Chapter 2, Building Abstractions with Data

Section - 2.5 - Systems with Generic Operations

Exercise 2.92


It took quite some time to get it working. At first glance it looked simple but :) This exercise felt boring as it was just a tedious task without any new underlying concepts. Perhaps I should not have done it but when I realised this I found myself in the middle of the solution so went ahead.

The outline of my approach:

There are two parts (i) Convert polynomial cannonical (ii) Changing operations add and mul to work with cannonical types.

Note: To decide priority - alphabetic ordering of the variable names is used.

(i) Convert to cannonial form:

This is easy. Lets assume we have add and mul that supports canonical form. Since our polynomial are represented as a sum of product form - we take every product, lets call it term, we just need to convert each of these terms(product) into canonical forms and add them! Now think of each term - it is a product of cofficient and power/order of the variable. Just convert the coefficient into cannonical and mul the converted coefficient with our variable raised to appropriate power/order.

The next problem is how to integrate this conversion in our existing system - Just convert to canonical form in the public api to make-polynomial. This should not be in the internal api to make polynomial because then it will go into loop.

Thats it! Here is the code:

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(define (make-canonical var terms)   
      (let ((list-type (type-tag terms)))
         (if (empty-termlist? terms)
             (make-poly var (empty-termlist-of-type list-type))
             (let ((term (first-term terms)))
                (add-poly
                      (make-canonical var (rest-terms terms))
                      (let ((cof (coeff term))
                            (ord (order term)))
                         (let ((type (type-tag term)))
                            (if (equal? type 'polynomial)
                                (let ((cof-poly (contents cof)))
                                   (let ((cof-can (make-canonical (variable cof-poly) (term-list cof-poly))))
                                      (mul-poly cof-can (make-poly-with-term ord 1 list-type var))
                                   )
                                )
                                (make-poly-with-term ord cof list-type var)
                            )
                         )
                      )
                )
             )
         )  
      )
)

(ii) Changing operations add and mul to work with canonical forms.

This is also easy:

We assume that all polynomicals passed to add or mul are canonical forms.

For add: if two polynomials are in different variables we just need to add the polynomial with lower priority variable as a constant in polynomial with higher priority variable. This can be done easily just be converting or raising the lower type polynomial into higher one. For eg: To add and . Here we first convert the lower priority polynomial(with variable ) into . Now we can just add them using our old procedure because now both polynomials are of same variable. The rest of the things will be taken cared by recursion :) Except for the case when we add/mul a polynomial and a number - This operation is not supported with our older add method.

  • One way to do this is to raise number into polynomial. I did not feel correct to raise numbers as it raises the question of which type of variable should the number be raised into.
  • Since we are using add which is internally calls apply-generic - it is difficult to do without raising the number to polynomial. (Note that in a way it appears a limitation of apply-generic. It would have been better if we have an apply-generic where we should be able to add polynomial and any type of number. Thus a way to identify types of number. Also another problem to notice is that suppose even if we allowed to raise a number to polynomial and we wrote a raise to raise complex to polynomial then all our numbers will be raised to complex types and in few situation they will not be dropped to basic type(when there is no arithmetic operation is performed)).
  • Now, To avoid raising types I need to identify the places in polynomial package where a number and polynomial are added or multiplied. This is easy to spot - Its only the places where we are adding coefficients by calling generic add.
  • Now instead of using generic add we use our new hybrid-add that checks if both arguments are number or polynomial then it calls generic add else if we add/mul polynomial by a constant it creates a new polynomial with this constant and then performs the actual operation(add-poly/mul-poly). Note that another approach can be to still use apply-generic by installing/putting appropriate procedures in the hash table. The problem is apply-generic strips off type-tag before calling the installed procedure. This is a probelm - consider for eg when we add the number to the polynomial - we are not adding directly - we are adding this number by adding this as a term with 0 order to the polynomial. Thus the actual number addition may not even happen if the polynomial does not have any constant term before. Now suppose later on this polynomial is added to another polynomial(both having constant terms) then addition of the last term/constant term will happen again by add. But at this point this add will not work because - we stripped the type-tag from the number in the last step!
  • The above point brings an interesting question - How to handle such situation where all we need to an operation that can take any type of number and do not strips the type-tags because we are not actually doing any number arithmetic in there? Hope this gets answered in later part of the book. Perhaps one solution can be that apply-generic does not do any untagging/tagging - its just left to the individual packages. -As of now I just went ahead with mixed-add which does some work that should actually be done by apply-generic.

For mul: this is similar to add - just convert lower variable polynomial into higher and multiply!

Both steps look really simple but it took me sometime to arrive at these steps as well as some time also spent in problems with tagging. Internal procedures in general do not need to tag the types but we can not avoid them completely even in the internal procedures and need to pay attention to every operation whether we need to remove type-tag before proceeding or not. This made me realise that how easy it is in languages where late-binding is supported.

Another problem arises while adding if some coefficient becomes zero or while multiplying by a constant - these details are taken cared in procedure make-poly-with-term.

