Chapter 24  User defined equalities and relations

• Relations and morphisms
• Adding new relations and morphisms
• Rewriting and non reflexive relations
• Rewriting and non symmetric relations
• Rewriting in ambiguous setoid contexts
• First class setoids and morphisms
• Tactics enabled on user provided relations
• Printing relations and morphisms
• Deprecated syntax and backward incompatibilities
• Rewriting under binders
• Sub-relations
• Constant unfolding

Matthieu Sozeau

This chapter presents the extension of several equality related tactics to work over user-defined structures (called setoids) that are equipped with ad-hoc equivalence relations meant to behave as equalities. Actually, the tactics have also been generalized to relations weaker then equivalences (e.g. rewriting systems).

This documentation is adapted from the previous setoid documentation by Claudio Sacerdoti Coen (based on previous work by Clément Renard). The new implementation is a drop-in replacement for the old one ¹, hence most of the documentation still applies.

The work is a complete rewrite of the previous implementation, based on the type class infrastructure. It also improves on and generalizes the previous implementation in several ways:

• User-extensible algorithm. The algorithm is separated in two parts: generations of the rewriting constraints (done in ML) and solving of these constraints using type class resolution. As type class resolution is extensible using tactics, this allows users to define general ways to solve morphism constraints.
• Sub-relations. An example extension to the base algorithm is the ability to define one relation as a subrelation of another so that morphism declarations on one relation can be used automatically for the other. This is done purely using tactics and type class search.
• Rewriting under binders. It is possible to rewrite under binders in the new implementation, if one provides the proper morphisms. Again, most of the work is handled in the tactics.
• First-class morphisms and signatures. Signatures and morphisms are ordinary Coq terms, hence they can be manipulated inside Coq, put inside structures and lemmas about them can be proved inside the system. Higher-order morphisms are also allowed.
• Performance. The implementation is based on a depth-first search for the first solution to a set of constraints which can be as fast as linear in the size of the term, and the size of the proof term is linear in the size of the original term. Besides, the extensibility allows the user to customize the proof-search if necessary.

24.1  Relations and morphisms

A parametric relation R is any term of type forall (x₁:T₁) …(x_n:T_n), relation A. The expression A, which depends on x₁ …x_n, is called the carrier of the relation and R is said to be a relation over A; the list x₁,…,x_n is the (possibly empty) list of parameters of the relation.

An instance of a parametric relation R with n parameters is any term (R t₁ …t_n).

Let R be a relation over A with n parameters. A term is a parametric proof of reflexivity for R if it has type forall (x₁:T₁) …(x_n:T_n), reflexive (R x₁ …x_n). Similar definitions are given for parametric proofs of symmetry and transitivity.

A parametric unary function f of type forall (x₁:T₁) …(x_n:T_n), A₁ -> A₂ covariantly respects two parametric relation instances R₁ and R₂ if, whenever x, y satisfy R₁ x y, their images (f x) and (f y) satisfy R₂ (f x) (f y) . An f that respects its input and output relations will be called a unary covariant morphism. We can also say that f is a monotone function with respect to R₁ and R₂. The sequence x₁,… x_n represents the parameters of the morphism.

Let R₁ and R₂ be two parametric relations. The signature of a parametric morphism of type forall (x₁:T₁) …(x_n:T_n), A₁ -> A₂ that covariantly respects two instances I_R1 and I_R2 of R₁ and R₂ is written I_R1 ++> I_R2. Notice that the special arrow ++>, which reminds the reader of covariance, is placed between the two relation instances, not between the two carriers. The signature relation instances and morphism will be typed in a context introducing variables for the parameters.

The previous definitions are extended straightforwardly to n-ary morphisms, that are required to be simultaneously monotone on every argument.

Morphisms can also be contravariant in one or more of their arguments. A morphism is contravariant on an argument associated to the relation instance R if it is covariant on the same argument when the inverse relation R⁻¹ (inverse R in Coq) is considered. The special arrow --> is used in signatures for contravariant morphisms.

Functions having arguments related by symmetric relations instances are both covariant and contravariant in those arguments. The special arrow ==> is used in signatures for morphisms that are both covariant and contravariant.

An instance of a parametric morphism f with n parameters is any term f t₁ …t_n.

