INTRODUCTION

introduction to ACL2
```Major Section:  ACL2-TUTORIAL
```

This section contains introductory material on ACL2 including what ACL2 is, how to get started using the system, how to read the output, and other introductory topics. It was written almost entirely by Bill Young of Computational Logic, Inc.

You might also find CLI Technical Report 101 helpful, especially if you are familiar with Nqthm. If you would like more familiarity with Nqthm, we suggest CLI Technical Report 100.

OVERVIEW

ACL2 is an automated reasoning system developed (for the first 9 years) at Computational Logic, Inc. and (from January, 1997) at the University of Texas at Austin. It is the successor to the Nqthm (or Boyer-Moore) logic and proof system and its Pc-Nqthm interactive enhancement. The acronym ACL2 actually stands for ``A Computational Logic for Applicative Common Lisp''. This title suggests several distinct but related aspects of ACL2.

We assume that readers of the ACL2 documentation have at least a very slight familiarity with some Lisp-like language. We will address the issue of prerequisites further, in ``ABOUT THIS TUTORIAL'' below.

As a logic, ACL2 is a formal system with rigorously defined syntax and semantics. In mathematical parlance, the ACL2 logic is a first-order logic of total recursive functions providing mathematical induction on the ordinals up to epsilon-0 and two extension principles: one for recursive definition and one for constrained introduction of new function symbols, here called encapsulation. The syntax of ACL2 is that of Common Lisp; ACL2 specifications are ``also'' Common Lisp programs in a way that we will make clear later. In less formal language, the ACL2 logic is an integrated collection of rules for defining (or axiomatizing) recursive functions, stating properties of those functions, and rigorously establishing those properties. Each of these activities is mechanically supported.

As a specification language, ACL2 supports modeling of systems of various kinds. An ACL2 function can equally be used to express purely formal relationships among mathematical entities, to describe algorithms, or to capture the intended behavior of digital systems. For digital systems, an ACL2 specification is a mathematical model that is intended to formalize relevant aspects of system behavior. Just as physics allows us to model the behavior of continuous physical systems, ACL2 allows us to model digital systems, including many with physical realizations such as computer hardware. As early as the 1930's Church, Kleene, Turing and others established that recursive functions provide an expressive formalism for modeling digital computation. Digital computation should be understood in a broad sense, covering a wide variety of activities including almost any systematic or algorithmic activity, or activity that can be reasonably approximated in that way. This ranges from the behavior of a digital circuit to the behavior of a programming language compiler to the behavior of a controller for a physical system (as long as the system can be adequately modeled discretely). All of these have been modeled using ACL2 or its predecessor Nqthm.

ACL2 is a computational logic in at least three distinct senses. First, the theory of recursive functions is often considered the mathematics of computation. Church conjectured that any ``effective computation'' can be modeled as a recursive function. Thus, ACL2 provides an expressive language for modeling digital systems. Second, many ACL2 specifications are executable. In fact, recursive functions written in ACL2 are Common Lisp functions that can be submitted to any compliant Common Lisp compiler and executed (in an environment where suitable ACL2-specific macros and functions are defined). Third, ACL2 is computational in the sense that calculation is heavily integrated into the reasoning process. Thus, an expression with explicit constant values but no free variables can be simplified by calculation rather than by complex logical manipulations.

ACL2 is a powerful, automated theorem prover or proof checker. This means that a competent user can utilize the ACL2 system to discover proofs of theorems stated in the ACL2 logic or to check previously discovered proofs. The basic deductive steps in an ACL2-checked proof are often quite large, due to the sophisticated combination of decision procedures, conditional rewriting, mathematical and structural induction, propositional simplification, and complex heuristics to orchestrate the interactions of these capabilities. Unlike some automated proof systems, ACL2 does not produce a formal proof. However, we believe that if ACL2 certifies the ``theoremhood'' of a given conjecture, then such a formal proof exists and, therefore, the theorem is valid. The ultimate result of an ACL2 proof session is a collection of ``events,'' possibly grouped into ``books,'' that can be replayed in ACL2. Therefore, a proof can be independently validated by any ACL2 user.

