\[\begin{split}\newcommand{\as}{\kw{as}} \newcommand{\case}{\kw{case}} \newcommand{\cons}{\textsf{cons}} \newcommand{\consf}{\textsf{consf}} \newcommand{\emptyf}{\textsf{emptyf}} \newcommand{\End}{\kw{End}} \newcommand{\kwend}{\kw{end}} \newcommand{\even}{\textsf{even}} \newcommand{\evenO}{\textsf{even}_\textsf{O}} \newcommand{\evenS}{\textsf{even}_\textsf{S}} \newcommand{\Fix}{\kw{Fix}} \newcommand{\fix}{\kw{fix}} \newcommand{\for}{\textsf{for}} \newcommand{\forest}{\textsf{forest}} \newcommand{\Functor}{\kw{Functor}} \newcommand{\In}{\kw{in}} \newcommand{\ind}[3]{\kw{Ind}~[#1]\left(#2\mathrm{~:=~}#3\right)} \newcommand{\Indp}[4]{\kw{Ind}_{#4}[#1](#2:=#3)} \newcommand{\Indpstr}[5]{\kw{Ind}_{#4}[#1](#2:=#3)/{#5}} \newcommand{\injective}{\kw{injective}} \newcommand{\kw}[1]{\textsf{#1}} \newcommand{\length}{\textsf{length}} \newcommand{\letin}[3]{\kw{let}~#1:=#2~\kw{in}~#3} \newcommand{\List}{\textsf{list}} \newcommand{\lra}{\longrightarrow} \newcommand{\Match}{\kw{match}} \newcommand{\Mod}[3]{{\kw{Mod}}({#1}:{#2}\,\zeroone{:={#3}})} \newcommand{\ModImp}[3]{{\kw{Mod}}({#1}:{#2}:={#3})} \newcommand{\ModA}[2]{{\kw{ModA}}({#1}=={#2})} \newcommand{\ModS}[2]{{\kw{Mod}}({#1}:{#2})} \newcommand{\ModType}[2]{{\kw{ModType}}({#1}:={#2})} \newcommand{\mto}{.\;} \newcommand{\nat}{\textsf{nat}} \newcommand{\Nil}{\textsf{nil}} \newcommand{\nilhl}{\textsf{nil\_hl}} \newcommand{\nO}{\textsf{O}} \newcommand{\node}{\textsf{node}} \newcommand{\nS}{\textsf{S}} \newcommand{\odd}{\textsf{odd}} \newcommand{\oddS}{\textsf{odd}_\textsf{S}} \newcommand{\ovl}[1]{\overline{#1}} \newcommand{\Pair}{\textsf{pair}} \newcommand{\plus}{\mathsf{plus}} \newcommand{\SProp}{\textsf{SProp}} \newcommand{\Prop}{\textsf{Prop}} \newcommand{\return}{\kw{return}} \newcommand{\Set}{\textsf{Set}} \newcommand{\Sort}{\mathcal{S}} \newcommand{\Str}{\textsf{Stream}} \newcommand{\Struct}{\kw{Struct}} \newcommand{\subst}[3]{#1\{#2/#3\}} \newcommand{\tl}{\textsf{tl}} \newcommand{\tree}{\textsf{tree}} \newcommand{\trii}{\triangleright_\iota} \newcommand{\Type}{\textsf{Type}} \newcommand{\WEV}[3]{\mbox{$#1[] \vdash #2 \lra #3$}} \newcommand{\WEVT}[3]{\mbox{$#1[] \vdash #2 \lra$}\\ \mbox{$ #3$}} \newcommand{\WF}[2]{{\mathcal{W\!F}}(#1)[#2]} \newcommand{\WFE}[1]{\WF{E}{#1}} \newcommand{\WFT}[2]{#1[] \vdash {\mathcal{W\!F}}(#2)} \newcommand{\WFTWOLINES}[2]{{\mathcal{W\!F}}\begin{array}{l}(#1)\\\mbox{}[{#2}]\end{array}} \newcommand{\with}{\kw{with}} \newcommand{\WS}[3]{#1[] \vdash #2 <: #3} \newcommand{\WSE}[2]{\WS{E}{#1}{#2}} \newcommand{\WT}[4]{#1[#2] \vdash #3 : #4} \newcommand{\WTE}[3]{\WT{E}{#1}{#2}{#3}} \newcommand{\WTEG}[2]{\WTE{\Gamma}{#1}{#2}} \newcommand{\WTM}[3]{\WT{#1}{}{#2}{#3}} \newcommand{\zeroone}[1]{[{#1}]} \end{split}\]

