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Dependent types are a very useful feature - the gold standard of enforcing invariants at compile time. However, they are still very much not practical, especially considering inference for unrestricted dependent types is equivalent to higher-order unification, which was proven to be undecidable.
Fortunately, many of the benefits that dependent types bring aren’t because of dependent products themselves, but instead because of associated features commonly present in those programming languages. One of these, which also happens to be especially easy to mimic, are inductive families, a generalisation of inductive data types: instead of defining a single type inductively, one defines an entire family of related types.
Many use cases for inductive families are actually instances of a rather less general concept, that of generalised algebraic data types, or GADTs: Contrary to the indexed data types of full dependently typed languages, these can and are implemented in several languages with extensive inference, such as Haskell, OCaml and, now, Amulet.
Before I can talk about their implementation, I am legally obligated to present the example of length indexed vectors, linked structures whose size is known at compile time—instead of carrying around an integer representing the number of elements, it is represented in the type-level by a Peano1 natural number, as an index to the vector type. By universally quantifying over the index, we can guarantee by parametricity2 that functions operating on these don’t do inappropriate things to the sizes of vectors.
type z ;; type s 'k ;; type vect 'n 'a = | Nil : vect z 'a | Cons : 'a * vect 'k 'a -> vect (s 'k) 'a
Since the argument
vect (its length) varies with the constructor one
chooses, we call it an index; On the other hand,
'a, being uniform over all
constructors, is called a parameter (because the type is parametric over
the choice of
'a). These definitions bake the measure of length into
the type of vectors: an empty vector has length 0, and adding an element
to the front of some other vector increases the length by 1.
Matching on a vector reveals its index: in the
Nil case, it’s possible
to (locally) conclude that it had length
z. Meanwhile, the
lets us infer that the length was the successor of some other natural
s 'k, and that the tail itself has length
If one were to write a function to
map a function over a
they would be bound by the type system to write a correct implementation
- well, either that or going out of their way to make a bogus one. It
would be possible to enforce total correctness of a function such as
this one, by adding linear types and making the vector parameter linear.
let map (f : 'a -> 'b) (xs : vect 'n 'a) : vect 'n 'b = match xs with | Nil -> Nil | Cons (x, xs) -> Cons (f x, map f xs) ;;
If we were to, say, duplicate every element in the list, an error would be reported. Unlike some others, this one is not very clear, and it definitely could be improved.
Occurs check: The type variable jx occurs in the type s 'jx · Arising from use of the expression Cons (f x, Cons (f x, map f xs)) │ 33 │ | Cons (x, xs) -> Cons (f x, Cons (f x, map f xs)) ;; │ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
This highlights the essence of GADTs: pattern matching on them reveals equalities about types that the solver can later exploit. This is what allows the programmer to write functions that vary their return types based on their inputs - a very limited form of type-term dependency, which brings us ever closer to the Calculus of Constructions corner of Barendregt’s lambda cube3.
The addition of generalised algebraic data types has been in planning for over two years—it was in the original design document. In a mission that not even my collaborator noticed, all of the recently-added type system and IR features were directed towards enabling the GADT work: bidirectional type checking, rank-N polymorphism and coercions.
All of these features had cover stories: higher-ranked polymorphism was motivated by monadic regions; bidirectional type checking was motivated by the aforementioned polymorphism; and coercions were motivated by newtype optimisation. But, in reality, it was a conspiracy to make GADTs possible: having support for these features simplified implementing our most recent form of fancy types, and while adding all of these in one go would be possible, doing it incrementally was a lot saner.
While neither higher-ranked types nor GADTs technically demand a bidirectional type system, implementing them with such a specification is considerably easier, removing the need for workarounds such as boxy types and a distinction between rigid/wobbly type variables. Our algorithm for GADT inference rather resembles Richard Eisenberg’s Bake4, in that it only uses local equalities in checking mode.
