universes.md

Notes on universe polymorphism

The implementation of universe polymorphism introduces a few changes to the API of Coq. First and foremost, the term language changes, as global references now carry a universe level substitution:

type 'a puniverses = 'a * Univ.Instance.t
type pconstant = constant puniverses
type pinductive = inductive puniverses
type pconstructor = constructor puniverses

type constr = ...
  | Const of puniverses
  | Ind of pinductive
  | Constr of pconstructor

Universes

Universe instances (an array of levels) gets substituted when unfolding definitions, are used to typecheck and are unified according to the rules in the ITP'14 paper on universe polymorphism in Coq.

type Level.t = Set | Prop | Level of int * dirpath (* hashconsed *)
type Instance.t = Level.t array
type Universe.t = Level.t list (* hashconsed *)

The universe module defines modules and abstract types for levels, universes etc.. Structures are hashconsed (with a hack to take care of the fact that deserialization breaks sharing).

Definitions (constants, inductives) now carry around not only constraints but also the universes they introduced (a Univ.UContext.t). There is another kind of contexts Univ.ContextSet.t, the latter has a set of universes, while the former has serialized the levels in an array, and is used for polymorphic objects. Both have "reified" constraints depending on global and local universes.

A polymorphic definition is abstract w.r.t. the variables in this context, while a monomorphic one (or template polymorphic) just adds the universes and constraints to the global universe context when it is put in the environment. No other universes than the global ones and the declared local ones are needed to check a declaration, hence the kernel does not produce any constraints anymore, apart from module subtyping.... There are hence two conversion functions now: check_conv and infer_conv: the former just checks the definition in the current env (in which we usually push_universe_context of the associated context), and infer_conv which produces constraints that were not implied by the ambient constraints. Ideally, that one could be put out of the kernel, but currently module subtyping needs it.

Inference of universes is now done during refinement, and the evar_map carries the incrementally built universe context, starting from the global universe constraints (see Evd.from_env). Evd.conversion is a wrapper around infer_conv that will do the bookkeeping for you, it uses evar_conv_x. There is a universe substitution being built incrementally according to the constraints, so one should normalize at the end of a proof (or during a proof) with that substitution just like we normalize evars. There are some nf_* functions in library/universes.ml to do that. Additionally, there is a minimization algorithm in there that can be applied at the end of a proof to simplify the universe constraints used in the term. It is heuristic but validity-preserving. No user-introduced universe (i.e. coming from a user-written anonymous Type) gets touched by this, only the fresh universes generated for each global application. Using

val pf_constr_of_global : Globnames.global_reference -> (constr -> tactic) -> tactic

Is the way to make a constr out of a global reference in the new API. If they constr is polymorphic, it will add the necessary constraints to the evar_map. Even if a constr is not polymorphic, we have to take care of keeping track of its universes. Typically, using:

  mkApp (coq_id_function, [| A; a |])

and putting it in a proof term is not enough now. One has to somehow show that A's type is in cumululativity relation with id's type argument, incurring a universe constraint. To do this, one can simply call Typing.resolve_evars env evdref c which will do some infer_conv to produce the right constraints and put them in the evar_map. Of course in some cases you might know from an invariant that no new constraint would be produced and get rid of it. Anyway the kernel will tell you if you forgot some. As a temporary way out, Universes.constr_of_global allows you to make a constr from any non-polymorphic constant, but it will fail on polymorphic ones.

Other than that, unification (w_unify and evarconv) now take account of universes and produce only well-typed evar_maps.

Some syntactic comparisons like the one used in change have to be adapted to allow identification up-to-universes (when dealing with polymorphic references), make_eq_univs_test is there to help. In constr, there are actually many new comparison functions to deal with that:

(** [equal a b] is true if [a] equals [b] modulo alpha, casts,
   and application grouping *)
val equal : constr -> constr -> bool

(** [eq_constr_univs u a b] is [true] if [a] equals [b] modulo alpha, casts,
   application grouping and the universe equalities in [u]. *)
val eq_constr_univs : constr Univ.check_function

(** [leq_constr_univs u a b] is [true] if [a] is convertible to [b] modulo
    alpha, casts, application grouping and the universe inequalities in [u]. *)
val leq_constr_univs : constr Univ.check_function

