Convert an untyped DSL view to a typed view

Question

Given a simple language, let's say

data E where
  ValE :: Typeable a => a -> E
  AppE :: E -> E -> E

Is it possible to convert it to a typed representation:

data T a where
  ValT :: Typeable a => a -> T a
  AppT :: T (a -> b) -> T a -> T b
  deriving Typeable

I tried various approaches, for example. following:

e2t :: Typeable a => E -> Maybe (T a)
e2t (ValE x) = cast (ValT x)
e2t (AppE e1 e2) = liftM2 AppT (e2t e1) (e2t e2)

This does not work, and I get the following error message:

Undefined variable of type 'a' in constraint:
'Typeable a'
arising from the use of 'e2t' in ...
Probable fix: add a type signature that corrects these type variables

However, if I like it

e2t :: Typeable a => E -> Maybe (T a)
e2t (ValE x) = cast (ValT x)
e2t (AppE e1 e2) = liftM2 AppT (e2t e1) (e2t e2 :: Maybe (T Int))

it compiles.

Answer 1

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    AnyT :: Typeable a => T a -> AnyT

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