REMOVE-DUPLICATES

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Remove-duplicates returns the result of deleting duplicate elements from the beginning of the given string or true list, i.e., leaving the last element in place. For example,

(remove-duplicates '(1 2 3 2 4))

The guard for Remove-duplicates requires that its argument is a string or a true-list of eqlablep objects. It uses the function eql to test for equality between elements of its argument.

Remove-duplicates is a Common Lisp function. See any Common Lisp documentation for more information. Note that we do not allow keyword arguments (such as test) in ACL2 functions, in particular, in remove-duplicates. But see remove-duplicates-equal, which is similar but uses the function equal to test for duplicate elements.

REMOVE-DUPLICATES-EQUAL

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Remove-duplicates-equal is the same as remove-duplicates, except that its argument must be a true list (not a string), and equal is used to check membership rather than eql. See remove-duplicates.

The guard for Remove-duplicates-equal requires that its argument is a true list. Note that unlike remove-duplicates, it does not allow string arguments.

REST

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In the logic, rest is just a macro for cdr.

Rest is a Common Lisp function. See any Common Lisp documentation for more information.

REVAPPEND

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(Revappend x y) concatenates the reverse of the list x to y, which is also typically a list.

The following theorem characterizes this English description.

(equal (revappend x y)
       (append (reverse x) y))

(defthm revappend-append
  (equal (append (revappend x y) z)
         (revappend x (append y z))))

The guard for (revappend x y) requires that x is a true list.

Revappend is defined in Common Lisp. See any Common Lisp documentation for more information.

REVERSE

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(Reverse x) is the result of reversing the order of the elements of the list x.

The guard for reverse requires that its argument is a true list.

Reverse is defined in Common Lisp. See any Common Lisp documentation for more information.

RFIX

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Rfix simply returns any rational number argument unchanged, returning 0 on a non-rational argument. Also see nfix, see ifix, and see fix for analogous functions that coerce to a natural number, an integer, and a number, respectively.

Rfix has a guard of t.

ROUND

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Example Forms:
ACL2 !>(round 14 3)
5
ACL2 !>(round -14 3)
-5
ACL2 !>(round 14 -3)
-5
ACL2 !>(round -14 -3)
5
ACL2 !>(round 13 3)
4
ACL2 !>(round -13 3)
-4
ACL2 !>(round 13 -3)
-4
ACL2 !>(round -13 -3)
4
ACL2 !>(round -15 -3)
5
ACL2 !>(round 15 -2)
-8

The guard for (round i j) requires that i and j are rational numbers and j is non-zero.

Round is a Common Lisp function. See any Common Lisp documentation for more information.

SECOND

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See any Common Lisp documentation for details.

SET-DIFFERENCE-EQUAL

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(Set-difference l1 l2) equals a list whose members (see member-equal) contains the members of x that are not members of y. More precisely, the resulting list is the same as one gets by deleting the members of y from x, leaving the remaining elements in the same order as they had in x.

The guard for set-difference-equal requires both arguments to be true lists. Essentially, set-difference-equal has the same functionality as the Common Lisp function set-difference, except that it uses the equal function to test membership rather than eql. However, we do not include the function set-difference in ACL2, because the Common Lisp language does not specify the order of the elements in the list that it returns.

SEVENTH

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See any Common Lisp documentation for details.

SIGNUM

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(Signum x) is 0 if x is 0, -1 if x is negative, and is 1 otherwise.

The guard for signum requires its argument to be rational.

Signum is a Common Lisp function. See any Common Lisp documentation for more information.

From ``Common Lisp the Language'' page 206, we see a definition of signum in terms of abs. As explained elsewhere (see abs), the guard for abs requires its argument to be a rational number; hence, we make the same restriction for signum.

SIXTH

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See any Common Lisp documentation for details.

STANDARD-CHAR-LISTP

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(standard-char-listp x) is true if and only if x is a null-terminated list all of whose members are standard characters. See standard-char-p.

Standard-char-listp has a guard of t.

