REM

remainder using truncate 
Major Section:  PROGRAMMING


ACL2 !>(rem 14 3)
2
ACL2 !>(rem -14 3)
-2
ACL2 !>(rem 14 -3)
2
ACL2 !>(rem -14 -3)
-2
ACL2 !>(rem -15 -3)
0
ACL2 !>
(Rem i j) is that number k that (* j (truncate i j)) added to k equals i.

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

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












































































REMOVE

remove all occurrences, testing using eql 
Major Section:  PROGRAMMING

(remove x l) is l if x is not a member of l, else is the result of removing all occurrences of x from l.

The guard for (remove x l) requires l to be a true list and moreover, either x is eqlablep or all elements of l are eqlablep.

Remove 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.












































































REMOVE-DUPLICATES

remove duplicates from a string or (using eql) a list 
Major Section:  PROGRAMMING

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))
is equal to '(1 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

remove duplicates from a list 
Major Section:  PROGRAMMING

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

rest (cdr) of the list 
Major Section:  PROGRAMMING

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

concatentate the reverse of one list to another 
Major Section:  PROGRAMMING

(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))
Hint: This lemma follows immediately from the definition of reverse and the following lemma. 
(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

reverse a list 
Major Section:  PROGRAMMING

(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

coerce to a rational number 
Major Section:  PROGRAMMING

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

division returning an integer by rounding off 
Major Section:  PROGRAMMING


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
(Round i j) is the result of taking the quotient of i and j and rounding off to the nearest integer. When the quotient is exactly halfway between consecutive integers, it rounds to the even one.

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

second member of the list 
Major Section:  PROGRAMMING

See any Common Lisp documentation for details.














































































SET-DIFFERENCE-EQUAL

elements of one list that are not elements of another 
Major Section:  PROGRAMMING

(Set-difference-equal x y) 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

seventh member of the list 
Major Section:  PROGRAMMING

See any Common Lisp documentation for details.














































































SIGNUM

indicator for positive, negative, or zero 
Major Section:  PROGRAMMING

(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

sixth member of the list 
Major Section:  PROGRAMMING

See any Common Lisp documentation for details.














































































STANDARD-CHAR-LISTP

recognizer for standard characters 
Major Section:  PROGRAMMING

(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

recognizer for standard characters 
Major Section:  PROGRAMMING

(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

the character output channel to which ld prints 
Major Section:  PROGRAMMING

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

the standard object input ``channel'' 
Major Section:  PROGRAMMING

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

coerce to a string 
Major Section:  PROGRAMMING

(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

recognizer for association lists with strings as keys 
Major Section:  PROGRAMMING

(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

concatenate two strings 
Major Section:  PROGRAMMING

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

(string-append str1 str2)
is equal to 
(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

in a given string, turn upper-case characters into lower-case 
Major Section:  PROGRAMMING

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

string equality without regard to case 
Major Section:  PROGRAMMING

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

recognizer for a true list of strings 
Major Section:  PROGRAMMING

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














































































STRING-UPCASE

in a given string, turn lower-case characters into upper-case 
Major Section:  PROGRAMMING

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<

less-than test for strings 
Major Section:  PROGRAMMING

(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<=

less-than-or-equal test for strings 
Major Section:  PROGRAMMING

(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>

greater-than test for strings 
Major Section:  PROGRAMMING

(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>=

less-than-or-equal test for strings 
Major Section:  PROGRAMMING

(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

recognizer for strings 
Major Section:  PROGRAMMING

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














































































STRIP-CARS

collect up all first components of pairs in a list 
Major Section:  PROGRAMMING

(strip-cars x) is the list obtained by walking through the list x and collecting up all first components (cars). This function is implemented in a tail-recursive way, despite its logical definition.

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












































































STRIP-CDRS

collect up all second components of pairs in a list 
Major Section:  PROGRAMMING

(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). This function is implemented in a tail-recursive way, despite its logical definition.














































































SUBLIS

substitute an alist into a tree 
Major Section:  PROGRAMMING

(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

subsequence of a string or list 
Major Section:  PROGRAMMING

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

test if every member of one list is a member of the other 
Major Section:  PROGRAMMING

(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

check if all members of one list are members of the other 
Major Section:  PROGRAMMING

(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

a single substitution into a tree 
Major Section:  PROGRAMMING

(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-<

less-than test for symbols 
Major Section:  PROGRAMMING

(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

recognizer for association lists with symbols as keys 
Major Section:  PROGRAMMING

(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

recognizer for a true list of symbols 
Major Section:  PROGRAMMING

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














































































SYMBOL-NAME

the name of a symbol (a string) 
Major Section:  PROGRAMMING

Completion Axiom: 

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

Guard for (symbol-name x): 

(symbolp x)













































































SYMBOL-PACKAGE-NAME

the name of the package of a symbol (a string) 
Major Section:  PROGRAMMING

Completion Axiom: 

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

Guard for (symbol-package-name x): 

(symbolp x)













































































SYMBOLP

recognizer for symbols 
Major Section:  PROGRAMMING

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














































































TAKE

initial segment of a list 
Major Section:  PROGRAMMING

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))
If n is is an integer greater than the length of l, then take pads the list with the appropriate number of nil elements. Thus, the following is also a theorem. 
(implies (and (integerp n)
              (true-listp l)
              (<= (length l) n))
         (equal (take n l)
                (append l (make-list (- n (length l))))))
For related functions, see nthcdr and see butlast.