Leibniz equality is a relation and every function is a morphism that respects Leibniz equality. Unfortunately, Leibniz equality is not always the intended equality for a given structure.

In the next section we will describe the commands to register terms as parametric relations and morphisms. Several tactics that deal with equality in Coq can also work with the registered relations. The exact list of tactic will be given in Sect. 24.7. For instance, the tactic reflexivity can be used to close a goal R n n whenever R is an instance of a registered reflexive relation. However, the tactics that replace in a context C[] one term with another one related by R must verify that C[] is a morphism that respects the intended relation. Currently the verification consists in checking whether C[] is a syntactic composition of morphism instances that respects some obvious compatibility constraints.

24.2  Adding new relations and morphisms

A parametric relation Aeq: forall (y₁ : β_! …y_m : β_m), relation (A t₁ …t_n) over (A : α_i -> …α_n -> Type) can be declared with the following command:

Add Parametric Relation (x₁ : T₁) …(x_n : T_k) : (A t₁ …t_n) (Aeq t′₁ …t′_m)
 [reflexivity proved by refl]
 [symmetry proved by sym]
 [transitivity proved by trans]
 as id.

after having required the Setoid module with the Require Setoid command.

The identifier id gives a unique name to the morphism and it is used by the command to generate fresh names for automatically provided lemmas used internally.

Notice that the carrier and relation parameters may refer to the context of variables introduced at the beginning of the declaration, but the instances need not be made only of variables. Also notice that A is not required to be a term having the same parameters as Aeq, although that is often the case in practice (this departs from the previous implementation).

In case the carrier and relations are not parametric, one can use the command Add Relation instead, whose syntax is the same except there is no local context.

The proofs of reflexivity, symmetry and transitivity can be omitted if the relation is not an equivalence relation. The proofs must be instances of the corresponding relation definitions: e.g. the proof of reflexivity must have a type convertible to reflexive (A t₁ …t_n) (Aeq t′₁ …t′_n). Each proof may refer to the introduced variables as well.

Some tactics (reflexivity, symmetry, transitivity) work only on relations that respect the expected properties. The remaining tactics (replace, rewrite and derived tactics such as autorewrite) do not require any properties over the relation. However, they are able to replace terms with related ones only in contexts that are syntactic compositions of parametric morphism instances declared with the following command.

Add Parametric Morphism (x₁ : T_!) …(x_k : T_k)
(f t₁ …t_n)
 with signature sig
 as id.
Proof
 …
Qed

The command declares f as a parametric morphism of signature sig. The identifier id gives a unique name to the morphism and it is used as the base name of the type class instance definition and as the name of the lemma that proves the well-definedness of the morphism. The parameters of the morphism as well as the signature may refer to the context of variables. The command asks the user to prove interactively that f respects the relations identified from the signature.

Is is possible to reduce the burden of specifying parameters using (maximally inserted) implicit arguments. If A is always set as maximally implicit in the previous example, one can write:

We proceed now by proving a simple lemma performing a rewrite step and then applying reflexivity, as we would do working with Leibniz equality. Both tactic applications are accepted since the required properties over eq_set and union can be established from the two declarations above.

The tables of relations and morphisms are managed by the type class instance mechanism. The behavior on section close is to generalize the instances by the variables of the section (and possibly hypotheses used in the proofs of instance declarations) but not to export them in the rest of the development for proof search. One can use the Existing Instance command to do so outside the section, using the name of the declared morphism suffixed by _Morphism, or use the Global modifier for the corresponding class instance declaration (see §24.6) at definition time. When loading a compiled file or importing a module, all the declarations of this module will be loaded.

24.3  Rewriting and non reflexive relations

To replace only one argument of an n-ary morphism it is necessary to prove that all the other arguments are related to themselves by the respective relation instances.

When the relations associated to some arguments are not reflexive, the tactic cannot automatically prove the reflexivity goals, that are left to the user.

Setoids whose relation are partial equivalence relations (PER) are useful to deal with partial functions. Let R be a PER. We say that an element x is defined if R x x. A partial function whose domain comprises all the defined elements only is declared as a morphism that respects R. Every time a rewriting step is performed the user must prove that the argument of the morphism is defined.

