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haskelltype-familiesdata-kinds

Folding recursive type family

I'm trying to fold a data with a phantom type of kind [*]. Here is simplified version of my code

{-# LANGUAGE DataKinds, KindSignatures #-}

module Stack where
import Data.HList
import Data.Foldable as F

data T (a :: [*]) = T (Tagged a String)
(!++!) :: T a -> T b -> T (HAppendList a b)
(T a) !++! (T b) = T (Tagged (untag a ++ untag b))


a = T (Tagged "1") :: T '[Int]
b = T (Tagged "-- ") :: T '[]
ab = a !++! b :: T '[Int]

I would like a fold operator

(!++*) :: (Foldable t ) =>  T a -> t (T '[]) -> T a
a !++* t = F.foldl (!++!) a t

But that doesn't work. The compiler that a and HAppendList a '[] are different, even though they are not.

Why can't the compile unify HAppendList a '[] and a ?

(I can't do the fold manually in ghci though :t a !++! b !++! b !++! b => T '[Int]

Solution

  • Note the definition of HAppendList:

    type family HAppendList (l1 :: [k]) (l2 :: [k]) :: [k]
type instance HAppendList '[] l = l
type instance HAppendList (e ': l) l' = e ': HAppendList l l'

    You and I know that [] is the left and right identity of ++, but the compiler is only aware of the left identity:

    happend' :: T a -> T b -> T (HAppendList a b)
happend' (T (Tagged a)) (T (Tagged b)) = (T (Tagged (a++b)))

-- Doesn't typecheck 
leftIdentity' :: T a -> T '[] -> T a 
leftIdentity' x y = happend' x y 

rightIdentity' :: T '[] -> T a -> T a 
rightIdentity' x y = happend' x y

    You would need to have 

    type instance HAppendList '[] l = l
type instance HAppendList l '[] = l
type instance HAppendList (e ': l) l' = e ': HAppendList l l'

    to have the compiler know about left and right identity; but these would be overlapping, so it doesn't type check. You can just flip the arguements, however:

    (!+++!) :: T a -> T b -> T (HAppendList a b) 
(!+++!) (T (Tagged x)) (T (Tagged y)) = T (Tagged (y ++ x))

(!++*) :: Foldable t => T a -> t (T '[]) -> T a
a !++* t = F.foldl (flip (!+++!)) a t

    With closed type families introduced in ghc 7.8, you can fix this problem:

    type family (++) (a :: [k]) (b :: [k]) :: [k] where 
  '[] ++ x = x
  x ++ '[] = x
  (x ': xs) ++ ys = x ': (xs ++ ys)

happend :: T a -> T b -> T (a ++ b)
happend (T (Tagged a)) (T (Tagged b)) = (T (Tagged (a++b)))

leftIdentity :: T a -> T '[] -> T a 
leftIdentity x y = happend x y 

rightIdentity :: T '[] -> T a -> T a 
rightIdentity x y = happend x y