# Recursive procedures without using letrec

Amazingly, Louis's intuition in exercise 4.20 is correct. It is indeed possible to specify recursive procedures without using letrec (or even define), although the method for accomplishing this is much more subtle than Louis imagined. The following expression computes 10 factorial by applying a recursive factorial procedure:

``````((lambda (n)
((lambda (fact)
(fact fact n))
(lambda (ft k)
(if (= k 1)
1
(* k (ft ft (- k 1)))))))
10)``````

a. Check (by evaluating the expression) that this really does compute factorials. Devise an analogous expression for computing Fibonacci numbers.

b. Consider the following procedure, which includes mutually recursive internal definitions:

``````(define (f x)
(define (even? n)
(if (= n 0)
true
(odd? (- n 1))))
(define (odd? n)
(if (= n 0)
false
(even? (- n 1))))
(even? x))``````

Fill in the missing expressions to complete an alternative definition of f, which uses neither internal definitions nor letrec:

``````(define (f x)
((lambda (even? odd?)
(even? even? odd? x))
(lambda (ev? od? n)
(if (= n 0) true (od? ?? ?? ??)))
(lambda (ev? od? n)
(if (= n 0) false (ev? ?? ?? ??)))))``````