Version 10 (modified by simonpj, 4 years ago) (diff) 

Patternmatching axioms
Thhis page describes an extension to type families that supports overlap.
 See this Github repo for a Latex draft of the design
 Here is a cached pdf of the current state
 We'll use GHC branch ghcaxioms for development work.
 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.
Status (Jan 12): the groundwork is done, in HEAD; mainly making CoAxiom a more fundamental data type. Not yet started on the details.
Background
One might imagine that it would be a simple matter to have a typelevel 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 satisfiablethe 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 sourcelanguage extension, that does allow overlapping type families, with care. Here's the idea:
 A type instance declaration can define multiple equations, not just one:
type instance Eq where Equal a a = True Equal a b = False
 Patterns within a single type instance declaration (henceforth "group") may overlap, and are matched top to bottom.
 A single type family may, as now, have multiple type instance declarations:
type family F a :: * type instance F where F [Int] = Int F [a] = Bool type instance F where F (Int,b) = Char F (a,b) = [Char]
 The groups for F may not overlap. That is, there must be no type t such that (F t) matches both groups.
 The groups do not need to be exhaustive. If there is no equation that matches, the call is stuck. (This is exactly as at present.)
 It would perhaps be possible to emit warnings for equations that are shadowed:
type instance F where F (a,b) = [Char] F (Int,b) = Char
Here the second equation can never match.
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axioms.pdf
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added by goldfire 3 years ago.
Description of FC extension to support overlapping type family instances
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