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












































































TENTH

tenth member of the list 
Major Section:  PROGRAMMING

See any Common Lisp documentation for details.














































































THE

run-time type check 
Major Section:  PROGRAMMING

(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
See type-spec for a discussion of the legal type specifications.

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












































































THIRD

third member of the list 
Major Section:  PROGRAMMING

See any Common Lisp documentation for details.














































































TRUE-LIST-LISTP

recognizer for true (proper) lists of true lists 
Major Section:  PROGRAMMING

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

recognizer for proper (null-terminated) lists 
Major Section:  PROGRAMMING

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














































































TRUNCATE

division returning an integer by truncating toward 0 
Major Section:  PROGRAMMING


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
(Truncate i j) is the result of taking the quotient of i and j and dropping the fraction. For example, the quotient of -14 by 3 is -4 2/3, so dropping the fraction 2/3, we obtain a result for (truncate -14 3) of -4.

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

type specifiers in declarations 
Major Section:  PROGRAMMING


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 (COMPLEX-RATIONALP X)
(COMPLEX RATIONAL)        (RATIONALP (REALPART X))
                          (RATIONALP (IMAGPART X)))
(COMPLEX type)       (AND (COMPLEX-RATIONALP 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)
Notes:

In general, (integerp i j) means 

     (AND (INTEGERP X)
          (<= i X)
          (<= X j)).
But if i is the symbol *, the first inequality is omitted. If j is the symbol *, the second inequality is omitted. If instead of being an integer, the second element of the type specification is a list containing an integer, (i), then the first inequality is made strict. An analogous remark holds for the (j) case. The RATIONAL type specifier is similarly generalized.












































































UNARY--

arithmetic negation function 
Major Section:  PROGRAMMING

Completion Axiom: 

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

Guard for (unary-- x): 

(acl2-numberp x)
Notice that like all arithmetic functions, unary-- treats non-numeric inputs as 0.

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












































































UNARY-/

reciprocal function 
Major Section:  PROGRAMMING

Completion 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)))
Notice that like all arithmetic functions, unary-/ treats non-numeric inputs as 0.

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












































































UNION-EQ

union of two lists of symbols 
Major Section:  PROGRAMMING

(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

union of two lists 
Major Section:  PROGRAMMING

(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

modify a list by putting the given value at the given position 
Major Section:  PROGRAMMING

(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)
We have the following theorem. 
(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

recognizer for upper case characters 
Major Section:  PROGRAMMING

(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

how to test for 0 
Major Section:  PROGRAMMING

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)
*See guards-and-evaluation, especially the subsection titled ``Guards and evaluation V: efficiency issues''. Primary code is relevant only if guards are verified. The ``compiled code'' shown is only suggestive.

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)))))
Observe the classic recursion scheme: a test against 0 and recursion by 1-. Note however that the test against 0 is expressed with the zp idiom. Note also the absence of a guard making explicit our intention that n is a natural number.

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)))))
This guarded definition has the same termination conditions as before -- termination is not sensitive to the guard. But the guards can be verified. This makes the primary raw lisp definition accessible during execution. In that definition, the (zp n) above is compiled as (= n 0), because n will always be a natural number when the primary code is executed. Thus, by adding a guard and verifying it, the elegant and easily used definition of factorial is also efficiently executed on natural numbers.

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)))))
This definition does not terminate. For example (fact -1) gives rise to a call of (fact -2), etc. Hence, this alternative is inadmissible. A plausible response is the addition of a guard restricting n to the naturals: 
(defun fact (n)
 (declare (xargs :guard (and (integerp n) (>= n 0))))
 (if (= n 0) 1 (* n (fact (1- n)))))
But because the termination argument is not sensitive to the guard, it is still impossible to admit this definition. To influence the termination argument one must change the conditions tested. Adding a runtime test that n is a natural number would suffice and allow both admission and guard verification. But such a test would slow down the execution of the compiled function.

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))))))
Observe that the function recurses by (floor i 2). Hence, calling the function on 25 causes calls on 12, 6, 3, 1, and 0, while calling it on -25 generates calls on -13, -7, -4, -2, and -1. By making (zip i) the first test, we terminate the recursion immediately on non-integers. The guard, if present, can be verified and allows the primary raw lisp definition to check (= i 0) as the first terminating condition (because the primary code is executed only on integers).