How to use `local` and a `Reader` monad with Scrap Your Boilerplate (SYB)?

Question

How can I use SYB (or some other Haskell generics package), to write a transformation in a Reader monad that uses local to modify the environment for subcomputations? The types of GenericM and everywhereM (with a -> m a) don't seem to support the use of local (of type m a -> m a) to wrap a subcomputation. I would like a solution that uses "standard" / "off the shelf" transformations if possible.

A Representative Example

A (mysterious) recursive data type:

{-# LANGUAGE DeriveDataTypeable , Rank2Types , ViewPatterns #-}
import Data.Generics
import Control.Applicative
import Control.Monad.Reader
import Control.Arrow

data Exp = Var Int | Exp :@ Exp | Lam (Binder Exp)
  deriving (Eq , Show , Data , Typeable)
newtype Binder a = Binder a
  deriving (Eq , Show , Data , Typeable)

A recursive function that increments all the embedded Ints with values larger than the number of Binders that wrap them:

-- Increment all free variables:
-- If G |- e:B then G,A |- weaken e:B.
weaken1 :: Exp -> Exp
weaken1 = w 0
  where
  w :: Int -> Exp -> Exp
  -- Base case: use the environment ('i'):
  w i (Var j)          = wVar i j
  -- Boilerplate recursive case:
  w i (e1 :@ e2)       = w i e1 :@ w i e2
  -- Interesting recursive case: modify the environment:
  w i (Lam (Binder e)) = Lam (Binder (w (succ i) e))

wVar :: Int -> Int -> Exp
wVar i j | i <= j    = Var (succ j)
         | otherwise = Var j

The goal is to put the i parameter to weaken1 in a Reader environment and use SYB to automatically handle the boilerplate recursive case for (:@).

A rewrite of weaken1, using a Reader environment, but not SYB:

weaken2 :: Exp -> Exp
weaken2 e = runReader (w e) 0
  where
  w :: Exp -> Reader Int Exp
  w (Var j) = do
    i <- ask
    return $ wVar i j
  w (e1 :@ e2)       = (:@) <$> w e1 <*> w e2
  w (Lam (Binder e)) = Lam . Binder <$> local succ (w e)

The point of the example:

• the (:@) case is typical boilerplate recursion: everywhereM works here automatically.
• the Var case uses the environment, but does not modify it: everywhereM works here, by applying mkM to a Exp -> Reader Int Exp specific to the Var case.
• the Lam case modifies the environment before recursion: everywhereM does not work here (as far as I can tell). The Binder type tells us where we need to use local, so that we might hope to apply mkM to a Binder Exp -> Reader Int (Binder Exp) specific case, but I can't figure out how.

Here is a Gist with more examples, including the above code.

Answer 1

Here is a solution by creating a new SYB traversal, everywhereMM:

newtype MM m x = MM { unMM :: m x -> m x }
mkMM :: (Typeable a , Typeable b) => (m a -> m a) -> m b -> m b
mkMM t = maybe id unMM (gcast (MM t))

-- Apply a 'GenericM' everywhere, transforming the results with a
-- 'GenericMM'.
type GenericMM m = Data a => m a -> m a
everywhereMM :: Monad m => GenericMM m -> GenericM m -> GenericM m
everywhereMM mm m x = mm (m =<< gmapM (everywhereMM mm m) x)

The definitions of everywhereMM and mkMM are similar to everywhereM and mkM in Data.Generics.Schemes and Data.Generics.Aliases; the difference is that everywhereMM uses mm, a GenericMM, to transform the result.

Now we can write a weaken3. The idea is to combine a standard GenericM for the Var case of Exp with a new GenericMM that applies local in the Binder case:

type W = Reader Int
weaken3 :: Exp -> Exp
weaken3 e = runReader (w e) 0
  where
  w :: GenericM W
  w = everywhereMM (mkMM b) (mkM v)

  b :: W (Binder Exp) -> W (Binder Exp)
  b = local succ

  v :: Exp -> W Exp
  v (Var j) = do
    i <- ask
    return $ wVar i j
  v e = return e

But this solution is somewhat unsatisfying: creating the new traversal required digging deeply into the SYB library code.

Again, here is a Gist with more examples, including the above code.