Generalized rewriting

Author

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 than equivalences (e.g. rewriting systems). The toolbox also extends the automatic rewriting capabilities of the system, allowing the specification of custom strategies for rewriting.

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 typeclass infrastructure. It also improves on and generalizes the previous implementation in several ways:

• User-extensible algorithm. The algorithm is separated into two parts: generation of the rewriting constraints (written in ML) and solving these constraints using typeclass resolution. As typeclass resolution is extensible using tactics, this allows users to define general ways to solve morphism constraints.

• Subrelations. 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 typeclass 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.

1

Nicolas Tabareau helped with the gluing.

Introduction to generalized rewriting

Relations and morphisms

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

Example: Parametric relation

It is possible to implement finite sets of elements of type A as unordered lists of elements of type A. The function set_eq: forall (A : Type), relation (list A) satisfied by two lists with the same elements is a parametric relation over (list A) with one parameter A. The type of set_eq is convertible with forall (A : Type), list A -> list A -> Prop.

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

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) ... (xn : Tn), reflexive (R x1 ... xn). Similar definitions are given for parametric proofs of symmetry and transitivity.

Example: Parametric relation (continued)

The set_eq relation of the previous example can be proved to be reflexive, symmetric and transitive. A parametric unary function f of type forall (x1 : T1) ... (xn : Tn), 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 ... xn 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) ... (xn : Tn), A1 -> A2 that covariantly respects two instances \(I_{R_1}\) and \(I_{R_2}\) of R1 and R2 is written \(I_{R_1} ++> I_{R_2}\). 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 with 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 \(f \, t_1 \ldots t_n\).

Example: Morphisms

Continuing the previous example, let union: forall (A : Type), list A -> list A -> list A perform the union of two sets by appending one list to the other. union is a binary morphism parametric over A that respects the relation instance (set_eq A). The latter condition is proved by showing:

forall (A: Type) (S1 S1' S2 S2': list A),   set_eq A S1 S1' ->   set_eq A S2 S2' ->   set_eq A (union A S1 S2) (union A S1' S2').

The signature of the function union A is set_eq A ==> set_eq A ==> set_eq A for all A.

Example: Contravariant morphisms

The division function Rdiv : R -> R -> R is a morphism of signature le ++> le --> le where le is the usual order relation over real numbers. Notice that division is covariant in its first argument and contravariant in its second argument.

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 tactics will be given in this section. For instance, the tactic reflexivity can be used to solve 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 of checking whether C[] is a syntactic composition of morphism instances that respects some obvious compatibility constraints.

Example: Rewriting

Continuing the previous examples, suppose that the user must prove set_eq int (union int (union int S1 S2) S2) (f S1 S2) under the hypothesis H : set_eq int S2 (@nil int). It is possible to use the rewrite tactic to replace the first two occurrences of S2 with @nil int in the goal since the context set_eq int (union int (union int S1 nil) nil) (f S1 S2), being a composition of morphisms instances, is a morphism. However the tactic will fail replacing the third occurrence of S2 unless f has also been declared as a morphism.

Adding new relations and morphisms

These commands support the local and global locality attributes. The default is local if the command is used inside a section, global otherwise. They also support the universes(polymorphic) attributes.

