to print (hide ...) as <hidden>

(assign eviscerate-hide-terms t)
(assign eviscerate-hide-terms nil)

Eviscerate-hide-terms is a state global variable whose value is either t or nil. The variable affects how terms are displayed. If t, terms of the form (hide ...) are printed as <hidden>. Otherwise, they are printed normally.


a rule for computing the value of a function

(:executable-counterpart length)
which may be abbreviated in theories as

Every defun introduces at least two rules used by the theorem prover. Suppose fn is the name of a defun'd function. Then (:definition fn) is the rune (see rune) naming the rule that allows the simplifier to replace calls of fn by its instantiated body. (:executable-counterpart fn) is the rune for the rule for how to evaluate the function on known constants.

When typing theories it is convenient to know that (fn) is a runic designator that denotes (:executable-counterpart fn). See theories.

If (:executable-counterpart fn) is enabled, then when applications of fn to known constants are seen by the simplifier they are computed out by executing the Common Lisp code for fn (with the appropriate handling of guards). Suppose fact is defined as the factorial function. If the executable counterpart rune of fact, (:executable-counterpart fact), is enabled when the simplifier encounters (fact 12), then that term will be ``immediately'' expanded to 479001600.

Such one-step expansions are sometimes counterproductive because they prevent the anticipated application of certain lemmas about the subroutines of the expanded function. Such computed expansions can be prevented by disabling the executable counterpart rune of the relevant function. For example, if (:executable-counterpart fact) is disabled, (fact 12) will not be expanded by computation. In this situation, (fact 12) may be rewritten to (* 12 (fact 11)), using the rule named (:definition fact), provided the system's heuristics permit the introduction of the term (fact 11). Note that lemmas about multiplication may then be applicable (while such lemmas would be inapplicable to 479001600). In many proofs it is desirable to disable the executable counterpart runes of certain functions to prevent their expansion by computation. See executable-counterpart-theory.

Finally: What do we do about functions that are ``constrained'' rather than defined, such as the following? (See encapsulate.)

(encapsulate ((foo (x) t))
             (local (defun foo (x) x)))
Does foo have an executable counterpart? Yes: since the vast majority of functions have sensible executable counterparts, it was decided that all functions, even such ``constrained'' ones, have executable counterparts. We essentially ``trap'' when such calls are inappropriate. Thus, consider for example:
(defun bar (x)
  (if (rationalp x)
      (+ x 1)
    (foo x)))
If the term (bar '3) is encountered by the ACL2 rewriter during a proof, and if the :executable-counterpart of bar is enabled, then it will be invoked to reduce this term to '4. However, if the term (bar 'a) is encountered during a proof, then since 'a is not a rationalp and since the :executable-counterpart of foo is only a ``trap,'' then this call of the :executable-counterpart of bar will result in a ``trap.'' In that case, the rewriter will return the term (hide (bar 'a)) so that it never has to go through this process again. See hide.


the end of pre-history

Exit-boot-strap-mode is the last step in creating the ACL2 world in which the user lives. It has command number 0. Commands before it are part of the system initialization and extend all the way back to :min. Commands after it are those of the user.

Exit-boot-strap-mode is a Common Lisp function but not an ACL2 function. It is called when every defconst, defun, etc., in our source code has been processed under ACL2 and the world is all but complete. exit-boot-strap-mode has only one job: to signal the completion of the boot-strapping.


how to deal with a proof failure in a forcing round

See forcing-round for a background discussion of the notion of forcing rounds. When a proof fails during a forcing round it means that the ``gist'' of the proof succeeded but some ``technical detail'' failed. The first question you must ask yourself is whether the forced goals are indeed theorems. We discuss the possibilities below.

If you believe the forced goals are theorems, you should follow the usual methodology for ``fixing'' failed ACL2 proofs, e.g., the identification of key lemmas and their timely and proper use as rules. See failure and see proof-tree.