ACL2 may be used in purely automated mode in the shallow sense that conjectures are submitted to the prover and the user does not interact with the proof attempt (except possibly to stop it) until the proof succeeds or fails. However, any non-trivial proof attempt is actually interactive, since successful proof ``events'' influence the subsequent behavior of the prover. For example, proving a lemma may introduce a rule that subsequently is used automatically by the prover. Thus, any realistic proof attempt, even in ``automatic'' mode, is really an interactive dialogue with the prover to craft a sequence of events building an appropriate theory and proof rules leading up to the proof of the desired result. Also, ACL2 supports annotating a theorem with ``hints'' designed to guide the proof attempt. By supplying appropriate hints, the user can suggest proof strategies that the prover would not discover automatically. There is a ``proof-tree'' facility (see proof-tree) that allows the user to monitor the progress and structure of a proof attempt in real-time. Exploring failed proof attempts is actually where heavy-duty ACL2 users spend most of their time.

ACL2 can also be used in a more explicitly interactive mode. The ``proof-checker'' subsystem of ACL2 allows exploration of a proof on a fairly low level including expanding calls of selected function symbols, invoking specific rewrite rules, and selectively navigating around the proof. This facility can be used to gain sufficient insight into the proof to construct an automatic version, or to generate a detailed interactive-style proof that can be replayed in batch mode.

Because ACL2 is all of these things -- computational logic, specification language, programming system, and theorem prover -- it is more than the sum of its parts. The careful integration of these diverse aspects has produced a versatile automated reasoning system suitable for building highly reliable digital systems. In the remainder of this tutorial, we will illustrate some simple uses of this automated reasoning system.

ACL2 is a complex system with a vast array of features, bells and whistles. However, it is possible to perform productive work with the system using only a small portion of the available functionality. The goals of this tutorial are to:

familiarize the new user with the most basic features of and modes of interaction with ACL2;

familiarize her with the form of output of the system; and

work through a graduated series of examples.

The more knowledge the user brings to this system, the easier it will be to become proficient. On one extreme: the ideal user of ACL2 is an expert Common Lisp programmer, has deep understanding of automated reasoning, and is intimately familiar with the earlier Nqthm system. Such ideal users are unlikely to need this tutorial. However, without some background knowledge, the beginning user is likely to become extremely confused and frustrated by this system. We suggest that a new user of ACL2 should:

(a) have a little familiarity with Lisp, including basic Lisp programming and prefix notation (a Lisp reference manual such as Guy Steele's ``Common Lisp: The Language'' is also helpful);

(b) be convinced of the utility of formal modeling; and

(c) be willing to gain familiarity with basic automated theorem proving topics such as rewriting and algebraic simplification.

We will not assume any deep familiarity with Nqthm (the so-called ``Boyer-Moore Theorem Prover''), though the book ``A Computational Logic Handbook'' by Boyer and Moore (Academic Press, 1988) is an extremely useful reference for many of the topics required to become a competent ACL2 user. We'll refer to it as ACLH below.

As we said in the introduction, ACL2 has various facets. For example, it can be used as a Common Lisp programming system to construct application programs. In fact, the ACL2 system itself is a large Common Lisp program constructed almost entirely within ACL2. Another use of ACL2 is as a specification and modeling tool. That is the aspect we will concentrate on in the remainder of this tutorial.

GETTING STARTED

This section is an abridged version of what's available elsewhere; feel free to see startup for more details.

How you start ACL2 will be system dependent, but you'll probably type something like ``acl2'' at your operating system prompt. Consult your system administrator for details.

When you start up ACL2, you'll probably find yourself inside the ACL2 command loop, as indicated by the following prompt.

```

ACL2 !>

```
If not, you should type `(LP)`. See lp, which has a lot more information about the ACL2 command loop.

