # |
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 #
ComonadTrans Cofree # | 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 #
Functor f => Generic1 (Cofree f :: Type -> Type) # |
Instance details Defined in Control.Comonad.Cofree Associated Types
type Rep1 (Cofree f :: Type -> Type) |
Instance details Defined in Control.Comonad.Cofree type Rep1 (Cofree f :: Type -> Type) = D1 ('MetaData "Cofree" "Control.Comonad.Cofree" "free-5.2-5ZpHY3BiLl3JKmP2qxELyz" '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))))
Methods from1 :: Cofree f a -> Rep1 (Cofree f) a # to1 :: Rep1 (Cofree f) a -> Cofree f a #
ComonadEnv e w => ComonadEnv e (Cofree w) # |
Instance details Defined in Control.Comonad.Cofree Methods ask :: Cofree w a -> e #
ComonadStore s w => ComonadStore s (Cofree w) # |
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) # |
Instance details Defined in Control.Comonad.Cofree Methods trace :: m -> Cofree w a -> a #
Functor f => ComonadCofree f (Cofree f) # |
Instance details Defined in Control.Comonad.Cofree Methods unwrap :: Cofree f a -> f (Cofree f a) #
(Alternative f, MonadZip f) => MonadZip (Cofree f) # |
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) #
Foldable f => Foldable (Cofree f) # |
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 #
Foldable1 f => Foldable1 (Cofree f) # |
Instance details Defined in Control.Comonad.Cofree Methods fold1 :: Semigroup m => Cofree f m -> m # foldMap1 :: Semigroup m => (a -> m) -> Cofree f a -> m # foldMap1' :: Semigroup m => (a -> m) -> Cofree f a -> m # toNonEmpty :: Cofree f a -> NonEmpty a # maximum :: Ord a => Cofree f a -> a # minimum :: Ord a => Cofree f a -> a # head :: Cofree f a -> a # last :: Cofree f a -> a # foldrMap1 :: (a -> b) -> (a -> b -> b) -> Cofree f a -> b # foldlMap1' :: (a -> b) -> (b -> a -> b) -> Cofree f a -> b # foldlMap1 :: (a -> b) -> (b -> a -> b) -> Cofree f a -> b # foldrMap1' :: (a -> b) -> (a -> b -> b) -> Cofree f a -> b #
Eq1 f => Eq1 (Cofree f) # |
Instance details Defined in Control.Comonad.Cofree Methods liftEq :: (a -> b -> Bool) -> Cofree f a -> Cofree f b -> Bool #
Ord1 f => Ord1 (Cofree f) # |
Instance details Defined in Control.Comonad.Cofree Methods liftCompare :: (a -> b -> Ordering) -> Cofree f a -> Cofree f b -> Ordering #
Read1 f => Read1 (Cofree f) # |
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) # |
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 #
Traversable f => Traversable (Cofree f) # |
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) #
Alternative f => Applicative (Cofree f) # |
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 #
Functor f => Functor (Cofree f) # |
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 => Monad (Cofree f) # |
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 => Comonad (Cofree f) # |
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) # |
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) # |
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) #
Apply f => Apply (Cofree f) # |
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 #
Functor f => Extend (Cofree f) # |
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 #
Traversable1 f => Traversable1 (Cofree f) # |
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) #
FoldableWithIndex i f => FoldableWithIndex [i] (Cofree f) # |
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 #
FunctorWithIndex i f => FunctorWithIndex [i] (Cofree f) # |
Instance details Defined in Control.Comonad.Cofree Methods imap :: ([i] -> a -> b) -> Cofree f a -> Cofree f b #
TraversableWithIndex i f => TraversableWithIndex [i] (Cofree f) # |
Instance details Defined in Control.Comonad.Cofree Methods itraverse :: Applicative f0 => ([i] -> a -> f0 b) -> Cofree f a -> f0 (Cofree f b) #
(Typeable f, Data (f (Cofree f a)), Data a) => Data (Cofree f a) # |
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) #
Generic (Cofree f a) # |
Instance details Defined in Control.Comonad.Cofree Associated Types
type Rep (Cofree f a) |
Instance details Defined in Control.Comonad.Cofree type Rep (Cofree f a) = D1 ('MetaData "Cofree" "Control.Comonad.Cofree" "free-5.2-5ZpHY3BiLl3JKmP2qxELyz" '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)))))
Methods from :: Cofree f a -> Rep (Cofree f a) x # to :: Rep (Cofree f a) x -> Cofree f a #
(Read1 f, Read a) => Read (Cofree f a) # |
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) # |
Instance details Defined in Control.Comonad.Cofree Methods showsPrec :: Int -> Cofree f a -> ShowS # show :: Cofree f a -> String # showList :: [Cofree f a] -> ShowS #
(Eq1 f, Eq a) => Eq (Cofree f a) # |
Instance details Defined in Control.Comonad.Cofree Methods (==) :: Cofree f a -> Cofree f a -> Bool # (/=) :: Cofree f a -> Cofree f a -> Bool #
(Ord1 f, Ord a) => Ord (Cofree f a) # |
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 #
type Rep1 (Cofree f :: Type -> Type) # |
Instance details Defined in Control.Comonad.Cofree type Rep1 (Cofree f :: Type -> Type) = D1 ('MetaData "Cofree" "Control.Comonad.Cofree" "free-5.2-5ZpHY3BiLl3JKmP2qxELyz" '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) # |
Instance details Defined in Control.Comonad.Cofree type Rep (Cofree f a) = D1 ('MetaData "Cofree" "Control.Comonad.Cofree" "free-5.2-5ZpHY3BiLl3JKmP2qxELyz" '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 :: Type -> Type) (w :: Type -> Type) | w -> f where #

Allows you to peel a layer off a cofree comonad.

Methods

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

Remove a layer.

#### Instances

Instances details

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

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

    lower . section = id

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

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 #

Like coiter for comonadic values.

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

Unfold a cofree comonad from a seed.

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

Unfold a cofree comonad from a seed, monadically.

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

# Lenses into cofree comonads

_extract :: forall f a (g :: Type -> Type). Functor f => (a -> f a) -> Cofree g a -> f (Cofree g a) #

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) #

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) #

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) #

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 :: forall f (g :: Type -> Type) a. (Applicative f, Traversable g) => (a -> f a) -> Cofree g a -> f (Cofree g a) #

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 :: forall f (g :: Type -> Type) a. (Applicative f, Traversable g) => (a -> f a) -> Cofree g a -> f (Cofree g a) #

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.