How can I recover sharing in a GADT?

Question

In Type-Safe Observable Sharing in Haskell Andy Gill shows how to recover sharing that existed on the Haskell level, in a DSL. His solution is implemented in the data-reify package. Can this approach be modified to work with GADTs? For example, given this GADT:

data Ast e where
  IntLit :: Int -> Ast Int
  Add :: Ast Int -> Ast Int -> Ast Int
  BoolLit :: Bool -> Ast Bool
  IfThenElse :: Ast Bool -> Ast e -> Ast e -> Ast e

I'd like to recover sharing by transforming the above AST to

type Name = Unique

data Ast2 e where
  IntLit2 :: Int -> Ast2 Int
  Add2 :: Ast2 Int -> Ast2 Int -> Ast2 Int
  BoolLit2 :: Bool -> Ast2 Bool
  IfThenElse2 :: Ast2 Bool -> Ast2 e -> Ast2 e -> Ast2 e
  Var :: Name -> Ast2 e

by the way of a function

recoverSharing :: Ast -> (Map Name, Ast2 e1, Ast2 e2)

(I'm not sure about the type of recoverSharing.)

Note that I don't care about introducing new bindings via a let construct, but only in recovering the sharing that existed on the Haskell level. That's why I have recoverSharing return a Map.

If it can't be done as reusable package, can it at least be done for specific GADT?

Answer 1

Interesting puzzle! It turns out you can use data-reify with GADTs. What you need is a wrapper that hides the type in an existential. The type can later be retrieved by pattern matching on the Type data type.

data Type a where
  Bool :: Type Bool
  Int :: Type Int

data WrappedAst s where
  Wrap :: Type e -> Ast2 e s -> WrappedAst s

instance MuRef (Ast e) where
  type DeRef (Ast e) = WrappedAst
  mapDeRef f e = Wrap (getType e) <$> mapDeRef' f e
    where
      mapDeRef' :: Applicative f => (forall b. (MuRef b, WrappedAst ~ DeRef b) => b -> f u) -> Ast e -> f (Ast2 e u)
      mapDeRef' f (IntLit i) = pure $ IntLit2 i
      mapDeRef' f (Add a b) = Add2 <$> (Var Int <$> f a) <*> (Var Int <$> f b)
      mapDeRef' f (BoolLit b) = pure $ BoolLit2 b
      mapDeRef' f (IfThenElse b t e) = IfThenElse2 <$> (Var Bool <$> f b) <*> (Var (getType t) <$> f t) <*> (Var (getType e) <$> f e)

getVar :: Map Name (WrappedAst Name) -> Type e -> Name -> Maybe (Ast2 e Name)
getVar m t n = case m ! n of Wrap t' e -> (\Refl -> e) <$> typeEq t t'

Here's the whole code: https://gist.github.com/3590197

Edit: I like the use of Typeable in the other answer. So I did a version of my code with Typeable too: https://gist.github.com/3593585. The code is significantly shorter. Type e -> is replaced by Typeable e =>, which also has a downside: we no longer know that the possible types are limited to Int and Bool, which means there has to be a Typeable e constraint in IfThenElse.