Complex
Create an ACL2 number
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 x
The 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)))