Command Add Parametric Relation binder* : one_term_A one_term_Aeq reflexivity proved by one_term? symmetry proved by one_term? transitivity proved by one_term? as ident

Declares a parametric relation of one_termA, which is a Type, say T, with one_termAeq, which is a relation on T, i.e. of type (T -> T -> Prop). Thus, if one_termA is A: forall α1 … αn, Type then one_termAeq is Aeq: forall α1 … αn, (A α1 … αn) -> (A α1 … αn) -> Prop, or equivalently, Aeq: forall α1 … αn, relation (A α1 … αn).

one_termA and one_termAeq must be typeable under the context binders. In practice, the binders usually correspond to the αs

The final ident 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).

To use this command, you need to first import the module Setoid using the command Require Import Setoid.

Command Add Relation one_term one_term reflexivity proved by one_term? symmetry proved by one_term? transitivity proved by one_term? as ident

If the carrier and relations are not parametric, use this command 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 … tn) (Aeq t′ 1 … t′ n). Each proof may refer to the introduced variables as well.

Example: Parametric relation

For Leibniz equality, we may declare:

Add Parametric Relation (A : Type) : A (@eq A)   [reflexivity proved by @refl_equal A] ...

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.

Command Add Parametric Morphism binder* : one_term with signature term as ident

Declares a parametric morphism one_term of signature term. The final identifier ident gives a unique name to the morphism and it is used as the base name of the typeclass 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 the function denoted by the first ident respects the relations identified from the signature.

Example

We start the example by assuming a small theory over homogeneous sets and we declare set equality as a parametric equivalence relation and union of two sets as a parametric morphism.

Require Export Setoid.

Require Export Relation_Definitions.

Set Implicit Arguments.

Parameter set : Type -> Type.

set is declared

Parameter empty : forall A, set A.

empty is declared

Parameter eq_set : forall A, set A -> set A -> Prop.

eq_set is declared

Parameter union : forall A, set A -> set A -> set A.

union is declared

Axiom eq_set_refl : forall A, reflexive _ (eq_set (A:=A)).

eq_set_refl is declared

Axiom eq_set_sym : forall A, symmetric _ (eq_set (A:=A)).

eq_set_sym is declared

Axiom eq_set_trans : forall A, transitive _ (eq_set (A:=A)).

eq_set_trans is declared

Axiom empty_neutral : forall A (S : set A), eq_set (union S (empty A)) S.

empty_neutral is declared

Axiom union_compat :   forall (A : Type),     forall x x' : set A, eq_set x x' ->     forall y y' : set A, eq_set y y' ->       eq_set (union x y) (union x' y').

union_compat is declared

Add Parametric Relation A : (set A) (@eq_set A)   reflexivity proved by (eq_set_refl (A:=A))   symmetry proved by (eq_set_sym (A:=A))   transitivity proved by (eq_set_trans (A:=A))   as eq_set_rel.

eq_set_rel_relation is defined eq_set_rel_Reflexive is defined eq_set_rel_Symmetric is defined eq_set_rel_Transitive is defined eq_set_rel is defined

Add Parametric Morphism A : (@union A)   with signature (@eq_set A) ==> (@eq_set A) ==> (@eq_set A) as union_mor.

1 goal A : Type ============================ forall x y : set A, eq_set x y -> forall x0 y0 : set A, eq_set x0 y0 -> eq_set (union x x0) (union y y0)

Proof.

  exact (@union_compat A).

No more goals.

Qed.

It 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:

Add Parametric Relation A : (set A) eq_set   reflexivity proved by eq_set_refl   symmetry proved by eq_set_sym   transitivity proved by eq_set_trans   as eq_set_rel. Add Parametric Morphism A : (@union A) with   signature eq_set ==> eq_set ==> eq_set as union_mor. Proof. exact (@union_compat A). Qed.

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.

Goal forall (S : set nat),   eq_set (union (union S (empty nat)) S) (union S S).

1 goal ============================ forall S : set nat, eq_set (union (union S (empty nat)) S) (union S S)

Proof. intros. rewrite empty_neutral. reflexivity. Qed.

