UseTactics: Tactic Library for Coq: A Gentle Introduction

Coq comes with a set of builtin tactics, such as reflexivity, intros, inversion and so on. While it is possible to conduct proofs using only those tactics, you can significantly increase your productivity by working with a set of more powerful tactics. This chapter describes a number of such very useful tactics, which, for various reasons, are not yet available by default in Coq. These tactics are defined in the LibTactics.v file.

Remark: SSReflect is another package providing powerful tactics. The library "LibTactics" differs from "SSReflect" in two respects:

• "SSReflect" was primarily developed for proving mathematical theorems, whereas "LibTactics" was primarily developed for proving theorems on programming languages. In particular, "LibTactics" provides a number of useful tactics that have no counterpart in the "SSReflect" package.
• "SSReflect" entirely rethinks the presentation of tactics, whereas "LibTactics" mostly stick to the traditional presentation of Coq tactics, simply providing a number of additional tactics. For this reason, "LibTactics" is probably easier to get started with than "SSReflect".

This chapter is a tutorial focusing on the most useful features from the "LibTactics" library. It does not aim at presenting all the features of "LibTactics". The detailed specification of tactics can be found in the source file LibTactics.v. Further documentation as well as demos can be found at http://www.chargueraud.org/softs/tlc/ . In this tutorial, tactics are presented using examples taken from the core chapters of the "Software Foundations" course. To illustrate the various ways in which a given tactic can be used, we use a tactic that duplicates a given goal. More precisely, dup produces two copies of the current goal, and dup n produces n copies of it.

Tactics for introduction and case analysis

This section presents the following tactics:

• introv, for naming hypotheses more efficiently,
• inverts, for improving the inversion tactic,
• cases, for performing a case analysis without losing information,
• cases_if, for automating case analysis on the argument of if.

The tactic introv

The tactic introv allows to automatically introduce the variables of a theorem and explicitly name the hypotheses involved. In the example shown next, the variables c, st, st1 and st2 involved in the statement of determinism need not be named explicitly, because their name where already given in the statement of the lemma. On the contrary, it is useful to provide names for the two hypotheses, which we name E1 and E2, respectively.

When there is no hypothesis to be named, one can call introv without any argument.

The tactic introv also applies to statements in which ∀ and → are interleaved.

Like the arguments of intros, the arguments of introv can be structured patterns.

Remark: the tactic introv works even when definitions need to be unfolded in order to reveal hypotheses.

The tactic inverts

The inversion tactic of Coq is not very satisfying for three reasons. First, it produces a bunch of equalities which one typically wants to substitute away, using subst. Second, it introduces meaningless names for hypotheses. Third, a call to inversion H does not remove H from the context, even though in most cases an hypothesis is no longer needed after being inverted. The tactic inverts address all of these three issues. It is intented to be used in place of the tactic inversion. The following example illustrates how the tactic inverts H behaves mostly like inversion H except that it performs some substitutions in order to eliminate the trivial equalities that are being produced by inversion.

A slightly more interesting example appears next.

The tactic inverts H as. is like inverts H except that the variables and hypotheses being produced are placed in the goal rather than in the context. This strategy allows naming those new variables and hypotheses explicitly, using either intros or introv.

In the particular case where a call to inversion produces a single subgoal, one can use the syntax inverts H as H1 H2 H3 for calling inverts and naming the new hypotheses H1, H2 and H3. In other words, the tactic inverts H as H1 H2 H3 is equivalent to inverts H as; introv H1 H2 H3. An example follows.

A more involved example appears next. In particular, this example shows that the name of the hypothesis being inverted can be reused.

Note: in the rare cases where one needs to perform an inversion on an hypothesis H without clearing H from the context, one can use the tactic inverts keep H, where the keyword keep indicates that the hypothesis should be kept in the context.

