# scopedSort and kind variable left-biasing

Ryan Scott ryan.gl.scott at gmail.com
Thu Feb 14 15:34:52 UTC 2019

```Consider this function:

f :: Proxy (a :: j) -> Proxy (b :: k)

If you just collect the free type variables of `f`'s type in left-to-right
order, you'd be left with [a,j,b,k]. But the type of `f` is not `forall (a
:: j) j (b :: k) k. Proxy a -> Proxy b`, as that would be ill scoped. `j`
must come before `a`, since `j` appears in `a`'s kind, and similarly, `k`
must come before `b`.

Fortunately, GHC is quite smart about sorting free variables such that they
respect dependency order. If you ask GHCi what the type of `f` is (with
-fprint-explicit-foralls enabled), it will tell you this:

λ> :type +v f
f :: forall j k (a :: j) (b :: k). Proxy a -> Proxy b

As expected, `j` appears before `a`, and `k` appears before `b`.

In a different context, I've been trying to implement a type variable
sorting algorithm similar to the one that GHC is using. My previous
understanding was that the entirely of this sorting algorithm was
implemented in `Type.scopedSort`. To test my understanding, I decided to
write a program using the GHC API which directly uses `scopedSort` on the
example above:

main :: IO ()
main = do
let tv :: String -> Int -> Type -> TyVar
tv n uniq ty = mkTyVar (mkSystemName (mkUniqueGrimily uniq)
(mkTyVarOcc n)) ty
j = tv "j" 0 liftedTypeKind
a = tv "a" 1 (TyVarTy j)
k = tv "k" 2 liftedTypeKind
b = tv "b" 3 (TyVarTy k)
sorted = scopedSort [a, j, b, k]
putStrLn \$ showSDocUnsafe \$ ppr sorted

To my surprise, however, running this program does /not/ give the answer
[j,k,a,b], like what :type reported:

λ> main
[j_0, a_1, k_2, b_3]

meets the specification of ScopedSort, since it respects dependency order
and preserves the left-to-right ordering of variables that don't depend on
each other (i.e., `j` appears to the left of `k`, and `a` appears to the
left of `b`). But it's noticeably different that what :type reports. The
order that :type reports, [j,k,a,b], appears to bias kind variables to the
left such that all kind variables (`j` and `k`) appear before any type
variables (`a` and `b`).

>From what I can tell, scopedSort isn't the full story here. That is,
something else appears to be left-biasing the kind variables. My question
is: which part of GHC is doing this left-biasing?

Ryan S.
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