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Haftungsausschluß.rst KontaktText {Text[84] Latech[106] Ada[227]}diese Dinge liegen außhalb unserer Verantwortung .. _polymorphicuniverses: Polymorphic Universes ====================== :Author: Matthieu Sozeau General Presentation --------------------- .. warning:: The status of Universe Polymorphism is experimental. This section describes the universe polymorphic extension of |Coq|. Universe polymorphism makes it possible to write generic definitions making use of universes and reuse them at different and sometimes incompatible universe levels. A standard example of the difference between universe *polymorphic* and *monomorphic* definitions is given by the identity function: .. coqtop:: in Definition identity {A : Type} (a : A) := a. By default, constant declarations are monomorphic, hence the identity function declares a global universe (say ``Top.1``) for its domain. Subsequently, if we try to self-apply the identity, we will get an error: .. coqtop:: all Fail Definition selfid := identity (@identity). Indeed, the global level ``Top.1`` would have to be strictly smaller than itself for this self-application to type check, as the type of :g:`(@identity)` is :g:`forall (A : Type@{Top.1}), A -> A` whose type is itself :g:`Type@{Top.1+1}`. A universe polymorphic identity function binds its domain universe level at the definition level instead of making it global. .. coqtop:: in Polymorphic Definition pidentity {A : Type} (a : A) := a. .. coqtop:: all About pidentity. It is then possible to reuse the constant at different levels, like so: .. coqtop:: in Definition selfpid := pidentity (@pidentity). Of course, the two instances of :g:`pidentity` in this definition are different. This can be seen when the :flag:`Printing Universes` flag is on: .. coqtop:: none Set Printing Universes. .. coqtop:: all Print selfpid. Now :g:`pidentity` is used at two different levels: at the head of the application it is instantiated at ``Top.3`` while in the argument position it is instantiated at ``Top.4``. This definition is only valid as long as ``Top.4`` is strictly smaller than ``Top.3``, as shown by the constraints. Note that this definition is monomorphic (not universe polymorphic), so the two universes (in this case ``Top.3`` and ``Top.4``) are actually global levels. When printing :g:`pidentity`, we can see the universes it binds in the annotation :g:`@{Top.2}`. Additionally, when :flag:`Printing Universes` is on we print the "universe context" of :g:`pidentity` consisting of the bound universes and the constraints they must verify (for :g:`pidentity` there are no constraints). Inductive types can also be declared universes polymorphic on universes appearing in their parameters or fields. A typical example is given by monoids: .. coqtop:: in Polymorphic Record Monoid := { mon_car :> Type; mon_unit : mon_car; mon_op : mon_car -> mon_car -> mon_car }. .. coqtop:: in Print Monoid. The Monoid's carrier universe is polymorphic, hence it is possible to instantiate it for example with :g:`Monoid` itself. First we build the trivial unit monoid in :g:`Set`: .. coqtop:: in Definition unit_monoid : Monoid := {| mon_car := unit; mon_unit := tt; mon_op x y := tt |}. From this we can build a definition for the monoid of :g:`Set`\-monoids (where multiplication would be given by the product of monoids). .. coqtop:: in Polymorphic Definition monoid_monoid : Monoid. refine (@Build_Monoid Monoid unit_monoid (fun x y => x)). Defined. .. coqtop:: all Print monoid_monoid. As one can see from the constraints, this monoid is “large”, it lives in a universe strictly higher than :g:`Set`. Polymorphic, Monomorphic ------------------------- .. cmd:: Polymorphic @definition As shown in the examples, polymorphic definitions and inductives can be declared using the ``Polymorphic`` prefix. .. flag:: Universe Polymorphism Once enabled, this option will implicitly prepend ``Polymorphic`` to any definition of the user. .. cmd:: Monomorphic @definition When the :flag:`Universe Polymorphism` option is set, to make a definition producing global universe constraints, one can use the ``Monomorphic`` prefix. Many other commands support the ``Polymorphic`` flag, including: .. TODO add links on each of these? - ``Lemma``, ``Axiom``, and all the other “definition” keywords support polymorphism. - ``Variables``, ``Context``, ``Universe`` and ``Constraint`` in a section support polymorphism. This means that the universe variables (and associated constraints) are discharged polymorphically over definitions that use them. In other words, two definitions in the section sharing a common variable will both get parameterized by the universes produced by the variable declaration. This is in contrast to a “mononorphic” variable which introduces global universes and constraints, making the two definitions depend on the *same* global universes associated to the variable. - :cmd:`Hint Resolve` and :cmd:`Hint Rewrite` will use the auto/rewrite hint polymorphically, not at a single instance. Cumulative, NonCumulative ------------------------- Polymorphic inductive types, coinductive types, variants and records can be declared cumulative using the :g:`Cumulative` prefix. .. cmd:: Cumulative @inductive Declares the inductive as cumulative Alternatively, there is a flag :flag:`Polymorphic Inductive Cumulativity` which when set, makes all subsequent *polymorphic* inductive definitions cumulative. When set, inductive types and the like can be enforced to be non-cumulative using the :g:`NonCumulative` prefix. .. cmd:: NonCumulative @inductive Declares the inductive as non-cumulative .. flag:: Polymorphic Inductive Cumulativity When this option is on, it sets all following polymorphic inductive types as cumulative (it is off by default). Consider the examples below. .. coqtop:: in Polymorphic Cumulative Inductive list {A : Type} := | nil : list | cons : A -> list -> list. .. coqtop:: all Print list. When printing :g:`list`, the universe context indicates the subtyping constraints by prefixing the level names with symbols. Because inductive subtypings are only produced by comparing inductives to themselves with universes changed, they amount to variance information: each universe is either invariant, covariant or irrelevant (there are no contravariant subtypings in Coq), respectively represented by the symbols `=`, `+` and `*`. Here we see that :g:`list` binds an irrelevant universe, so any two instances of :g:`list` are convertible: :math:`E[Γ] ⊢ \mathsf{list}@\{i\}~A =_{βδιζη} \mathsf{list}@\{j\}~B` whenever :math:`E[Γ] ⊢ A =_{βδιζη} B` and this applies also to their corresponding constructors, when they are comparable at the same type. See :ref:`Conversion-rules` for more details on convertibility and subtyping. The following is an example of a record with non-trivial subtyping relation: .. coqtop:: all Polymorphic Cumulative Record packType := {pk : Type}. :g:`packType` binds a covariant universe, i.e. .. math:: E[Γ] ⊢ \mathsf{packType}@\{i\} =_{βδιζη} \mathsf{packType}@\{j\}~\mbox{ whenever }~i ≤ j Cumulative inductive types, coinductive types, variants and records only make sense when they are universe polymorphic. Therefore, an error is issued whenever the user uses the :g:`Cumulative` or :g:`NonCumulative` prefix in a monomorphic context. Notice that this is not the case for the option :flag:`Polymorphic Inductive Cumulativity`. That is, this option, when set, makes all subsequent *polymorphic* inductive declarations cumulative (unless, of course the :g:`NonCumulative` prefix is used but has no effect on *monomorphic* inductive declarations. Consider the following examples. .. coqtop:: all reset Fail Monomorphic Cumulative Inductive Unit := unit. .. coqtop:: all reset Fail Monomorphic NonCumulative Inductive Unit := unit. .. coqtop:: all reset Set Polymorphic Inductive Cumulativity. Inductive Unit := unit. An example of a proof using cumulativity ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. coqtop:: in Set Universe Polymorphism. Set Polymorphic Inductive Cumulativity. Inductive eq@{i} {A : Type@{i}} (x : A) : A -> Type@{i} := eq_refl : eq x x. Definition funext_type@{a b e} (A : Type@{a}) (B : A -> Type@{b}) := forall f g : (forall a, B a), (forall x, eq@{e} (f x) (g x)) -> eq@{e} f g. Section down. Universes a b e e'. Constraint e' < e. Lemma funext_down {A B} (H : @funext_type@{a b e} A B) : @funext_type@{a b e'} A B. Proof. exact H. Defined. End down. Cumulativity Weak Constraints ----------------------------- .. flag:: Cumulativity Weak Constraints When set, which is the default, causes "weak" constraints to be produced when comparing universes in an irrelevant position. Processing weak constraints is delayed until minimization time. A weak constraint between `u` and `v` when neither is smaller than the other and one is flexible causes them to be unified. Otherwise the constraint is silently discarded. This heuristic is experimental and may change in future versions. Disabling weak constraints is more predictable but may produce arbitrary numbers of universes. Global and local universes --------------------------- Each universe is declared in a global or local environment before it can be used. To ensure compatibility, every *global* universe is set to be strictly greater than :g:`Set` when it is introduced, while every *local* (i.e. polymorphically quantified) universe is introduced as greater or equal to :g:`Set`. Conversion and unification --------------------------- The semantics of conversion and unification have to be modified a little to account for the new universe instance arguments to polymorphic references. The semantics respect the fact that definitions are transparent, so indistinguishable from their bodies during conversion. This is accomplished by changing one rule of unification, the first- order approximation rule, which applies when two applicative terms with the same head are compared. It tries to short-cut unfolding by comparing the arguments directly. In case the constant is universe polymorphic, we allow this rule to fire only when unifying the universes results in instantiating a so-called flexible universe variables (not given by the user). Similarly for conversion, if such an equation of applicative terms fail due to a universe comparison not being satisfied, the terms are unfolded. This change implies that conversion and unification can have different unfolding behaviors on the same development with universe polymorphism switched on or off. Minimization ------------- Universe polymorphism with cumulativity tends to generate many useless inclusion constraints in general. Typically at each application of a polymorphic constant :g:`f`, if an argument has expected type :g:`Type@{i}` and is given a term of type :g:`Type@{j}`, a :math:`j ≤ i` constraint will