[Haskell-cafe] Why doesn't GHC derive these types

[Clinton Mead]

Consider the following program:

{-# LANGUAGE TypeFamilyDependencies #-}

data D x

type family F t = s | s -> t
type instance F (D t) = D (F t)

f :: F s -> ()
f _ = ()

g :: D (F t) -> ()
g x = f x

main = return ()


The problem seems to be the call from "g" to "f". We're calling "f" with an
argument of type "D (F t)". "f" then has to determine what "s" is in it's
signature. We know:

1. "F s ~ D (F t)" (from function call)
2. "D (F t) ~ F (D t)" (from the right hand side of the injective type
definition)

Therefore we should be able to derive:

3. "F s ~ F (D t)" (type equality is transitive)
4. "s ~ D t" (as F is injective)

I suspect the part we're missing in GHC is step 4. I recall reading this
somewhere but I can't find where now.

However, the paper about injective types says that this style of inference,
namely "F a ~ F b => a ~ b" should occur. I quote (
https://www.microsoft.com/en-us/research/wp-content/uploads/2016/07/injective-type-families-acm.pdf
section
5.1 p125):

So, faced with the constraint F α ∼ F β, the inference engine does not in
> general unify α := β; so the constraint F α ∼ F β is not solved, and hence
> f (g 3) will be rejected. But if we knew that F was injective, we can unify
> α := β without guessing.



> Improvement (a term due to Mark Jones (Jones 1995, 2000)) is a process
> that adds extra "derived" equality constraints that may make some extra
> unifications apparent, thus allowing inference to proceed further without
> having to make guesses. In the case of an injective F, improvement adds α ∼
> β, which the constraint solver can solve by unification. In general,
> improvement of wanted constraint is extremely simple:



> Definition 11 (Wanted improvement). Given the wanted constraint F σ ∼ F τ
> , add the derived wanted constraint σn ∼ τn for each n-injective argument
> of F.



> Why is this OK? Because if it is possible to prove the original constraint
> F σ ∼ F τ , then (by Definition 1) we will also have a proof of σn ∼ τn. So
> adding σn ∼ τn as a new wanted constraint does not constrain the solution
> space. Why is it beneficial? Because, as we have seen, it may expose
> additional guess-free unification opportunities that that solver can
> exploit.


Am I correct in my assessment of what is happening here with GHC? Is there
anyway to get it to compile this program, perhaps with an extension?
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