ACL2 !>(app nil '(x y z))(X Y Z)ACL2 !>

(app '(1 2 3) '(4 5 6 7))(1 2 3 4 5 6 7)ACL2 !>

(app '(a b c d e f g) '(x y z)); click here for an explanation (A B C D E F G X Y Z)ACL2 !>

(app (app '(1 2) '(3 4)) '(5 6))(1 2 3 4 5 6)ACL2 !>

(app '(1 2) (app '(3 4) '(5 6)))(1 2 3 4 5 6)ACL2!>

(let ((a '(1 2))(b '(3 4))(c '(5 6)))(equal (app (app a b) c)(app a (app b c))))T

As we can see from these examples, ACL2 functions can be executed more or less as Common Lisp.

The last three examples suggest an interesting property of `app`

.

Induction on **a** is unflawed: every occurrence of **a** in
the conjecture

(equal (app (appis in a position being recursively decomposed!ab) c) (appa(app b c)))

Now look at the occurrences of `b`

. The first (shown in **bold** below)
is in a position that is held constant in the recursion of `(app a b)`

.
It would be ``bad'' to induct on `b`

here.

(equal (app (app ab) c) (app a (app b c)))

ACL2 !>(equal (app (app a b) c)(app a (app b c))))ACL2 Error in TOP-LEVEL: Global variables, such as C, B, and A, are not allowed. See :DOC ASSIGN and :DOC @.

ACL2 does not allow ``global variables'' in top-level input. There is no ``top-level binding environment'' to give meaning to these variables.

Thus, expressions involving no variables can generally be evaluated,

ACL2 !>but expressions containing variables cannot.(equal (app (app '(1 2) '(3 4)) '(5 6))(app '(1 2) (app '(3 4) '(5 6))))(1 2 3 4 5 6)

There is an exception to rule. References to ``single-threaded objects'' may appear in top-level forms. See stobj . A single-threaded object is an ACL2 object, usually containing many fields, whose use is syntactically restricted so that it may be given as input only to certain functions and must be returned as output by certain functions. These restrictions allow single- threaded objects to be efficiently manipulated. For example, only a single copy of the object actually exists, even though from a logical perspective one might expect the object to be ``copied on write.''

The most commonly used single-threaded object in ACL2 is the ACL2
system state, whose current value is always held in the variable
`state`

.

ACL2 provides a way for you to use `state`

to save values of
computations at the top-level and refer to them later. See
assign and @ .

Consider a typical ``stack'' of control frames.

Suppose the model required that we express the idea of ``the most recent
frame whose return program counter points into `MAIN`

.''

The natural expression of this notion involves

**function application** -- ``fetch the `return-pc`

of this frame''

**case analysis** -- ``**if** the pc is `MAIN`

, **then** ...''

**iteration** or **recursion** -- ``pop this frame off and repeat.''

The designers of ACL2 have taken the position that a **programming**
**language** is the natural language in which to define such notions,
**provided** the language has a mathematical foundation so that
models can be analyzed and properties derived logically.

is the language supported by ACL2. Click here for an optional and very brief introduction to Common Lisp.

Common Lisp functions are partial; they are not defined for all
possible inputs. But ACL2 functions are total. Roughly speaking,
the logical function of a given name in ACL2 is a **completion** of
the Common Lisp function of the same name obtained by adding some
arbitrary but ``natural'' values on arguments outside the ``intended
domain'' of the Common Lisp function.

ACL2 requires that every ACL2 function symbol have a ``guard,''
which may be thought of as a predicate on the formals of the
function describing the intended domain. The guard on the primitive
function `car`

, for example, is `(or (consp x) (equal x nil))`

,
which requires the argument to be either an ordered pair or
`nil`

. We will discuss later how to specify a guard for a
defined function; when one is not specified, the guard is `t`

which
is just to say all arguments are allowed.

**But guards are entirely extra-logical**: they are not involved in
the axioms defining functions. If you put a guard on a defined
function, the defining axiom added to the logic defines the function
on **all** arguments, not just on the guarded domain.

So what is the purpose of guards?

The key to the utility of guards is that we provide a mechanism,
called ``guard verification,'' for checking that all the guards in a
formula are true. See `VERIFY-GUARDS`

. This mechanism will attempt
to prove that all the guards encountered in the evaluation of a
guarded function are true every time they are encountered.

