We have seen many examples of proof automation so far. This chapter aims to give a principled presentation of the features of Ltac, focusing in particular on the Ltac match construct, which supports a novel approach to backtracking search. First, though, we will run through some useful automation tactics that are built into Coq. They are described in detail in the manual, so we only outline what is possible.

Some Built-In Automation Tactics

A number of tactics are called repeatedly by crush. intuition simplifies propositional structure of goals. congruence applies the rules of equality and congruence closure, plus properties of constructors of inductive types. The omega tactic provides a complete decision procedure for a theory that is called quantifier-free linear arithmetic or Presburger arithmetic, depending on whom you ask. That is, omega proves any goal that follows from looking only at parts of that goal that can be interpreted as propositional formulas whose atomic formulas are basic comparison operations on natural numbers or integers.

The ring tactic solves goals by appealing to the axioms of rings or semi-rings (as in algebra), depending on the type involved. Coq developments may declare new types to be parts of rings and semi-rings by proving the associated axioms. There is a simlar tactic field for simplifying values in fields by conversion to fractions over rings. Both ring and field can only solve goals that are equalities. The fourier tactic uses Fourier's method to prove inequalities over real numbers, which are axiomatized in the Coq standard library.

The setoid facility makes it possible to register new equivalence relations to be understood by tactics like rewrite. For instance, Prop is registered as a setoid with the equivalence relation "if and only if." The ability to register new setoids can be very useful in proofs of a kind common in math, where all reasoning is done after "modding out by a relation."

Hint Databases

Another class of built-in tactics includes auto, eauto, and autorewrite. These are based on hint databases, which we have seen extended in many examples so far. These tactics are important, because, in Ltac programming, we cannot create "global variables" whose values can be extended seamlessly by different modules in different source files. We have seen the advantages of hints so far, where crush can be defined once and for all, while still automatically applying the hints we add throughout developments.

The basic hints for auto and eauto are Hint Immediate lemma, asking to try solving a goal immediately by applying a lemma and discharging any hypotheses with a single proof step each; Resolve lemma, which does the same but may add new premises that are themselves to be subjects of nested proof search; Constructor type, which acts like Resolve applied to every constructor of an inductive type; and Unfold ident, which tries unfolding ident when it appears at the head of a proof goal. Each of these Hint commands may be used with a suffix, as in Hint Resolve lemma : my_db. This adds the hint only to the specified database, so that it would only be used by, for instance, auto with my_db. An additional argument to auto specifies the maximum depth of proof trees to search in depth-first order, as in auto 8 or auto 8 with my_db. The default depth is 5.

All of these Hint commands can be issued alternatively with a more primitive hint kind, Extern. A few examples should do best to explain how Hint Extern works.

crush would have discharged this goal, but the default hint database for auto contains no hint that applies.

It is hard to come up with a bool-specific hint that is not just a restatement of the theorem we mean to prove. Luckily, a simpler form suffices.

Our hint says: "whenever the conclusion matches the pattern _ <> _, try applying congruence." The 1 is a cost for this rule. During proof search, whenever multiple rules apply, rules are tried in increasing cost order, so it pays to assign high costs to relatively expensive Extern hints.

Extern hints may be implemented with the full Ltac language. This example shows a case where a hint uses a match.

crush makes no progress beyond what intros would have accomplished. auto will not apply the hypothesis both to prove the goal, because the conclusion of both does not unify with the conclusion of the goal. However, we can teach auto to handle this kind of goal.

We see that an Extern pattern may bind unification variables that we use in the associated tactic. proj1 is a function from the standard library for extracting a proof of R from a proof of R /\ S.

After our success on this example, we might get more ambitious and seek to generalize the hint to all possible predicates P.


  Hint Extern 1 (?P ?X) =>
    match goal with
      | [ H : forall x, P x /\ _ |- _ ] => apply (proj1 (H X))
    end.

User error: Bound head variable
 

Coq's auto hint databases work as tables mapping head symbols to lists of tactics to try. Because of this, the constant head of an Extern pattern must be determinable statically. In our first Extern hint, the head symbol was not, since x <> y desugars to not (eq x y); and, in the second example, the head symbol was P.

This restriction on Extern hints is the main limitation of the auto mechanism, preventing us from using it for general context simplifications that are not keyed off of the form of the conclusion. This is perhaps just as well, since we can often code more efficient tactics with specialized Ltac programs, and we will see how in later sections of the chapter.