STANDARD-CHAR-P

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(Standard-char-p x) is true if and only if x is a ``standard'' character, i.e., a member of the list *standard-chars*. This list includes #Newline and #Space characters, as well as the usual punctuation and alphanumeric characters.

Standard-char-p has a guard requiring its argument to be a character.

Standard-char-p is a Common Lisp function. See any Common Lisp documentation for more information.

STANDARD-CO

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Standard-co is an ld special (see ld). The accessor is (standard-co state) and the updater is (set-standard-co val state). Standard-co must be an open character output channel. It is to this channel that ld prints the prompt, the form to be evaluated, and the results. The event commands such as defun, defthm, etc., which print extensive commentary do not print to standard-co but rather to a different channel, proofs-co, so that you may redirect this commentary while still interacting via standard-co. See proofs-co.

``Standard-co'' stands for ``standard character output.'' The initial value of standard-co is the same as the value of *standard-co* (see *standard-co*).

STANDARD-OI

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Standard-oi is an ld special (see ld). The accessor is (standard-oi state) and the updater is (set-standard-oi val state). Standard-oi must be an open object input channel, a true list of objects, or a list of objects whose last cdr is an open object input channel. It is from this source that ld takes the input forms to process. When ld is called, if the value specified for standard-oi is a string or a list of objects whose last cdr is a string, then ld treats the string as a file name and opens an object input channel from that file. The channel opened by ld is closed by ld upon termination.

``Standard-oi'' stands for ``standard object input.'' The read-eval-print loop in ld reads the objects in standard-oi and treats them as forms to be evaluated. The initial value of standard-oi is the same as the value of *standard-oi* (see *standard-oi*).

STRING

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(String x) coerces x to a string. If x is already a string, then it is returned unchanged; if x is a symbol, then its symbol-name is returned; and if x is a character, the corresponding one-character string is returned.

The guard for string requires its argument to be a string, a symbol, or a character.

String is a Common Lisp function. See any Common Lisp documentation for more information.

STRING-ALISTP

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(String-alistp x) is true if and only if x is a list of pairs of the form (cons key val) where key is a string.

String-alistp has a guard of t.

STRING-APPEND

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NOTE: It is probably more efficient to use the Common Lisp function concatenate in place of string-append. That is,

(string-append str1 str2)

(concatenate 'string str1 str2).

At any rate, string-append takes two arguments, which are both strings (if the guard is to be met), and returns a string obtained concatenating together the characters in the first string followed by those in the second. See concatenate.

STRING-DOWNCASE

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For a string x, (string-downcase x) is the result of applying char-downcase to each character in x.

The guard for string-downcase requires its argument to be a string.

String-downcase is a Common Lisp function. See any Common Lisp documentation for more information.

STRING-EQUAL

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For strings str1 and str2, (string-equal str1 str2) is true if and only str1 and str2 are the same except perhaps for the cases of their characters.

The guard on string-equal requires that its arguments are strings.

String-equal is a Common Lisp function. See any Common Lisp documentation for more information.

STRING-LISTP

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The predicate string-listp tests whether its argument is a true-listp of strings.

STRING-UPCASE

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For a string x, (string-upcase x) is the result of applying char-upcase to each character in x.

The guard for string-upcase requires its argument to be a string.

String-upcase is a Common Lisp function. See any Common Lisp documentation for more information.

STRING<

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(String< str1 str2) is non-nil if and only if the string str1 precedes the string str2 lexicographically, where character inequalities are tested using char<. When non-nil, (string< str1 str2) is the first position (zero-based) at which the strings differ. Here are some examples.

ACL2 !>(string< "abcd" "abu")
2
ACL2 !>(string< "abcd" "Abu")
NIL
ACL2 !>(string< "abc" "abcde")
3
ACL2 !>(string< "abcde" "abc")
NIL

The guard for string< specifies that its arguments are strings.

String< is a Common Lisp function. See any Common Lisp documentation for more information.

