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 complete documentation of the tactics can be found in the source file LibTactics.v. Moreover, demos illustrating the working of all the tactics can be found in the file LibTacticsDemos.v. The tactics are illustrated mostly through 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'll 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 saving time in the naming of hypotheses,
• 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 determinacy 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, E1 and E2.

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 the appropriate substitutions.

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 invert H; introv H1 H2 H3. An example follows.

A more involved example follows. In particular, it shows that the name of an inverted hypothesis can be reused.

Note: in the rare cases where one need 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.

The tactics cases and cases_if

The tactic cases E is a shorthand for remember E as x; destruct x, with the difference that it generates the symmetric of the equality produced by remember. For example, cases would produce the equality beq_id k1 k2 = true rather than the equality true = beq_id k1 k2, the latter being fairly awkward to read. The tactics cases E as H allows specifying the name to be used for the generated equality. Remark: cases is quite similar to case_eq. For the sake of compatibility with remember and case_eq, the library "LibTactics" provides a tactic called cases' that generates exactly the same equalities as remember or case_eq would, i.e., producing an equality in the form true = beq_id k1 k2 rather than beq_id k1 k2 = true. The following examples illustrate the behavior of the tactic cases' E as H.

The tactic cases_if calls cases E on the first expression E that appears in the goal or in the context as argument of a if. So, the tactic cases_if saves the need to copy-past an expression that already occurs in the goal. Here again, for compatibility reasons, the library provides a tactic called cases_if', and the syntax cases_if' as H allows specifying the name of the generated equalities.

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 turn out to 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 proved 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, except that this facility only works up to arity 4, whereas LibTactics supports arities up to 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 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 generating the equality y = z.

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 (works with existential variables),
• sort for moving propositions at the bottom of the proof context.

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 unfolds all the head constants, unfolds performs only a single 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 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

The ability to admit 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, and 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 (typing_inversion_var _ _ _ Htypt) as [T [Hctx Hsub]]. can be rewritten in the form destruct (typing_inversion_var _ _ _ 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 (tm_var 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 (tm_var 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 (tm_var 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 could be even more extreme and not give 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, we want m to be instantiated as the value 3, but we want n to be instantiated using an existential variable.

Note: one can write lets: E0 E1 E2 in place of lets H: E0 E1 E2 when the name H needs not be mentioned in the proof script. Note: the tactics lets accepts up to five arguments. Another syntax if available for providing more than five arguments. It consists in using a list introduced with the keyword >>, 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 specializes and forwards can be found in the tutorial file UseAuto.v.

Example of instantiations

In the following proof, lets is used instead of destruct and applys is used instead of apply in several places. Those places indicated by a comment starting with "old:". The exercise consists in also using lets where indicated.