Subsumption in polymorphic types

Question

In 'Practical type inference for arbitrary-rank types' , the authors talk about subsumption :

I try to test things in GHCi as I read, but even though g k2 is meant to typecheck, it doesn't when I try with GHC 7.8.3:

λ> :set -XRankNTypes
λ> let g :: ((forall b. [b] -> [b]) -> Int) -> Int; g = undefined
λ> let k1 :: (forall a. a -> a) -> Int; k1 = undefined
λ> let k2 :: ([Int] -> [Int]) -> Int; k2 = undefined
λ> :t g k1

<interactive>:1:3: Warning:
    Couldn't match type ‘a’ with ‘[a]’
      ‘a’ is a rigid type variable bound by
          the type forall a1. a1 -> a1 at <interactive>:1:3
    Expected type: (forall b. [b] -> [b]) -> Int
      Actual type: (forall a. a -> a) -> Int
    In the first argument of ‘g’, namely ‘k1’
    In the expression: g k1
g k1 :: Int
λ> :t g k2

<interactive>:1:3: Warning:
    Couldn't match type ‘[Int] -> [Int]’ with ‘forall b. [b] -> [b]’
    Expected type: (forall b. [b] -> [b]) -> Int
      Actual type: ([Int] -> [Int]) -> Int
    In the first argument of ‘g’, namely ‘k2’
    In the expression: g k2
g k2 :: Int

I haven't really gotten to the point where I understand the paper, yet, but still, I worry that I have misunderstood something. Should this typecheck? Are my Haskell types wrong?

Answer 1

The typechecker doesn't know when to apply the subsumption rule.

You can tell it when with the following function.

Prelude> let u :: ((f a -> f a) -> c) -> ((forall b. f b -> f b) -> c); u f n = f n

This says, given a function from a transformation for a specific type, we can make a function from a natural transformation forall b. fb -> fb forall b. fb -> fb .

We can then try it successfully on the second example.

Prelude> :t g (u k2)
g (u k2) :: Int

The first example now gives a more informative error as well.

Prelude> :t g (u k1)
    Couldn't match type `forall a. a -> a' with `[a0] -> [a0]'
    Expected type: ([a0] -> [a0]) -> Int
      Actual type: (forall a. a -> a) -> Int
    In the first argument of `u', namely `k1'
    In the first argument of `g', namely `(u k1)'

I don't know if we can write a more general version of u ; we'd need a constraint-level notion of less polymorphic to write something like let s :: (a :<: b) => (a -> c) -> (b -> c); sfx = fx let s :: (a :<: b) => (a -> c) -> (b -> c); sfx = fx