Note: I dont support operation for adding a number and a polynomial directly from outside - Outer api only supports addition/multiplication of polynomials but if situation happens when carrying out the operation - number and polynomial are added/multiplied then it works as mentioned in the above approach.

So here goes the complete code(Examples are at the bottom):

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#lang sicp

(#%require (only racket/base error))
(#%require (only racket/base make-hash))
(#%require (only racket/base hash-set!))
(#%require (only racket/base hash-ref))

;Polynomial Package



(define (install-polynomial-package)
  (define (make-canonical var terms)   
      (let ((list-type (type-tag terms)))
         (if (empty-termlist? terms)
             (make-poly var (empty-termlist-of-type list-type))
             (let ((term (first-term terms)))
                (add-poly
                      (make-canonical var (rest-terms terms))
                      (let ((cof (coeff term))
                            (ord (order term)))
                         (let ((type (type-tag term)))
                            (if (equal? type 'polynomial)
                                (let ((cof-poly (contents cof)))
                                   (let ((cof-can (make-canonical (variable cof-poly) (term-list cof-poly))))
                                      (mul-poly cof-can (make-poly-with-term ord 1 list-type var))
                                   )
                                )
                                (make-poly-with-term ord cof list-type var)
                            )
                         )
                      )
                )
             )
         )  
      )
  )
                            
  ;; internal procedures
  ;; representation of poly
  (define (make-poly variable term-list)
    (cons variable term-list)
  )
  
  (define (variable p) (car p))
  (define (term-list p) (cdr p))
  (define (variable? x) (symbol? x))
  (define (same-variable? v1 v2) (and (variable? v1) (variable? v2) (eq? v1 v2)))
  ;; representation of terms and term lists
  
  (define (adjoin-term term list)
       ((apply-generic 'adjoin-term-proc list) term)
  )
  (define (first-term terms)
    (apply-generic 'first-term terms)
  )
  (define (rest-terms terms)
    (apply-generic 'rest-terms terms)
  )
  (define (empty-termlist? terms)
    (apply-generic 'empty-termlist? terms)
  )
  (define (empty-termlist-of-type type) ((get 'empty-termlist type)))
  (define (make-term order coeff) (list order coeff))
  (define (order term) (car term))
  (define (coeff term) (cadr term))

  (define (order-by-priority p1 p2)
    (let ((var1 (variable p1))
          (var2 (variable p2)))
      (if (string<? (symbol->string var1) (symbol->string var2))
          (cons p1 p2)
          (cons p2 p1)
      )
    )
  )

  (define (make-poly-with-term ord coff list-type var)
   (let ((empty-list ((get 'empty-termlist list-type))))
     (if (=zero? coff)
         (make-poly var empty-list)
         (let ((type (type-tag coff)))
           (if (equal? type 'polynomial)
               (let ((poly (contents coff)))
                 (let ((terms (term-list poly)))
                   (let ((first (first-term terms)))
                     (let ((first-term-order (order first)))
                       (if (= first-term-order 0)
                           (let ((tl (adjoin-term (make-term ord (coeff first)) empty-list)))
                             (make-poly var tl)
                           )
                           (let ((tl (adjoin-term (make-term ord coff) empty-list)))
                             (make-poly var tl)
                           )
                       ) 
                     )
                   )
                 )
               )
               (let ((tl (adjoin-term (make-term ord coff) empty-list)))
                 (make-poly var tl)
               )
           )
         )
      )
   )
 )
  
  (define (add-poly p1 p2)
      (if (same-variable? (variable p1) (variable p2))
          (make-poly (variable p1)
                     (add-terms (term-list p1)
                            (term-list p2)
                     )
          )
          (let ((ordered (order-by-priority p1 p2)))
            (let ((high (car ordered))
                  (low (cdr ordered))
                  (list-type (type-tag (term-list p1))))
              (let ((raised-low (make-poly-with-term 0 (tag low) list-type (variable high))))
                (add-poly high raised-low)
              )
            )
          )
      )
  )
  (define (mul-poly p1 p2)
      (if (same-variable? (variable p1) (variable p2))
          (make-poly (variable p1)
                     (mul-terms (term-list p1)
                                (term-list p2)
                     )
          )
          (let ((ordered (order-by-priority p1 p2)))
            (let ((high (car ordered))
                  (low (cdr ordered))
                  (list-type (type-tag (term-list p1))))
              (let ((raised-low (make-poly-with-term 0 (tag low) list-type (variable high))))
                (mul-poly high raised-low)
              )
            )
          )
      )
  )
  (define (div-poly p1 p2)
      (if (same-variable? (variable p1) (variable p2))
          (make-poly (variable p1)
                     (div-terms (term-list p1)
                                (term-list p2)
                     )
          )
      )
  )
  (define (div-terms L1 L2)
    (let ((empty-lst (empty-termlist-of-type (type-tag (term-list L1)))))
      (if (empty-termlist? L1)
          (list empty-lst empty-lst)
          (let ((t1 (first-term L1))
                (t2 (first-term L2))
               )
               (if (> (order t2) (order t1))
                   (list empty-lst L1)
                   (let ((new-c (div (coeff t1) (coeff t2)))
                        (new-o (- (order t1) (order t2))))
                        (let ((rest-of-result
                                  (div-terms
                                       (add-terms
                                            L1
                                            (negate-termlist
                                                 (mul-term-by-all-terms (make-term new-o new-c) L2)
                                            )
                                       )
                                       L2
                                  )
                             ))
                             (let ((quotient (car rest-of-result))
                                   (remainder (cdr rest-of-result))
                                   )
                                   (list (adjoin-term (make-term new-o new-c) quotient) remainder)
                             )
                        )  
                   )
               )
          )
      )
    )
  )
  (define (sub-poly p1 p2) (add-poly p1 (negate-poly p2)))
  (define (negate-poly p) (make-poly (variable p) (negate-termlist (term-list p))))

