Exercise 2.92 polynomials in different variables.rkt

#lang racket

; Exercise 2.92.  By imposing an ordering on variables, extend the polynomial package so that addition and
; multiplication of polynomials works for polynomials in different variables. (This is not easy!)

; S O L U T I O N

; Assumed ordering of variables is (in increasing order of priority):
; p, q, r, s, t, u, v, w, x, y, z
; I have implemented a generic print procedure so that polynomials and other types can be printed
; in an easy to read manner.
; The main work in this exercise lies in the 'convert-polynomial' procedure that transforms a polynomial
; from one variable to another

; This problem was quite hard and took me several days to solve correctly. Right in the beginning, I decided to write a 'convert' procedure that 'casts' a polynomial in one variable to the same polynomial in a different variable by re-arranging the terms. In this procedure I needed to account for the fact that the coefficients of a polynomial can themselves be polynomials. This 'polynomials inside a polynomial' can go to any depth.

; The hardest part was to find a clean, generic way to do the multiplying and re-arranging of variables that we so easily do by hand. 

; At the very beginning I decided that a pretty-print procedure was needed for the polynomials because it is very hard to read the internal list representation (sparse or dense) of a polynomial. This print-poly procedure really saved my life especially while testing my program.

; I knew that the convert-polynomial procedure had to be recursive. If a polynomial in x needs to be converted to a polynomial in y, then we can convert each term of the x-polynomial to a y-polynomial and then add them all together using ordinary polynomial addition. So this was the main driving recursive logic for the conversion process. The 'first-terms' and 'rest-terms' procedures came in handy here.

; Once this main driving logic was in place, I started thinking about the conditions where the recursion terminates and the call-stack unwinds. These conditions are:

; 1. If we have an empty polynomial in x, then convert-polynomial simply returns an empty polynomial in y.

; 2. If the polynomial we are trying to convert is already in the target variable, then there is nothing to do. Simply return the same polynomial.

; 3. If the polynomial has only one term then we have to account for the following possibilities:

; 3a. The coefficient of this term is a non-empty polynomial: In this case, first convert the coefficient so that it is in the target variable. Then do something like this (In this example, the converted coefficient is Cx^2):

; Cx^2y^4 needs to be transformed to Cy^4x^2 i.e.

; ('polynomial y (4 (polynomial x (2 C))))
; needs to become
; ('polynomial x (2 (polynomial y (4 C))))

; Note that the converted coefficient can be a polynomial with multiple terms itself so the above variable swapping transformation needs to be done for all the terms and after that these terms need to be added back together. Again, the 'first-term' and 'rest-terms' procedures are useful for this.

; 3b. Coefficient is an empty polynomial: Return an empty polynomial in the new variable.

; 3c. Coefficient in the term is not a polynomial:  Example: In 7y^4, the coefficient is not a polynomial but just a constant. Assuming that we are converting polynomials in y to polynomials in x, this needs to be transformed to (7y^4)x^0.

; It took multiple rounds of analysis and thought to arrive at this structure.

; Other procedures I needed to write to support this conversion process:

; higher-in-hierarchy?: This is a yes/no procedure that tells me if the variable specified by the first argument is higher in the hierarchy than the variable specified by the second argument. When we have to combine two polynomials in different variables, this procedure is used to determine which one to convert to the other.

; Generic =zero? procedure that checks if a polynomial is equal t zero. For this we basically check if the term list is empty.

; equal-polynomial-real?: Aids in simplifying polynomials. I found that when 'convert-polynomial' runs, it invariably ends up with 'polynomials' that are really just ordinary numbers. This procedure can determine if this is the case and after that we can just replace the polynomial with the number it really is.

; After this, I had to do the following:

; 1. Modify the 'add' and 'mul' procedures for polynomial to polynomial operations to check if the variables are different and if they are, simply call the old 'add-poly' or 'mul-poly' and if not, then determine which variable is higher in the hierarchy, do polynomial conversion in the right direction and then call 'add-poly' or 'mul-poly'.

; 2. I found that there is a need to support the addition and multiplication of polynomials to real numbers and complex numbers. This is because, when we combine terms, the coefficient of one term may be a polynomial and the coefficient of the other term may be an ordinary number. So I implemented a generic procedure for this. Again, the method is to convert the real number to a polynomial in the same variable as the other polynomial and then do ordinary polynomial addition or multiplication.

; During testing, I encountered infinite loop issues multiple times because the 'raises' and 'drops' in the generic procedure framework cause this when crucial combinations are missing in the op-table. I broke these infinite loops by supporting operations/type combinations that I had missed earlier.

; GENERIC PROCEDURES

; Generic Polynomial procedures
; Note: I have designed this with the assumption that the procedures adjoin-term, first-term and rest-terms
; though generic, will still be used only internally by the polynomial procedures.
; These three procedures are generic but not exposed to the outside world.
(define (variable p) (apply-generic 'variable p))
(define (term-list p) (apply-generic 'term-list p))
(define (adjoin-term term term-list) (apply-generic 'adjoin-term term term-list))
(define (first-term term-list) (apply-generic 'first-term term-list))
(define (rest-terms term-list) (apply-generic 'rest-terms term-list))
(define (order term) (apply-generic 'order term))
(define (coeff term) (apply-generic 'coeff term))
(define (negate-term term) (apply-generic 'negate-term term))

; Generic Print procedure
(define (print x) (apply-generic 'print x))