STRING<=

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(String<= str1 str2) is non-nil if and only if the string str1 precedes the string str2 lexicographically or the strings are equal. When non-nil, (string<= str1 str2) is the first position (zero-based) at which the strings differ, if they differ, and otherwise is their common length. See string<.

The guard for string<= specifies that its arguments are strings.

String<= is a Common Lisp function. See any Common Lisp documentation for more information.

STRING>

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(String> str1 str2) is non-nil if and only if str2 precedes str1 lexicographically. When non-nil, (string> str1 str2) is the first position (zero-based) at which the strings differ. See string<.

String> is a Common Lisp function. See any Common Lisp documentation for more information.

STRING>=

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(String>= str1 str2) is non-nil if and only if the string str2 precedes the string str1 lexicographically or the strings are equal. When non-nil, (string>= str1 str2) is the first position (zero-based) at which the strings differ, if they differ, and otherwise is their common length. See string>.

The guard for string>= specifies that its arguments are strings.

String>= is a Common Lisp function. See any Common Lisp documentation for more information.

STRINGP

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(stringp x) is true if and only if x is a string.

STRIP-CARS

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(strip-cars x) is the list obtained by walking through the list x and collecting up all first components (cars).

(strip-cars x) has a guard of (alistp x).

STRIP-CDRS

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(strip-cdrs x) has a guard of (alistp x), and returns the list obtained by walking through the list x and collecting up all second components (cdrs).

SUBLIS

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(Sublis alist tree) is obtained by replacing every leaf of tree with the result of looking that leaf up in the association list alist. However, a leaf is left unchanged if it is not found as a key in alist.

Leaves are lookup up using the function assoc. The guard for (subsetp alist tree) requires (eqlable-alistp alist). This guard ensures that the guard for assoc will be met for each lookup generated by sublis.

Sublis is defined in Common Lisp. See any Common Lisp documentation for more information.

SUBSEQ

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For any natural numbers start and end, where start <= end <= (length seq), (subseq start end) is the subsequence of seq from index start up to, but not including, index end. End may be nil, which which case it is treated as though it is (length seq), i.e., we obtain the subsequence of seq from index start all the way to the end.

The guard for (subseq seq start end) is that seq is a true list or a string, start and end are integers (except, end may be nil, in which case it is treated as (length seq) for ther rest of this discussion), and 0 <= start <= end <= (length seq).

Subseq is a Common Lisp function. See any Common Lisp documentation for more information. Note: In Common Lisp the third argument of subseq is optional, but in ACL2 it is required, though it may be nil as explained above.

SUBSETP

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(Subsetp x y) is true if and only if every member of the list x is a member of the list y.

Membership is tested using the function member. Thus, the guard for subsetp requires that its arguments are true lists, and moreover, at least one satisfies eqlable-listp. This guard ensures that the guard for member will be met for each call generated by subsetp.

Subsetp is defined in Common Lisp. See any Common Lisp documentation for more information.

SUBSETP-EQUAL

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(Subsetp-equal x y) returns t if every member of x is a member of y, where membership is tested using member-equal.

The guard for subsetp-equal requires both arguments to be true lists. Subsetp-equal has the same functionality as the Common Lisp function subsetp, except that it uses the equal function to test membership rather than eql.

SUBST

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(Subst new old tree) is obtained by substituting new for every occurence of old in the given tree.

Equality to old is determined using the function eql. The guard for (subst new old tree) requires (eqlablep old), which ensures that the guard for eql will be met for each comparison generated by subst.

Subst is defined in Common Lisp. See any Common Lisp documentation for more information.

SYMBOL-<

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(symbol-< x y) is non-nil if and only if either the symbol-name of the symbol x lexicographially precedes the symbol-name of the symbol y (in the sense of string<) or else the symbol-names are equal and the symbol-package-name of x lexicographically precedes that of y (in the same sense). So for example, (symbol-< 'abcd 'abce) and (symbol-< 'acl2::abcd 'foo::abce) are true.