For a thorough discussion of guards, see the paper [km97] in the
ACL2 bibliography.

When a function is admitted to the logic, ACL2 tries to ``guess''
what type of object it returns. This guess is codified as a term
that expresses a property of the value of the function. For `app`

the term is

(OR (CONSP (APP X Y)) (EQUAL (APP X Y) Y))which says that

`app`

returns either a cons or its second argument.
This formula is added to ACL2's rule base as a `type-prescription`

rule. Later we will discuss how rules are used by the ACL2
theorem prover. The point here is just that when you add a definition,
the data base of rules is updated, not just by the addition of the
definitional axiom, but by several new rules.
You should now return to the Walking Tour.

Now that you have seen the theorem prover in action you might be curious as to how you guide it.

The idea is that the user submits conjectures and advice and reads the proof attempts as they are produced.

Most of the time, the conjectures submitted are **lemmas** to be
used in the proofs of other theorems.

The example

ACL2 !>illustrates the fact that while ACL2 is an untyped language the ACL2 evaluator can be configured so as to check ``types'' at runtime. We should not say ``types'' here but ``guards.'' Click here for a discussion of guards.(app 7 27)ACL2 Error in TOP-LEVEL: The guard for the function symbol ENDP, which is (OR (CONSP X) (EQUAL X NIL)), is violated by the arguments in the call (ENDP 7).

The guard on `endp`

requires its argument to be a true
list. Since 7 is not a true list, and since ACL2 is checking guards
in this example, an error is signaled by ACL2. How do you know
ACL2 is checking guards? Because the prompt tells us
(click here) with its ``!''.

The training time depends primarily on the background of the user.

We expect that a user who

* has a bachelor's degree inwill probably takecomputer scienceormathematics,* has some experience with

formal methods,* has had some exposure to

Lispprogramming and iscomfortablewith the Lispnotation,* is

familiarwith and hasunlimited accessto a Common Lisphostprocessor,operating system, andtext editor(we use Sun workstations running Unix and GNU Emacs),* is willing to read and study the

ACL2 documentation, and* is given the opportunity to start with ``

toy'' projectsbeforebeing expected to tackle the company's Grand Challenge,

Most ACL2 primitives are documented. Here is the definition of
`app`

again, with the documented topics highlighted. All of the
links below lead into the ACL2 online documentation, 1.5 megabytes of
hyperlinked text. So follow these links around, but remember to come
back here!

(defun app (x y) (cond ((endp x) y) (t (cons (car x) (app (cdr x) y)))))

By following the link on endp we see that it is a
Common Lisp function and is defined to be the same as atom
, which recognizes non-conses. But `endp`

has a guard.
Since we are ignorning guards for now, we'll ignore the guard issue
on `endp`

.

So this definition reads ``to `app`

`x`

and `y`

: if `x`

is an
atom, return `y`

; otherwise, `app`

`(cdr x)`

and `y`

and then
cons `(car x)`

onto that.''

You can always use the Index to find the documentation of functions. Try it. Click on the Index icon below. Then use the Find command of your browser to find ``endp'' in that document and follow the link. But remember to come back here.

The ACL2 documentation is also available via Emacs' TexInfo, allowing you to explore the hyperlinked documentation in the comfort of a text editor that can also interact with ACL2.

In addition, some runtime images of ACL2 (the so-called ``large
image'') have the hyperlinked text as a large ACL2 data structure
that can be explored with ACL2's **:doc** command. If you have ACL2
running, try the command **:doc endp**. If that doesn't work, you
are running the ``small image.''

Another way to find out about ACL2 functions, if you have an ACL2
image available, is to use the command :`args`

which
prints the formals, type, and guard of a function symbol.

Of course, the ACL2 documentation can also be printed out as 700 page book. See the ACL2 Home Page to download the Postscript.

Now let's continue with `app`

.

Below we define **mc(s,n)** to be the function that **single-step**s **n**
times from a given starting state, **s**.
In Common Lisp, ``mc(s,n)'' is written `(mc s n)`

.

(defun mc (s n); To stepsntimes:(if (zp n); Ifnis 0s; then returns(mc (single-step s) (- n 1)))); else stepsingle-step(s)n-1times.