Tactics for n-ary connectives

Because Coq encodes conjunctions and disjunctions using binary constructors ∧ and ∨, working with a conjunction or a disjunction of N facts can sometimes be quite cumbursome. For this reason, "LibTactics" provides tactics offering direct support for n-ary conjunctions and disjunctions. It also provides direct support for n-ary existententials. This section presents the following tactics:

• splits for decomposing n-ary conjunctions,
• branch for decomposing n-ary disjunctions,
• ∃ for proving n-ary existentials.

The tactic splits

The tactic splits applies to a goal made of a conjunction of n propositions and it produces n subgoals. For example, it decomposes the goal G1 ∧ G2 ∧ G3 into the three subgoals G1, G2 and G3.

The tactic branch

The tactic branch k can be used to prove a n-ary disjunction. For example, if the goal takes the form G1 ∨ G2 ∨ G3, the tactic branch 2 leaves only G2 as subgoal. The following example illustrates the behavior of the branch tactic.

The tactic ∃

The library "LibTactics" introduces a notation for n-ary existentials. For example, one can write ∃ x y z, H instead of ∃ x, ∃ y, ∃ z, H. Similarly, the library provides a n-ary tactic ∃ a b c, which is a shorthand for ∃ a; ∃ b; ∃ c. The following example illustrates both the notation and the tactic for dealing with n-ary existentials.

Remark: a similar facility for n-ary existentials is provided by the module Coq.Program.Syntax from the standard library. (Coq.Program.Syntax supports existentials up to arity 4; LibTactics supports them up to arity 10.

Tactics for working with equality

One of the major weakness of Coq compared with other interactive proof assistants is its relatively poor support for reasoning with equalities. The tactics described next aims at simplifying pieces of proof scripts manipulating equalities. This section presents the following tactics:

• asserts_rewrite for introducing an equality to rewrite with,
• cuts_rewrite, which is similar except that its subgoals are swapped,
• substs for improving the subst tactic,
• fequals for improving the f_equal tactic,
• applys_eq for proving P x y using an hypothesis P x z, automatically producing an equality y = z as subgoal.

The tactics asserts_rewrite and cuts_rewrite

The tactic asserts_rewrite (E1 = E2) replaces E1 with E2 in the goal, and produces the goal E1 = E2.

The tactic cuts_rewrite (E1 = E2) is like asserts_rewrite (E1 = E2), except that the equality E1 = E2 appears as first subgoal.

More generally, the tactics asserts_rewrite and cuts_rewrite can be provided a lemma as argument. For example, one can write asserts_rewrite (∀ a b, a*(S b) = a×b+a). This formulation is useful when a and b are big terms, since there is no need to repeat their statements.

The tactic substs

The tactic substs is similar to subst except that it does not fail when the goal contains "circular equalities", such as x = f x.

The tactic fequals

The tactic fequals is similar to f_equal except that it directly discharges all the trivial subgoals produced. Moreover, the tactic fequals features an enhanced treatment of equalities between tuples.

The tactic applys_eq

The tactic applys_eq is a variant of eapply that introduces equalities for subterms that do not unify. For example, assume the goal is the proposition P x y and assume we have the assumption H asserting that P x z holds. We know that we can prove y to be equal to z. So, we could call the tactic assert_rewrite (y = z) and change the goal to P x z, but this would require copy-pasting the values of y and z. With the tactic applys_eq, we can call applys_eq H 1, which proves the goal and leaves only the subgoal y = z. The value 1 given as argument to applys_eq indicates that we want an equality to be introduced for the first argument of P x y counting from the right. The three following examples illustrate the behavior of a call to applys_eq H 1, a call to applys_eq H 2, and a call to applys_eq H 1 2.

If the mismatch was on the first argument of P instead of the second, we would have written applys_eq H 2. Recall that the occurences are counted from the right.

When we have a mismatch on two arguments, we want to produce two equalities. To achieve this, we may call applys_eq H 1 2. More generally, the tactic applys_eq expects a lemma and a sequence of natural numbers as arguments.

