Haskell: Heterogeneous list for data with phantom variable

Question

I'm learning about existential quantification, phantom types, and GADTs at the moment. How do I go about creating a heterogeneous list of a data type with a phantom variable? For example:

{-# LANGUAGE GADTs #-}
{-# LANGUAGE ExistentialQuantification #-}

data Toy a where
  TBool :: Bool -> Toy Bool
  TInt  :: Int  -> Toy Int

instance Show (Toy a) where
  show (TBool b) = "TBool " ++ show b
  show (TInt  i) = "TInt "  ++ show i

bools :: [Toy Bool]
bools = [TBool False, TBool True]

ints  :: [Toy Int]
ints  = map TInt [0..9]

Having functions like below are OK:

isBool :: Toy a -> Bool
isBool (TBool _) = True
isBool (TInt  _) = False

addOne :: Toy Int -> Toy Int
addOne (TInt a) = TInt $ a + 1

However, I would like to be able to declare a heterogeneous list like so:

zeros :: [Toy a]
zeros =  [TBool False, TInt 0]

I tried using an empty type class to restrict the type on a by:

class Unify a
instance Unify Bool
instance Unify Int

zeros :: Unify a => [Toy a]
zeros =  [TBool False, TInt 0]

But the above would fail to compile. I was able to use existential quantification to do get the following:

data T = forall a. (Forget a, Show a) => T a

instance Show T where
  show (T a) = show a

class (Show a) => Forget a
instance Forget (Toy a)
instance Forget T

zeros :: [T]
zeros = [T (TBool False), T (TInt 0)]

But this way, I cannot apply a function that was based on the specific type of a in Toy a to T e.g. addOne above.

In conclusion, what are some ways I can create a heterogeneous list without forgetting/losing the phantom variable?

Answer 1

Start with the Toy type:

data Toy a where
  TBool :: Bool -> Toy Bool
  TInt :: Int -> Toy Int

Now you can wrap it up in an existential without over-generalizing with the class system:

data WrappedToy where
  Wrap :: Toy a -> WrappedToy

Since the wrapper only holds Toys, we can unwrap them and get Toys back:

incIfInt :: WrappedToy -> WrappedToy
incIfInt (Wrap (TInt n)) = Wrap (TInt (n+1))
incIfInt w = w

And now you can distinguish things within the list:

incIntToys :: [WrappedToy] -> [WrappedToy]
incIntToys = map incIfInt

Edit

As Cirdec points out, the different pieces can be teased apart a bit:

onInt :: (Toy Int -> WrappedToy) -> WrappedToy -> WrappedToy
onInt f (Wrap t@(TInt _)) = f t
onInt _ w = w

mapInt :: (Int -> Int) -> Toy Int -> Toy Int
mapInt f (TInt x) = TInt (f x)

incIntToys :: [WrappedToy] -> [WrappedToy]
incIntToys = map $ onInt (Wrap . mapInt (+1))

I should also note that nothing here so far really justifies the Toy GADT. bheklilr's simpler approach of using a plain algebraic datatype should work just fine.

Answer 2

There was a very similar question a few days ago.

In your case it would be

{-# LANGUAGE GADTs, PolyKinds, Rank2Types #-}

data Exists :: (k -> *) -> * where
  This :: p x -> Exists p

type Toys = [Exists Toy]

zeros :: Toys
zeros = [This (TBool False), This (TInt 0)]

It's easy to eliminate an existential:

recEx :: (forall x. p x -> c) -> Exists p -> c
recEx f (This x) = f x

Then if you have a recursor for the Toy datatype

recToy :: (Toy Bool -> c) -> (Toy Int -> c) -> Toy a -> c
recToy f g x@(TBool _) = f x
recToy f g x@(TInt  _) = g x

you can map a wrapped toy:

mapToyEx :: (Toy Bool -> p x) -> (Toy Int -> p y) -> Exists Toy -> Exists p
mapToyEx f g = recEx (recToy (This . f) (This . g))

For example

non_zeros :: Toys
non_zeros = map (mapToyEx (const (TBool True)) addOne) zeros

This approach is similar to one in @dfeuer's answer, but it's less ad hoc.

Answer 3

The ordinary heterogeneous list indexed by a list of the types of its elements is

{-# LANGUAGE DataKinds #-}
{-# LANGUAGE TypeOperators #-}
{-# LANGUAGE GADTs #-}

data HList l where
    HNil :: HList '[]
    HCons :: a -> HList l -> HList (a ': l)

We can modify this to hold values inside some f :: * -> *.

data HList1 f l where
    HNil1 :: HList1 f '[]
    HCons1 :: f a -> HList1 f l -> HList1 f (a ': l)

Which you can use to write zeros without forgetting the type variables.

zeros :: HList1 Toy [Bool, Int]  
zeros = HCons1 (TBool False) $ HCons1 (TInt 0) $ HNil1

Answer 4

Have you played with Data.Typeable? A Typeable constraint allows you to make guesses at the type hidden by the existential, and cast to that type when you guess right.

Not your example, but some example code I have lying around:

{-# LANGUAGE GADTs, ScopedTypeVariables, TypeOperators #-}

import Data.Typeable

data Showable where
    -- Note that this is an existential defined in GADT form
    Showable :: (Typeable a, Show a) => a -> Showable

instance Show Showable where
    show (Showable value) = "Showable " ++ show value

-- Example of casting Showable to Integer
castToInteger :: Showable -> Maybe Integer
castToInteger (Showable (value :: a)) = 
    case eqT :: Maybe (a :~: Integer) of
      Just Refl -> Just value
      Nothing -> Nothing

example1 = [Showable "foo", Showable 5]
example2 = map castToInteger example1