1 goal S : set nat ============================ eq_set (union (union S (empty nat)) S) (union S S) 1 goal S : set nat ============================ eq_set (union S S) (union S S) No more goals.

The tables of relations and morphisms are managed by the typeclass 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 cmd: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 First Class Setoids and Morphisms) at definition time. When loading a compiled file or importing a module, all the declarations of this module will be loaded.

Rewriting and nonreflexive 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.

Example

To replace (union S empty) with S in (union (union S empty) S) (union S S) the rewrite tactic must exploit the monotony of union (axiom union_compat in the previous example). Applying union_compat by hand we are left with the goal eq_set (union S S) (union S S).

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

Setoids whose relations are partial equivalence relations (PER) are useful for dealing 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 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.

Example

Let eqO be fun x y => x = y /\ x <> 0 (the smallest PER over nonzero elements). Division can be declared as a morphism of signature eq ==> eq0 ==> eq. Replacing x with y in div x n = div y n opens an additional goal eq0 n n which is equivalent to n = n /\ n <> 0.

Rewriting and nonsymmetric 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.

Example: Covariance and contravariance

Suppose that division over real numbers has been defined as a morphism of signature Z.div : Z.lt ++> Z.lt --> Z.lt (i.e. Z.div is increasing in its first argument, but decreasing on the second one). Let < denote Z.lt. Under the hypothesis H : x < y we have k < x / y -> k < x / x, but not k < y / x -> k < x / x. Dually, under the same hypothesis k < x / y -> k < y / y holds, but k < y / x -> k < y / y does not. Thus, if the current goal is k < x / x, it is possible to replace only the second occurrence of x (in contravariant position) with y since the obtained goal must imply the current one. On the contrary, if k < x / x is an hypothesis, it is possible to replace only the first occurrence of x (in covariant position) with y since the current hypothesis must imply the obtained one.

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 occurrences should be rewritten.

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

Example

Let us continue the previous example and let us consider the goal x / (x / x) < k. The first and third occurrences of x are in a contravariant position, while the second one is in covariant position. More in detail, the second occurrence of x occurs covariantly in (x / x) (since division is covariant in its first argument), and thus contravariantly in x / (x / x) (since division is contravariant in its second argument), and finally covariantly in x / (x / x) < k (since <, as every transitive relation, is contravariant in its first argument with respect to the relation itself).

Rewriting in ambiguous setoid contexts

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

Example

Union over homogeneous lists can be given all the following signatures: eq ==> eq ==> eq (eq being the equality over ordered lists) set_eq ==> set_eq ==> set_eq (set_eq being the equality over unordered lists up to duplicates), multiset_eq ==> multiset_eq ==> multiset_eq (multiset_eq being the equality over unordered lists).

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 the possible solutions to warn the user about them.

Rewriting with Type valued relations

Definitions in Classes.Relations, Classes.Morphisms and Classes.Equivalence are based on Prop. Analogous definitions with the same names based on Type are in Classes.CRelations, Classes.CMorphisms and Classes.CEquivalence. The C identifies the "computational" versions.

Importing these modules allows for generalized rewriting with relations of the form R : A -> A -> Type together with support for universe polymorphism.

Declaring rewrite relations

The RewriteRelation A R typeclass, indexed by a type and relation, registers relations that generalized rewriting handles. The default instances of this class are the iff`, impl and flip impl relations on Prop, any declared Equivalence on a type A (including Leibniz equality), and pointwise extensions of declared relations for function types. Users can simply add new instances of this class to register relations with the generalized rewriting machinery. It is used in two cases:

• Inference of morphisms: In some cases, generalized rewriting might face constraints of the shape Proper (S ==> ?R) f for a function f with no matching Proper instance. In this situation, the RewriteRelation instances are used to instantiate the relation ?R. If the instantiated relation is reflexive, then the Proper constraint can be automatically discharged.

• Compatibility with ssreflect's rewrite: The rewrite (ssreflect) tactic uses generalized rewriting when possible, by checking that a RewriteRelation R instance exists when rewriting with a term of type R t u.