There are two ``modes'' for using ACL2, `:``logic` and `:``program`. When you begin ACL2, you will ordinarily be in the `:``logic` mode. This means that any new function defined is not only executable but also is axiomatically defined in the ACL2 logic. (See defun-mode and see default-defun-mode.) Roughly speaking, `:``program` mode is available for using ACL2 as a programming language without some of the logical burdens necessary for formal reasoning. In this tutorial we will assume that we always remain in `:``logic` mode and that our purpose is to write formal models of digital systems and to reason about them.

Now, within the ACL2 command loop you can carry out various kinds of activities, including the folllowing. (We'll see examples later of many of these.)

define new functions (see defun);

execute functions on concrete data;

pose and attempt to prove conjectures about previously defined functions (see defthm);

query the ACL2 ``world'' or database (e.g., see pe); and

numerous other things.

In addition, there is extensive on-line documentation, of which this tutorial introduction is a part.

INTERACTING WITH ACL2

The standard means of interacting with ACL2 is to submit a sequence of forms for processing by the ACL2 system. These forms are checked for syntactic and semantic acceptability and appropriately processed by the system. These forms can be typed directly at the ACL2 prompt. However, most successful ACL2 users prefer to do their work using the Emacs text editor, maintaining an Emacs ``working'' buffer in which forms are edited. Those forms are then copied to the ACL2 interaction buffer, which is often the `"*shell*"` buffer.

In some cases, processing succeeds and makes some change to the ACL2 ``logical world,'' which affects the processing of subsequent forms. How can this processing fail? For example, a proposed theorem will be rejected unless all function symbols mentioned have been previously defined. Also the ability of ACL2 to discover the proof of a theorem may depend on the user previously having proved other theorems. Thus, the order in which forms are submitted to ACL2 is quite important. Maintaining forms in an appropriate order in your working buffer will be helpful for re-playing the proof later.

One of the most common events in constructing a model is introducing new functions. New functions are usually introduced using the `defun` form; we'll encounter some exceptions later. Proposed function definitions are checked to make sure that they are syntactically and semantically acceptable (e.g., that all mentioned functions have been previously defined) and, for recursive functions, that their recursive calls terminate. A recursive function definition is guaranteed to terminate if there is some some ``measure'' of the arguments and a ``well-founded'' ordering such that the arguments to the function get smaller in each recursive call. See well-founded-relation.

For example, suppose that we need a function that will append two lists together. (We already have one in the ACL2 `append` function; but suppose perversely that we decide to define our own.) Suppose we submit the following definition (you should do so as well and study the system output):

```

(defun my-app (x y)
(if (atom x)
y
(cons (car x) (my-app x y))))

```
The system responds with the following message:
```

ACL2 Error in ( DEFUN MY-APP ...):  No :MEASURE was supplied with
the definition of MY-APP.  Our heuristics for guessing one have not
made any suggestions.  No argument of the function is tested along
every branch and occurs as a proper subterm at the same argument
position in every recursive call.  You must specify a :MEASURE.  See
:DOC defun.

```
This means that the system could not find an expression involving the formal parameters `x` and `y` that decreases under some well-founded order in every recursive call (there is only one such call). It should be clear that there is no such measure in this case because the only recursive call doesn't change the arguments at all. The definition is obviously flawed; if it were accepted and executed it would loop forever. Notice that a definition that is rejected is not stored in the system database; there is no need to take any action to have it ``thrown away.'' Let's try again with the correct definition. The interaction now looks like (we're also putting in the ACL2 prompt; you don't type that):
```

ACL2 !>(defun my-app (x y)
(if (atom x)
y
(cons (car x) (my-app (cdr x) y))))

The admission of MY-APP is trivial, using the relation O<
(which is known to be well-founded on the domain recognized by
O-P) and the measure (ACL2-COUNT X).  We observe that the
type of MY-APP is described by the theorem
(OR (CONSP (MY-APP X Y)) (EQUAL (MY-APP X Y) Y)).
We used primitive type reasoning.

Summary
Form:  ( DEFUN MY-APP ...)
Rules: ((:FAKE-RUNE-FOR-TYPE-SET NIL))
Warnings:  None
Time:  0.07 seconds (prove: 0.00, print: 0.00, other: 0.07)
MY-APP

```
Notice that this time the function definition was accepted. We didn't have to supply a measure explicitly; the system inferred one from the form of the definition. On complex functions it may be necessary to supply a measure explicitly. (See xargs.)

