This page summarizes the design behind injective type families (#6018). It is a
work in progress. This page will evolve to reflect the progress made.
Person responsible for this page and the implementation: Jan Stolarek (just so
you now who is meant by "I").
Forms of injectivity
The idea behind #6018 is to allow users to declare that a type family is
injective. So far we have identified these forms of injectivity:
Injectivity in all the arguments, where knowing the result (right-hand
side) of a type family determines all the arguments on the left-hand
typefamilyId a whereId a = a
typefamilyF a b ctypeinstanceFIntCharBool=BooltypeinstanceFCharBoolInt=InttypeinstanceFBoolIntChar=Char
Injectivity in some of the arguments, where knowing the RHS of a type
family determines only some of the arguments on the LHS. Example:
typefamilyG a b ctypeinstanceGIntCharBool=BooltypeinstanceGIntCharInt=BooltypeinstanceGBoolIntInt=Int
Here knowing the RHS allows us to determine first two arguments, but not the
Injectivity in some of the arguments, where knowing the RHS of a type
family and some of the LHS arguments determines other (possibly not all)
LHS arguments. Examples:
typefamilyPlus a b wherePlusZ n = n Plus(S m ) n =S(Plus n m)
Here knowing the RHS and any single argument uniquely determines the
typefamilyH a b ctypeinstanceHIntCharDouble=InttypeinstanceHBoolDoubleDouble=Int
Knowing the RHS and either a or b allows to uniquely determine the
remaining two arguments, but knowing the RHS and c gives us no information
about a or b.
In the following text I will refer to these three forms of injectivity as A, B
Note that at the moment we only have practical use cases for injectivity of
form A. Because of that I propose that we implement only this form of injectivity.
We can implement B and C when somebody comes up with compelling use cases.
Two important things to consider when deciding on a concrete syntax are:
Backwards compatibility: at the moment no decision has been made on
whether injectivity will become a part of existing TypeFamilies extension
or if we create a new InjectiveTypeFamilies extension that implies
TypeFamilies. If we choose the former then we need to come up with syntax
that is backwards compatible. (Perhaps this is the other way around: if we
end up having backwards incompatible syntax then we must create a new
Future extensibility: if we only implement A we still want to be able
to add B and C in the future without breaking A.
We decided to use syntax similar to functional dependencies. The injectivity
declaration begins with | following type family declaration head. | is
followed by a list of comma-separated injectivity conditions. Each injectivity
condition has the form:
where A and B are non-empty lists of type variables declared in type family
head. Things on the left and right of -> are called LHS and RHS of an
injectivity condition, respectively. The user must be able to name the result of
a type family. To achieve that we extend syntax of type family head by allowing
to write = tyvar or = (tyvar :: kind) annotations in addition to already
allowed :: kind annotation. In other words all these declaration are
but the third or fourth form is required if the user wants to introduce
injectivity declaration. Here are examples of injectivity declarations using
typefamilyId a = result | result -> a wheretypefamilyF a b c = d | d -> a b ctypefamilyG a b c = foo | foo -> a b wheretypefamilyPlus a b =(sum ::Nat)| sum a -> b, sum b -> a wheretypefamilyH a b c = xyz | xyz a -> b c, xyz b -> a c
Note that above examples demonstrate all three types of injectivivity but for
now we only plan to implement injectivity A.
natural reading: res a -> b reads as "knowing res and a determines b".
extensible for the future
since we only plan to implement injectivity of form A we require that there
is only one injectivity condition, its LHS contains only a type variable
naming the result and its RHS contains all type variables naming the
arguments to a type family. This might be a bit annoying.
