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Numbers

Numbers-introduction

Numbers in ACL2

ACL2 numbers are precisely represented and unbounded. They can be partitioned into the following subtypes:

ACL2 Numbers
 |
 |- 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) ; i.e., 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. 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)))