The rules designed for the goals of forcing rounds are often just what is needed to prove the forced hypothesis at the time it is forced. Thus, you may find that when the system has been ``taught'' how to prove the goals of the forcing round no forcing round is needed. This is intended as a feature to help structure the discovery of the necessary rules.

If a hint must be provided to prove a goal in a forcing round, the appropriate ``goal specifier'' (the string used to identify the goal to which the hint is to be applied) is just the text printed on the line above the formula, e.g., "[1]Subgoal *1/3''". See goal-spec.

If you solve a forcing problem by giving explicit hints for the goals of forcing rounds, you might consider whether you could avoid forcing the assumption in the first place by giving those hints in the appropriate places of the main proof. This is one reason that we print out the origins of each forced assumption. An argument against this style, however, is that an assumption might be forced in hundreds of places in the main goal and proved only once in the forcing round, so that by delaying the proof you actually save time.

We now turn to the possibility that some goal in the forcing round is not a theorem.

There are two possibilities to consider. The first is that the original theorem has insufficient hypotheses to insure that all the forced hypotheses are in fact always true. The ``fix'' in this case is to amend the original conjecture so that it has adequate hypotheses.

A more difficult situation can arise and that is when the conjecture has sufficient hypotheses but they are not present in the forcing round goal. This can be caused by what we call ``premature'' forcing.

Because ACL2 rewrites from the inside out, it is possible that it will force hypotheses while the context is insufficient to establish them. Consider trying to prove (p x (foo x)). We first rewrite the formula in an empty context, i.e., assuming nothing. Thus, we rewrite (foo x) in an empty context. If rewriting (foo x) forces anything, that forced assumption will have to be proved in an empty context. This will likely be impossible.

On the other hand, suppose we did not attack (foo x) until after we had expanded p. We might find that the value of its second argument, (foo x), is relevant only in some cases and in those cases we might be able to establish the hypotheses forced by (foo x). Our premature forcing is thus seen to be a consequence of our ``over eager'' rewriting.

Here, just for concreteness, is an example you can try. In this example, (foo x) rewrites to x but has a forced hypothesis of (rationalp x). P does a case split on that very hypothesis and uses its second argument only when x is known to be rational. Thus, the hypothesis for the (foo x) rewrite is satisfied. On the false branch of its case split, p simplies to (p1 x) which can be proved under the assumption that x is not rational.

(defun p1 (x) (not (rationalp x)))
(defun p (x y)(if (rationalp x) (equal x y) (p1 x)))
(defun foo (x) x)
(defthm foo-rewrite (implies (force (rationalp x)) (equal (foo x) x)))
(in-theory (disable foo))
The attempt then to do (thm (p x (foo x))) forces the unprovable goal (rationalp x).

Since all ``formulas'' are presented to the theorem prover as single terms with no hypotheses (e.g., since implies is a function), this problem would occur routinely were it not for the fact that the theorem prover expands certain ``simple'' definitions immediately without doing anything that can cause a hypothesis to be forced. See simple. This does not solve the problem, since it is possible to hide the propositional structure arbitrarily deeply. For example, one could define p, above, recursively so that the test that x is rational and the subsequent first ``real'' use of y occurred arbitrarily deeply.

Therefore, the problem remains: what do you do if an impossible goal is forced and yet you know that the original conjecture was adequately protected by hypotheses?

One alternative is to disable forcing entirely. See disable-forcing. Another is to disable the rule that caused the force.