The system output provides several pieces of information.

The revised definition is acceptable. The system realized that there is a particular measure (namely, `(acl2-count x)`) and a well-founded relation (`o<`) under which the arguments of `my-app` get smaller in recursion. Actually, the theorem prover proved several theorems to admit `my-app`. The main one was that when `(atom x)` is false the `acl2-count` of `(cdr x)` is less than (in the `o<` sense) the `acl2-count` of `x`. `Acl2-count` is the most commonly used measure of the ``size`` of an ACL2 object. `o<` is the ordering relation on ordinals less than epsilon-0. On the natural numbers it is just ordinary ``<''.

The observation printed about ``the type of MY-APP'' means that calls of the function `my-app` will always return a value that is either a cons pair or is equal to the second parameter.

The summary provides information about which previously introduced definitions and lemmas were used in this proof, about some notable things to watch out for (the Warnings), and about how long this event took to process.

Usually, it's not important to read this information. However, it is a good habit to scan it briefly to see if the type information is surprising to you or if there are Warnings. We'll see an example of them later.

After a function is accepted, it is stored in the database and available for use in other function definitions or lemmas. To see the definition of any function use the `:``pe` command (see pe). For example,

```

ACL2 !>:pe my-app
L       73:x(DEFUN MY-APP (X Y)
(IF (ATOM X)
Y (CONS (CAR X) (MY-APP (CDR X) Y))))

```
This displays the definition along with some other relevant information. In this case, we know that this definition was processed in `:``logic` mode (the ```L`'') and was the 73rd command processed in the current session.

We can also try out our newly defined function on some sample data. To do that, just submit a form to be evaluated to ACL2. For example,

```

ACL2 !>(my-app '(0 1 2) '(3 4 5))
(0 1 2 3 4 5)
ACL2 !>(my-app nil nil)
NIL
ACL2 !>

```

Now suppose we want to prove something about the function just introduced. We conjecture, for example, that the length of the append of two lists is the sum of their lengths. We can formulate this conjecture in the form of the following ACL2 `defthm` form.

```

(defthm my-app-length
(equal (len (my-app x y))
(+ (len x) (len y))))

```
First of all, how did we know about the functions `len` and `+`, etc.? The answer to that is somewhat unsatisfying -- we know them from our past experience in using Common Lisp and ACL2. It's hard to know that a function such as `len` exists without first knowing some Common Lisp. If we'd guessed that the appropriate function was called `length` (say, from our knowledge of Lisp) and tried `:pe length`, we would have seen that `length` is defined in terms of `len`, and we could have explored from there. Luckily, you can write a lot of ACL2 functions without knowing too many of the primitive functions.

Secondly, why don't we need some ``type'' hypotheses? Does it make sense to append things that are not lists? Well, yes. ACL2 and Lisp are both quite weakly typed. For example, inspection of the definition of `my-app` shows that if `x` is not a cons pair, then `(my-app x y)` always returns `y`, no matter what `y` is.

Thirdly, would it matter if we rewrote the lemma with the equality reversed, as follows?

```

(defthm my-app-length2
(equal (+ (len x) (len y))
(len (my-app x y)))).

```
The two are logically equivalent, but...yes, it would make a big difference. Recall our remark that a lemma is not only a ``fact'' to be proved; it also is used by the system to prove other later lemmas. The current lemma would be stored as a rewrite rule. (See rule-classes.) For a rewrite rule, a conclusion of the form `(EQUAL LHS RHS)` means to replace instances of the `LHS` by the appropriate instance of the `RHS`. Presumably, it's better to rewrite `(len (my-app x y))` to `(+ (len x) (len y))` than the other way around. The reason is that the system ``knows'' more about `+` than it does about the new function symbol `my-app`.