(** [eq_constr_universes a b] [true, c] if [a] equals [b] modulo alpha, casts,
   application grouping and the universe equalities in [c]. *)
val eq_constr_universes : constr -> constr -> bool Univ.universe_constrained

(** [leq_constr_universes a b] [true, c] if [a] is convertible to [b] modulo
    alpha, casts, application grouping and the universe inequalities in [c]. *)
val leq_constr_universes : constr -> constr -> bool Univ.universe_constrained

(** [eq_constr_univs a b] [true, c] if [a] equals [b] modulo alpha, casts,
   application grouping and ignoring universe instances. *)
val eq_constr_nounivs : constr -> constr -> bool

The _univs versions are doing checking of universe constraints according to a graph, while the _universes are producing (non-atomic) universe constraints. The non-atomic universe constraints include the ULub constructor: when comparing f (* u1 u2 *) c and f (* u1' u2' *) c we add ULub constraints on u1, u1' and u2, u2'. These are treated specially: as unfolding f might not result in these unifications, we need to keep track of the fact that failure to satisfy them does not mean that the term are actually equal. This is used in unification but probably not necessary to the average programmer.

Another issue for ML programmers is that tables of constrs now usually need to take a constr Univ.in_universe_context_set instead, and properly refresh the universes context when using the constr, e.g. using Clenv.refresh_undefined_univs clenv or:

(** Get fresh variables for the universe context.
    Useful to make tactics that manipulate constrs in universe contexts polymorphic. *)
val fresh_universe_context_set_instance : universe_context_set ->
  universe_level_subst * universe_context_set

The substitution should be applied to the constr(s) under consideration, and the context_set merged with the current evar_map with:

val merge_context_set : rigid -> evar_map -> Univ.universe_context_set -> evar_map

The rigid flag here should be Evd.univ_flexible most of the time. This means the universe levels of polymorphic objects in the constr might get instantiated instead of generating equality constraints (Evd.univ_rigid does that).

On this issue, I recommend forcing commands to take global_references only, the user can declare his specialized terms used as hints as constants and this is cleaner. Alas, backward-compatibility-wise, this is the only solution I found. In the case of global_references only, it's just a matter of using Evd.fresh_global / pf_constr_of_global to let the system take care of universes.

The universe graph

To accommodate universe polymorphic definitions, the graph structure in kernel/univ.ml was modified. The new API forces every universe to be declared before it is mentioned in any constraint. This forces to declare every universe to be >= Set or > Set. Every universe variable introduced during elaboration is >= Set. Every global universe is now declared explicitly > Set, after typechecking the definition. In polymorphic definitions Type@{i} ranges over Set and any other universe j. However, at instantiation time for polymorphic references, one can try to instantiate a universe parameter with Prop as well, if the instantiated constraints allow it. The graph invariants ensure that no universe i can be set lower than Set, so the chain of universes always bottoms down at Prop < Set.

Modules

One has to think of universes in modules as being globally declared, so when including a module (type) which declares a type i (e.g. through a parameter), we get back a copy of i and not some fresh universe.

Incompatibilities

Old-style universe polymorphic definitions were implemented by taking advantage of the fact that elaboration (i.e., pretyping and unification) were not universe aware, so some of the constraints generated during pretypechecking would be forgotten. In the current setting, this is not possible, as unification ensures that the substitution is built is entirely well-typed, even w.r.t universes. This means that some terms that type-checked before no longer do, especially projections of the pair:

@fst ?x ?y : prod ?x ?y : Type (max(Datatypes.i, Datatypes.j)).

The "template universe polymorphic" variables i and j appear during typing without being refreshed, meaning that they can be lowered (have upper constraints) with user-introduced universes. In most cases this won't work, so ?x and ?y have to be instantiated earlier, either from the type of the actual projected pair term (some t : prod A B) or the typing constraint. Adding the correct type annotations will always fix this.

Unification semantics

In Ltac, matching with:

• a universe polymorphic constant c matches any instance of the constant.
• a variable ?x already bound to a term t (non-linear pattern) uses strict equality of universes (e.g., Type@{i} and Type@{j} are not equal).

In tactics:

• change foo with bar, pattern foo will unify all instances of foo (and convert them with bar). This might incur unifications of universes. change uses conversion while pattern only does syntactic matching up-to unification of universes.
• apply, refine use unification up to universes.