We have used Hint Rewrite in many examples so far. crush uses these hints by calling autorewrite. Our rewrite hints have taken the form Hint Rewrite lemma : cpdt, adding them to the cpdt rewrite database. This is because, in contrast to auto, autorewrite has no default database. Thus, we set the convention that crush uses the cpdt database.

This example shows a direct use of autorewrite.

There are a few ways in which autorewrite can lead to trouble when insufficient care is taken in choosing hints. First, the set of hints may define a nonterminating rewrite system, in which case invocations to autorewrite may not terminate. Second, we may add hints that "lead autorewrite down the wrong path." For instance:

============================
 g (g (g x)) = g x

Our new hint was used to rewrite the goal into a form where the old hint could no longer be applied. This "non-monotonicity" of rewrite hints contrasts with the situation for auto, where new hints may slow down proof search but can never "break" old proofs.

autorewrite works with quantified equalities that include additional premises, but we must be careful to avoid similar incorrect rewritings.

  ============================
   g (g (g x)) = g x

subgoal 2 is:
 P x
subgoal 3 is:
 P (f x)
subgoal 4 is:
 P (f x)

The inappropriate rule fired the same three times as before, even though we know we will not be able to prove the premises.

Our final, successful, attempt uses an extra argument to Hint Rewrite that specifies a tactic to apply to generated premises.

autorewrite will still use f_g when the generated premise is among our assumptions.

It can also be useful to use the autorewrite with db in * form, which does rewriting in hypotheses, as well as in the conclusion.

Ltac Programming Basics

We have already seen many examples of Ltac programs. In the rest of this chapter, we attempt to give a more principled introduction to the important features and design patterns.

One common use for match tactics is identification of subjects for case analysis, as we see in this tactic definition.

The tactic checks if the conclusion is an if, destructing the test expression if so. Certain classes of theorem are trivial to prove automatically with such a tactic.

The repeat that we use here is called a tactical, or tactic combinator. The behavior of repeat t is to loop through running t, running t on all generated subgoals, running t on their generated subgoals, and so on. When t fails at any point in this search tree, that particular subgoal is left to be handled by later tactics. Thus, it is important never to use repeat with a tactic that always succeeds.

Another very useful Ltac building block is context patterns.

The behavior of this tactic is to find any subterm of the conclusion that is an if and then destruct the test expression. This version subsumes find_if.

We can also use find_if_inside to prove goals that find_if does not simplify sufficiently.

Many decision procedures can be coded in Ltac via "repeat match loops." For instance, we can implement a subset of the functionality of tauto.

Since match patterns can share unification variables between hypothesis and conclusion patterns, it is easy to figure out when the conclusion matches a hypothesis. The exact tactic solves a goal completely when given a proof term of the proper type.

It is also trivial to implement the "introduction rules" for a few of the connectives. Implementing elimination rules is only a little more work, since we must give a name for a hypothesis to destruct.

The last rule implements modus ponens. The most interesting part is the use of the Ltac-level let with a fresh expression. fresh takes in a name base and returns a fresh hypothesis variable based on that name. We use the new name variable H as the name we assign to the result of modus ponens. The use of generalize changes our conclusion to be an implication from Q. We clear the original hypothesis and move Q into the context with name H.

It was relatively easy to implement modus ponens, because we do not lose information by clearing every implication that we use. If we want to implement a similarly-complete procedure for quantifier instantiation, we need a way to ensure that a particular proposition is not already included among our hypotheses. To do that effectively, we first need to learn a bit more about the semantics of match.

It is tempting to assume that match works like it does in ML. In fact, there are a few critical differences in its behavior. One is that we may include arbitrary expressions in patterns, instead of being restricted to variables and constructors. Another is that the same variable may appear multiple times, inducing an implicit equality constraint.

There is a related pair of two other differences that are much more important than the others. match has a backtracking semantics for failure. In ML, pattern matching works by finding the first pattern to match and then executing its body. If the body raises an exception, then the overall match raises the same exception. In Coq, failures in case bodies instead trigger continued search through the list of cases.

For instance, this (unnecessarily verbose) proof script works:

The first case matches trivially, but its body tactic fails, since the conclusion does not begin with a quantifier or implication. In a similar ML match, that would mean that the whole pattern-match fails. In Coq, we backtrack and try the next pattern, which also matches. Its body tactic succeeds, so the overall tactic succeeds as well.