Some convenient shorthands

This section of the tutorial introduces a few tactics that help make proof scripts shorter and more readable:

• unfolds (without argument) for unfolding the head definition,
• false for replacing the goal with False,
• gen as a shorthand for dependent generalize,
• skip for skipping a subgoal even if it contains existential variables,
• sort for re-ordering the proof context by moving moving all propositions at the bottom.

The tactic unfolds

The tactic unfolds (without any argument) unfolds the head constant of the goal. This tactic saves the need to name the constant explicitly.

Remark: contrary to the tactic hnf, which may unfold several constants, unfolds performs only a single step of unfolding. Remark: the tactic unfolds in H can be used to unfold the head definition of the hypothesis H.

The tactics false and tryfalse

The tactic false can be used to replace any goal with False. In short, it is a shorthand for apply ex_falso_quodlibet. Moreover, false proves the goal if it contains an absurd assumption, such as False or 0 = S n, or if it contains contradictory assumptions, such as x = true and x = false.

The tactic false can be given an argument: false H replace the goals with False and then applies H.

The tactic tryfalse is a shorthand for try solve [false]: it tries to find a contradiction in the goal. The tactic tryfalse is generally called after a case analysis.

The tactic gen

The tactic gen is a shortand for generalize dependent that accepts several arguments at once. An invokation of this tactic takes the form gen x y z.

The tactics skip, skip_rewrite and skip_goal

Temporarily admitting a given subgoal is very useful when constructing proofs. It gives the ability to focus first on the most interesting cases of a proof. The tactic skip is like admit except that it also works when the proof includes existential variables. Recall that existential variables are those whose name starts with a question mark, e.g. ?24, and which are typically introduced by eapply.

The tactic skip H: P adds the hypothesis H: P to the context, without checking whether the proposition P is true. It is useful for exploiting a fact and postponing its proof. Note: skip H: P is simply a shorthand for assert (H:P). skip.

The tactic skip_rewrite (E1 = E2) replaces E1 with E2 in the goal, without checking that E1 is actually equal to E2.

Remark: the tactic skip_rewrite can in fact be given a lemma statement as argument, in the same way as asserts_rewrite. The tactic skip_goal adds the current goal as hypothesis. This cheat is useful to set up the structure of a proof by induction without having to worry about the induction hypothesis being applied only to smaller arguments. Using skip_goal, one can construct a proof in two steps: first, check that the main arguments go through without waisting time on fixing the details of the induction hypotheses; then, focus on fixing the invokations of the induction hypothesis.

The tactic sort

The tactic sort reorganizes the proof context by placing all the variables at the top and all the hypotheses at the bottom, thereby making the proof context more readable.

Tactics for advanced lemma instantiation

This last section describes a mechanism for instantiating a lemma by providing some of its arguments and leaving other implicit. Variables whose instantiation is not provided are turned into existentential variables, and facts whose instantiation is not provided are turned into subgoals. Remark: this instantion mechanism goes far beyond the abilities of the "Implicit Arguments" mechanism. The point of the instantiation mechanism described in this section is that you will no longer need to spend time figuring out how many underscore symbols you need to write. In this section, we'll use a useful feature of Coq for decomposing conjunctions and existentials. In short, a tactic like intros or destruct can be provided with a pattern (H1 & H2 & H3 & H4 & H5), which is a shorthand for [H1 [H2 [H3 [H4 H5]]]]]. For example, destruct (H _ _ _ Htypt) as [T [Hctx Hsub]]. can be rewritten in the form destruct (H _ _ _ Htypt) as (T & Hctx & Hsub).

Working of lets

When we have a lemma (or an assumption) that we want to exploit, we often need to explicitly provide arguments to this lemma, writing something like: destruct (typing_inversion_var _ _ _ Htypt) as (T & Hctx & Hsub). The need to write several times the "underscore" symbol is tedious. Not only we need to figure out how many of them to write down, but it also makes the proof scripts look prettly ugly. With the tactic lets, one can simply write: lets (T & Hctx & Hsub): typing_inversion_var Htypt. In short, this tactic lets allows to specialize a lemma on a bunch of variables and hypotheses. The syntax is lets I: E0 E1 .. EN, for building an hypothesis named I by applying the fact E0 to the arguments E1 to EN. Not all the arguments need to be provided, however the arguments that are provided need to be provided in the correct order. The tactic relies on a first-match algorithm based on types in order to figure out how the to instantiate the lemma with the arguments provided.

