Major Section: ACL2-BUILT-INS
Examples: (complex x 3) ; x + 3i, where i is the principal square root of -1 (complex x y) ; x + yi (complex x 0) ; same as x, for rational numbers xThe function
complex takes two rational number arguments and
returns an ACL2 number. This number will be of type
(complex rational) [as defined in the Common Lisp language], except
that if the second argument is zero, then complex returns its first
argument. The function complex-rationalp is a recognizer for
complex rational numbers, i.e. for ACL2 numbers that are not
rational numbers.The reader macro #C (which is the same as #c) provides a convenient
way for typing in complex numbers. For explicit rational numbers x
and y, #C(x y) is read to the same value as (complex x y).
The functions realpart and imagpart return the real and imaginary
parts (respectively) of a complex (possibly rational) number. So
for example, (realpart #C(3 4)) = 3, (imagpart #C(3 4)) = 4,
(realpart 3/4) = 3/4, and (imagpart 3/4) = 0.
The following built-in axiom may be useful for reasoning about complex numbers.
(defaxiom complex-definition
(implies (and (real/rationalp x)
(real/rationalp y))
(equal (complex x y)
(+ x (* #c(0 1) y))))
:rule-classes nil)
A completion axiom that shows what complex returns on arguments
violating its guard (which says that both arguments are rational
numbers) is the following, named completion-of-complex.
(equal (complex x y)
(complex (if (rationalp x) x 0)
(if (rationalp y) y 0)))