  (define (negate-termlist terms)
    (if (empty-termlist? terms)
        terms        
        (adjoin-term (make-term (order (first-term terms)) (apply-generic 'negate (coeff (first-term terms)))) (negate-termlist (rest-terms terms)))
    )
  )

  (define (add-poly-to-number poly n)
    (let ((var (variable poly))
          (list-type (type-tag (term-list poly)))
         )
      (add-poly (make-poly-with-term 0 n list-type var)
                poly
      )
    )
  )
  
  (define (mul-poly-by-number poly n)
    (let ((var (variable poly))
          (list-type (type-tag (term-list poly)))
         )
      (mul-poly (make-poly-with-term 0 n list-type var)
                poly
      )
    )
  )
  
  (define (hybrid-add arg1 arg2)
    (let ((type1 (type-tag arg1))
          (type2 (type-tag arg2)))
      (cond
        ((and (equal? 'polynomial type1) (not (equal? 'polynomial type2))) (tag (add-poly-to-number (contents arg1) arg2)))
        ((and (equal? 'polynomial type2) (not (equal? 'polynomial type1))) (tag (add-poly-to-number (contents arg2) arg1)))
        (else (add arg1 arg2))
      )
    )
  )
  
  (define (hybrid-mul arg1 arg2)
    (let ((type1 (type-tag arg1))
          (type2 (type-tag arg2)))
      (cond
        ((and (equal? 'polynomial type1) (not (equal? 'polynomial type2))) (tag (mul-poly-by-number (contents arg1) arg2)))
        ((and (equal? 'polynomial type2) (not (equal? 'polynomial type1))) (tag (mul-poly-by-number (contents arg2) arg1)))
        (else (mul arg1 arg2))
      )
    )
  )
          
  
  (define (add-terms L1 L2)
     (cond ((empty-termlist? L1) L2)
        ((empty-termlist? L2) L1)
        (else
           (let ((t1 (first-term L1)) (t2 (first-term L2)))
                (cond ((> (order t1) (order t2))
                           (adjoin-term
                                t1 (add-terms (rest-terms L1) L2))
                      )
                      ((< (order t1) (order t2))
                      (adjoin-term t2 (add-terms L1 (rest-terms L2))))
                      (else
                           (adjoin-term
                                (make-term (order t1)
                                           (hybrid-add (coeff t1) (coeff t2))
                                )
                                (add-terms (rest-terms L1)
                                     (rest-terms L2)
                                )
                           )
                      )
                 )
             )
          )
        )
    )

    (define (mul-terms L1 L2)
        (if (empty-termlist? L1)
            L1
            (add-terms (mul-term-by-all-terms (first-term L1) L2)
                       (mul-terms (rest-terms L1) L2)
            )
        )
    )
    (define (mul-term-by-all-terms t1 L)
        (if (empty-termlist? L)
            L
            (let ((t2 (first-term L)))
                 (adjoin-term
                     (make-term (+ (order t1) (order t2))
                                (hybrid-mul (coeff t1) (coeff t2))
                     )
                     (mul-term-by-all-terms t1 (rest-terms L))
                 )
            )
        )
    )

    
   (define (=is-zero? poly)
       (define (iter terms)
         (if (apply-generic 'empty-termlist? terms)
             #t
             (and (=zero? (coeff (first-term terms))) (iter (rest-terms terms)))
         )
       )
       (iter (term-list poly))
   )
  
   ;; interface to rest of the system
   (define (tag p) (attach-tag 'polynomial p))
   (put 'add '(polynomial polynomial)
        (lambda (p1 p2) (tag (add-poly p1 p2))))
   (put 'sub '(polynomial polynomial)
        (lambda (p1 p2) (tag (add-poly p1 (negate-poly p2)))))
   (put 'mul '(polynomial polynomial)
        (lambda (p1 p2) (tag (mul-poly p1 p2))))
   (put 'div '(polynomial polynomial) (lambda (p1 p2) (tag (div-poly p1 p2))))
   (put 'make 'polynomial
        (lambda (var terms) (tag (make-poly var terms))))
   (put 'make-term 'polynomial make-term)
   (put 'order 'polynomial order)
   (put 'coeff 'polynomial coeff)
   (put 'negate '(polynomial) (lambda (p) (tag (negate-poly p))))
   (put 'make-canonical 'polynomial (lambda(var terms) (tag (make-canonical var terms))))
 
   (put '=zero? '(polynomial) =is-zero?)
  