A third alternative is to prove that the negation of the main goal implies the forced hypothesis. For example,

(defthm not-p-implies-rationalp
  (implies (not (p x (foo x))) (rationalp x))
  :rule-classes nil)
Observe that we make no rules from this formula. Instead, we merely :use it in the subgoal where we must establish (rationalp x).
(thm (p x (foo x))
     :hints (("Goal" :use not-p-implies-rationalp)))
When we said, above, that (p x (foo x)) is first rewritten in an empty context we were misrepresenting the situation slightly. When we rewrite a literal we know what literal we are rewriting and we implicitly assume it false. This assumption is ``dangerous'' in that it can lead us to simplify our goal to nil and give up -- we have even seen people make the mistake of assuming the negation of what they wished to prove and then via a very complicated series of transformations convince themselves that the formula is false. Because of this ``tail biting'' we make very weak use of the negation of our goal. But the use we make of it is sufficient to establish the forced hypothesis above.

A fourth alternative is to weaken your desired theorem so as to make explicit the required hypotheses, e.g., to prove

(defthm rationalp-implies-main
  (implies (rationalp x) (p x (foo x)))
  :rule-classes nil)
This of course is unsatisfying because it is not what you originally intended. But all is not lost. You can now prove your main theorem from this one, letting the implies here provide the necessary case split.
(thm (p x (foo x))
     :hints (("Goal" :use rationalp-implies-main)))


how to deal with a proof failure

When ACL2 gives up it does not mean that the submitted conjecture is invalid, even if the last formula ACL2 printed in its proof attempt is manifestly false. Since ACL2 sometimes generalizes the goal being proved, it is possible it adopted an invalid subgoal as a legitimate (but doomed) strategy for proving a valid goal. Nevertheless, conjectures submitted to ACL2 are often invalid and the proof attempt often leads the careful reader to the realization that a hypothesis has been omitted or that some special case has been forgotten. It is good practice to ask yourself, when you see a proof attempt fail, whether the conjecture submitted is actually a theorem.

If you think the conjecture is a theorem, then you must figure out from ACL2's output what you know that ACL2 doesn't about the functions in the conjecture and how to impart that knowledge to ACL2 in the form of rules. Books could be written about this, but they haven't been yet. However, see proof-tree for a utility that may be very helpful in locating parts of the failed proof that are of particular interest. See also the discussion of how to read Nqthm proofs and how to use Nqthm rules in ``A Computational Logic Handbook'' by Boyer and Moore (Academic Press, 1988).

If the failure occurred during a forcing round, see failed-forcing.


find the rules named rune

General Form:
(find-rules-of-rune rune wrld)

This function finds all the rules in wrld with :rune rune. It returns a list of rules in their internal form (generally as described by the corresponding defrec). Decyphering these rules is difficult since one cannot always look at a rule object and decide what kind of record it is without exploiting many system invariants (e.g., that :rewrite runes only name rewrite-rules).

At the moment this function returns nil if the rune in question is a :refinement rune, because there is no object representing :refinement rules. (:refinement rules cause changes in the 'coarsenings properties.) In addition, if the rune is an :equivalence rune, then congruence rules with that rune will be returned -- because :equivalence lemmas generate some congruence rules -- but the fact that a certain function is now known to be an equivalence relation is not represented by any rule object and so no such rule is returned. (The fact that the function is an equivalence relation is encoded entirely in its presence as a 'coarsening of equal.)


identity function used to force a case split

When a hypothesis of a conditional rule has the form (force hyp) it is logically equivalent to hyp but has a pragmatic effect. In particular, when the rule is considered, the needed instance of the hypothesis, hyp', is assumed and a special case is generated, requiring the system to prove that hyp' is true in the current context. The proofs of all such ``forced assumptions'' are delayed until the successful completion of the main goal. See forcing-round.

Forcing should only be used on hypotheses that are always expected to be true, such as the guards of functions. All the power of the theorem prover is brought to bear on a forced hypothesis and no backtracking is possible. If the :executable-counterpart of the function force is disabled, then no hypothesis is forced. See enable-forcing and see disable-forcing.

