Closed type families with overlapping equations
This page describes an extension to type families that supports overlap.
 See also the Discussion Page added May 2012 (and now rather outofdate), for comment/suggestions/requests for clarification/alternative solutions, to explore the design space.
 See also the Coincident Overlap page (added August 2012) for a discussion around the usefulness of allowing certain overlaps when the righthand sides coincide.
 See also the Template Haskell page (added December 2012) for a proposal for the Template Haskell changes necessary to support this change.
 See also the Nonlinearity and Closed Type Families pages (added May 2013) for discussion and a proposal around type unsoundness that can be caused by repeated variables on the lefthand side of an instance. The proposal on that page will likely be implemented and will then be copied here.
See also
 #8154
 #8155
 Email thread on overlap restrictions for open families
 Draft paper about closed type families
Status: A working implementation with closed type families has been pushed to HEAD. The description of the feature below is accurate as of Jun 24, 2013.
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
A new version of axioms is now implemented. The formal treatment can be found in docs/corespec/corespec.pdf.
Here are the changes to source Haskell.
 A
type family
declaration can now define equations, making a closed type family:
type family Equal a b :: Bool where
Equal a a = True
Equal a b = False

Patterns within a single closed declaration may overlap, and are matched top to bottom.

Open type families are unchanged, except that a certain corner case of instances with a nontrivial overlap is disallowed. See here. I (Richard) do not expect this change to break much code.

A closed family does not need to be exhaustive. If there is no equation that matches, the call is stuck. (This is exactly as at present.)

An error is issued when a later equation is matched by a former, making the later one inaccessible.
type family F a where F (a,b) = [Char] F (Int,b) = Char
Here the second equation can never match.
For closed kinds (and maybe for open ones, but I can't unravel it), it seems possible to write a set of equations that will catch all possible cases but doesn't match the general case. This situation is currently (Dec 2012) undetected, because I (Richard,
eir
atcis.upenn.edu
) am unconvinced I have a strong enough handle on the details. For example, what aboutAny
? 
The equations do not need to share a common pattern:
type family F a where F Int = Char F (a,b) = Int

When matching a use of a closed type family, special care must be taken (by GHC) not to accidentally introduce incoherence. Consider the following example:
type family F a where F Int = Bool F a = Char
and we try to simplify the type
F b
. The naive implementation would just simplifyF b
toChar
, but this would be wrong. The problem is thatb
may later be unified withInt
, meaningF b
should simplify toBool
, notChar
. So, the correct behavior is not to simplifyF b
at all; it is stuck for now. Note that the second equation above is not useless: we will still simplify, say,F Double
toChar
. 
More formally, we only match a type (called the target) against an equation in a closed type family when, for all previous equations: the LHS is apart from the target, or the equation is compatible with the chosen one. Apartness and compatibility are defined below.

Two types are apart when they cannot simplify to a common reduct, even after arbitrary substitutions. This is undecidable, in general, so we implement a conservative check as follows: two types are considered to be apart when they cannot unify, omitting the occurs check in the unification algorithm.

Two equations are compatible if, either, their LHSs are apart, or their RHSs are syntactically equal after substitution with the unifier of their LHSs. This last condition allows closed type families to participate in the coincident overlap that open families have supported for years.
For example:
type family And (a :: Bool) (b :: Bool) where
And False a = False
And True b = b
And c False = False
And d True = d
And e e = e
All of these equations are compatible, meaning that GHC does not have to do the apartness check against the target.
Limitations
The implementation described above does not address all desired use cases. In particular, it does not work with associated types at all. The biggest reason not to add associated types into the mix is that it will be a confusing feature. Overlap among class instances is directed by specificity; overlap among family instances is ordered by the programmer. Users would likely expect the two to coincide, but they don't and can't, as it would not be type safe:
It seems that intermodule overlapping noncoincident associated types are a Bad Idea, but please add comments if you think otherwise and/or need such a feature. Why is it a Bad Idea? Because it would violate type safety: different modules with different visible instances could simplify type family applications to different ground types, perhaps concluding True ~ False
, and the world would immediately cease to exist.
This last point doesn't apply to overlapping type class instances because type class instance selection compiles to a termlevel thing (a dictionary). Using two different dictionaries for the same constraint in different places may be silly, but it won't end the world.