The example shows how failure can move to a different pattern within a match. Failure can also trigger an attempt to find a different way of matching a single pattern. Consider another example:

Coq prints "H1". By applying idtac with an argument, a convenient debugging tool for "leaking information out of matches," we see that this match first tries binding H to H1, which cannot be used to prove Q. Nonetheless, the following variation on the tactic succeeds at proving the goal:

The tactic first unifies H with H1, as before, but exact H fails in that case, so the tactic engine searches for more possible values of H. Eventually, it arrives at the correct value, so that exact H and the overall tactic succeed.

Now we are equipped to implement a tactic for checking that a proposition is not among our hypotheses:

We use the equality checking that is built into pattern-matching to see if there is a hypothesis that matches the proposition exactly. If so, we use the fail tactic. Without arguments, fail signals normal tactic failure, as you might expect. When fail is passed an argument n, n is used to count outwards through the enclosing cases of backtracking search. In this case, fail 1 says "fail not just in this pattern-matching branch, but for the whole match." The second case will never be tried when the fail 1 is reached.

This second case, used when P matches no hypothesis, checks if P is a conjunction. Other simplifications may have split conjunctions into their component formulas, so we need to check that at least one of those components is also not represented. To achieve this, we apply the first tactical, which takes a list of tactics and continues down the list until one of them does not fail. The fail 2 at the end says to fail both the first and the match wrapped around it.

The body of the ?P1 /\ ?P2 case guarantees that, if it is reached, we either succeed completely or fail completely. Thus, if we reach the wildcard case, P is not a conjunction. We use idtac, a tactic that would be silly to apply on its own, since its effect is to succeed at doing nothing. Nonetheless, idtac is a useful placeholder for cases like what we see here.

With the non-presence check implemented, it is easy to build a tactic that takes as input a proof term and adds its conclusion as a new hypothesis, only if that conclusion is not already present, failing otherwise.

We see the useful type of operator of Ltac. This operator could not be implemented in Gallina, but it is easy to support in Ltac. We end up with t bound to the type of pf. We check that t is not already present. If so, we use a generalize/intro combo to add a new hypothesis proved by pf.

With these tactics defined, we can write a tactic completer for adding to the context all consequences of a set of simple first-order formulas.

We use the same kind of conjunction and implication handling as previously. Note that, since -> is the special non-dependent case of forall, the fourth rule handles intro for implications, too.

In the fifth rule, when we find a forall fact H with a premise matching one of our hypotheses, we add the appropriate instantiation of H's conclusion, if we have not already added it.

We can check that completer is working properly:

  x : A
  H : P x
  H0 : Q x
  H3 : R x
  H4 : S x
  ============================
   S x

We narrowly avoided a subtle pitfall in our definition of completer. Let us try another definition that even seems preferable to the original, to the untrained eye.

The only difference is in the modus ponens rule, where we have replaced an unused unification variable ?Q with a wildcard. Let us try our example again with this version:

    completer'.
 

Coq loops forever at this point. What went wrong?

A few examples should illustrate the issue. Here we see a match-based proof that works fine:

This one fails.

  match goal with
    | [ |- forall x, ?P ] => trivial
  end.

User error: No matching clauses for match goal

The problem is that unification variables may not contain locally-bound variables. In this case, ?P would need to be bound to x = x, which contains the local quantified variable x. By using a wildcard in the earlier version, we avoided this restriction.

The Coq 8.2 release includes a special pattern form for a unification variable with an explicit set of free variables. That unification variable is then bound to a function from the free variables to the "real" value. In Coq 8.1 and earlier, there is no such workaround.

No matter which version you use, it is important to be aware of this restriction. As we have alluded to, the restriction is the culprit behind the infinite-looping behavior of completer'. We unintentionally match quantified facts with the modus ponens rule, circumventing the "already present" check and leading to different behavior.

Functional Programming in Ltac

Ltac supports quite convenient functional programming, with a Lisp-with-syntax kind of flavor. However, there are a few syntactic conventions involved in getting programs to be accepted. The Ltac syntax is optimized for tactic-writing, so one has to deal with some inconveniences in writing more standard functional programs.

To illustrate, let us try to write a simple list length function. We start out writing it just like in Gallina, simply replacing Fixpoint (and its annotations) with Ltac.


Ltac length ls :=
  match ls with
    | nil => O
    | _ :: ls' => S (length ls')
  end.

Error: The reference ls' was not found in the current environment
 

At this point, we hopefully remember that pattern variable names must be prefixed by question marks in Ltac.


Ltac length ls :=
  match ls with
    | nil => O
    | _ :: ?ls' => S (length ls')
  end.