It sometimes happens that a conditional rule is not applied because some hypothesis, hyp, could not be relieved, even though the required instance of hyp, hyp', can be shown true in the context. This happens when insufficient resources are brought to bear on hyp' at the time we try to relieve it. A sometimes desirable alternative behavior is for the system to assume hyp', apply the rule, and to generate explicitly a special case to show that hyp' is true in the context. This is called ``forcing'' hyp. It can be arranged by restating the rule so that the offending hypothesis, hyp, is embedded in a call of force, as in (force hyp). By using the :corollary field of the rule-classes entry, a hypothesis can be forced without changing the statement of the theorem from which the rule is derived.

Technically, force is just a function of one argument that returns that argument. It is generally enabled and hence evaporates during simplification. But its presence among the hypotheses of a conditional rule causes case splitting to occur if the hypothesis cannot be conventionally relieved.

Since a forced hypothesis must be provable whenever the rule is otherwise applicable, forcing should be used only on hypotheses that are expected always to be true. A common situation is when the hypothesis is in fact a guard (or part of a guard) of some function involved in the pattern that triggers the rule. Intuitively, if that pattern term occurs in the current conjecture, then its guards had better be true, since otherwise nothing is known about the term.

A particularly common situation in which some hypotheses should be forced is in ``most general'' type-prescription lemmas. If a single lemma describes the ``expected'' type of a function, for all ``expected'' arguments, then it is probably a good idea to force the hypotheses of the lemma. Thus, every time a term involving the function arises, the term will be given the expected type and its arguments will be required to be of the expected type. In applying this advice it might be wise to avoid forcing those hypotheses that are in fact just type predicates on the arguments, since the routine that applies type-prescription lemmas has fairly thorough knowledge of the types of all terms.

Force can have the additional benefit of causing the ACL2 typing mechanism to interact with the ACL2 rewriter to establish the hypotheses of type-prescription rules. To understand this remark, think of the ACL2 type reasoning system as a rather primitive rule-based theorem prover for questions about Common Lisp types, e.g., ``does this expression produce a consp?'' ``does this expression produce some kind of ACL2 number, e.g., an integerp, a rationalp, or a complex-rationalp?'' etc. It is driven by type-prescription rules. To relieve the hypotheses of such rules, the type system recursively invokes itself. This can be done for any hypothesis, whether it is ``type-like'' or not, since any proposition, p, can be phrased as the type-like question ``does p produce an object of type nil?'' However, as you might expect, the type system is not very good at establishing hypotheses that are not type-like, unless they happen to be assumed explicitly in the context in which the question is posed, e.g., ``If p produces a consp then does p produce nil?'' If type reasoning alone is insufficient to prove some instance of a hypothesis, then the instance will not be proved by the type system and a type-prescription rule with that hypothesis will be inapplicable in that case. But by embedding such hypotheses in force expressions you can effectively cause the type system to ``punt'' them to the rest of the theorem prover. Of course, as already noted, this should only be done on hypotheses that are ``always true.'' In particular, if rewriting is required to establish some hypothesis of a type-prescription rule, then the rule will be found inapplicable because the hypothesis will not be established by type reasoning alone.

The ACL2 rewriter uses the type reasoning system as a subsystem. It is therefore possible that the type system will force a hypothesis that the rewriter could establish. Before a forced hypothesis is reported out of the rewriter, we try to establish it by rewriting.

This makes the following surprising behavior possible: A type-prescription rule fails to apply because some true hypothesis is not being relieved. The user changes the rule so as to force the hypothesis. The system then applies the rule but reports no forcing. How can this happen? The type system ``punted'' the forced hypothesis to the rewriter, which established it.

