How to use `local` and a `Reader` monad with Scrap Your Boilerplate (SYB)?
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 -> ma ) don't seem to support the use of local (of type ma -> ma ) 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 Int s with values larger than the number of Binder s 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:
(:@) case is typical boilerplate recursion: everywhereM works here automatically. 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. 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.
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.
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