wiki:NewAxioms

Version 8 (modified by simonpj, 23 months ago) (diff)

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Pattern-matching axioms

Thhis page describes an extension to type families that supports overlap.

  • See also the Discussion Page added May 2012, for comment/suggestions/requests for clarification/alternative solutions, to explore the design space.
  • We'll need some concrete syntax for the discussion, so we'll follow the cached pdf, but note that the syntax there is not final.

Background

One might imagine that it would be a simple matter to have a type-level function

type family Equal a b :: Bool

so that (Equal t1 t2) was True if t1=t2 and False otherwise. But it isn't.

You can't write

type instance Equal a a = True
type instance Equal a b = False

because System FC (rightly) prohibits overlapping family instances.

Expanding this out, you can do it for a fixed collection of types thus:

type instance Equal Int Int = True
type instance Equal Bool Bool = True
type instance Equal Int Bool = False
type instance Equal Bool Int = False

but this obviously gets stupid as you add more types.

Furthermore, this is not what you want. Even if we restrict the equality function to booleans

type family Equal (a :: Bool) (b :: Bool) :: Bool

we can't define instances of Equal so that a constraint like this one

Equal a a ~ True

is satisfiable---the type instances only reduce if a is known to True or False. GHC doesn't reason by cases. (Nor should it, |Any| also inhabits |Bool|. No kinds really are closed.)

The only way to work with this sort of reasoning is to use Overlapping Instances, as suggested in the HList paper.

What to do about it

So the deficiency is in System FC, and it seems fundamental. We've been working on an extension to System FC, with a corresponding source-language extension, that does allow overlapping type families, with care. You would write something like this:

type instance where
  Equal a a = True
  Equal a b = False

This wiki page is a stub:

Status (Jan 12): the groundwork is done, in HEAD; mainly making CoAxiom a more fundamental data type. Not yet started on the details.

Attachments (1)

  • axioms.pdf (212.0 KB) - added by goldfire 20 months ago. Description of FC extension to support overlapping type family instances

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