DataKind Unions

Question

I'm not sure if it is the right terminology, but is it possible to declare function types that take in an 'union' of datakinds?

For example, I know I can do the following:

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

...

data Shape'
  = Circle'
  | Square'
  | Triangle'

data Shape :: Shape' -> * where
  Circle :: { radius :: Int} -> Shape Circle'
  Square :: { side :: Int} -> Shape Square'
  Triangle
    :: { a :: Int
       , b :: Int
       , c :: Int}
    -> Shape Triangle'

test1 :: Shape Circle' -> Int
test1 = undefined

However, what if I want to take in a shape that is either a circle or a square? What if I also want to take in all shapes for a separate function?

Is there a way for me to either define a set of Shape' kinds to use, or a way for me to allow multiple datakind definitions per data?

Edit:

The usage of unions doesn't seem to work:

{-# LANGUAGE ConstraintKinds #-}
{-# LANGUAGE DataKinds       #-}
{-# LANGUAGE GADTs           #-}
{-# LANGUAGE KindSignatures  #-}
{-# LANGUAGE PolyKinds       #-}
{-# LANGUAGE TypeFamilies    #-}
{-# LANGUAGE TypeOperators   #-}

...

type family Union (a :: [k]) (r :: k) :: Constraint where
  Union (x ': xs) x = ()
  Union (x ': xs) y = Union xs y

data Shape'
  = Circle'
  | Square'
  | Triangle'

data Shape :: Shape' -> * where
  Circle :: { radius :: Int} -> Shape Circle'
  Square :: { side :: Int} -> Shape Square'
  Triangle
    :: { a :: Int
       , b :: Int
       , c :: Int}
    -> Shape Triangle'

test1 :: Union [Circle', Triangle'] s => Shape s -> Int
test1 Circle {} = undefined
test1 Triangle {} = undefined
test1 Square {} = undefined

The part above compiles

Answer 1

You can accomplish something like this in (I think) a reasonably clean way using a type family together with ConstraintKinds and PolyKinds:

type family Union (a :: [k]) (r :: k) :: Constraint where
  Union (x ': xs) x = ()
  Union (x ': xs) y = Union xs y

test1 :: Union [Circle', Triangle'] s => Shape s -> Int
test1 = undefined

The () above is the empty constraint (it's like an empty "list" of type class constraints).

The first "equation" of the type family makes use of the nonlinear pattern matching available in type families (it uses x twice on the left hand side). The type family also makes use of the fact that if none of the cases match, it will not give you a valid constraint.

You should also be able to use a type-level Boolean instead of ConstraintKinds. That would be a bit more cumbersome and I think it would be best to avoid using a type-level Boolean here (if you can).

Side-note (I can never remember this and I had to look it up for this answer): You get Constraint in-scope by importing it from GHC.Exts.

Edit: Partially disallowing unreachable definitions

Here is a modification to get it to (partially) disallow unreachable definitions as well as invalid calls. It is slightly more roundabout, but it seems to work.

Modify Union to give a * instead of a constraint, like this:

type family Union (a :: [k]) (r :: k) :: * where
  Union (x ': xs) x = ()
  Union (x ': xs) y = Union xs y

It doesn't matter too much what the type is, as long as it has an inhabitant you can pattern match on, so I give back the () type (the unit type).

This is how you would use it:

test1 :: Shape s -> Union [Circle', Triangle'] s -> Int
test1 Circle {}   () = undefined
test1 Triangle {} () = undefined
-- test1 Square {} () = undefined -- This line won't compile

If you forget to match on it (like, if you put a variable name like x instead of matching on the () constructor), it is possible that an unreachable case can be defined. It will still give a type error at the call-site when you actually try to reach that case, though (so, even if you don't match on the Union argument, the call test1 (Square undefined) () will not type check).

Note that it seems the Union argument must come after the Shape argument in order for this to work (fully as described, anyway).

Answer 2

This is getting kind of awful, but I guess you could require a proof that it's either a circle or a square using Data.Type.Equality:

test1 :: Either (s :~: Circle') (s :~: Square') -> Shape s -> Int

Now the user has to give an extra argument (a "proof term") saying which one it is.

In fact you can use the proof term idea to "complete" bradm's solution, with:

class MyOpClass sh where
    myOp :: Shape sh -> Int
    shapeConstraint :: Either (sh :~: Circle') (sh :~: Square')

Now nobody can go adding any more instances (unless they use undefined, which would be impolite).

Answer 3

You could use typeclasses:

class MyOpClass sh where
    myOp :: Shape sh -> Int

instance MyOpClass Circle' where
    myOp (Circle r) = _

instance MyOpClass Square' where
    myOP (Square s) = _

This doesn't feel like a particularly 'complete' solution to me - anyone could go back and add another instance MyOpClass Triangle' - but I can't think of any other solution. Potentially you could avoid this problem simply by not exporting the typeclass however.

Answer 4

Another solution I've noticed, though pretty verbose, is to create a kind that has a list of feature booleans. You can then pattern match on the features when restricting the type:

-- [circleOrSquare] [triangleOrSquare]
data Shape' =
  Shape'' Bool
          Bool

data Shape :: Shape' -> * where
  Circle :: { radius :: Int} -> Shape (Shape'' True False)
  Square :: { side :: Int} -> Shape (Shape'' True True)
  Triangle
    :: { a :: Int
       , b :: Int
       , c :: Int}
    -> Shape (Shape'' False True)

test1 :: Shape (Shape'' True x) -> Int
test1 Circle {}   = 2
test1 Square {}   = 2
test1 Triangle {} = 2

Here, Triangle will fail to match:

    • Couldn't match type ‘'True’ with ‘'False’
      Inaccessible code in
        a pattern with constructor:
          Triangle :: Int -> Int -> Int -> Shape ('Shape'' 'False 'True),
        in an equation for ‘test1’
    • In the pattern: Triangle {}
      In an equation for ‘test1’: test1 Triangle {} = 2
   |
52 | test1 Triangle {} = 2
   |       ^^^^^^^^^^^

Unfortunately, I don't think you can write this as a record, which may be clearer and avoids the ordering of the features.

This might be usable in conjunction with the class examples for readability.