This is an example of a formal model in ACL2.

Frequently, engineers use mathematical models. Use of such models frequently lead to

**better designs**,

**faster completion of acceptable products**, and

**reduced overall cost**,

because models allow the trained user to study the design before it is built and analyze its properties. Usually, testing and analyzing a model is cheaper and faster than fabricating and refabricating the product.

Click here to continue.

Computing machines, whether hardware or software or some combintation, are frequently modeled as ``state machines.''

To so model a computing machine we must represent its **states** as objects
in our mathematical framework.

**Transitions** are functions or relations on state objects.

In what language shall we define these objects, functions, and relations?

The mathematical languages we were taught in high school

**algebra**,

**geometry**,

**trignometry**, and

**calculus**

are inappropriate for modeling digital systems. They primarily let us talk about numbers and continuous functions.

To see what kind of expressive power we need, take a closer look at what a typical state contains.

When the theorem prover explicitly assigns a name, like `*1`

, to a
formula, it has decided to prove the formula by induction.

The numbers in ACL2 can be partitioned into the following subtypes:

Rationals Integers Positive integers`3`

Zero`0`

Negative Integers`-3`

Non-Integral Rationals Positive Non-Integral Rationals`19/3`

Negative Non-Integral Rationals`-22/7`

Complex Rational Numbers`#c(3 5/2) ; = 3+(5/2)i`

Signed integer constants are usually written (as illustrated above)
as sequences of decimal digits, possibly preceded by `+`

or `-`

.
Decimal points are not allowed. Integers may be written in binary,
as in `#b1011`

(= 23) and `#b-111`

(= -7). Octal may also be
used, `#o-777`

= -511. Non-integral rationals are written as a
signed decimal integer and an unsigned decimal integer, separated by
a slash. Complex rationals are written as #c(rpart ipart) where
rpart and ipart are rationals.

Of course, 4/2 = 2/1 = 2 (i.e., not every rational written with a slash is a non-integer). Similarly, #c(4/2 0) = #c(2 0) = 2.

The common arithmetic functions and relations are denoted by `+`

,
`-`

, `*`

, `/`

, `=`

, `<`

, `<=`

, `>`

and `>=`

. However there
are many others, e.g., `floor`

, `ceiling`

, and `lognot`

. We
suggest you see programming where we list all of the primitive
ACL2 functions. Alternatively, see any Common Lisp language
documentation.

The primitive predicates for recognizing numbers are illustrated below. The following ACL2 function will classify an object, x, according to its numeric subtype, or else return 'NaN (not a number). We show it this way just to illustrate programming in ACL2.

(defun classify-number (x) (cond ((rationalp x) (cond ((integerp x) (cond ((< 0 x) 'positive-integer) ((= 0 x) 'zero) (t 'negative-integer))) ((< 0 x) 'positive-non-integral-rational) (t 'negative-non-integral-rational))) ((complex-rationalp x) 'complex-rational) (t 'NaN)))

`Subgoal *1/2`

is the **induction step** from the scheme, obtained by
instantiating the scheme with our conjecture.

We number the cases ``backward'', so this is case ``2'' of the
proof of ``*1''. We number them backward so you can look at a subgoal
number and get an estimate for how close you are to the end.

ACL2 is distributed on the Web without fee.

There is a **License** agreement, which is available via the ACL2 Home
Page link below.

ACL2 currently runs on the **Unix**, **Linux**, and **Macintosh**
operating systems.

It can be built in any of the following Common Lisps:

*Allegro, *GCL(Gnu Common Lisp) [or, AKCL], *Lispworks, *Lucid, and *MCL(Macintosh Common Lisp).

We recommend running ACL2 with at least **16MB** of memory and prefer
significantly more (e.g., **64MB**).

`Subgoal *1/1`

is the **Base Case** of our induction. It
simplifies to `Subgoal *1/1'`

by expanding the **ENDP** term in the
hypothesis, just as we saw in the earlier proof of `Subgoal *1/2`

.

In this message the system is saying that `Subgoal *1/2`

has been
rewritten to the `Subgoal *1/2'`

, by expanding the definition of **endp**.
This is an example of **simplification**, one of the main proof
techniques used by the theorem prover.