Finally, we should mention that the rewriter is never willing to force when there is an if term present in the goal being simplified. Since and and or terms are merely abbreviations for if terms, they also prevent forcing.


a section of a proof dealing with forced assumptions

If ACL2 ``forces'' some hypothesis of some rule to be true, it is obliged later to prove the hypothesis. See force. ACL2 delays the consideration of forced hypotheses until the main goal has been proved. It then undertakes a new round of proofs in which the main goal is essentially the conjunction of all hypotheses forced in the preceding proof. Call this round of proofs the ``Forcing Round.'' Additional hypotheses may be forced by the proofs in the Forcing Round. The attempt to prove these hypotheses is delayed until the Forcing Round has been successfully completed. Then a new Forcing Round is undertaken to prove the recently forced hypotheses and this continues until no hypotheses are forced. Thus, there is a succession of Forcing Rounds.

The Forcing Rounds are enumerated starting from 1. The Goals and Subgoals of a Forcing Round are printed with the round's number displayed in square brackets. Thus, "[1]Subgoal 1.3" means that the goal in question is Subgoal 1.3 of the 1st forcing round. To supply a hint for use in the proof of that subgoal, you should use the goal specifier "[1]Subgoal 1.3". See goal-spec.

When a round is successfully completed -- and for these purposes you may think of the proof of the main goal as being the 0th forcing round -- the system collects all of the assumptions forced by the just-completed round. Here, an assumption should be thought of as an implication, (implies context hyp), where context describes the context in which hyp was assumed true. Before undertaking the proofs of these assumptions, we try to ``clean them up'' in an effort to reduce the amount of work required. This is often possible because the forced assumptions are generated by the same rule being applied repeatedly in a given context.

For example, suppose the main goal is about some term (pred (xtrans i) i) and that some rule rewriting pred contains a forced hypothesis that the first argument is a good-inputp. Suppose that during the proof of Subgoal 14 of the main goal, (good-inputp (xtrans i)) is forced in a context in which i is an integerp and x is a consp. (Note that x is irrelevant.) Suppose finally that during the proof of Subgoal 28, (good-inputp (xtrans i)) is forced ``again,'' but this time in a context in which i is a rationalp and x is a symbolp. Since the forced hypothesis does not mention x, we deem the contextual information about x to be irrelevant and discard it from both contexts. We are then left with two forced assumptions: (implies (integerp i) (good-inputp (xtrans i))) from Subgoal 14, and (implies (rationalp i) (good-inputp (xtrans i))) from Subgoal 28. Note that if we can prove the assumption required by Subgoal 28 we can easily get that for Subgoal 14, since the context of Subgoal 28 is the more general. Thus, in the next forcing round we will attempt to prove just

(implies (rationalp i) (good-inputp (xtrans i)))
and ``blame'' both Subgoal 14 and Subgoal 28 of the previous round for causing us to prove this.

By delaying and collecting the forced assumptions until the completion of the ``main goal'' we gain two advantages. First, the user gets confirmation that the ``gist'' of the proof is complete and that all that remains are ``technical details.'' Second, by delaying the proofs of the forced assumptions ACL2 can undertake the proof of each assumption only once, no matter how many times it was forced in the main goal.

In order to indicate which proof steps of the previous round were responsible for which forced assumptions, we print a sentence explaining the origins of each newly forced goal. For example,

[1]Subgoal 1, below, will focus on
which was forced in
 Subgoal 14, above,
  by applying (:REWRITE PRED-CRUNCHER) to
 Subgoal 28, above,
  by applying (:REWRITE PRED-CRUNCHER) to

In this entry, ``[1]Subgoal 1'' is the name of a goal which will be proved in the next forcing round. On the next line we display the forced hypothesis, call it x, which is (good-inputp (xtrans i)) in this example. This term will be the conclusion of the new subgoal. Since the new subgoal will be printed in its entirety when its proof is undertaken, we do not here exhibit the context in which x was forced. The sentence then lists (possibly a succession of) a goal name from the just-completed round and some step in the proof of that goal that forced x. In the example above we see that Subgoals 14 and 28 of the just-completed proof forced (good-inputp (xtrans i)) by applying (:rewrite pred-cruncher) to the term (pred (xtrans i) i).