Adding GADTs also lead directly to a rewrite of the solver, which now
has to work with implication constraints, of the form
(Q₁, ..., Qₙ) => Q, which should be read as “Assuming
Q.” Pattern matching on generalised constructors, in checking mode,
captures every constraint generated by checking the right-hand side of a
clause and captures that as an implication constraint, with all the
constructor-bound equalities as assumptions. As an example, this lets us
write a type-safe cast function:
type eq 'a 'b = Refl : eq 'a 'a (* an inhabitant of eq 'a 'b is a proof that 'a and 'b are equal *) let subst (Refl : eq 'a 'b) (x : 'a) : 'b = x ;;
Unfortunately, to keep inference decidable, many functions that depend on generalised pattern matching need explicit type annotations, to guide the type checker.
When checking the body of the function, namely the variable reference
x, the solver is working under an assumption
'a ~ 'b (i.e.,
'b stand for the same type), which lets us unify the stated type of
'a, with the return type of the function,
If we remove the local assumption, say, by not matching on
Refl, the solver will not be allowed to unify the two type
'b, and an error message will be reported5:
examples/gadt/equality.ml[11:43 ..11:43]: error Can not unify rigid type variable b with the rigid type variable a · Note: the variable b was rigidified because of a type ascription against the type forall 'a 'b. t 'a 'b -> 'a -> 'b and is represented by the constant bq · Note: the rigid type variable a, in turn, was rigidified because of a type ascription against the type forall 'a 'b. t 'a 'b -> 'a -> 'b · Arising from use of the expression x │ 11 │ let subst (_ : t 'a 'b) (x : 'a) : 'b = x ;; │ ~
Our intermediate language was also extended, from a straightforward System F-like lambda calculus with type abstractions and applications, to a System FC-like system with coercions, casts, and coercion abstraction. Coercions are the evidence, produced by the solver, that an expression is usable as a given type—GADT patterns bind coercions like these, which are the “reification” of an implication constraint. This lets us make type-checking on the intermediate language fast and decidable6, as a useful sanity check.
The two new judgements for GADT inference correspond directly to new
cases in the
check functions, the latter of which I
present here for completeness. The simplicity of this change serves as
concrete evidence of the claim that bidirectional systems extend readily
to new, complex features, producing maintainable and readable code.
Match t ps a) ty = do check (<- infer t (t, tt) <- for ps $ \(p, e) -> do ps <- checkPattern p tt (p', ms, cs) let tvs = Set.map unTvName (boundTvs p' ms) <$> implies (Arm p e) tt cs (p',) %~ Set.union tvs) (local (typeVars (extendMany ms (check e ty)))pure (Match t ps (a, ty))
This corresponds to the checking judgement for matches, presented below.
Note that in my (rather informal) theoretical presentation of Amulet
typing judgements, we present implication constraints as a lexical scope
of equalities conjoined with the scope of variables; Inference
judgements (with double right arrows, ) correspond to uses of
infer, pattern checking judgements ()
checkPattern, which also doubles as and
, and the main checking judgement is the
Our implementation of the type checker is a bit more complex, because it also does (some) elaboration and bookkeeping: tagging terms with types, blaming type errors correctly, etc.
This new, complicated feature was a lot harder to implement than originally expected, but in the end it worked out. GADTs let us make the type system stronger, while maintaining the decidable inference that the non-fancy subset of the language enjoys.
The example presented here was the most boring one possible, mostly because two weeks ago I wrote about their impact on the language’s ability to make things safer.
Peano naturals are one particular formulation of the natural numbers, which postulates that zero (denoted
zabove) is a natural number, and any natural number’s successor (denoted
s 'kabove) is itself natural.↩︎
Amulet is currently somewhere on the edge between λ2 - the second order lambda calculus, System F, and λP2, a system that allows quantification over types and terms using the dependent product form, which subsumes both the ∀ binder and the → arrow. Our lack of type functions currently leaves us very far from the CoC.↩︎
Even if we don’t do it yet—work is still ongoing to make the type checker and solver sane.↩︎