  
  'done)


(define (make-polynomial var terms)
   ((get 'make-canonical 'polynomial) var terms)
)

(define (make-term order coeff)
  ((get 'make-term 'polynomial) order coeff))

(define (order term)
  ((get 'order 'polynomial) term))
    
(define (coeff term)
  ((get 'coeff 'polynomial) term))
        

;dense-termlist package
(define (install-dense-termlist-package)

    (define (adjoin-term-dense term term-list)
        (define (iter count terms)
            (if (= count 0)
                terms
                (iter (- count 1) (cons 0 terms))
            )
        )
        (let ((cof (coeff term))
              (count (- (order term) (length term-list)))
             )
             (cond
                ((=zero? cof) term-list)
                ((< count 0) (error "Can not add term - order of passed term is already present in the list"))
                (else (cons cof (iter count term-list)))
             )
        )
    )
    (define (first-term term-list) (make-term (- (length term-list) 1) (car term-list)))

       ;; interface to rest of the system
    (define (tag tl) (attach-tag 'dense-termlist tl))
  
    (put 'empty-termlist 'dense-termlist (lambda () (tag '())))    
    (put 'first-term '(dense-termlist) first-term)
    (put 'rest-terms '(dense-termlist) (lambda (tl) (tag (cdr tl))))
    (put 'empty-termlist? '(dense-termlist) null?)
    (put 'adjoin-term-proc '(dense-termlist) (lambda(tl) (lambda(term) (tag (adjoin-term-dense term tl)))))
  
  'done)

;sparse-termlist package
(define (install-sparse-termlist-package)
    (define (adjoin-term-sparse term term-list)
        (if (=zero? (coeff term))
            term-list
            (cons term term-list)
        )
    )
    
       ;; interface to rest of the system
    (define (tag tl) (attach-tag 'sparse-termlist tl))
  
    (put 'empty-termlist 'sparse-termlist (lambda () (tag '())))
    (put 'first-term '(sparse-termlist) (lambda (tl) (car tl)))
    (put 'rest-terms '(sparse-termlist) (lambda (tl) (tag (cdr tl))))
    (put 'empty-termlist? '(sparse-termlist) null?)
    (put 'adjoin-term-proc '(sparse-termlist) (lambda(tl) (lambda(term) (tag (adjoin-term-sparse term tl)))))
  
  'done)

; Code below this is the arithmetic package we built in the last section
; Arithmetic Package


(define (sine x) (sin (contents (raise-to-type 'scheme-number x))))
(define (cosine x) (cos (contents (raise-to-type 'scheme-number x))))
(define (arctan x) (atan (contents (raise-to-type 'scheme-number x)))) 

(define (drop arg)
 (if (has-type-tag? arg)
     (let ((type (type-tag arg)))
       (let ((proc (get-projection type)))
          (if (not (null? proc))
             (let ((projected (proc arg)))
               (let ((raised (raise-to-type type projected)))
                 (if ((get 'equ? (list type type)) (contents raised) (contents arg))
                       (drop projected)
                       arg
                 )
               )
             )                
             arg
          )
       )
     )
     arg
  )
)

(define (apply-generic op . args)
  (let ((type-tags (map type-tag args)))
    (let ((proc (get op type-tags)))
      (if (not (null? proc))
          (drop (apply proc (map contents args)))
          (let ((has-same-type
                     (accumulate
                          (lambda (a b) (and a b))
                          #t
                          (map
                             (lambda(x) (equal? (car type-tags) x))
                             (cdr type-tags)
                          )
                     )
               ))
               (let ((highest-type (if has-same-type
                                       (get-parent (car type-tags))
                                       (find-highest-type type-tags)
                                   )
                    ))
                    (if (null? highest-type)
                       (error "Could not find a suitable ancestor" op args)
                       (let ((raised-args (map-until
                                                    '()
                                                    (lambda(arg)
                                                        (raise-to-type highest-type arg)
                                                    )
                                                    args
                                           )
                            ))
                           (if (not (null? raised-args))
                               (apply apply-generic op raised-args)
                               (error "Error. No suitable raise method found to raise all types into " highest-type " for raising types " type-tags)
                           )
                       )
                    )     
              )
          )
      )
   )
 )
)  

(define *type-table* (make-hash))

(define (put-parent type parent)
   (let ((already-set (hash-ref *type-table* (list 'parent type) '())))
     (if (null? already-set)
         (hash-set! *type-table*  (list 'parent type) parent)
         (error "Parent already defined for :" type " parent currently *present* is " already-set " the passed parent " parent)
     )
   )
)

