free- Monads for free

Copyright(C) 2012-2013 Edward Kmett
LicenseBSD-style (see the file LICENSE)
MaintainerEdward Kmett <>
PortabilityGADTs, Rank2Types
Safe HaskellSafe-Inferred




Applicative functors for free



Compared to the free monad, they are less expressive. However, they are also more flexible to inspect and interpret, as the number of ways in which the values can be nested is more limited.

See Free Applicative Functors, by Paolo Capriotti and Ambrus Kaposi, for some applications.

data Ap f a where Source

The free Applicative for a Functor f.


Pure :: a -> Ap f a 
Ap :: f a -> Ap f (a -> b) -> Ap f b 


Functor (Ap f) 
Applicative (Ap f) 
Apply (Ap f) 
Typeable ((* -> *) -> * -> *) Ap 

runAp :: Applicative g => (forall x. f x -> g x) -> Ap f a -> g a Source

Given a natural transformation from f to g, this gives a canonical monoidal natural transformation from Ap f to g.

runAp t == retractApp . hoistApp t

runAp_ :: Monoid m => (forall a. f a -> m) -> Ap f b -> m Source

Perform a monoidal analysis over free applicative value.


count :: Ap f a -> Int
count = getSum . runAp_ (\_ -> Sum 1)

liftAp :: f a -> Ap f a Source

A version of lift that can be used with just a Functor for f.

hoistAp :: (forall a. f a -> g a) -> Ap f b -> Ap g b Source

Given a natural transformation from f to g this gives a monoidal natural transformation from Ap f to Ap g.

retractAp :: Applicative f => Ap f a -> f a Source

Interprets the free applicative functor over f using the semantics for pure and <*> given by the Applicative instance for f.

retractApp == runAp id