Commands and tactics

First class setoids and morphisms

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

Add Parametric Relation (x1 : T1) ... (xn : Tn) : (A t1 ... tn) (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 id (x1 : T1) ... (xn : Tn) : @Equivalence (A t1 ... tn) (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 of a typeclass named Proper` defined in Classes.Morphisms. See the documentation on Typeclasses and the theories files in Classes for further explanations.

One can inform the rewrite tactic about morphisms and relations just by using the typeclass mechanism to declare them using the Instance and Context commands. Any object of type Proper (the type of morphism declarations) 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.

Example: First class setoids

Require Import Relation_Definitions Setoid.

Record Setoid : Type := { car: Type;   eq: car -> car -> Prop;   refl: reflexive _ eq;   sym: symmetric _ eq;   trans: transitive _ eq }.

Setoid is defined car is defined eq is defined refl is defined sym is defined trans is defined

Add Parametric Relation (s : Setoid) : (@car s) (@eq s)   reflexivity proved by (refl s)   symmetry proved by (sym s)   transitivity proved by (trans s) as eq_rel.

eq_rel_relation is defined eq_rel_Reflexive is defined eq_rel_Symmetric is defined eq_rel_Transitive is defined eq_rel is defined

Record Morphism (S1 S2 : Setoid) : Type := { f: car S1 -> car S2;   compat: forall (x1 x2 : car S1), eq S1 x1 x2 -> eq S2 (f x1) (f x2) }.

Morphism is defined f is defined compat is defined

Add Parametric Morphism (S1 S2 : Setoid) (M : Morphism S1 S2) :   (@f S1 S2 M) with signature (@eq S1 ==> @eq S2) as apply_mor.

1 goal S1, S2 : Setoid M : Morphism S1 S2 ============================ forall x y : car S1, eq S1 x y -> eq S2 (f S1 S2 M x) (f S1 S2 M y)

Proof. apply (compat S1 S2 M). Qed.

No more goals.

Lemma test : forall (S1 S2 : Setoid) (m : Morphism S1 S2)   (x y : car S1), eq S1 x y -> eq S2 (f _ _ m x) (f _ _ m y).

1 goal ============================ forall (S1 S2 : Setoid) (m : Morphism S1 S2) (x y : car S1), eq S1 x y -> eq S2 (f S1 S2 m x) (f S1 S2 m y)

Proof. intros. rewrite H. reflexivity. Qed.

1 goal S1, S2 : Setoid m : Morphism S1 S2 x, y : car S1 H : eq S1 x y ============================ eq S2 (f S1 S2 m x) (f S1 S2 m y) 1 goal S1, S2 : Setoid m : Morphism S1 S2 x, y : car S1 H : eq S1 x y ============================ eq S2 (f S1 S2 m y) (f S1 S2 m y) No more goals.

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.

Tactic setoid_reflexivity

Tactic setoid_symmetry in ident?

Tactic setoid_transitivity one_term

Tactic setoid_etransitivity

Tactic setoid_rewrite ->​<-? one_term_with_bindings at rewrite_occs? in ident?

Tactic setoid_rewrite ->​<-? one_term_with_bindings in ident at rewrite_occs

Tactic setoid_replace one_term with one_term using relation one_term? in ident? at int_or_var+? by ltac_expr3?

rewrite_occs::=integer+|ident

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 global environment, but it can be customized by declaring new DefaultRelation instances. As Leibniz equality is a declared equivalence, it will fall back to it if no other relation is declared on a given 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 proofs of Leibniz equalities. In particular it is possible to exploit autorewrite to simulate normalization in a term rewriting system up to user defined equalities.

Printing relations and morphisms

Use the Print Instances command with the class names Reflexive, Symmetric or Transitive to print registered reflexive, symmetric or transitive relations and with the class name Proper to print morphisms.

When rewriting tactics refuse to replace a term in a context because the latter is not a composition of morphisms, this command can be useful to understand what additional morphisms should be registered.

