sol-1.18.tex

Assuming for simplicity that we are multiplying positive numbers, the requested procedure can be defined as
\begtt\scm
(define (* a b)
  (define (iter a b p)
    (cond ((= b 0) p)
          ((even? b) (iter (double a) (halve b) p))
          (else (iter a (- b 1) (+ p a)))))
  (iter a b 0))
\endtt
Here we added an additional state variable $p$ and defined the transformation in such a way that the quantity $ab + p$ is invariant.  At the beginning $p$ is set to be $0$, and the answer is the value of $p$ at the end of the process.