Click here if you would like to step through the
simplification of `Subgoal *1/2`

.

The **But** is our signal that the goal is proved.

Click here to step through the proof. It is very simple.

ACL2 !>(defthm trivial-consequence(equal (app (app (app (app x1 x2) (app x3 x4)) (app x5 x6)) x7)(app x1 (app (app x2 x3) (app (app x4 x5) (app x6 x7))))))ACL2 Warning [Subsume] in ( DEFTHM TRIVIAL-CONSEQUENCE ...): The previously added rule ASSOCIATIVITY-OF-APP subsumes the newly proposed :REWRITE rule TRIVIAL-CONSEQUENCE, in the sense that the old rule rewrites a more general target. Because the new rule will be tried first, it may nonetheless find application.

By the simple :rewrite rule ASSOCIATIVITY-OF-APP we reduce the conjecture to

Goal' (EQUAL (APP X1 (APP X2 (APP X3 (APP X4 (APP X5 (APP X6 X7)))))) (APP X1 (APP X2 (APP X3 (APP X4 (APP X5 (APP X6 X7))))))).

But we reduce the conjecture to T, by primitive type reasoning.

Q.E.D.

Summary Form: ( DEFTHM TRIVIAL-CONSEQUENCE ...) Rules: ((:REWRITE ASSOCIATIVITY-OF-APP) (:FAKE-RUNE-FOR-TYPE-SET NIL)) Warnings: Subsume Time: 0.20 seconds (prove: 0.02, print: 0.00, other: 0.18) TRIVIAL-CONSEQUENCE

You might explore the links before moving on.

Subgoal *1/1 (IMPLIES (ENDP A) (EQUAL (APP (APP A B) C) (APP A (APP B C)))).By the simple :definition ENDP we reduce the conjecture to

Subgoal *1/1' (IMPLIES (NOT (CONSP A)) (EQUAL (APP (APP A B) C) (APP A (APP B C)))).

But simplification reduces this to T, using the :definition APP and primitive type reasoning.

In this message the system is saying that `Subgoal *1/2'`

has been
rewritten T using the rules noted. The word ``**But**'' at the
beginning of the sentence is a signal that the goal has been proved.

Click here to step through the proof of `Subgoal *1/2'`

.

Subgoal *1/2 (IMPLIES (AND (NOT (ENDP A)) (EQUAL (APP (APP (CDR A) B) C) (APP (CDR A) (APP B C)))) (EQUAL (APP (APP A B) C) (APP A (APP B C)))).By the simple :definition ENDP we reduce the conjecture to

Subgoal *1/2' (IMPLIES (AND (CONSP A) (EQUAL (APP (APP (CDR A) B) C) (APP (CDR A) (APP B C)))) (EQUAL (APP (APP A B) C) (APP A (APP B C)))).

But simplification reduces this to T, using the :definition APP, the :rewrite rules CDR-CONS and CAR-CONS and primitive type reasoning.

The theorem prover's proof is printed in real time. At the time it
prints ``Perhaps'' it does not know the proof will succeed.

Recall that our induction scheme (click here to revisit it)
had two cases, the induction step (`Subgoal *1/2`

) and the base
case (`Subgoal *1/1`

). Both have been proved!

But ACL2 is a **logic**. We can **prove theorems about the model**.

Theorem. MC 'mult is a multiplier(implies (and (natp x) (natp y)) (equal (lookup 'z (mc (s 'mult x y) (mclk x))) (* x y))).

This theorem says that a certain program running on the **mc** machine
will correctly multiply **any two natural numbers**.

It is a statement about an **infinite** number of test cases!

We know it is true about the model because we **proved** it.

Here is the
definition of `app`

again with certain parts highlighted. If you
are taking the Walking Tour, please read the text carefully and
click on each of the links below, **except those marked** .
Then come **back** here.

ACL2 !>(defun app (x y)(cond ((endp x) y)(t (cons (car x)(app (cdr x) y)))))The admission of APP is trivial, using the relation E0-ORD-< (which is known to be well-founded on the domain recognized by E0-ORDINALP ) and the measure (ACL2-COUNT X). We observe that the type of APP is described by the theorem (OR (CONSP (APP X Y)) (EQUAL (APP X Y) Y)). We used primitive type reasoning.