(define (get-parent type)
  (hash-ref *type-table* (list 'parent type) '())
)

(define (put-child type child)
   (let ((already-set (hash-ref *type-table* (list 'child type) '())))
     (if (null? already-set)
         (hash-set! *type-table*  (list 'child type) child)
         (error "Child already defined for :" type " child currently *present* is " already-set " the passed child " child)
     )
   )
)

(define (get-child type)
  (hash-ref *type-table* (list 'child type) '())
)


(define *cast-table* (make-hash))

(define (put-raise type proc)
   (let ((already-set (hash-ref *cast-table* (list 'raise type) '())))
     (if (null? already-set)
         (hash-set! *cast-table*  (list 'raise type) proc)
         (error "Raise is already defined for :" type)
     )
   )
)

(define (get-raise type)
  (hash-ref *cast-table* (list 'raise type) '())
)

(define (put-projection type proc)
   (let ((already-set (hash-ref *cast-table* (list 'project type) '())))
     (if (null? already-set)
         (hash-set! *cast-table*  (list 'project type) proc)
         (error "Project is already defined for :" type)
     )
   )
)

(define (get-projection type)
  (hash-ref *cast-table* (list 'project type) '())
)

(define (get-ancestors type)
  (if (null? type)
      '()
       (cons type (get-ancestors (get-parent type)))
  )
)

(define (contains item item-list)
      (fold-left
         (lambda(result new)
             (or result (equal? item new))
         )
         #f
         item-list
      )
)
     

(define (find-highest-type type-tags)
  (let ((ancestors-of-each-type (map get-ancestors type-tags)))
     (let ((smallest-ancestor-set
               (accumulate (lambda(new remaining)
                               (if (< (size new) (size remaining))
                                   new
                                   remaining
                               )
                           )
                           (car ancestors-of-each-type)
                           ancestors-of-each-type
               )
          ))
          (define (find-type-present-in-each-ancestors type-list)
              (if (null? type-list)
                  '()
                  (let ((found (accumulate
                                      (lambda (a b) (and a b))
                                      #t
                                      (map
                                          (lambda(ancestors) (contains (car type-list) ancestors))
                                          ancestors-of-each-type)
                               )
                        ))
                        (if found
                            (car type-list)
                            (find-type-present-in-each-ancestors (cdr type-list))
                        )
                  )
              )
          )
          (find-type-present-in-each-ancestors smallest-ancestor-set)
       )
    )
)

(define (raise-to-type type arg)
     (cond ((null? arg) '())
           ((equal? type (type-tag arg)) arg)
           (else (let ((proc (get-raise (type-tag arg))))
                   (if (null? proc)
                       '()
                       (raise-to-type type (proc arg))
                   )
                 )
           )
    )       
)

(define (map-until until-val proc args)
   (if (null? args)
       '()
       (let ((arg (car args)))
           (let ((marg (proc arg)))
             (if (equal? marg until-val)
                 '()
                 (cons marg (map-until until-val proc (cdr args)))
             )
           )
      )
  )
)

(define (fold-left op initial sequence)
  (define (iter result rest)
    (if (null? rest)
        result
        (iter (op result (car rest))
              (cdr rest))))
  (iter initial sequence))


(define (raise n)
  ((get-raise (type-tag n)) n)
)

;use mul as x can be rational-number also
(define (square x) (mul x x))

(define *op-table* (make-hash))

(define (put op type proc)
  (hash-set! *op-table* (list op type) proc)
)

(define (get op type)
  (hash-ref *op-table* (list op type) '())
)

(define (attach-tag type-tag contents) 
   (if (equal? 'scheme-number type-tag)
       contents
       (cons type-tag contents)
   )
)

(define (has-type-tag? datum) 
  (cond 
        ((number? datum) #t)
        ((pair? datum)
                (contains (car datum) '(int complex rational))
        )
        (else #f)
  )
)

(define (type-tag datum) 
  (cond 
        ((number? datum) 'scheme-number)
        ((pair? datum) (car datum))
        (else (error "Bad tagged datum -- TYPE-TAG" datum))
  )
)
  
(define (contents datum)
  (cond 
        ((pair? datum) (cdr datum)) 
        (error "Bad tagged datum -- CONTENTS" datum)
        )
  )

(define (filter predicate sequence)
  (cond ((null? sequence) nil)
        ((predicate (car sequence))
         (cons (car sequence)
               (filter predicate (cdr sequence))))
        (else (filter predicate (cdr sequence)))))

(define (size list)
  (if (null? list) 0 (+ 1 (size (cdr list)))))

(define (list-and proc list)
    (define (iter list)
        (if (null? list)
            #t
            (let ((rs (proc (car list))))
              (if (boolean? rs)
                  (if rs (iter (cdr list)) #f)
                  (error "Proc should return boolean")
              )
            )
        )
    )
  
