We have seen many examples of proof automation so far, some with tantalizing code snippets from Ltac, Coq's domain-specific language for proof search procedures. This chapter aims to give a bottom-up 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. The intuition tactic simplifies propositional structure of goals. The congruence tactic 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 similar 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."

There are several other built-in "black box" automation tactics, which one can learn about by perusing the Coq manual. The real promise of Coq, though, is in the coding of problem-specific tactics with Ltac.

Ltac Programming Basics

We have already seen many examples of Ltac programs. In the rest of this chapter, we attempt to give a thorough 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 (in the sense of natural deduction) 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, using a tactic specialize which will replace a hypothesis with a version that is specialized to a provided set of arguments (for quantified variables or local hypotheses from implications).

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. The match construct 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. The tactic generalize takes as input a term t (for instance, a proof of some proposition) and then changes the conclusion from G to T -> G, where T is the type of t (for instance, the proposition proved by a proof given as argument).

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. To understand why this applies to the completer tactics, recall that, in Coq, implication is shorthand for degenerate universal quantification where the quantified variable is not used. Nonetheless, in an Ltac pattern, Coq is happy to match a wildcard implication against a universal quantification.

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, where the same fact may be added to the context repeatedly in an infinite loop. Our earlier completer tactic uses a modus ponens rule that matches the implication conclusion with a variable, which blocks matching against non-trivial universal quantifiers.

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 a particular term. We could also have used idtac n in place of pose n, which would have printed the result without changing the context.

The value of 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.

Reset length.

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. The function 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

One other gotcha shows up when we want to debug our Ltac functional programs. We might expect the following code to work, to give us a version of length that prints a debug trace of the arguments it is called with.

Reset length.

Coq accepts the tactic definition, but the code is fatally flawed and will always lead to dynamic type errors.

  let n := length (1 :: 2 :: 3 :: nil) in
    pose n.

Error: variable n should be bound to a term.

What is going wrong here? The answer has to do with the dual status of Ltac as both a purely functional and an imperative programming language. The basic programming language is purely functional, but tactic scripts are one "datatype" that can be returned by such programs, and Coq will run such a script using an imperative semantics that mutates proof states. Readers familiar with monadic programming in Haskell may recognize a similarity. Side-effecting Haskell programs can be thought of as pure programs that return the code of programs in an imperative language, where some out-of-band mechanism takes responsibility for running these derived programs. In this way, Haskell remains pure, while supporting usual input-output side effects and more. Ltac uses the same basic mechanism, but in a dynamically typed setting. Here the embedded imperative language includes all the tactics we have been applying so far.

Even basic idtac is an embedded imperative program, so we may not automatically mix it with purely functional code. In fact, a semicolon operator alone marks a span of Ltac code as an embedded tactic script. This makes some amount of sense, since pure functional languages have no need for sequencing: since they lack side effects, there is no reason to run an expression and then just throw away its value and move on to another expression.

The solution is like in Haskell: we must "monadify" our pure program to give it access to side effects. The trouble is that the embedded tactic language has no return construct. Proof scripts are about proving theorems, not calculating results. We can apply a somewhat awkward workaround that requires translating our program into continuation-passing style, a program structuring idea popular in functional programming.

Reset length.

The new length takes a new input: a continuation k, which is a function to be called to continue whatever proving process we were in the middle of when we called length. The argument passed to k may be thought of as the return value of length.

(1 :: 2 :: 3 :: nil)
(2 :: 3 :: nil)
(3 :: nil)
nil

We see exactly the trace of function arguments that we expected initially, and an examination of the proof state afterward would show that variable n has been added with value 3.

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.

The tactic 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. The key pieces of state include not only the form of the goal, but also decisions about the values of unification variables. These decisions are rolled back with all the other state after failure.

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. The call 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.

The search function 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
 



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

The matcher tactic 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 unfolding in the expression y, using the eval Ltac construct, which works with the same reduction strategies that we have seen in tactics (e.g., simpl, compute, etc.).

  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. We use the Ltac construct fresh to generate a hypothesis name that is not already used, based on a string suggesting a good name.

Neither eauto nor firstorder is clever enough to prove this goal. We can help out by doing some of the work with quantifiers ourselves, abbreviating the proof with the do tactical for repetition of a tactic a set number of times.

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. (Recall that ex is the underlying type family to which uses of the exists syntax are compiled.)

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.

Reset 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. Also recall that the tactic form solve [ t ] fails if t does not completely solve the goal.

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.

This tactic 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.

Exercises

1. An anonymous Coq fan from the Internet was excited to come up with this tactic definition shortly after getting started learning Ltac:

Without lifting a finger, exciting theorems can be proved:

Unfortunately, this tactic exhibits some degenerate behavior. Consider the following example:

This (failed) proof already takes about one second on my workstation. I hope a pattern in the theorem statement is clear; this is a representative of a class of theorems, where we may add more matched pairs of x and y variables, with equality hypotheses between them. The running time of deSome is exponential in the number of such hypotheses.

The task in this exercise is twofold. First, figure out why deSome exhibits exponential behavior for this class of examples and record your explanation in a comment. Second, write an improved version of deSome that runs in polynomial time.

Sometimes it can be convenient to know that a proof attempt is doomed because the theorem is false. For instance, here are three non-theorems about lists:

The task in this exercise is to write a tactic that disproves these and many other related "theorems" about lists. Your tactic should follow a simple brute-force enumeration strategy, considering all list bool values with length up to some bound given by the user, as a nat argument to the tactic. A successful invocation should add a new hypothesis of the negation of the theorem (guaranteeing that the tactic has made a sound decision about falsehood).

