Note [Splitting nested sigma types in class type signatures]

GHC/Tc/TyCl.hs:5598 compiler

Consider this type synonym and class definition:

  type Traversal s t a b = forall f. Applicative f => (a -> f b) -> s -> f t

  class Each s t a b where
    each         ::                                      Traversal s t a b
    default each :: (Traversable g, s ~ g a, t ~ g b) => Traversal s t a b

It might seem obvious that the tau types in both type signatures for `each`
are the same, but actually getting GHC to conclude this is surprisingly tricky.
That is because in general, the form of a class method's non-default type
signature is:

  forall a. C a => forall d. D d => E a b

And the general form of a default type signature is:

  forall f. F f => E a f -- The variable `a` comes from the class

So it you want to get the tau types in each type signature, you might find it
reasonable to call tcSplitSigmaTy twice on the non-default type signature, and
call it once on the default type signature. For most classes and methods, this
will work, but Each is a bit of an exceptional case. The way `each` is written,
it doesn't quantify any additional type variables besides those of the Each
class itself, so the non-default type signature for `each` is actually this:

  forall s t a b. Each s t a b => Traversal s t a b

Notice that there _appears_ to only be one forall. But there's actually another
forall lurking in the Traversal type synonym, so if you call tcSplitSigmaTy
twice, you'll also go under the forall in Traversal! That is, you'll end up
with:

  (a -> f b) -> s -> f t

A problem arises because you only call tcSplitSigmaTy once on the default type
signature for `each`, which gives you

  Traversal s t a b

Or, equivalently:

  forall f. Applicative f => (a -> f b) -> s -> f t

This is _not_ the same thing as (a -> f b) -> s -> f t! So now tcMatchTy will
say that the tau types for `each` are not equal.

A solution to this problem is to use tcSplitNestedSigmaTys instead of
tcSplitSigmaTy. tcSplitNestedSigmaTys will always split any foralls that it
sees until it can't go any further, so if you called it on the default type
signature for `each`, it would return (a -> f b) -> s -> f t like we desired.

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