Lecture 16: Functions are People, too

1. Returning functions from functions:

   #| Sigma : list-of-numbers -> number Purpose: add the nunbers in alon (define (Sigma alon) ... ) Examples: (Sigma (list 1 2 3)) = 6 (Sigma (list 10 2 13)) = 25 |#	#| Pi : list-of-numbers -> number Purpose: multiply the nunbers in alon (define (Pi alon) ... ) Examples: (Pi (list 1 2 3)) = 6 (Sigma (list 10 2 13)) = 260 |#
   (define (Sigma alon) (cond ((empty? alon) 0) (else (+ (first alon) (Sigma (rest alon))))))	(define (Pi alon) (cond ((empty? alon) 1) (else (* (first alon) (Pi (rest alon))))))
   Now abstract over the two:
   (define (make-pi/sigma base combine) (local ((define (F alon) (cond ((empty? alon) base) (else (combine (first alon) (F (rest alon))))))) F))
   How do we get back Sigma and Pi:
   (define Sigma (make-pi/sigma 0 +))	(define Pi (make-pi/sigma 1 *))

   Let's verify that Pi and Sigma are defined properly.

     (make-pi/sigma 1 *)
   = (local ((define (F alon)
   	    (cond
   	      ((empty? alon) base)
   	      (else (combine (first alon) (F (rest alon)))))))
       F)
   = F with
            (define (F alon)
   	    (cond
   	      ((empty? alon) base)
   	      (else (combine (first alon) (F (rest alon))))))
        added to the Definitions window
   
   ;; So:
   
   (define (F alon)
     (cond
       ((empty? alon) base)
       (else (combine (first alon) (F (rest alon))))))
   (define Pi F)
   

   Now do this for Sigma and we get the following Definitions Window:

   
   (define (F alon)
     (cond
       ((empty? alon) 0)
       (else (+ (first alon) (F (rest alon))))))
   
   (define (F alon)
     (cond
       ((empty? alon) 1)
       (else (* (first alon) (F (rest alon))))))
   
   (define Sigma F)
   
   (define Pi F)
   

   Which one is it?
   
   

2. LOCAL renames as we lift out the function definitions!

   Before we lift local-definitions to the Definitions Window, we systematically rename them -- according to the laws of bindings and scope

   Modify above derivations slightly!

     (make-pi/sigma 1 *)
   = (local ((define (F alon)
   	    (cond
   	      ((empty? alon) base)
   	      (else (combine (first alon) (F (rest alon)))))))
       F)
   = (local ((define (FPI alon)
   	    (cond
   	      ((empty? alon) base)
   	      (else (combine (first alon) (FPI (rest alon)))))))
       FPI) 
     because FPI does not occur at all in define's
   = FPI with
            (define (FPI alon)
   	    (cond
   	      ((empty? alon) base)
   	      (else (combine (first alon) (FPI (rest alon))))))
        added to the Definitions window
   
   ;; So now we have: 
   
   (define (FPI alon)
     (cond
       ((empty? alon) base)
       (else (combine (first alon) (FPI (rest alon))))))
   
   (define Pi F)
   
   ;; in the Definition's window.