How to derive Y combinator

y-combinator.scm

(define (add1 x) (+ 1 x))
(define l3 '(1 2 3))

(define length
  (lambda (l)
    (cond ((null? l) 0)
	  (else (add1 (length (cdr l)))))))


(lambda (length)
  (lambda (l)
    (cond ((null? l) 0)
	  (else (add1 (length (cd l)))))))

((lambda (length)
  (lambda (l)
    (cond ((null? l) 0)
	  (else (add1 (length (cd l)))))))
  (lambda (length)
  (lambda (l)
    (cond ((null? l) 0)
	  (else (add1 (length (cd l))))))))


((lambda (length)
  (lambda (l)
    (cond ((null? l) 0)
	  (else (add1 ((length length)(cd l)))))))
  (lambda (length)
  (lambda (l)
    (cond ((null? l) 0)
	  (else (add1 ((length  length)(cd l))))))))

((lambda (mk-length)
   (mk-length mk-length))
 (lambda (length)
   (lambda (l)
     (cond ((null? l) 0)
	   (else (add1 ((length length) (cdr l))))))))


((lambda (mk-length)
   (mk-length mk-length))
 (lambda (fx)
   (lambda (l)
     (cond ((null? l) 0)
	   (else (add1 ((fx fx) (cdr l))))))))




(define y0
((lambda (mk-length)
   (mk-length mk-length))
 (lambda (fx)
   ((lambda (F)
      (lambda (l)
	(cond ((null? l) 0)
	      (else (add1 (F (cdr l)))))))
    (lambda (x)
      ((fx fx) x))))))

(define y1
  (lambda (work)
    ((lambda (mk-length)
       (mk-length mk-length))
     (lambda (fx)
       (work (lambda (x) ((fx fx) x)))))))

(define work
  (lambda (F)
    (lambda (l)
      (cond ((null? l) 0)
	    (else (add1 (F (cdr l))))))))