24.4  Rewriting and non symmetric relations

When the user works up to relations that are not symmetric, it is no longer the case that any covariant morphism argument is also contravariant. As a result it is no longer possible to replace a term with a related one in every context, since the obtained goal implies the previous one if and only if the replacement has been performed in a contravariant position. In a similar way, replacement in an hypothesis can be performed only if the replaced term occurs in a covariant position.

Contrary to the previous implementation, no specific error message will be raised when trying to replace a term that occurs in the wrong position. It will only fail because the rewriting constraints are not satisfiable. However it is possible to use the at modifier to specify which occurences should be rewritten.

As expected, composing morphisms together propagates the variance annotations by switching the variance every time a contravariant position is traversed.

24.5  Rewriting in ambiguous setoid contexts

One function can respect several different relations and thus it can be declared as a morphism having multiple signatures.

To declare multiple signatures for a morphism, repeat the Add Morphism command.

When morphisms have multiple signatures it can be the case that a rewrite request is ambiguous, since it is unclear what relations should be used to perform the rewriting. Contrary to the previous implementation, the tactic will always choose the first possible solution to the set of constraints generated by a rewrite and will not try to find all possible solutions to warn the user about.

24.6  First class setoids and morphisms

The implementation is based on a first-class representation of properties of relations and morphisms as type classes. That is, the various combinations of properties on relations and morphisms are represented as records and instances of theses classes are put in a hint database. For example, the declaration:

Add Parametric Relation (x₁ : T₁) …(x_n : T_k) : (A t₁ …t_n) (Aeq t′₁ …t′_m)
 [reflexivity proved by refl]
 [symmetry proved by sym]
 [transitivity proved by trans]
 as id.

is equivalent to an instance declaration:

Instance (x₁ : T₁) …(x_n : T_k) => id : @Equivalence (A t₁ …t_n) (Aeq t′₁ …t′_m) :=
 [Equivalence_Reflexive := refl]
 [Equivalence_Symmetric := sym]
 [Equivalence_Transitive := trans].

The declaration itself amounts to the definition of an object of the record type Coq.Classes.RelationClasses.Equivalence and a hint added to the typeclass_instances hint database. Morphism declarations are also instances of a type class defined in Classes.Morphisms. See the documentation on type classes 18 and the theories files in Classes for further explanations.

One can inform the rewrite tactic about morphisms and relations just by using the typeclass metchanism to declare them using Instance and Context vernacular commands. Any object of type Morphism in the local context will also be automatically used by the rewriting tactic to solve constraints.

Other representations of first class setoids and morphisms can also be handled by encoding them as records. In the following example, the projections of the setoid relation and of the morphism function can be registered as parametric relations and morphisms.

24.7  Tactics enabled on user provided relations

The following tactics, all prefixed by setoid_, deal with arbitrary registered relations and morphisms. Moreover, all the corresponding unprefixed tactics (i.e. reflexivity, symmetry, transitivity, replace, rewrite) have been extended to fall back to their prefixed counterparts when the relation involved is not Leibniz equality. Notice, however, that using the prefixed tactics it is possible to pass additional arguments such as using relation.

setoid_reflexivity

setoid_symmetry [in ident]

setoid_transitivity

setoid_rewrite [orientation] term  [at occs]  [in ident]

setoid_replace term with term  [in ident]  [using relation term]  [by tactic]

The using relation arguments cannot be passed to the unprefixed form. The latter argument tells the tactic what parametric relation should be used to replace the first tactic argument with the second one. If omitted, it defaults to the DefaultRelation instance on the type of the objects. By default, it means the most recent Equivalence instance in the environment, but it can be customized by declaring new DefaultRelation instances. As leiniz equality is a declared equivalence, it will fall back to it if no other relation is declared on a type.

Every derived tactic that is based on the unprefixed forms of the tactics considered above will also work up to user defined relations. For instance, it is possible to register hints for autorewrite that are not proof of Leibniz equalities. In particular it is possible to exploit autorewrite to simulate normalization in a term rewriting system up to user defined equalities.

24.8  Printing relations and morphisms

The Print Instances command can be used to show the list of currently registered Reflexive (using Print Instances Reflexive), Symmetric or Transitive relations, Equivalences, PreOrders, PERs, and Morphisms. When the rewriting tactics refuse to replace a term in a context because the latter is not a composition of morphisms, the Print Instances commands can be useful to understand what additional morphisms should be registered.