The guard for symbol specifies that its arguments are symbols.

SYMBOL-ALISTP

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(Symbol-alistp x) is true if and only if x is a list of pairs of the form (cons key val) where key is a symbolp.

SYMBOL-LISTP

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The predicate symbol-listp tests whether its argument is a true list of symbols.

SYMBOL-NAME

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Basic axiom:

(equal (symbol-name x)
       (if (symbolp x)
           (symbol-name x)
         ""))

Guard for (symbol-name x):

(symbolp x)

SYMBOL-PACKAGE-NAME

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Basic axiom:

(equal (symbol-package-name x)
       (if (symbolp x)
           (symbol-package-name x)
         ""))

Guard for (symbol-package-name x):

(symbolp x)

SYMBOLP

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(symbolp x) is true if and only if x is a symbol.

TAKE

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For any natural number n not exceeding the length of l, (take n l) collects the first n elements of the list l.

The following is a theorem (though it takes some effort, including lemmas, to get ACL2 to prove it):

(equal (length (take n l)) (nfix n))

(implies (and (integerp n)
              (true-listp l)
              (<= (length l) n))
         (equal (take n l)
                (append l (make-list (- n (length l))))))

The guard for (take n l) is that n is a nonnegative integer and l is a true list.

TENTH

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See any Common Lisp documentation for details.

THE

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(The typ val) checks that val satisfies the type specification typ (see type-spec). An error is caused if the check fails, and otherwise, val is the value of this expression. Here are some examples.

(the integer 3)       ; returns 3
(the (integer 0 6) 3) ; returns 3
(the (integer 0 6) 7) ; causes an error

The is defined in Common Lisp. See any Common Lisp documentation for more information.

THIRD

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See any Common Lisp documentation for details.

TRUE-LIST-LISTP

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True-list-listp is the function that checks whether its argument is a list that ends in, or equals, nil, and furthermore, all of its elements have that property. Also see true-listp.

TRUE-LISTP

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True-listp is the function that checks whether its argument is a list that ends in, or equals, nil.

TRUNCATE

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Example Forms:
ACL2 !>(truncate 14 3)
4
ACL2 !>(truncate -14 3)
-4
ACL2 !>(truncate 14 -3)
-4
ACL2 !>(truncate -14 -3)
4
ACL2 !>(truncate -15 -3)
5

The guard for (truncate i j) requires that i and j are rational numbers and j is non-zero.

Truncate is a Common Lisp function. See any Common Lisp documentation for more information.

TYPE-SPEC

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Examples:
The symbol INTEGER in (declare (type INTEGER i j k)) is a type-spec.  Other
type-specs supported by ACL2 include RATIONAL, COMPLEX, (INTEGER 0 127),
(RATIONAL 1 *), CHARACTER, and ATOM.  

The type-specs and their meanings (when applied to the variable x as in (declare (type type-spec x)) are given below.

type-spec                 meaning
ATOM                 (ATOM X)
BIT                  (OR (EQUAL X 1) (EQUAL X 0))
CHARACTER            (CHARACTERP X)
COMPLEX,             (AND (COMPLEXP X)
(COMPLEX RATIONAL)        (RATIONALP (REALPART X))
                          (RATIONALP (IMAGPART X)))
(COMPLEX type)       (AND (COMPLEXP X) (p (REALPART X)) (p (IMAGPART X)))
                          where (p x) is the meaning for type-spec type
CONS                 (CONSP X)
INTEGER              (INTEGERP X)
(INTEGER i j)        (AND (INTEGERP X)   ; See notes below
                          (<= i X)
                          (<= X j))
(MEMBER x1 ... xn)   (MEMBER X '(x1 ... xn))
(MOD i)              same as (INTEGER 0 i-1)
NIL                  NIL
NULL                 (EQ X NIL)
RATIO                (AND (RATIONALP X) (NOT (INTEGERP X)))
RATIONAL             (RATIONALP X)
(RATIONAL i j)       (AND (RATIONALP X)  ; See notes below
                          (<= i X)
                          (<= X j))
(SATISFIES pred)     (pred X)
SIGNED-BYTE          (INTEGERP X)
(SIGNED-BYTE i)      same as (INTEGER -2**i-1 (2**i-1)-1)
STANDARD-CHAR        (STANDARD-CHARP X)
STRING               (STRINGP X)
(STRING max)         (AND (STRINGP X) (EQUAL (LENGTH X) max))
SYMBOL               (SYMBOLP X)
T                    T
UNSIGNED-BYTE        same as (INTEGER 0 *)
(UNSIGNED-BYTE i)    same as (INTEGER 0 (2**i)-1)

