# SICP - Solution: Exercise 1.9

##### October 1, 2018

**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?

**Solution**

## First procedure #

The procedure:

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

have a **recursive** process:

```
(+ 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
```

## Second procedure #

The procedure:

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

have an **iterative** process:

```
(+ 4 5)
(+ 3 6)
(+ 2 7)
(+ 1 8)
(+ 0 9)
9
```