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.Trans.Coiter

Contents

- The coiterative comonad transformer
- The coiterative comonad
- Generating coiterative comonads
- Cofree comonads
- Examples

Description

The coiterative comonad generated by a comonad

Synopsis

- newtype CoiterT (w :: Type -> Type) a = CoiterT {
  - runCoiterT :: w (a, CoiterT w a)
  }
- type Coiter = CoiterT Identity
- coiter :: a -> Coiter a -> Coiter a
- runCoiter :: Coiter a -> (a, Coiter a)
- unfold :: Comonad w => (w a -> a) -> w a -> CoiterT w a
- class (Functor f, Comonad w) => ComonadCofree (f :: Type -> Type) (w :: Type -> Type) | w -> f where
  - unwrap :: w a -> f (w a)

# Documentation

Coiterative comonads represent non-terminating, productive computations.

They are the dual notion of iterative monads. While iterative computations produce no values or eventually terminate with one, coiterative computations constantly produce values and they never terminate.

It's simpler form, `Coiter`, is an infinite stream of data. `CoiterT` extends this so that each step of the computation can be performed in a comonadic context.

# The coiterative comonad transformer

newtype CoiterT (w :: Type -> Type) a #

This is the coiterative comonad generated by a comonad

Constructors

CoiterT |
Fields runCoiterT :: w (a, CoiterT w a)