    (cond ((null? list) (error "Operation not defined for 0 items"))
          ((not (pair? list)) (error "Operation works only for lists"))
          (else (iter list))
    )
)                              
(define (accumulate op initial sequence)
  (if (null? sequence)
      initial
      (op (car sequence)
          (accumulate op initial (cdr sequence)))
  )
)


;;;;;;;;;;;;;;;;;;;;;;;;

(define (install-int-package)
  (define (tag x)
    (attach-tag 'int x))
  (put 'add '(int int)
       (lambda (x y) (tag (+ x y))))
  (put 'sub '(int int)
       (lambda (x y) (tag (- x y))))
  (put 'mul '(int int)
       (lambda (x y) (tag (* x y))))
  (put 'div '(int int)
       (lambda (x y) (tag (/ x y))))
  (put 'make 'int
       (lambda (x) (tag x)))  
  ;equ?
  (put 'equ? '(int int) =)
  (put 'negate '(int) -)
  
  ; raise operation
  (put-raise 'int (lambda(x) (make-rational (contents x) 1)))
  ; set parent
  (put-parent 'int 'rational)  
 
  (put '=zero? '(int) (lambda (x) (= x 0)))

  'done
)

(define (make-integer n)
  ((get 'make 'int) n)
)



;;;;;;;;;;;;;;;;;;;;;;;;
(define (install-scheme-number-package)
  (define (tag x)
    (attach-tag 'scheme-number x))
  (put 'add '(scheme-number scheme-number)
       (lambda (x y) (tag (+ x y))))
  (put 'sub '(scheme-number scheme-number)
       (lambda (x y) (tag (- x y))))
  (put 'mul '(scheme-number scheme-number)
       (lambda (x y) (tag (* x y))))
  (put 'div '(scheme-number scheme-number)
       (lambda (x y) (tag (/ x y))))
  (put 'make 'scheme-number
       (lambda (x) (tag x)))  
  ;equ?
  (put 'equ? '(scheme-number scheme-number) =)
  (put 'negate '(scheme-number) -)
  
  ;; following added to Scheme-number package
  (put 'exp '(scheme-number scheme-number)
       (lambda (x y) (tag (expt x y)))) ; using primitive expt

  ; raise operation
  (put-raise 'scheme-number (lambda(x) (make-complex-from-real-imag (contents x) 0)))
  ; set parent
  (put-parent 'scheme-number 'complex)  

  ;a simple way to convert from real to rational is by multiplying and dividing by a
  ; constant but it will only be correct to 4 places of decimal(4 zeroes in 10000)
  (define multiplier 10000)
  ; project operation
  (put-projection 'scheme-number (lambda(c) (make-rational (floor (* (contents c) multiplier)) multiplier)))
  ; set child
  (put-child 'scheme-number 'rational)
 
  (put '=zero? '(scheme-number) (lambda (x) (= x 0)))

  'done
)

(define (make-number n)
  ((get 'make 'scheme-number) n)
)  

(define (install-rational-package)
  ;; internal procedures
  (define (numer x) (car x))
  (define (denom x) (cdr x))
  (define (make-rat n d)
    (if (and (integer? n) (integer? d))
        (let ((nn (inexact->exact n))
              (dd (inexact->exact d))
             )
             (let ((g (gcd nn dd)))
                 (cons (/ nn g) (/ dd g))
             )
        )
        (error "Number and Denom shoud be integers" n d)
     )
  )
  (define (add-rat x y)
    (make-rat (+ (* (numer x) (denom y))
                 (* (numer y) (denom x)))
              (* (denom x) (denom y))))
  (define (sub-rat x y)
    (make-rat (- (* (numer x) (denom y))
                 (* (numer y) (denom x)))
              (* (denom x) (denom y))))
  (define (mul-rat x y)
    (make-rat (* (numer x) (numer y))
              (* (denom x) (denom y))))
  (define (div-rat x y)
    (make-rat (* (numer x) (denom y))
              (* (denom x) (numer y))))

  ;; equ?
  (define (equ? x y) 
    (and (= (numer x) (numer y)) (= (denom x) (denom y)))) 
     
  ;; interface to rest of the system
  (define (tag x) (attach-tag 'rational x))
  (put 'add '(rational rational)
       (lambda (x y) (tag (add-rat x y))))
  ; comment this to check the example suggested above for tree structure support
  (put 'sub '(rational rational)
       (lambda (x y) (tag (sub-rat x y))))
  (put 'mul '(rational rational)
       (lambda (x y) (tag (mul-rat x y))))
  (put 'div '(rational rational)
       (lambda (x y) (tag (div-rat x y))))
  (put 'make 'rational
       (lambda (n d) (tag (make-rat n d))))
  (put 'equ? '(rational rational) equ?)
  (put 'negate '(rational) (lambda (r) (make-rat (- (numer r)) (denom r))))