Deprecated syntax and backward incompatibilities

Command Add Setoid one_term_carrier one_term_congruence one_term_proofs as ident

This command for declaring setoids and morphisms is also accepted due to backward compatibility reasons.

Here one_termcongruence is a congruence relation without parameters, one_termcarrier is its carrier and one_termproofs is an object of type (Setoid_Theory one_termcarrier one_termcongruence) (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.

Command Add Parametric Setoid binder* : one_term one_term one_term as ident

Command Add Morphism one_term : ident

Command Add Morphism one_term with signature term as ident

This command 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.

Command Declare Morphism one_term : ident

Declares a parameter in a module type that is a morphism.

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 tactic differs slightly from the old one and rewrite.

Tactic head_of_constr ident one_term

For internal use only. It may be removed without warning. Do not use.

Extensions

Rewriting under binders

Warning

Due to compatibility issues, this feature is enabled only when calling the setoid_rewrite tactic 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:

Require Import Morphisms.

Instance all_iff_morphism (A : Type) :          Proper (pointwise_relation A iff ==> iff) (@all A).

1 goal A : Type ============================ Proper (pointwise_relation A iff ==> iff) (all (A:=A))

Proof. simpl_relation.

1 goal A : Type x, y : A -> Prop H : pointwise_relation A iff x y ============================ all x <-> all y

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 implicitly 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 Proper (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:

Instance map_morphism `{Equivalence A eqA, Equivalence B eqB} :          Proper ((eqA ==> eqB) ==> list_equiv eqA ==> list_equiv eqB) (@map A B).

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 context, 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.

Subrelations

Subrelations can be used to specify that one relation is included in another, so that morphism 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: Proper (impl ==> impl ==> impl) and and Proper (iff ==> iff ==> iff) and. This is sufficient to satisfy any rewriting constraints arising from a rewrite using iff, impl or inverse impl through and.

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

Constant unfolding

The resolution tactic is based on typeclasses 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 Proper 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.

Strategies for rewriting

Usage

Tactic rewrite_strat rewstrategy in ident?

Rewrite using rewstrategy in the conclusion or in the hypothesis ident.

Error Nothing to rewrite.

The strategy didn't find any matches.

Error No progress made.

If the strategy succeeded but made no progress.

Error Unable to satisfy the rewriting constraints.

If the strategy succeeded and made progress but the corresponding rewriting constraints are not satisfied.

setoid_rewrite one_term is basically equivalent to rewrite_strat outermost one_term.

Tactic rewrite_db ident₁ in ident₂?

Equivalent to rewrite_strat (topdown (hints ident1)) in ident2?

Definitions

The generalized rewriting tactic is based on a set of strategies that can be combined to create custom rewriting procedures. Its set of strategies is based on the programmable rewriting strategies with generic traversals by Visser et al. [LV97] [VBT98], which formed the core of the Stratego transformation language [Vis01]. Rewriting strategies are applied using the rewrite_strat tactic.

rewstrategy::=rewstrategy1+;rewstrategy1::=<- one_term|progress rewstrategy1|try rewstrategy1|choice rewstrategy0+|repeat rewstrategy1|any rewstrategy1|subterm rewstrategy1|subterms rewstrategy1|innermost rewstrategy1|outermost rewstrategy1|bottomup rewstrategy1|topdown rewstrategy1|hints ident|terms one_term*|eval red_expr|fold one_term|rewstrategy0|old_hints identrewstrategy0::=one_term|fail|id|refl|( rewstrategy )

one_term

lemma, left to right

fail

failure

id

identity

refl

reflexivity

<- one_term

lemma, right to left

progress rewstrategy1

progress

try rewstrategy1

try catch

rewstrategy ; rewstrategy1

composition

choice rewstrategy0+

first successful strategy

repeat rewstrategy1

one or more

any rewstrategy1

zero or more

subterm rewstrategy1

one subterm

subterms rewstrategy1

all subterms

innermost rewstrategy1

Innermost first. When there are multiple nested matches in a subterm, the innermost subterm is rewritten. For example, rewriting (a + b) + c with Nat.add_comm gives (b + a) + c.