#### Instances

Instances details

ComonadHoist CoiterT # |
Instance details Defined in Control.Comonad.Trans.Coiter Methods cohoist :: (Comonad w, Comonad v) => (forall x. w x -> v x) -> CoiterT w a -> CoiterT v a #
ComonadTrans CoiterT # |
Instance details Defined in Control.Comonad.Trans.Coiter Methods lower :: Comonad w => CoiterT w a -> w a #
ComonadEnv e w => ComonadEnv e (CoiterT w) # |
Instance details Defined in Control.Comonad.Trans.Coiter Methods ask :: CoiterT w a -> e #
ComonadStore s w => ComonadStore s (CoiterT w) # |
Instance details Defined in Control.Comonad.Trans.Coiter Methods pos :: CoiterT w a -> s # peek :: s -> CoiterT w a -> a # peeks :: (s -> s) -> CoiterT w a -> a # seek :: s -> CoiterT w a -> CoiterT w a # seeks :: (s -> s) -> CoiterT w a -> CoiterT w a # experiment :: Functor f => (s -> f s) -> CoiterT w a -> f a #
ComonadTraced m w => ComonadTraced m (CoiterT w) # |
Instance details Defined in Control.Comonad.Trans.Coiter Methods trace :: m -> CoiterT w a -> a #
Comonad w => ComonadCofree Identity (CoiterT w) # |
Instance details Defined in Control.Comonad.Trans.Coiter Methods unwrap :: CoiterT w a -> Identity (CoiterT w a) #
Foldable w => Foldable (CoiterT w) # |
Instance details Defined in Control.Comonad.Trans.Coiter Methods fold :: Monoid m => CoiterT w m -> m # foldMap :: Monoid m => (a -> m) -> CoiterT w a -> m # foldMap' :: Monoid m => (a -> m) -> CoiterT w a -> m # foldr :: (a -> b -> b) -> b -> CoiterT w a -> b # foldr' :: (a -> b -> b) -> b -> CoiterT w a -> b # foldl :: (b -> a -> b) -> b -> CoiterT w a -> b # foldl' :: (b -> a -> b) -> b -> CoiterT w a -> b # foldr1 :: (a -> a -> a) -> CoiterT w a -> a # foldl1 :: (a -> a -> a) -> CoiterT w a -> a # toList :: CoiterT w a -> [a] # null :: CoiterT w a -> Bool # length :: CoiterT w a -> Int # elem :: Eq a => a -> CoiterT w a -> Bool # maximum :: Ord a => CoiterT w a -> a # minimum :: Ord a => CoiterT w a -> a # sum :: Num a => CoiterT w a -> a # product :: Num a => CoiterT w a -> a #
Eq1 w => Eq1 (CoiterT w) # |
Instance details Defined in Control.Comonad.Trans.Coiter Methods liftEq :: (a -> b -> Bool) -> CoiterT w a -> CoiterT w b -> Bool #
Ord1 w => Ord1 (CoiterT w) # |
Instance details Defined in Control.Comonad.Trans.Coiter Methods liftCompare :: (a -> b -> Ordering) -> CoiterT w a -> CoiterT w b -> Ordering #
Read1 w => Read1 (CoiterT w) # |
Instance details Defined in Control.Comonad.Trans.Coiter Methods liftReadsPrec :: (Int -> ReadS a) -> ReadS [a] -> Int -> ReadS (CoiterT w a) # liftReadList :: (Int -> ReadS a) -> ReadS [a] -> ReadS [CoiterT w a] # liftReadPrec :: ReadPrec a -> ReadPrec [a] -> ReadPrec (CoiterT w a) # liftReadListPrec :: ReadPrec a -> ReadPrec [a] -> ReadPrec [CoiterT w a] #
Show1 w => Show1 (CoiterT w) # |
Instance details Defined in Control.Comonad.Trans.Coiter Methods liftShowsPrec :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> Int -> CoiterT w a -> ShowS # liftShowList :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> [CoiterT w a] -> ShowS #
Traversable w => Traversable (CoiterT w) # |
Instance details Defined in Control.Comonad.Trans.Coiter Methods traverse :: Applicative f => (a -> f b) -> CoiterT w a -> f (CoiterT w b) # sequenceA :: Applicative f => CoiterT w (f a) -> f (CoiterT w a) # mapM :: Monad m => (a -> m b) -> CoiterT w a -> m (CoiterT w b) # sequence :: Monad m => CoiterT w (m a) -> m (CoiterT w a) #
Functor w => Functor (CoiterT w) # |
Instance details Defined in Control.Comonad.Trans.Coiter Methods fmap :: (a -> b) -> CoiterT w a -> CoiterT w b # (<$) :: a -> CoiterT w b -> CoiterT w a #
Comonad w => Comonad (CoiterT w) # |
Instance details Defined in Control.Comonad.Trans.Coiter Methods extract :: CoiterT w a -> a # duplicate :: CoiterT w a -> CoiterT w (CoiterT w a) # extend :: (CoiterT w a -> b) -> CoiterT w a -> CoiterT w b #
(Typeable w, Data (w (a, CoiterT w a)), Data a) => Data (CoiterT w a) # |
Instance details Defined in Control.Comonad.Trans.Coiter Methods gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> CoiterT w a -> c (CoiterT w a) # gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (CoiterT w a) # toConstr :: CoiterT w a -> Constr # dataTypeOf :: CoiterT w a -> DataType # dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c (CoiterT w a)) # dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (CoiterT w a)) # gmapT :: (forall b. Data b => b -> b) -> CoiterT w a -> CoiterT w a # gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> CoiterT w a -> r # gmapQr :: forall r r'. (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> CoiterT w a -> r # gmapQ :: (forall d. Data d => d -> u) -> CoiterT w a -> [u] # gmapQi :: Int -> (forall d. Data d => d -> u) -> CoiterT w a -> u # gmapM :: Monad m => (forall d. Data d => d -> m d) -> CoiterT w a -> m (CoiterT w a) # gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> CoiterT w a -> m (CoiterT w a) # gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> CoiterT w a -> m (CoiterT w a) #
(Read1 w, Read a) => Read (CoiterT w a) # |
Instance details Defined in Control.Comonad.Trans.Coiter Methods readsPrec :: Int -> ReadS (CoiterT w a) # readList :: ReadS [CoiterT w a] # readPrec :: ReadPrec (CoiterT w a) # readListPrec :: ReadPrec [CoiterT w a] #
(Show1 w, Show a) => Show (CoiterT w a) # |
Instance details Defined in Control.Comonad.Trans.Coiter Methods showsPrec :: Int -> CoiterT w a -> ShowS # show :: CoiterT w a -> String # showList :: [CoiterT w a] -> ShowS #
(Eq1 w, Eq a) => Eq (CoiterT w a) # |
Instance details Defined in Control.Comonad.Trans.Coiter Methods (==) :: CoiterT w a -> CoiterT w a -> Bool # (/=) :: CoiterT w a -> CoiterT w a -> Bool #
(Ord1 w, Ord a) => Ord (CoiterT w a) # |
Instance details Defined in Control.Comonad.Trans.Coiter Methods compare :: CoiterT w a -> CoiterT w a -> Ordering # (<) :: CoiterT w a -> CoiterT w a -> Bool # (<=) :: CoiterT w a -> CoiterT w a -> Bool # (>) :: CoiterT w a -> CoiterT w a -> Bool # (>=) :: CoiterT w a -> CoiterT w a -> Bool # max :: CoiterT w a -> CoiterT w a -> CoiterT w a # min :: CoiterT w a -> CoiterT w a -> CoiterT w a #

# The coiterative comonad

type Coiter = CoiterT Identity #

The coiterative comonad

coiter :: a -> Coiter a -> Coiter a #

Prepends a result to a coiterative computation.

    runCoiter . uncurry coiter == id

runCoiter :: Coiter a -> (a, Coiter a) #

Extracts the first result from a coiterative computation.

    uncurry coiter . runCoiter == id

# Generating coiterative comonads

unfold :: Comonad w => (w a -> a) -> w a -> CoiterT w a #

Unfold a `CoiterT` comonad transformer from a cokleisli arrow and an initial comonadic seed.

# Cofree comonads

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) #

# Examples

Newton's method