be generated. It is however often the case that an equation :math:`j = i` would be more appropriate, when :g:`f`\'s universes are fresh for example. Consider the following example: .. coqtop:: none Polymorphic Definition pidentity {A : Type} (a : A) := a. Set Printing Universes. .. coqtop:: in Definition id0 := @pidentity nat 0. .. coqtop:: all Print id0. This definition is elaborated by minimizing the universe of :g:`id0` to level :g:`Set` while the more general definition would keep the fresh level :g:`i` generated at the application of :g:`id` and a constraint that :g:`Set` :math:`≤ i`. This minimization process is applied only to fresh universe variables. It simply adds an equation between the variable and its lower bound if it is an atomic universe (i.e. not an algebraic max() universe). .. flag:: Universe Minimization ToSet Turning this flag off (it is on by default) disallows minimization to the sort :g:`Set` and only collapses floating universes between themselves. Explicit Universes ------------------- The syntax has been extended to allow users to explicitly bind names to universes and explicitly instantiate polymorphic definitions. .. cmd:: Universe @ident Polymorphic Universe @ident In the monorphic case, this command declares a new global universe named :g:`ident`, which can be referred to using its qualified name as well. Global universe names live in a separate namespace. The command supports the ``Polymorphic`` flag only in sections, meaning the universe quantification will be discharged on each section definition independently. One cannot mix polymorphic and monomorphic declarations in the same section. .. cmd:: Constraint @universe_constraint Polymorphic Constraint @universe_constraint This command declares a new constraint between named universes. .. productionlist:: coq universe_constraint : `qualid` < `qualid` : `qualid` <= `qualid` : `qualid` = `qualid` If consistent, the constraint is then enforced in the global environment. Like :cmd:`Universe`, it can be used with the ``Polymorphic`` prefix in sections only to declare constraints discharged at section closing time. One cannot declare a global constraint on polymorphic universes. .. exn:: Undeclared universe @ident. :undocumented: .. exn:: Universe inconsistency. :undocumented: Polymorphic definitions ~~~~~~~~~~~~~~~~~~~~~~~ For polymorphic definitions, the declaration of (all) universe levels introduced by a definition uses the following syntax: .. coqtop:: in Polymorphic Definition le@{i j} (A : Type@{i}) : Type@{j} := A. .. coqtop:: all Print le. During refinement we find that :g:`j` must be larger or equal than :g:`i`, as we are using :g:`A : Type@{i} <= Type@{j}`, hence the generated constraint. At the end of a definition or proof, we check that the only remaining universes are the ones declared. In the term and in general in proof mode, introduced universe names can be referred to in terms. Note that local universe names shadow global universe names. During a proof, one can use :cmd:`Show Universes` to display the current context of universes. Definitions can also be instantiated explicitly, giving their full instance: .. coqtop:: all Check (pidentity@{Set}). Monomorphic Universes k l. Check (le@{k l}). User-named universes and the anonymous universe implicitly attached to an explicit :g:`Type` are considered rigid for unification and are never minimized. Flexible anonymous universes can be produced with an underscore or by omitting the annotation to a polymorphic definition. .. coqtop:: all Check (fun x => x) : Type -> Type. Check (fun x => x) : Type -> Type@{_}. Check le@{k _}. Check le. .. flag:: Strict Universe Declaration Turning this option off allows one to freely use identifiers for universes without declaring them first, with the semantics that the first use declares it. In this mode, the universe names are not associated with the definition or proof once it has been defined. This is meant mainly for debugging purposes. .. flag:: Private Polymorphic Universes This option, on by default, removes universes which appear only in the body of an opaque polymorphic definition from the definition's universe arguments. As such, no value needs to be provided for these universes when instantiating the definition. Universe constraints are automatically adjusted. Consider the following definition: .. coqtop:: all Lemma foo@{i} : Type@{i}. Proof. exact Type. Qed. Print foo. The universe :g:`Top.xxx` for the :g:`Type` in the body cannot be accessed, we only care that one exists for any instantiation of the universes appearing in the type of :g:`foo`. This is guaranteed when the transitive constraint ``Set <= Top.xxx < i`` is verified. Then when using the constant we don't need to put a value for the inner universe: .. coqtop:: all Check foo@{_}. and when not looking at the body we don't mention the private universe: .. coqtop:: all About foo. To recover the same behaviour with regard to universes as :g:`Defined`, the option :flag:`Private Polymorphic Universes` may be unset: .. coqtop:: all Unset Private Polymorphic Universes. Lemma bar : Type. Proof. exact Type. Qed. About bar. Fail Check bar@{_}. Check bar@{_ _}. Note that named universes are always public. .. coqtop:: all Set Private Polymorphic Universes. Unset Strict Universe Declaration. Lemma baz : Type@{outer}. Proof. exact Type@{inner}. Qed. About baz. [ Verzeichnis aufwärts0.117unsichere Verbindung ] |