Summary Form: ( DEFUN APP ...) Rules: ((:FAKE-RUNE-FOR-TYPE-SET NIL)) Warnings: None Time: 0.03 seconds (prove: 0.00, print: 0.00, other: 0.03) APP

By reading the documentation of `defthm`

(and
especially of its :rule-classes argument)
you would learn that
when we submitted the command

we not only command the system to prove that(defthm associativity-of-app(equal (app (app a b) c)(app a (app b c))))

`app`

is an associative
function but
*we commanded it to use that fact as a rewrite rule.

That means that every time the system encounters a term of the form

(app (appit will replace it withxy)z)

(appx(appyz))!

Suppose the machine being modeled is some kind of arithmetic unit.
Suppose the model can be initialized so as to **multiply** **x** times
**y** and leave the answer in **z**. Then if we initialize **s** to
**multiply** with **x=5** and **y=7** and run the machine long enough, we can
read the answer **35** in the final state.

Because ACL2 is a programming language, our
model can be **run** or **executed**.

If you defined the model in ACL2 and then typed

> (lookup 'z (mc (s 'mult 5 7) 29))

then ACL2 would compute 35. You can **emulate** or **test** the
model of your machine.

This is **obvious** because ACL2 is Common Lisp; and Common Lisp is a
**programming language**.

After collecting induction suggestions from these three terms

(appthe system notices that the first and last suggest the same decomposition ofab)(app

bc)(app

a(app b c))

`a`

. So we are left with two ideas about how to induct:
Decompose **a** as we would to unwind (app **a** b).

Decompose **b** as we would to unwind (app **b** c).

To find a plausible induction argument, the system studies the recursions exhibited by the terms in the conjecture.

Roughly speaking, a call of a recursive function ``suggests'' an induction if the argument position decomposed in recursion is occupied by a variable.

In this conjecture, three terms suggest inductions:

(appThe variable recursively decomposed is indicated inab)(app

bc)(app

a(app b c))

But ACL2 is more than a programming language.

Initialize **x** to 5 and let **y** be **any legal value**.

Because ACL2 is a mathematical language, we can simplify the expression

(lookup 'z (mc (s 'mult 5 y) 29))

and get (+ y y y y y). This is **symbolic execution** because not all
of the parameters are known.

Here is what it looks like to submit the definition of `app`

to ACL2:

ACL2 !>(defun app (x y)(cond ((endp x) y)(t (cons (car x)(app (cdr x) y)))))The admission of APP is trivial, using the relation E0-ORD-< (which is known to be well-founded on the domain recognized by E0-ORDINALP) and the measure (ACL2-COUNT X). We observe that the type of APP is described by the theorem (OR (CONSP (APP X Y)) (EQUAL (APP X Y) Y)). We used primitive type reasoning.

Summary Form: ( DEFUN APP ...) Rules: ((:FAKE-RUNE-FOR-TYPE-SET NIL)) Warnings: None Time: 0.03 seconds (prove: 0.00, print: 0.00, other: 0.03) APP

The text between the lines above is one interaction with the ACL2 command loop.

Above you see the user's **input** and how the system responds.
This little example shows you what the syntax looks like and is a
very typical **successful** interaction with the definitional
principle.

Let's look at it a little more closely.

ACL2!>(let ((a '(1 2))(b '(3 4))(c '(5 6)))(equal (app (app a b) c)(app a (app b c))))T

Observe that, for the particular `a`

, `b`

, and `c`

above,
`(app (app a b) c)`

returns the same thing as `(app a (app b c))`

.
Perhaps `app`

is **associative**. Of course, to be associative means
that the above property must hold for all values of `a`

, `b`

, and `c`

,
not just the ones **tested** above.

Wouldn't it be cool if you could type

ACL2!>and have ACL2 compute the value(equal (app (app a b) c)(app a (app b c))))

`T`

? Well, We cannot evaluate a form on an infinite number of cases. But we can prove that a form is a theorem and hence know that it will always evaluate to true.

This formula is the **Base Case**. It consists of two parts, a test
identifying the non-inductive case and the conjecture to prove.

(IMPLIES (ENDP A)When we prove this we can assume; Test(:P A B C)); Conjecture

` * ``A`

is empty

and we have to prove the conjecture for `A`

.