  ; raise operation
  (put-raise 'rational (lambda(r) (make-number (/ (numer (contents r)) (denom (contents r))))))
  ; set parent
  (put-parent 'rational 'scheme-number)

  ; project operation
  (put-projection 'rational (lambda(r) (make-integer (floor (/ (numer (contents r)) (denom (contents r)))))))
  ; set child
  (put-child 'rational 'int)
  
  ; uncomment this and comment above raise to make rational as child of complex instead of number
  ; (put-raise 'rational (lambda(r) (make-complex-from-real-imag (/ (numer (contents r)) (denom (contents r))) 0)))
  ; uncomment this and comment above raise to make rational as child of complex instead of number
  ; (put-parent 'rational 'complex)

  (put '=zero? '(rational) (lambda (x) (= (numer x) 0)))
  
  'done
  )

(define (make-rational n d)
  ((get 'make 'rational) n d)
  )

(define (install-complex-package)
  ;; imported procedures from rectangular and polar packages
  (define (make-from-real-imag x y)
    ((get 'make-from-real-imag 'rectangular) x y))
  (define (make-from-mag-ang r a)
    ((get 'make-from-mag-ang 'polar) r a))
  ;; internal procedures
  (define (add-complex z1 z2)
    (make-from-real-imag (add (apply-generic 'real-part z1) (apply-generic 'real-part z2))
                         (add (apply-generic 'imag-part z1) (apply-generic 'imag-part z2))))
  (define (sub-complex z1 z2)
    (make-from-real-imag (sub (apply-generic 'real-part z1) (apply-generic 'real-part z2))
                         (sub (apply-generic 'imag-part z1) (apply-generic 'imag-part z2))))
  (define (mul-complex z1 z2)
    (make-from-mag-ang (mul (apply-generic 'magnitude z1) (apply-generic 'magnitude z2))
                       (add (apply-generic 'angle z1) (apply-generic 'angle z2))))
  (define (div-complex z1 z2)
    (make-from-mag-ang (div (apply-generic 'magnitude z1) (apply-generic 'magnitude z2))
                       (sub (apply-generic 'angle z1) (apply-generic 'angle z2))))
  (define (equ? z1 z2) 
    (and
     (apply-generic 'equ? (apply-generic 'real-part z1) (apply-generic 'real-part z2))
     (apply-generic 'equ? (apply-generic 'imag-part z1) (apply-generic 'imag-part z2))
     )
  )

  (define (complex-negate z)
      (make-from-real-imag
           (apply-generic 'negate (apply-generic 'real-part z))
           (apply-generic 'negate (apply-generic 'real-part z))
      )
  )


  ;; interface to rest of the system
  (define (tag z) (attach-tag 'complex z))
  (put 'add '(complex complex)
       (lambda (z1 z2) (tag (add-complex z1 z2))))
  (put 'sub '(complex complex)
       (lambda (z1 z2) (tag (sub-complex z1 z2))))
  (put 'mul '(complex complex)
       (lambda (z1 z2) (tag (mul-complex z1 z2))))
  (put 'div '(complex complex)
       (lambda (z1 z2) (tag (div-complex z1 z2))))
  (put 'make-from-real-imag 'complex
       (lambda (x y) (tag (make-from-real-imag x y))))
  (put 'make-from-mag-ang 'complex
       (lambda (r a) (tag (make-from-mag-ang r a))))
  (put 'equ? '(complex complex) equ?)
  (put `nagate '(complex) complex-negate)

  ; raise operation is not defined for complex
  ; parent not defined for complex

  ; project operation
  (put-projection
          'complex
           (lambda(c)
               ;note that we can not call apply-generic here because this method is used by drop and drop is used in generic
               ;thus it can go into a loop
               ;so need to take care of type-tags here only
               ; c is complex number(includes tag complex) cc is rectangular/angular including the type-tag
               ; Note that, the repurcurssions of drop lead to the point that real and imag can be any number except number
               ; thus raising it to the real number type (because apart from complex numbers, my understanding is any number can be raised to real number
               (let ((cc (contents c)))
                     (raise-to-type 'scheme-number ((get 'real-part (list (type-tag cc))) (contents cc)))
               )
           )
  )
  ; set child
  (put-child 'complex 'scheme-number)

   ;; complex package:
   (define (=is-zero? z) 
       (and
           (= (apply-generic 'real-part z) 0)
           (= (apply-generic 'imag-part z) 0)
       )
   )
  (put '=zero? '(complex) =is-zero?)
  