Error: The reference S was not found in the current environment
 

The problem is that Ltac treats the expression S (length ls') as an invocation of a tactic S with argument length ls'. We need to use a special annotation to "escape into" the Gallina parsing nonterminal.

This definition is accepted. It can be a little awkward to test Ltac definitions like this. Here is one method.

  n := S (length (2 :: 3 :: nil)) : nat
  ============================
   False
 

We use the pose tactic, which extends the proof context with a new variable that is set equal to particular a term. We could also have used idtac n in place of pose n, which would have printed the result without changing the context.

n only has the length calculation unrolled one step. What has happened here is that, by escaping into the constr nonterminal, we referred to the length function of Gallina, rather than the length Ltac function that we are defining.

The thing to remember is that Gallina terms built by tactics must be bound explicitly via let or a similar technique, rather than inserting Ltac calls directly in other Gallina terms.

  n := 3 : nat
  ============================
   False

We can also use anonymous function expressions and local function definitions in Ltac, as this example of a standard list map function shows.

Ltac functions can have no implicit arguments. It may seem surprising that we need to pass T, the carried type of the output list, explicitly. We cannot just use type of f, because f is an Ltac term, not a Gallina term, and Ltac programs are dynamically typed. f could use very syntactic methods to decide to return differently typed terms for different inputs. We also could not replace constr:(@nil T) with constr:nil, because we have no strongly-typed context to use to infer the parameter to nil. Luckily, we do have sufficient context within constr:(x' :: ls'').

Sometimes we need to employ the opposite direction of "nonterminal escape," when we want to pass a complicated tactic expression as an argument to another tactic, as we might want to do in invoking map.

  l := (1, 1) :: (2, 2) :: (3, 3) :: nil : list (nat * nat)
  ============================
   False

Recursive Proof Search

Deciding how to instantiate quantifiers is one of the hardest parts of automated first-order theorem proving. For a given problem, we can consider all possible bounded-length sequences of quantifier instantiations, applying only propositional reasoning at the end. This is probably a bad idea for almost all goals, but it makes for a nice example of recursive proof search procedures in Ltac.

We can consider the maximum "dependency chain" length for a first-order proof. We define the chain length for a hypothesis to be 0, and the chain length for an instantiation of a quantified fact to be one greater than the length for that fact. The tactic inster n is meant to try all possible proofs with chain length at most n.

inster begins by applying propositional simplification. Next, it checks if any chain length remains. If so, it tries all possible ways of instantiating quantified hypotheses with properly-typed local variables. It is critical to realize that, if the recursive call inster n' fails, then the match goal just seeks out another way of unifying its pattern against proof state. Thus, this small amount of code provides an elegant demonstration of how backtracking match enables exhaustive search.

We can verify the efficacy of inster with two short examples. The built-in firstorder tactic (with no extra arguments) is able to prove the first but not the second.

The style employed in the definition of inster can seem very counterintuitive to functional programmers. Usually, functional programs accumulate state changes in explicit arguments to recursive functions. In Ltac, the state of the current subgoal is always implicit. Nonetheless, in contrast to general imperative programming, it is easy to undo any changes to this state, and indeed such "undoing" happens automatically at failures within matches. In this way, Ltac programming is similar to programming in Haskell with a stateful failure monad that supports a composition operator along the lines of the first tactical.

Functional programming purists may react indignantly to the suggestion of programming this way. Nonetheless, as with other kinds of "monadic programming," many problems are much simpler to solve with Ltac than they would be with explicit, pure proof manipulation in ML or Haskell. To demonstrate, we will write a basic simplification procedure for logical implications.

This procedure is inspired by one for separation logic, where conjuncts in formulas are thought of as "resources," such that we lose no completeness by "crossing out" equal conjuncts on the two sides of an implication. This process is complicated by the fact that, for reasons of modularity, our formulas can have arbitrary nested tree structure (branching at conjunctions) and may include existential quantifiers. It is helpful for the matching process to "go under" quantifiers and in fact decide how to instantiate existential quantifiers in the conclusion.

To distinguish the implications that our tactic handles from the implications that will show up as "plumbing" in various lemmas, we define a wrapper definition, a notation, and a tactic.

These lemmas about imp will be useful in the tactic that we will write.

The first order of business in crafting our matcher tactic will be auxiliary support for searching through formula trees. The search_prem tactic implements running its tactic argument tac on every subformula of an imp premise. As it traverses a tree, search_prem applies some of the above lemmas to rewrite the goal to bring different subformulas to the head of the goal. That is, for every subformula P of the implication premise, we want P to "have a turn," where the premise is rearranged into the form P /\ Q for some Q. The tactic tac should expect to see a goal in this form and focus its attention on the first conjunct of the premise.