In general, (integerp i j) means

     (AND (INTEGERP X)
          (<= i X)
          (<= X j)).

UNARY--

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Basic axiom:

(equal (unary-- x)
       (if (acl2-numberp x)
           (unary-- x)
         0))

Guard for (unary-- x):

(acl2-numberp x)

Calls of the macro - on one argument expand to calls of unary--; see -.

UNARY-/

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Basic axiom:

(equal (unary-/ x)
       (if (and (acl2-numberp x)
                (not (equal x 0)))
           (unary-/ x)
         0))

Guard for (unary-/ x):

(and (acl2-numberp x)
     (not (equal x 0)))

Calls of the macro / on one argument expand to calls of unary-/; see /.

UNION-EQ

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(Union-eq x y) equals a list whose members (see member-eq) contains the members of x and the members of y. More precisely, the resulting list is the same as one would get by first deleting the members of y from x, and then concatenating the result to the front of y.

The guard for union-eq requires both arguments to be true lists, but in fact further requires the first list to contain only symbols, as the function member-eq is used to test membership (with eq). See union-equal.

UNION-EQUAL

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(Union-equal x y) equals a list whose members (see member-equal) contains the members of x and the members of y. More precisely, the resulting list is the same as one would get by first deleting the members of y from x, and then concatenating the result to the front of y.

The guard for union-equal requires both arguments to be true lists. Essentially, union-equal has the same functionality as the Common Lisp function union, except that it uses the equal function to test membership rather than eql. However, we do not include the function union in ACL2, because the Common Lisp language does not specify the order of the elements in the list that it returns.

UPDATE-NTH

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(Update-nth key val l) returns a list that is the same as the list l, except that the value at the 0-based position key (a natural number) is val.

If key is an integer at least as large as the length of l, then l will be padded with the appropriate number of nil elements, as illustrated by the following example.

ACL2 !>(update-nth 8 'z '(a b c d e))
(A B C D E NIL NIL NIL Z)

(implies (and (true-listp l)
              (integerp key)
              (<= 0 key))
         (equal (length (update-nth key val l))
                (if (< key (length l))
                    (length l)
                  (+ 1 key))))

The guard of update-nth requires that its first (position) argument is a natural number and its last (list) argument is a true list.

UPPER-CASE-P

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(Upper-case-p x) is true if and only if x is an upper case character, i.e., a member of the list #A, #B, ..., #Z.

The guard for upper-case-p requires its argument to be a character.

Upper-case-p is a Common Lisp function. See any Common Lisp documentation for more information.

ZERO-TEST-IDIOMS

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Below are six commonly used idioms for testing whether x is 0. Zip and zp are the preferred termination tests for recursions down the integers and naturals, respectively.

idiom       logical              guard              primary
            meaning                              compiled code*


(equal x 0)(equal x 0)          t                (equal x 0)


(eql x 0)  (equal x 0)          t                (eql x 0)


(zerop x)  (equal x 0)          x is a number    (= x 0)


(= x 0)    (equal x 0)          x is a number    (= x 0)


(zip x)    (equal (ifix x) 0)   x is an integer  (= x 0)


(zp x)     (equal (nfix x) 0)   x is a natural   (= x 0)

The first four idioms all have the same logical meaning and differ only with respect to their executability and efficiency. In the absence of compiler optimizing, (= x 0) is probably the most efficient, (equal x 0) is probably the least efficient, and (eql x 0) is in between. However, an optimizing compiler could always choose to compile (equal x 0) as (eql x 0) and, in situations where x is known at compile-time to be numeric, (eql x 0) as (= x 0). So efficiency considerations must, of course, be made in the context of the host compiler.