  'done
)

(define (sqrt x)
  ;note the magic of apply-generic
  ;it will convert that rational into real because expt is only defined in
  ;scheme-number package and rational can be raised to scheme-number
  (exp x (make-rational 1 2))
)
        

(define (install-rectangular-package)
  ;; internal procedures
  (define (real-part z) (car z))
  (define (imag-part z) (cdr z))
  (define (make-from-real-imag x y) (cons x y))
  (define (magnitude z)
    (sqrt (add (square (real-part z))
             (square (imag-part z)))))
  (define (angle z)
    (arctan (div (imag-part z) (real-part z))))
  (define (make-from-mag-ang r a)
    (cons (mul r (cosine a)) (mul r (sine a))))
  ;; interface to the rest of the system
  (define (tag x) (attach-tag 'rectangular x))
  (put 'real-part '(rectangular) real-part)
  (put 'imag-part '(rectangular) imag-part)
  (put 'magnitude '(rectangular) magnitude)
  (put 'angle '(rectangular) angle)
  (put 'make-from-real-imag 'rectangular
       (lambda (x y) (tag (make-from-real-imag x y))))
  (put 'make-from-mag-ang 'rectangular
       (lambda (r a) (tag (make-from-mag-ang r a))))
  'done)
 
(define (install-polar-package)
  ;; internal procedures
  (define (magnitude z) (car z))
 
  (define (angle z) (cdr z))
  (define (make-from-mag-ang r a) (cons r a))
  (define (real-part z)
    (mul (magnitude z) (cosine (angle z))))
  (define (imag-part z)
    (mul (magnitude z) (sine (angle z))))
  (define (make-from-real-imag x y)
    (cons (sqrt (add (square x) (square y)))
          (arctan (div y x))))
  ;; interface to the rest of the system
  (define (tag x) (attach-tag 'polar x))
  (put 'real-part '(polar) real-part)
  (put 'imag-part '(polar) imag-part)
  (put 'magnitude '(polar) magnitude)
  (put 'angle '(polar) angle)
  (put 'make-from-real-imag 'polar
       (lambda (x y) (tag (make-from-real-imag x y))))
  (put 'make-from-mag-ang 'polar
       (lambda (r a) (tag (make-from-mag-ang r a))))
  'done)

(define (make-complex-from-real-imag x y)
  ((get 'make-from-real-imag 'complex) x y))
(define (make-complex-from-mag-ang r a)
  ((get 'make-from-mag-ang 'complex) r a))

(define (equ? x y)
  (apply-generic 'equ? x y)
)

(define (exp x y) (apply-generic 'exp x y))
(define (add x y) (apply-generic 'add x y))
(define (sub x y) (apply-generic 'sub x y))
(define (mul x y) (apply-generic 'mul x y))
(define (div x y) (apply-generic 'div x y))

(define (=zero? x) (apply-generic '=zero? x))
(define (negate x) (apply-generic 'negate x))


(install-rectangular-package)
(install-polar-package)
(install-scheme-number-package)
(install-rational-package)
(install-int-package)
(install-complex-package)
(install-sparse-termlist-package)
(install-dense-termlist-package)
(install-polynomial-package)

Examples/Output

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> (display (add (add (make-polynomial 'y (attach-tag 'sparse-termlist '((5 2) (3 2)))) (make-polynomial 'x (attach-tag 'sparse-termlist '((5 3) (3 6))))) (make-polynomial 'x (attach-tag 'sparse-termlist '((5 9) (0 1000)) ))))
(polynomial x sparse-termlist (5 (int . 12)) (3 6) (0 (polynomial y sparse-termlist (5 2) (3 2) (0 1000))))
> (display (mul (make-polynomial 'x (attach-tag 'sparse-termlist '((5 2) (3 2)))) (make-polynomial 'x (attach-tag 'sparse-termlist '((5 3) (3 6))))))
(polynomial x sparse-termlist (10 (int . 6)) (8 (int . 18)) (6 (int . 12)))
> (display (mul (make-polynomial 'y (attach-tag 'sparse-termlist '((5 2) (3 2)))) (make-polynomial 'x (attach-tag 'sparse-termlist '((5 3) (3 6))))))
(polynomial x sparse-termlist (5 (polynomial y sparse-termlist (5 (int . 6)) (3 (int . 6)))) (3 (polynomial y sparse-termlist (5 (int . 12)) (3 (int . 12)))))
> (display (add (make-polynomial 'x (attach-tag 'sparse-termlist '((2 1) (0 2)))) (make-polynomial 'y (attach-tag 'sparse-termlist '((0 8)))) ))
(polynomial x sparse-termlist (2 1) (0 (int . 10)))
> (display (add (make-polynomial 'x (attach-tag 'sparse-termlist '((2 1) (0 2)))) (make-polynomial 'x (attach-tag 'sparse-termlist '((0 8)))) ))
(polynomial x sparse-termlist (2 1) (0 (int . 10)))
; Note this below example where all the terms of polynomial in x are gone in addition except of the constant part. This can be further improved - see the procedure make-poly-with-term
> (display (add (add (make-polynomial 'y (attach-tag 'sparse-termlist '((5 2) (3 2)))) (make-polynomial 'x (attach-tag 'sparse-termlist '((5 3) (3 6))))) (make-polynomial 'x (attach-tag 'sparse-termlist '((5 -3) (3 -6) (0 1000)) ))))
(polynomial x sparse-termlist (0 (polynomial y sparse-termlist (5 2) (3 2) (0 1000))))
>