To understand how search_prem works, we turn first to the final match. If the premise begins with a conjunction, we call the search procedure on each of the conjuncts, or only the first conjunct, if that already yields a case where tac does not fail. search P expects and maintains the invariant that the premise is of the form P /\ Q for some Q. We pass P explicitly as a kind of decreasing induction measure, to avoid looping forever when tac always fails. The second match case calls a commutativity lemma to realize this invariant, before passing control to search. The final match case tries applying tac directly and then, if that fails, changes the form of the goal by adding an extraneous True conjunct and calls tac again.

search itself tries the same tricks as in the last case of the final match. Additionally, if neither works, it checks if P is a conjunction. If so, it calls itself recursively on each conjunct, first applying associativity lemmas to maintain the goal-form invariant.

We will also want a dual function search_conc, which does tree search through an imp conclusion.

Now we can prove a number of lemmas that are suitable for application by our search tactics. A lemma that is meant to handle a premise should have the form P /\ Q --> R for some interesting P, and a lemma that is meant to handle a conclusion should have the form P --> Q /\ R for some interesting Q.

We will also want a "base case" lemma for finishing proofs where cancelation has removed every constituent of the conclusion.

Our final matcher tactic is now straightforward. First, we intros all variables into scope. Then we attempt simple premise simplifications, finishing the proof upon finding False and eliminating any existential quantifiers that we find. After that, we search through the conclusion. We remove True conjuncts, remove existential quantifiers by introducing unification variables for their bound variables, and search for matching premises to cancel. Finally, when no more progress is made, we see if the goal has become trivial and can be solved by imp_True. In each case, we use the tactic simple apply in place of apply to use a simpler, less expensive unification algorithm.

Our tactic succeeds at proving a simple example.

In the generated proof, we find a trace of the workings of the search tactics.

t2 =
fun P Q : Prop =>
comm_prem (assoc_prem1 (assoc_prem2 (False_prem (P:=P /\ P /\ Q) (P /\ Q))))
     : forall P Q : Prop, Q /\ (P /\ False) /\ P --> P /\ Q
 

We can also see that matcher is well-suited for cases where some human intervention is needed after the automation finishes.

  ============================
   True --> R
 

matcher canceled those conjuncts that it was able to cancel, leaving a simplified subgoal for us, much as intuition does.

matcher even succeeds at guessing quantifier instantiations. It is the unification that occurs in uses of the Match lemma that does the real work here.

t4 =
fun (P : nat -> Prop) (Q : Prop) =>
and_True_prem
  (ex_prem (P:=fun x : nat => P x /\ Q)
     (fun x : nat =>
      assoc_prem2
        (Match (P:=Q)
           (and_True_conc
              (ex_conc (fun x0 : nat => P x0) x
                 (Match (P:=P x) (imp_True (P:=True))))))))
     : forall (P : nat -> Prop) (Q : Prop),
       (exists x : nat, P x /\ Q) --> Q /\ (exists x : nat, P x)

Creating Unification Variables

A final useful ingredient in tactic crafting is the ability to allocate new unification variables explicitly. Tactics like eauto introduce unification variable internally to support flexible proof search. While eauto and its relatives do backward reasoning, we often want to do similar forward reasoning, where unification variables can be useful for similar reasons.

For example, we can write a tactic that instantiates the quantifiers of a universally-quantified hypothesis. The tactic should not need to know what the appropriate instantiantiations are; rather, we want these choices filled with placeholders. We hope that, when we apply the specialized hypothesis later, syntactic unification will determine concrete values.

Before we are ready to write a tactic, we can try out its ingredients one at a time.

  H : forall x : nat, S x > x
  ============================
   2 > 1
 

To instantiate H generically, we first need to name the value to be used for x.

  H : forall x : nat, S x > x
  y := ?279 : nat
  ============================
   2 > 1
 

The proof context is extended with a new variable y, which has been assigned to be equal to a fresh unification variable ?279. We want to instantiate H with ?279. To get ahold of the new unification variable, rather than just its alias y, we perform a trivial call-by-value reduction in the expression y. In particular, we only request the use of one reduction rule, delta, which deals with definition unfolding. We pass a flag further stipulating that only the definition of y be unfolded. This is a simple trick for getting at the value of a synonym variable.