Real-life use cases
If you can supply more examples (type family declarations + their usage that
currently doesn't work but should work with injectivity) please add them here.
typefamilyF a | result -> a whereFChar=BoolFBool=IntFInt=CharidChar::(F a ~Bool)=> a ->CharidChar a = a
GHC should infer that a is in fact Char. Right now this program is rejected.
classManifold' a wheretypeField a typeBase a typeTangent a typeTangentBundle a typeDimension a ::NattypeUsesMetric a ::Symbol project :: a ->Base a unproject ::Base a -> a tangent :: a ->TangentBundle a cotangent :: a ->(TangentBundle a ->Field a)-- this worksid':: forall a.(Manifold' a )=>Base a ->Base aid' input = project out where out :: a out = unproject input-- but this requires injective type familiesid':: forall a.(Manifold' a )=>Base a ->Base aid'= project . unproject
Verifying correctness of user's injectivity declaration
Once the user declares type family to be injective GHC must verify that this
declaration is correct, ie. type family really is injective. Below is an outline
of the algorithm. It is then follow with a discussion and rationale. Note: I
am not too familiar with GHC's type checking so please tell me if I'm making any
wrong assumptions here. Also, I might be using incorrect terminology. If so
please complain or fix.
Important: this algorithm is only for the injectivity of type A. RHS
refers to the right-hand side of the type family being checked for
injectivity. LHS refers to the arguments of that type family.
If a RHS contains a call to a type family we conclude that the type family is
not injective. I am not certain if this extends to self-recursion -- see
Open type families:
When checking equation of an open type family we try to unify its RHS with
RHSs of all equations we've seen so far. If we succeed then LHSs of unified
equations must be identical under that substitution. If they are not identical
then we declare that a type family is not injective.
RAE: I believe this would also have to check that all variables mentioned in the LHS are mentioned in the RHS. Or, following along with Simon's idea in comment:76:ticket:6018, those variables that appear in the injectivity annotation must be mentioned in the RHS. End RAE
Closed type families
If a type family has only one equation we declare it to be injective (unless
case 1 above applies). RAE or there are variables missing from the RHS End RAE
If a type family has more equations then we check them one by one. Checking
each equation consists of these steps:
verify 1) for the current equation.
a) attempt to unify RHS with RHSs of all previous equations. If this does not
succeed proceed to next equation. If this was the last equation then type
family is injective.
b) if unification in a) succeeds and no substitutions are required then type
family is not injective and we report an error to the user. By "no
substitution" I mean situation when there are no type variables involved
ie. both sides declare the same concrete type (eg. Int). RAE This seems to be a straightforward special case of (c). Omit? End RAE
c) if unification succeeds and there are type variables involved we substitute
unified type variables on the LHS and check whether this LHS overlaps with
any of the previous equations. If it does we proceed to the next equation
(if this was the last equation then type family is injective). If it
doesn't then type family is not injective and we report an error to the
user. See examples and discussion below for more details.
RAE: We have to be careful with the word overlap here. I think we want "overlaps" to be "is subsumed by". Basically, you want to know of the equation at hand is reachable, given the LHS substitution. Even if it is reachable, the (substituted) LHS may coincide with the LHS of the earlier equation whose RHS unified with the current RHS. So, just because the current LHS is reachable doesn't mean the family is not injective.
Even subtler, it's possible that certain values for type variables are excluded if the current LHS is reachable (for example, some variable a couldn't be Int, because if a were Int, then a previous equation would have triggered). Perhaps these exclusions can also be taken into account. Thankfully, forgetting about the exclusions is conservative -- the exclusions can only make more families injective, not fewer. End RAE
Case 1 above is conservative and may incorrectly reject injectivity declaration
for some type families. For example type family I:
typefamilyI a whereI a =Id a -- Id defined earlier
is injective but restriction in case 1 will reject it. Why such a restriction?
Consider this example. Let us assume that we have type families X and Y that
we already know are injective, and type constructors Bar :: * -> * and
Baz :: * -> * -> *. Now we declare:
typefamilyFoo a whereFoo(Bar a)=X a Foo(Baz a b)=Y a b
Here we would need to check whether results of X a and Y a b can overlap and
I don't see a good way of doing this once the RHS has nested calls to type
families. Second of all if we see Foo a ~ Bool during type checking GHC can't
solve that without trying each equation one by one to see which one returns
Bool. This takes a lot of guessing and I believe GHC should refuse to do that.
See also #4259.