So let's see if we can prove this lemma. Submitting our preferred `defthm` to ACL2 (do it!), we get the following interaction:

```          --------------------------------------------------
ACL2 !>(defthm my-app-length
(equal (len (my-app x y))
(+ (len x) (len y))))

Name the formula above *1.

Perhaps we can prove *1 by induction.  Three induction schemes are
suggested by this conjecture.  These merge into two derived
induction schemes.  However, one of these is flawed and so we are
left with one viable candidate.

We will induct according to a scheme suggested by (LEN X), but
modified to accommodate (MY-APP X Y).  If we let (:P X Y) denote *1
above then the induction scheme we'll use is
(AND (IMPLIES (NOT (CONSP X)) (:P X Y))
(IMPLIES (AND (CONSP X) (:P (CDR X) Y))
(:P X Y))).
This induction is justified by the same argument used to admit LEN,
namely, the measure (ACL2-COUNT X) is decreasing according to the
relation O< (which is known to be well-founded on the domain
recognized by O-P).  When applied to the goal at hand the
above induction scheme produces the following two nontautological
subgoals.

Subgoal *1/2
(IMPLIES (NOT (CONSP X))
(EQUAL (LEN (MY-APP X Y))
(+ (LEN X) (LEN Y)))).

But simplification reduces this to T, using the :definitions of FIX,
LEN and MY-APP, the :type-prescription rule LEN, the :rewrite rule
UNICITY-OF-0 and primitive type reasoning.

Subgoal *1/1
(IMPLIES (AND (CONSP X)
(EQUAL (LEN (MY-APP (CDR X) Y))
(+ (LEN (CDR X)) (LEN Y))))
(EQUAL (LEN (MY-APP X Y))
(+ (LEN X) (LEN Y)))).

This simplifies, using the :definitions of LEN and MY-APP, primitive
type reasoning and the :rewrite rules COMMUTATIVITY-OF-+ and
CDR-CONS, to

Subgoal *1/1'
(IMPLIES (AND (CONSP X)
(EQUAL (LEN (MY-APP (CDR X) Y))
(+ (LEN Y) (LEN (CDR X)))))
(EQUAL (+ 1 (LEN (MY-APP (CDR X) Y)))
(+ (LEN Y) 1 (LEN (CDR X))))).

But simplification reduces this to T, using linear arithmetic,
primitive type reasoning and the :type-prescription rule LEN.

That completes the proof of *1.

Q.E.D.

Summary
Form:  ( DEFTHM MY-APP-LENGTH ...)
Rules: ((:REWRITE UNICITY-OF-0)
(:DEFINITION FIX)
(:REWRITE COMMUTATIVITY-OF-+)
(:DEFINITION LEN)
(:REWRITE CDR-CONS)
(:DEFINITION MY-APP)
(:TYPE-PRESCRIPTION LEN)
(:FAKE-RUNE-FOR-TYPE-SET NIL)
(:FAKE-RUNE-FOR-LINEAR NIL))
Warnings:  None
Time:  0.30 seconds (prove: 0.13, print: 0.05, other: 0.12)
MY-APP-LENGTH
--------------------------------------------------
```

Wow, it worked! In brief, the system first tried to rewrite and simplify as much as possible. Nothing changed; we know that because it said ``Name the formula above *1.'' Whenever the system decides to name a formula in this way, we know that it has run out of techniques to use other than proof by induction.

The induction performed by ACL2 is structural or ``Noetherian'' induction. You don't need to know much about that except that it is induction based on the structure of some object. The heuristics infer the structure of the object from the way the object is recursively decomposed by the functions used in the conjecture. The heuristics of ACL2 are reasonably good at selecting an induction scheme in simple cases. It is possible to override the heuristic choice by providing an `:induction` hint (see hints). In the case of the theorem above, the system inducts on the structure of `x` as suggested by the decomposition of `x` in both `(my-app x y)` and `(len x)`. In the base case, we assume that `x` is not a `consp`. In the inductive case, we assume that it is a `consp` and assume that the conjecture holds for `(cdr x)`.