  H : forall x : nat, S x > x
  ============================
   S ?279 > ?279 -> 2 > 1
 

Our instantiation was successful. We can finish by using the refined formula to replace the original.

  H : S ?281 > ?281
  ============================
   2 > 1
 

We can finish the proof by using apply's unification to figure out the proper value of ?281. (The original unification variable was replaced by another, as often happens in the internals of the various tactics' implementations.)

Now we can write a tactic that encapsulates the pattern we just employed, instantiating all quantifiers of a particular hypothesis.

This particular example is somewhat silly, since apply by itself would have solved the goal originally. Separate forward reasoning is more useful on hypotheses that end in existential quantifications. Before we go through an example, it is useful to define a variant of insterU that does not clear the base hypothesis we pass to it.

Neither eauto nor firstorder is clever enough to prove this goal. We can help out by doing some of the work with quantifiers ourselves.

Our proof state is extended with two generic instances of H1.


  H' : exists u : B, P ?4289 u
  H'0 : exists u : B, P ?4288 u
  ============================
   exists u1 : B, exists u2 : B, P (f v1 v2) (g u1 u2)
 

eauto still cannot prove the goal, so we eliminate the two new existential quantifiers.

Now the goal is simple enough to solve by logic programming.

Our insterU tactic does not fare so well with quantified hypotheses that also contain implications. We can see the problem in a slight modification of the last example. We introduce a new unary predicate Q and use it to state an additional requirement of our hypothesis H1.

This proof script does not hit any errors until the very end, when an error message like this one is displayed.


No more subgoals but non-instantiated existential variables :
Existential 1 =
?4384 : [A : Type
         B : Type
         Q : A -> Prop
         P : A -> B -> Prop
         f : A -> A -> A
         g : B -> B -> B
         H1 : forall v : A, Q v -> exists u : B, P v u
         H2 : forall (v1 : A) (u1 : B) (v2 : A) (u2 : B),
              P v1 u1 -> P v2 u2 -> P (f v1 v2) (g u1 u2)
         v1 : A
         v2 : A
         H : Q v1
         H0 : Q v2
         H' : Q v2 -> exists u : B, P v2 u |- Q v2]
 

There is another similar line about a different existential variable. Here, "existential variable" means what we have also called "unification variable." In the course of the proof, some unification variable ?4384 was introduced but never unified. Unification variables are just a device to structure proof search; the language of Gallina proof terms does not include them. Thus, we cannot produce a proof term without instantiating the variable.

The error message shows that ?4384 is meant to be a proof of Q v2 in a particular proof state, whose variables and hypotheses are displayed. It turns out that ?4384 was created by insterU, as the value of a proof to pass to H1. Recall that, in Gallina, implication is just a degenerate case of forall quantification, so the insterU code to match against forall also matched the implication. Since any proof of Q v2 is as good as any other in this context, there was never any opportunity to use unification to determine exactly which proof is appropriate. We expect similar problems with any implications in arguments to insterU.

We can redefine insterU to treat implications differently. In particular, we pattern-match on the type of the type T in forall x : ?T, .... If T has type Prop, then x's instantiation should be thought of as a proof. Thus, instead of picking a new unification variable for it, we instead apply a user-supplied tactic tac. It is important that we end this special Prop case with || fail 1, so that, if tac fails to prove T, we abort the instantiation, rather than continuing on to the default quantifier handling.

We can prove the goal by calling insterKeep with a tactic that tries to find and apply a Q hypothesis over a variable about which we do not yet know any P facts. We need to begin this tactic code with idtac; to get around a strange limitation in Coq's proof engine, where a first-class tactic argument may not begin with a match.

It is often useful to instantiate existential variables explicitly. A built-in tactic provides one way of doing so.

The 1 above is identifying an existential variable appearing in the current goal, with the last existential appearing assigned number 1, the second last assigned number 2, and so on. The named existential is replaced everywhere by the term to the right of the :=.

The instantiate tactic can be convenient for exploratory proving, but it leads to very brittle proof scripts that are unlikely to adapt to changing theorem statements. It is often more helpful to have a tactic that can be used to assign a value to a term that is known to be an existential. By employing a roundabout implementation technique, we can build a tactic that generalizes this functionality. In particular, our tactic equate will assert that two terms are equal. If one of the terms happens to be an existential, then it will be replaced everywhere with the other term.

equate fails if it is not possible to prove x = y by reflexivity. We perform the proof only for its unification side effects, clearing the fact x = y afterward. With equate, we can build a less brittle version of the prior example.