What about self-recursion? Consider this type family:
-- assumes promoted Maybe and NattypefamilyNatToMaybe a whereNatToMaybeZ=NothingNatToMaybe(S n)=Just(NatToMaybe n)
Using Case 4a we will infer correctly that this type family is injective. Now
typefamilyBan a whereBanZ=ZBan(S n)=Ban n
To verify injectivity in this case we need to ask whether Z ~ Ban n (case 4a
above). We can only answer such question when a type family is injective. Here
we have not confirmed that Ban is injective so we may rightly refuse to answer
Z ~ Ban n and conclude (correctly) that this type family is not injective. So it
seems to me that we could allow self-recursion - I have not yet identified any
corner cases that would prevent us from doing so.
RAE: But it seems that, under this treatment, any self-recursion would automatically lead to a conclusion of "not injective", just like any other use of a type family. For example:
typefamilyIdNat n whereIdNatZ=ZIdNat(S n)=S(IdNat n)
IdNat is injective. But, following the algorithm above, GHC would see the recursive use of IdNat, not know whether IdNat is injective, and then give up, concluding "not injective". Is there a case where the special treatment of self-recursion leads to a conclusion of "injective"? End RAE
Here's an example of case 4c in action:
typefamilyBak a whereBakInt=CharBakChar=IntBak a = a
Equation 2 falls under case 4a. When we reach equation 3 we attempt to verify
it's RHS with each RHS of the previous equations. Attempt to unify a with
Char succeeds. Since unification was successful we substitute Char on the LHS
of the 3rd equation and get Bak Char. Now we must check if this equation
overlaps with any of the previous equations. It does overlap with the second
one, so everything is fine: we know that 3rd equation will never return a Char
(like 1st equation) because this case will be caught by the 2nd equation. We
know repeat the same steps for the second equations and conclude that 3rd
equation will never produce an Int because this will be caught by the 1st
equation. We thus conclude that Bak is injective.
Other implementation details
The implementation needs to check the correctness of injectivity conditions
declarations. This includes checking that:
only in-scope type variables are used. For example
type family F a | result -> b should result with "not in scope: b" error.
there are no identical conditions (this wouldn't hurt, but the user deserves
a warning about this)
type variables are not repeated on either LHS or RHS of the injectivity
condition. For example result a a -> ... or ... -> a b a should generate
a warning. Note that it probably is OK to have the same variable both on the
LHS and RHS of an injectivity condition: in the above examples it is true
that type family G a b c | result c -> a b c. The question is whether this
has any practical relevance.
injectivity conditions don't overlap (eg. result -> a b overlaps
result -> a). This probably deserves a warning.
I am not certain at the moment how to treat these injectivity conditions
result -> a, result -> b is technically correct but we could just say
result -> a b. Do the two separate declarations have the same power as the
Here it was suggested by Simon that we could infer
injectivity for closed type families. This is certainly possible for injectivity
A, but not for B or C, where the number of possible injectivity conditions is
exponential in the number of arguments. I personally am against inferring
injectivity. The reasons are:
Before 7.10 comes out there will already be some code in the wild that uses
closed type families introduced in GHC 7.8. None of these type families
require injectivity to work because GHC 7.8 does not support injectivity. If
we attempt to infer injectivity for all these already existing closed type
families we will only increase compilation time of existing code with
absolutely no gain in functionality of the code. There were some complaints
about GHC's performance decreasing with each release and I don't want to add
I believe that requiring explicit injectivity annotations is a valuable
source code documentation for the programmer.
Annotations also make it explicit which code will compile with GHC 7.10 and
which will not.
I don't like the idea of mismatch between open type families and closed type
families, meaning that injectivity of open type families would be openly
documented whereas for closed type families it would be hidden.
This page does not mention anything about associated types. The intention is
that injectivity will also work for those. I just haven't thought about the
Here Richard mentions head-injectivity used by
Agda. We don't plan to follow that direction.
I'm not yet sure what is the exact scope of things we want to do with
injectivity. Certainly if we have F a ~ F b and F is injective then we
can declare a ~ b. (Note: Simon remarks that in such case a must be a
skolem constant and b a unification variable. I don't know the difference
between the two). Moreover if we have F a ~ Int then we can infer a. But
what if we also have injective type family G and see a constraint like
F a ~ G b c? For injective type families there exists at most one solution
but I think that GHC should refuse to solve such riddles.