Note also that (zerop x) and (= x 0) are indistinguishable. They have the same meaning and the same guard, and can reasonably be expected to generate equally efficient code.

Note that (zip x) and (zp x) do not have the same logical meanings as the others or each other. They are not simple tests for equality to 0. They each coerce x into a restricted domain, zip to the integers and zp to the natural numbers, choosing 0 for x when x is outside the domain. Thus, 1/2, #c(1 3), and 'abc, for example, are all ``recognized'' as zero by both zip and zp. But zip reports that -1 is different from 0 while zp reports that -1 ``is'' 0. More precisely, (zip -1) is nil while (zp -1) is t.

Note that the last four idioms all have guards that restrict their Common Lisp executability. If these last four are used in situations in which guards are to be verified, then proof obligations are incurred as the price of using them. If guard verification is not involved in your project, then the first four can be thought of as synonymous.

Zip and zp are not provided by Common Lisp but are ACL2-specific functions. Why does ACL2 provide these functions? The answer has to do with the admission of recursively defined functions and efficiency. Zp is provided as the zero-test in situations where the controlling formal parameter is understood to be a natural number. Zip is analogously provided for the integer case. We illustrate below.

Here is an admissible definition of factorial

(defun fact (n) (if (zp n) 1 (* n (fact (1- n)))))

This definition of factorial is readily admitted because when (zp n)

is false (i.e., nil) then n is a natural number other than 0 and so (1- n) is less than n. The base case, where (zp n) is true, handles all the ``unexpected'' inputs, such as arise with (fact -1) and (fact 'abc). When calls of fact are evaluated, (zp n) checks (integerp n) and (> n 0). Guard verification is unsuccessful for this definition of fact because zp requires its argument to be a natural number and there is no guard on fact, above. Thus the primary raw lisp for fact is inaccessible and only the :logic definition (which does runtime ``type'' checking) is used in computation. In summary, this definition of factorial is easily admitted and easily manipulated by the prover but is not executed as efficiently as it could be.

Runtime efficiency can be improved by adding a guard to the definition.

(defun fact (n)
  (declare (xargs :guard (and (integerp n) (>= n 0))))
  (if (zp n) 1 (* n (fact (1- n)))))

Now let us consider an alternative definition of factorial in which (= n 0) is used in place of (zp n).

(defun fact (n) (if (= n 0) 1 (* n (fact (1- n)))))

(defun fact (n)
 (declare (xargs :guard (and (integerp n) (>= n 0))))
 (if (= n 0) 1 (* n (fact (1- n)))))

The use of (zp n) as the test avoids this dilemma. Zp provides the logical equivalent of a runtime test that n is a natural number but the execution efficiency of a direct = comparison with 0, at the expense of a guard conjecture to prove. In addition, if guard verification and most-efficient execution are not needed, then the use of (zp n) allows the admission of the function without a guard or other extraneous verbiage.

While general rules are made to be broken, it is probably a good idea to get into the habit of using (zp n) as your terminating ``0 test'' idiom when recursing down the natural numbers. It provides the logical power of testing that n is a non-0 natural number and allows efficient execution.

We now turn to the analogous function, zip. Zip is the preferred 0-test idiom when recursing through the integers toward 0. Zip considers any non-integer to be 0 and otherwise just recognizes 0. A typical use of zip is in the definition of integer-length, shown below. (ACL2 can actually accept this definition, but only after appropriate lemmas have been proved.)

(defun integer-length (i)
  (declare (xargs :guard (integerp i)))
  (if (zip i)
      0
    (if (= i -1)
      0
      (+ 1 (integer-length (floor i 2))))))