First, assume we have an assumption H with the type of the form has_type G (tvar x) T. We can obtain the conclusion of the lemma typing_inversion_var by invoking the tactics lets K: typing_inversion_var H, as shown next.

Assume now that we know the values of G, x and T and we want to obtain S, and have has_type G (tvar x) T be produced as a subgoal. To indicate that we want all the remaining arguments of typing_inversion_var to be produced as subgoals, we use a triple-underscore symbol ___. (We'll later introduce a shorthand tactic called forwards to avoid writing triple underscores.)

Usually, there is only one context G and one type T that are going to be suitable for proving has_type G (tvar x) T, so we don't really need to bother giving G and T explicitly. It suffices to call lets (S & Eq & Sub): typing_inversion_var x. The variables G and T are then instantiated using existential variables.

We may go even further by not giving any argument to instantiate typing_inversion_var. In this case, three unification variables are introduced.

Note: if we provide lets with only the name of the lemma as argument, it simply adds this lemma in the proof context, without trying to instantiate any of its arguments.

A last useful feature of lets is the double-underscore symbol, which allows skipping an argument when several arguments have the same type. In the following example, our assumption quantifies over two variables n and m, both of type nat. We would like m to be instantiated as the value 3, but without specifying a value for n. This can be achieved by writting lets K: H __ 3.

Note: one can write lets: E0 E1 E2 in place of lets H: E0 E1 E2. In this case, the name H is chosen arbitrarily. Note: the tactics lets accepts up to five arguments. Another syntax is available for providing more than five arguments. It consists in using a list introduced with the special symbol >>, for example lets H: (>> E0 E1 E2 E3 E4 E5 E6 E7 E8 E9 10).

Working of applys, forwards and specializes

The tactics applys, forwards and specializes are shorthand that may be used in place of lets to perform specific tasks.

• forwards is a shorthand for instantiating all the arguments

of a lemma. More precisely, forwards H: E0 E1 E2 E3 is the same as lets H: E0 E1 E2 E3 ___, where the triple-underscore has the same meaning as explained earlier on.

• applys allows building a lemma using the advanced instantion

mode of lets, and then apply that lemma right away. So, applys E0 E1 E2 E3 is the same as lets H: E0 E1 E2 E3 followed with eapply H and then clear H.

• specializes is a shorthand for instantiating in-place

an assumption from the context with particular arguments. More precisely, specializes H E0 E1 is the same as lets H': H E0 E1 followed with clear H and rename H' into H. Examples of use of applys appear further on. Several examples of use of forwards can be found in the tutorial chapter UseAuto.

Example of instantiations

The following proof shows several examples where lets is used instead of destruct, as well as examples where applys is used instead of apply. The proof also contains some holes that you need to fill in as an exercise.

Summary

In this chapter we have presented a number of tactics that help make proof script more concise and more robust on change.

• introv and inverts improve naming and inversions.
• false and tryfalse help discarding absurd goals.
• unfolds automatically calls unfold on the head definition.
• gen helps setting up goals for induction.
• cases and cases_if help with case analysis.
• splits, branch and ∃ to deal with n-ary constructs.
• asserts_rewrite, cuts_rewrite, substs and fequals help working with equalities.
• lets, forwards, specializes and applys provide means of very conveniently instantiating lemmas.
• applys_eq can save the need to perform manual rewriting steps before being able to apply lemma.
• skip, skip_rewrite and skip_goal give the flexibility to choose which subgoals to try and discharge first.

Making use of these tactics can boost one's productivity in Coq proofs. If you are interested in using LibTactics.v in your own developments, make sure you get the lastest version from: http://www.chargueraud.org/softs/tlc/ .