If one were to inspect the theorem prover's description of the proof steps applied to Subgoals 14 and 28 one would find the word ``forced'' (or sometimes ``forcibly'') occurring in the commentary. Whenever you see that word in the output, you know you will get a subsequent forcing round to deal with the hypotheses forced. Similarly, if at the beginning of a forcing round a rune is blamed for causing a force in some subgoal, inspection of the commentary for that subgoal will reveal the word ``forced'' after the rule name blamed.

Most forced hypotheses come from within the prover's simplifier. When the simplifier encounters a hypothesis of the form (force hyp) it first attempts to establish it by rewriting hyp to, say, hyp'. If the truth or falsity of hyp' is known, forcing is not required. Otherwise, the simplifier actually forces hyp'. That is, the x mentioned above is hyp', not hyp, when the forced subgoal was generated by the simplifier.

Once the system has printed out the origins of the newly forced goals, it proceeds to the next forcing round, where those goals are individually displayed and attacked.

At the beginning of a forcing round, the enabled structure defaults to the global enabled structure. For example, suppose some rune, rune, is globally enabled. Suppose in some event you disable the rune at "Goal" and successfully prove the goal but force "[1]Goal". Then during the proof of "[1]Goal", rune is enabled ``again.'' The right way to think about this is that the rune is ``still'' enabled. That is, it is enabled globally and each forcing round resumes with the global enabled structure.


potential soundness issues related to ACL2 predicates

The discussion below outlines a potential source of unsoundness in ACL2. Although to our knowledge there is no existing Lisp implementation in which this problem can arise, nevertheless we feel compelled to explain this situation.

Consider for example the question: What is the value of (equal 3 3)? According to the ACL2 axioms, it's t. And as far as we know, based on considerable testing, the value is t in every Common Lisp implementation. However, according the Common Lisp draft proposed ANSI standard, any non-nil value could result. Thus for example, (equal 3 3) is allowed by the standard to be 17.

The Common Lisp standard specifies (or soon will) that a number of Common Lisp functions that one might expect to return Boolean values may, in fact, return arbitrary values. Examples of such functions are equal, rationalp, and <; a much more complete list is given below. Indeed, we dare to say that every Common Lisp function that one may believe returns only Booleans is, nevertheless, not specified by the standard to have that property, with the exceptions of the functions not and null. The standard refers to such arbitrary values as ``generalized Booleans,'' but in fact this terminology makes no restriction on those values. Rather, it merely specifies that they are to be viewed, in an informal sense, as being either nil or not nil.

This situation is problematic for ACL2, which axiomatizes these functions to return Booleans. The problem is that because (for efficiency and simplicity) ACL2 makes very direct use of compiled Common Lisp functions to support the execution of its functions, there is in principle a potential for unsoundness due to these ``generalized Booleans.'' For example, ACL2's equal function is defined to be Common Lisp's equal. Hence if in Common Lisp the form (equal 3 3) were to evaluate to 17, then in ACL2 we could prove (using the :executable-counterpart of equal) (equal (equal 3 3) 17). However, ACL2 can also prove (equal (equal x x) t), and these two terms together are contradictory, since they imply (instantiating x in the second term by 3) that (equal 3 3) is both equal to 17 and to t.

To make matters worse, the standard allows (equal 3 3) to evaluate to different non-nil values every time. That is: equal need not even be a function!

Fortunately, no existing Lisp implementation takes advantage of the flexibility to have (most of) its predicates return generalized Booleans, as far as we know. We may implement appropriate safeguards in future releases of ACL2, in analogy to (indeed, probably extending) the existing safeguards by way of guards (see guard). For now, we'll sleep a bit better knowing that we have been up-front about this potential problem.

The following list of functions contains all those that are defined in Common Lisp to return generalized Booleans but are assumed by ACL2 to return Booleans. In addition, the functions acl2-numberp and complex-rationalp are directly defined in terms of respective Common Lisp functions numberp and complexp.