Notes on universe polymorphism and primitive projections, M. Sozeau =================================================================== The new implementation of universe polymorphism and primitive projections 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 | Proj of constant * constr 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_reference]s 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 accomodate universe polymorphic definitions, the graph structure in kernel/univ.ml was modified. The new API forces every universe to be declared before it is mentionned 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. Projections =========== | Proj of constant * constr Projections are always applied to a term, which must be of a record type (i.e. reducible to an inductive type [I params]). Type-checking, reduction and conversion are fast (not as fast as they could be yet) because we don't keep parameters around. As you can see, it's currently a [constant] that is used here to refer to the projection, that will change to an abstract [projection] type in the future. Basically a projection constant records which inductive it is a projection for, the number of params and the actual position in the constructor that must be projected. For compatibility reason, we also define an eta-expanded form (accessible from user syntax @f). The constant_entry of a projection has both informations. Declaring a record (under [Set Primitive Projections]) will generate such definitions. The API to declare them is not stable at the moment, but the inductive type declaration also knows about the projections, i.e. a record inductive type decl contains an array of terms representing the projections. This is used to implement eta-conversion for record types (with at least one field and having all projections definable). The canonical value being [Build_R (pn x) ... (pn x)]. Unification and conversion work up to this eta rule. The records can also be universe polymorphic of course, and we don't need to keep track of the universe instance for the projections either. Projections are reduced _eagerly_ everywhere, and introduce a new Zproj constructor in the abstract machines that obeys both the delta (for the constant opacity) and iota laws (for the actual reduction). Refolding works as well (afaict), but there is a slight hack there related to universes (not projections). For the ML programmer, the biggest change is that pattern-matchings on kind_of_term require an additional case, that is handled usually exactly like an [App (Const p) arg]. There are slight hacks related to hints is well, to use the primitive projection form of f when one does [Hint Resolve f]. Usually hint resolve will typecheck the term, resulting in a partially applied projection (disallowed), so we allow it to take [constr_or_global_reference] arguments instead and special-case on projections. Other tactic extensions might need similar treatment. WIP === - [vm_compute] does not deal with universes and projections correctly, except when it goes to a normal form with no projections or polymorphic constants left (the most common case). E.g. Ring with Set Universe Polymorphism and Set Primitive Projections work (at least it did at some point, I didn't recheck yet). - [native_compute] works with universes and projections. 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.