outermost rewstrategy1

Outermost first. When there are multiple nested matches in a subterm, the outermost subterm is rewritten. For example, rewriting (a + b) + c with Nat.add_comm gives c + (a + b).

bottomup rewstrategy1

bottom-up

topdown rewstrategy1

top-down

hints ident

apply hints from hint database

terms one_term*

any of the terms

eval red_expr

apply reduction

fold term

unify

( rewstrategy )

to be documented

old_hints ident

to be documented

Conceptually, a few of these are defined in terms of the others using a primitive fixpoint operator fix, which the tactic doesn't currently support:

• try rewstrategy1 := choice (rewstrategy1) id

• any rewstrategy1 := fix ident. try (rewstrategy1 ; ident)

• repeat rewstrategy1 := rewstrategy1; any rewstrategy1

• bottomup rewstrategy1 := fix ident. (choice (progress (subterms ident)) rewstrategy1) ; try ident

• topdown rewstrategy1 := fix ident. (choice (rewstrategy1) (progress (subterms ident))) ; try ident

• innermost rewstrategy1 := fix ident. (choice (subterm ident) (rewstrategy1))

• outermost rewstrategy1 := fix ident. (choice (rewstrategy1) (subterm ident))

The basic control strategy semantics are straightforward: strategies are applied to subterms of the term to rewrite, starting from the root of the term. The lemma strategies unify the left-hand-side of the lemma with the current subterm and on success rewrite it to the right- hand-side. Composition can be used to continue rewriting on the current subterm. The fail strategy always fails while the identity strategy succeeds without making progress. The reflexivity strategy succeeds, making progress using a reflexivity proof of rewriting. progress tests progress of the argument rewstrategy1 and fails if no progress was made, while try always succeeds, catching failures. choice uses the first successful strategy in the list of rewstrategy0`s. One can iterate a strategy at least 1 time using ``repeat` and at least 0 times using any.

The subterm and subterms strategies apply their argument rewstrategy1 to respectively one or all subterms of the current term under consideration, left-to-right. subterm stops at the first subterm for which rewstrategy1 made progress. The composite strategies innermost and outermost perform a single innermost or outermost rewrite using their argument rewstrategy1. Their counterparts bottomup and topdown perform as many rewritings as possible, starting from the bottom or the top of the term.

Hint databases created for autorewrite can also be used by rewrite_strat using the hints strategy that applies any of the lemmas at the current subterm. The terms strategy takes the lemma names directly as arguments. The eval strategy expects a reduction expression (see Applying conversion rules) and succeeds if it reduces the subterm under consideration. The fold strategy takes a term and tries to unify it to the current subterm, converting it to term on success. It is stronger than the tactic fold.

Note

The symbol ';' is used to separate sequences of tactics as well as sequences of rewriting strategies. rewrite_strat s; fail is interpreted as rewrite_strat (s; fail), in which fail is a rewriting strategy. Use (rewrite_strat s); fail to make fail a tactic. rewrite_strat s; apply I gives a syntax error (apply is not a valid rewrite strategy).

Example: innermost and outermost

The type of Nat.add_comm is forall n m : nat, n + m = m + n.

Require Import Coq.Arith.Arith.

[Loading ML file ring_plugin.cmxs (using legacy method) ... done]

Set Printing Parentheses.

Goal forall a b c: nat, a + b + c = 0.

1 goal ============================ forall a b c : nat, ((a + b) + c) = 0

rewrite_strat innermost Nat.add_comm.

1 goal ============================ forall a b c : nat, ((b + a) + c) = 0

Abort.

Goal forall a b c: nat, a + b + c = 0.

1 goal ============================ forall a b c : nat, ((a + b) + c) = 0

Using outermost instead gives this result:

rewrite_strat outermost Nat.add_comm.

1 goal ============================ forall a b c : nat, (c + (a + b)) = 0

Abort.