24.9  Deprecated syntax and backward incompatibilities

Due to backward compatibility reasons, the following syntax for the declaration of setoids and morphisms is also accepted.

Add Setoid A Aeq ST as ident

where Aeq is a congruence relation without parameters, A is its carrier and ST is an object of type (Setoid_Theory A Aeq) (i.e. a record packing together the reflexivity, symmetry and transitivity lemmas). Notice that the syntax is not completely backward compatible since the identifier was not required.

Add Morphism f:ident.
Proof.
…
Qed.

The latter command also is restricted to the declaration of morphisms without parameters. It is not fully backward compatible since the property the user is asked to prove is slightly different: for n-ary morphisms the hypotheses of the property are permuted; moreover, when the morphism returns a proposition, the property is now stated using a bi-implication in place of a simple implication. In practice, porting an old development to the new semantics is usually quite simple.

Notice that several limitations of the old implementation have been lifted. In particular, it is now possible to declare several relations with the same carrier and several signatures for the same morphism. Moreover, it is now also possible to declare several morphisms having the same signature. Finally, the replace and rewrite tactics can be used to replace terms in contexts that were refused by the old implementation. As discussed in the next section, the semantics of the new setoid_rewrite command differs slightly from the old one and rewrite.

24.10  Rewriting under binders

Warning: Due to compatibility issues, this feature is enabled only when calling the setoid_rewrite tactics directly and not rewrite.

To be able to rewrite under binding constructs, one must declare morphisms with respect to pointwise (setoid) equivalence of functions. Example of such morphisms are the standard all and ex combinators for universal and existential quantification respectively. They are declared as morphisms in the Classes.Morphisms_Prop module. For example, to declare that universal quantification is a morphism for logical equivalence:

One then has to show that if two predicates are equivalent at every point, their universal quantifications are equivalent. Once we have declared such a morphism, it will be used by the setoid rewriting tactic each time we try to rewrite under an all application (products in Prop are implicitely translated to such applications).

Indeed, when rewriting under a lambda, binding variable x, say from P x to Q x using the relation iff, the tactic will generate a proof of pointwise_relation A iff (fun x => P x) (fun x => Q x) from the proof of iff (P x) (Q x) and a constraint of the form Morphism (pointwise_relation A iff ==> ?) m will be generated for the surrounding morphism m.

Hence, one can add higher-order combinators as morphisms by providing signatures using pointwise extension for the relations on the functional arguments (or whatever subrelation of the pointwise extension). For example, one could declare the map combinator on lists as a morphism:

where list_equiv implements an equivalence on lists parameterized by an equivalence on the elements.

Note that when one does rewriting with a lemma under a binder using setoid_rewrite, the application of the lemma may capture the bound variable, as the semantics are different from rewrite where the lemma is first matched on the whole term. With the new setoid_rewrite, matching is done on each subterm separately and in its local environment, and all matches are rewritten simultaneously by default. The semantics of the previous setoid_rewrite implementation can almost be recovered using the at 1 modifier.

24.11  Sub-relations

Sub-relations can be used to specify that one relation is included in another, so that morphisms signatures for one can be used for the other. If a signature mentions a relation R on the left of an arrow ==>, then the signature also applies for any relation S that is smaller than R, and the inverse applies on the right of an arrow. One can then declare only a few morphisms instances that generate the complete set of signatures for a particular constant. By default, the only declared subrelation is iff, which is a subrelation of impl and inverse impl (the dual of implication). That’s why we can declare only two morphisms for conjunction: Morphism (impl ==> impl ==> impl) and and Morphism (iff ==> iff ==> iff) and. This is sufficient to satisfy any rewriting constraints arising from a rewrite using iff, impl or inverse impl through and.

Sub-relations are implemented in Classes.Morphisms and are a prime example of a mostly user-space extension of the algorithm.

24.12  Constant unfolding

The resolution tactic is based on type classes and hence regards user-defined constants as transparent by default. This may slow down the resolution due to a lot of unifications (all the declared Morphism instances are tried at each node of the search tree). To speed it up, declare your constant as rigid for proof search using the command Typeclasses Opaque (see §18.5.4).

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Nicolas Tabareau helped with the glueing

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