There is a close connection between the analysis that goes on when a function like `my-app` is accepted and when we try to prove something inductively about it. That connection is spelled out well in Boyer and Moore's book ``A Computational Logic,'' if you'd like to look it up. But it's pretty intuitive. We accepted `my-app` because the ``size'' of the first argument `x` decreases in the recursive call. That tells us that when we need to prove something inductively about `my-app`, it's a good idea to try an induction on the size of the first argument. Of course, when you have a theorem involving several functions, it may be necessary to concoct a more complicated induction schema, taking several of them into account. That's what's meant by ``merging'' the induction schemas.

The proof involves two cases: the base case, and the inductive case. You'll notice that the subgoal numbers go down rather than up, so you always know how many subgoals are left to process. The base case (`Subgoal *1/2`) is handled by opening up the function definitions, simplifying, doing a little rewriting, and performing some reasoning based on the types of the arguments. You'll often encounter references to system defined lemmas (like `unicity-of-0`). You can always look at those with `:``pe`; but, in general, assume that there's a lot of simplification power under the hood that's not too important to understand fully.

The inductive case (`Subgoal *1/1`) is also dispatched pretty easily. Here we assume the conjecture true for the `cdr` of the list and try to prove it for the entire list. Notice that the prover does some simplification and then prints out an updated version of the goal (`Subgoal *1/1'`). Examining these gives you a pretty good idea of what's going on in the proof.

Sometimes one goal is split into a number of subgoals, as happened with the induction above. Sometimes after some initial processing the prover decides it needs to prove a subgoal by induction; this subgoal is given a name and pushed onto a stack of goals. Some steps, like generalization (see ACLH), are not necessarily validity preserving; that is, the system may adopt a false subgoal while trying to prove a true one. (Note that this is ok in the sense that it is not ``unsound.'' The system will fail in its attempt to establish the false subgoal and the main proof attempt will fail.) As you gain facility with using the prover, you'll get pretty good at recognizing what to look for when reading a proof script. The prover's proof-tree utility helps with monitoring an ongoing proof and jumping to designated locations in the proof (see proof-tree). See tips for a number of useful pointers on using the theorem prover effectively.

When the prover has successfully proved all subgoals, the proof is finished. As with a `defun`, a summary of the proof is printed. This was an extremely simple proof, needing no additional guidance. More realistic examples typically require the user to look carefully at the failed proof log to find ways to influence the prover to do better on its next attempt. This means either: proving some rules that will then be available to the prover, changing the global state in ways that will affect the proof, or providing some hints locally that will influence the prover's behavior. Proving this lemma (`my-app-length`) is an example of the first. Since this is a rewrite rule, whenever in a later proof an instance of the form `(LEN (MY-APP X Y))` is encountered, it will be rewritten to the corresponding instance of `(+ (LEN X) (LEN Y))`. Disabling the rule by executing the command

```

(in-theory (disable my-app-length)),

```
is an example of a global change to the behavior of the prover since this rewrite will not be performed subsequently (unless the rule is again enabled). Finally, we can add a (local) disable ``hint'' to a `defthm`, meaning to disable the lemma only in the proof of one or more subgoals. For example:
```
(defthm my-app-length-commutativity
(equal (len (my-app x y))
(len (my-app y x)))
:hints (("Goal" :in-theory (disable my-app-length))))

```
In this case, the hint supplied is a bad idea since the proof is much harder with the hint than without it. Try it both ways.

By the way, to undo the previous event use `:u` (see u). To undo back to some earlier event use `:ubt` (see ubt). To view the current event use `:pe :here`. To list several events use `:pbt` (see pbt).

Notice the form of the hint in the previous example (see hints). It specifies a goal to which the hint applies. `"Goal"` refers to the top-level goal of the theorem. Subgoals are given unique names as they are generated. It may be useful to suggest that a function symbol be disabled only for Subgoal 1.3.9, say, and a different function enabled only on Subgoal 5.2.8. Overuse of such hints often suggests a poor global proof strategy.

We now recommend that you visit documentation on additional examples. See tutorial-examples.  