A few hints: A good starting point is to pattern-match the conclusion formula and use the assert tactic on its negation. An assert invocation may include a by clause to specify a tactic to use to prove the assertion.

The idea in this exercise is to disprove a quantified formula by finding instantiations for the quantifiers that make it manifestly false. Recall the specialize tactic for specializing a hypothesis to particular quantifier instantiations. When you have instantiated quantifiers fully, discriminate is a good choice to derive a contradiction. (It at least works for the three examples above and is smart enough for this exercise's purposes.) The type of Ltac construct may be useful to analyze the type of a hypothesis to choose how to instantiate its quantifiers.

To enumerate all boolean lists up to a certain length, it will be helpful to write a recursive tactic in continuation-passing style, where the continuation is meant to be called on each candidate list.

Remember that arguments to Ltac functions may not be type-checked in contexts large enough to allow usual implicit argument inference, so instead of nil it will be useful to write @nil bool, which specifies the usually implicit argument explicitly.

Some theorems involving existential quantifiers are easy to prove with eauto.

Others are harder. The problem with the next theorem is that the existentially quantified variable does not appear in the rest of the theorem, so eauto has no way to deduce its value. However, we know that we had might as well instantiate that variable to tt, the only value of type unit.

We also run into trouble in the next theorem, because eauto does not understand the fst and snd projection functions for pairs.

Both problems show up in this monster example.

The task in this problem is to write a tactic that preprocesses such goals so that eauto can finish them. Your tactic should serve as a complete proof of each of the above examples, along with the wide class of similar examples. The key smarts that your tactic will bring are: first, it introduces separate unification variables for all the "leaf types" of compound types built out of pairs; and second, leaf unification variables of type unit are simply replaced by tt.

A few hints: The following tactic is more convenient than direct use of the built-in tactic evar, for generation of new unification variables:

This is a continuation-passing style tactic. For instance, when the goal begins with existential quantification over a type T, the following tactic invocation will create a new unification variable to use as the quantifier instantiation:

makeEvar T ltac:(fun x => exists x)

Recall that exists formulas are desugared to uses of the ex inductive family. In particular, a pattern like the following can be used to extract the domain of an exists quantifier into variable T:

| [ |- ex (A := ?T) _ ] => ...

The equate tactic used as an example in this chapter will probably be useful, to unify two terms, for instance if the first is a unification variable whose value you want to set.
Ltac equate E1 E2 := let H := fresh in
  assert (H : E1 = E2) by reflexivity; clear H.



Finally, there are some minor complications surrounding overloading of the * operator for both numeric multiplication and Cartesian product for sets (i.e., pair types). To ensure that an Ltac pattern is using the type version, write it like this:

| (?T1 * ?T2)%type => ...

An exercise in the last chapter dealt with automating proofs about rings using eauto, where we must prove some odd-looking theorems to push proof search in a direction where unification does all the work. Algebraic proofs consist mostly of rewriting in equations, so we might hope that the autorewrite tactic would yield more natural automated proofs. Indeed, consider this example within the same formulation of ring theory that we dealt with last chapter, where each of the three axioms has been added to the rewrite hint database cpdt using Hint Rewrite:
Theorem test1 : forall a b, a * b * i b = a.
  intros; autorewrite with cpdt; reflexivity.
Qed.



So far so good. However, consider this further example:
Theorem test2 : forall a, a * e * i a * i e = e.
  intros; autorewrite with cpdt.



The goal is merely reduced to a * (i a * i e) = e, which of course reflexivity cannot prove. The essential problem is that autorewrite does not do backtracking search. Instead, it follows a "greedy" approach, at each stage choosing a rewrite to perform and then never allowing that rewrite to be undone. An early mistake can doom the whole process.

The task in this problem is to use Ltac to implement a backtracking version of autorewrite that works much like eauto, in that its inputs are a database of hint lemmas and a bound on search depth. Here our search trees will have uses of rewrite at their nodes, rather than uses of eapply as in the case of eauto, and proofs must be finished by reflexivity.

An invocation to the tactic to prove test2 might look like this:
  rewriter (right_identity, (right_inverse, tt)) 3.



The first argument gives the set of lemmas to consider, as a kind of list encoded with pair types. Such a format cannot be analyzed directly by Gallina programs, but Ltac allows us much more freedom to deconstruct syntax. For example, to case analyze such a list found in a variable x, we need only write:
  match x with
    | (?lemma, ?more) => ...
  end



In the body of the case analysis, lemma will be bound to the first lemma, and more will be bound to the remaining lemmas. There is no need to consider a case for tt, our stand-in for nil. This is because lack of any matching pattern will trigger failure, which is exactly the outcome we would like upon reaching the end of the lemma list without finding one that applies. The tactic will fail, triggering backtracking to some previous match.

There are different kinds of backtracking, corresponding to different sorts of decisions to be made. The examples considered above can be handled with backtracking that only reconsiders decisions about the order in which to apply rewriting lemmas. A full-credit solution need only handle that kind of backtracking, considering all rewriting sequences up to the length bound passed to your tactic. A good test of this level of applicability is to prove both test1 and test2 above. However, some theorems could only be proved using a smarter tactic that considers not only order of rewriting lemma uses, but also choice of arguments to the lemmas. That is, at some points in a proof, the same lemma may apply at multiple places within the goal formula, and some choices may lead to stuck proof states while others lead to success. For an extra challenge (without any impact on the grade for the problem), you might try beefing up your tactic to do backtracking on argument choice, too.