Primitive recursion schemes and combinators specialised to Nat.

Base.Nat.hs

-- Folding

project :: Nat -> Maybe Nat

cata  ::                                (Maybe    a  -> a) -> Nat -> a
gcata :: Comonad w => Distributive w => (Maybe (w a) -> a) -> Nat -> a

para  ::                                (Maybe (Nat,   a) -> a) -> Nat -> a
gpara :: Comonad w => Distributive w => (Maybe (Nat, w a) -> a) -> Nat -> a

prepro  ::                                (Maybe ~> Maybe) -> (Maybe    a  -> a) -> Nat -> a
gprepro :: Comonad w => Distributive w => (Maybe ~> Maybe) -> (Maybe (w a) -> a) -> Nat -> a

zygo  ::                                (Maybe b -> b) -> (Maybe (b,   a) -> a) -> Nat -> a
gzygo :: Comonad w => Distributive w => (Maybe b -> b) -> (Maybe (b, w a) -> a) -> Nat -> a

histo  ::                   Maybe (Cofree Maybe a -> a) -> Nat -> a
ghisto :: Distributive f => Maybe (Cofree     f a -> a) -> Nat -> a

futu  ::                  (a -> Maybe (Free Maybe a)) -> a -> Nat
gfutu :: Traversable f => (a -> Maybe (Free     f a)) -> Nat -> a 

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

-- Distributive laws

distCata :: Functor f => f (Identity a) -> Identity (f a)

distPara  ::                                Maybe (Nat,   a) -> (Nat,    Maybe a)
distParaT :: Comonad w => Distributive w => Maybe (Nat, w a) -> (Nat, w (Maybe a))

distZygo  :: Functor f =>                                (f b -> b) -> f (b,   a) ->  b    (f, a)
distZygoT :: Functor f => Distributive w => Comonad w => (f b -> b) -> f (b, w a) -> (b, w (f a))

distHisto  :: Functor f =>                   f (Cofree f a) -> Cofree f (f a)
distGHisto :: Functor f => Distributive h => f (Cofree h a) -> Cofree h (f a)

distFutu  ::      Functor f =>              Free f (f a) -> f (Free f a)
distGFutu :: Distributive f => Functor h => Free h (f a) -> f (Free h a)

-- Unfolding

embed :: Maybe Nat -> Nat

ana  ::                  (a -> Maybe    a ) -> a -> Nat
gana :: Traversable m => (a -> Maybe (m a)) -> a -> Nat

apo  ::                   (a -> Maybe (Either Nat a)) -> a -> Nat
gapo :: (b -> Maybe b) -> (a -> Maybe (Either   b a)) -> a -> Nat

postpro  ::                             (Maybe ~> Maybe) -> (a -> Maybe    a ) -> a -> Nat
gpostpro :: Monad m => Traversable m => (Maybe ~> Maybe) -> (a -> Maybe (m a)) -> a -> Nat

-- Distributive laws

distAna :: Functor f => Identity (f a) -> f (Identity a)

distApo   ::                                                Either Nat (Maybe a)  -> Maybe (Either Nat    a)
distGApo  :: Functor f =>                  (b -> f b) ->    Either   b (    f a)  ->     f (Either   b    a)
distGApoT :: Functor f => Traversable m => (b -> f b) -> m (Either   b (    f a)) ->     f (Either   b (m a))

-- Zygohistomorphic prepromorphisms

zygoHistoPrepro :: (Maybe b -> b) -> (Maybe ~> Maybe) -> (Maybe (b, Cofree Maybe a) -> a) -> Nat -> a

----------

-- Refolding

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

-- OR traversable + distributive f???
ghylo :: Comonad w => Functor f => Monad m => Distributive w => Traversable m => (f (w b) -> b) -> (a -> f (m a)) -> a -> b

-- Elgot

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