How do I implement Reader using free monads?

Question

Ok, so I have figured out how to implement Reader (and ReaderT, not shown) using the operational package:

{-# LANGUAGE GADTs, ScopedTypeVariables #-}

import Control.Monad.Operational

data ReaderI r a where
    Ask :: ReaderI r r

type Reader r a = Program (ReaderI r) a

ask :: Reader r r
ask = singleton Ask

runReader :: forall r a. Reader r a -> r -> a
runReader = interpretWithMonad evalI
    where evalI :: forall b. ReaderI r b -> (r -> b)
          evalI Ask = id

But I can't figure out for my life how to do this with free monads (I'm using Edward Kmett's free package). The closest I've gotten is this, which I understand is cheating (something about how ((->) r) is already a monad):

import Control.Monad.Free

type Reader r a = Free ((->) r) a

ask :: Reader r r
ask = Free Pure

runReader :: Reader r a -> r -> a
runReader (Pure a) _ = a
runReader (Free k) r = runReader (k r) r

-- Or, more simply and tellingly:
--
-- > runReader = retract

Even if this wasn't as dumb as I suspect it is, it's not what I want because what I want, basically, is to be able to inspect a Reader as data...

Answer 1

I don't think it can be done except they way you have. But, I don't think this is unique to reader. Consider the free monad version of writer

data WriterF m a = WriterF m a deriving (Functor)

type Writer m = Free (WriterF m)

obviously, WriterF is isomorphic to writer, but this does behave the way we would expect with the simple algebra

algebraWriter :: Monoid m => WriterF m (m,a) -> (m,a)
algebraWriter (WriterF m1 (m2,a)) = (m1 <> m2,a)

thus

runWriter :: Monoid m => Writer m a -> (m,a)
runWriter (Pure a) = (mempty,a)
runWriter (Free x) = algebraWriter . fmap runWriter $ x

Similarly, I think of the Free reader as

type ReaderF r = (->) r

type Reader r = Free (ReaderF r)

I like this, because adding them gives you the state monad

type State x = Free ((ReaderF x) :+: (WriterF x))

runState :: State x a -> x -> (a,x)
runState (Pure a) x                    = (a,x)
runState (Free (Inl f)) x              = runState (f x) x
runState (Free (Inr (WriterF x f))) _  = runState f x

Note, that your operational solution could be made to work with Free by using the "free functor", as can anything that works with operational

data FreeFunctor f x = forall a. FreeFunctor (f a) (a -> x)

but, that FreeFunctor ReaderI is also isomorphic to (->).