1.9

Exercise 1.9: Each of the following two procedures defines a method for adding two positive integers in terms of the procedures inc, which increments its argument by 1, and dec, which decrements its argument by 1.

(define (+ a b)
            (if (= a 0)
                b
                (inc (+ (dec a) b))))

(define (+ a b)
    (if (= a 0)
    b
    (+ (dec a) (inc b))))

Using the substitution model, illustrate the process generated by each procedure in evaluating (+ 4 5). Are these processes iterative or recursive?

The former one is recursive:

(+ 4 5) ----------------------------\
(inc (+ 3 5))                       |
(inc (inc (+ 2 5)))                 |
(inc (inc (inc (+ 1 5))))           |
(inc (inc (inc (inc (+ 0 5)))))     |
(inc (inc (inc (inc 5))))           |
(inc (inc (inc 6)))                 |
(inc (inc 7))                       |
(inc 8)                             |
9       <---------------------------/

The latter is iterative:

(+ 4 5) --------------------------\
(+ 3 6)                           |
(+ 2 7)                           |
(+ 1 8)                           |
(+ 0 9)                           |
9       <-------------------------/