; Generic Logical procedures
(define (ABS x) (apply-generic 'abs x))
(define (=zero? x) (apply-generic '=zero? x))
(define (equ? x y) (apply-generic 'equal? x y))
(define (greater? x y) (apply-generic 'greater? x y))
(define (lesser? x y) (apply-generic 'lesser? x y))
(define (empty-term? t) (apply-generic 'empty-term? t))

(define (square x) (apply-generic 'square x))
(define (square-root x) (apply-generic 'square-root x))
(define (NEGATE x) (apply-generic 'negate x))

; Generic Arithmetic procedures with two arguments
(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 (exp x y) (apply-generic 'exp x y))

; Generic Trigonometric procedures
(define (sine x) (apply-generic 'sine x))
(define (cosine x) (apply-generic 'cosine x))
(define (tan-inverse x y) (apply-generic 'tan-inverse x y))

; Generic Contrived procedures
(define (add-four-quantities w x y z) (apply-generic 'add-four-quantities w x y z))
(define (mul-and-scale x y factor) (apply-generic 'mul-and-scale x y factor))
(define (mul-five-quantities v w x y z) (apply-generic 'mul-five-quantities v w x y z))

; Generic operation that 'projects' x one level lower in the tower
(define (project x) (apply-generic 'project x))

; Generic operation that raises x one level in the tower
(define (raise x) (apply-generic 'raise x))

; Generic Constructions of specific types of entities (ordinary, rational, complex etc.)
(define (make-scheme-number n) ((get 'make 'scheme-number) n))
(define (make-natural n) ((get 'make 'natural) n))
(define (make-integer n) ((get 'make 'integer) n))
(define (make-rational n d) ((get 'make 'rational) n d))
(define (make-real r) ((get 'make 'real) r))

(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 (make-polynomial var terms) ((get 'make 'polynomial) var terms))
(define (make-polynomial-dense-terms terms) ((get 'make 'polynomial-dense-terms) terms))
(define (make-polynomial-sparse-terms terms) ((get 'make 'polynomial-sparse-terms) terms))
(define (make-term order coeff) ((get 'make 'polynomial-term) order coeff))

(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))

; POLYNOMIAL PROCEDURES

(define (make-poly variable term-list) (cons variable term-list))
(define (make-dense-terms terms) terms)
(define (make-sparse-terms terms) terms)

(define (variable-poly p) (car p))
(define (term-list-poly p) (cdr p))

; Returns the polynomial constructed with the first-term of the supplied polynomial
(define (first-term-part-poly p)
	(make-polynomial (variable p) (make-polynomial-sparse-terms (list (contents (first-term (term-list p))))))
)

; Returns the polynomial constructed with the rest-terms of the supplied polynomial
(define (rest-terms-part-poly p)
	(let ((rt (rest-terms (term-list p))))
		(if (empty-termlist? rt)
			(make-polynomial (variable p) (the-empty-poly-termlist))
			(make-polynomial (variable p) (rest-terms (term-list p)))
		)
	)
)

(define (print-poly p)

	(define (print-poly-internal var terms first-time?)
		(cond
			((empty-termlist? terms) (display "}"))
			(else
				(let ((coeff-first-term (coeff (first-term terms))) (order-first-term (order (first-term terms))))
					; Print + or -
					(cond
						((=zero? coeff-first-term) (display ""))
						((and (pair? coeff-first-term) (lesser? coeff-first-term 0)) (display " ") (display '-) (display " "))
						((and (pair? coeff-first-term) (not first-time?)) (display " ") (display '+) (display " "))
						((and (not (pair? coeff-first-term)) (> coeff-first-term 0) (not first-time?)) (display " ") (display '+) (display " "))
						((and (not (pair? coeff-first-term)) (< coeff-first-term 0)) (display " ") (display '-) (display " "))
					)
					; Print the coefficient
					(cond
						((=zero? coeff-first-term) (display ""))
						((and (pair? coeff-first-term) (not (or (equ? coeff-first-term 1) (equ? coeff-first-term -1))))
							; Expecting this to be a polynomial
							(print (ABS coeff-first-term))
						)
						((not (equ? coeff-first-term 1))
							(display (ABS coeff-first-term))
						)
					)
					; Print the variable and order
					(cond
						((=zero? coeff-first-term) (display ""))
						((equal? order-first-term 1)
							(display var)
						)
						((not (=zero? order-first-term))
							(display var)
							(display '^)
							(print order-first-term)
						)
					)
					(if (and (=zero? coeff-first-term) first-time?)
						(print-poly-internal var (rest-terms terms) true)
						(print-poly-internal var (rest-terms terms) false)
					)
				)
			)
		)
	)

	; (display "Printing: ")
	; (display p)
	; (newline)
	(display "{")
	(print-poly-internal (variable-poly p) (term-list-poly p) true)
)

(define (is-poly? p)
	(if (pair? p)
		(equal? (type-tag p) 'polynomial)
		false
	)
)

(define (the-empty-poly-termlist) '())
; (define (the-empty-poly-termlist) (list 'polynomial-sparse-terms (list 0 0)))

(define (convert-polynomial p new-var)
	; Converts the polynomial by expanding and rearranging the terms so that 'new-var' becomes the variable
	; of the polynomial
	; Assumptions made in this implementation:
	; 1. The supplied object p should be tagged with the tag 'polynomial
	; 2. We expect that a polynomial in a variable say z, will not have any coefficient that itself contains z
	(cond
		; If the polynomial is empty, then return a new empty polynomial in the new-variable
		((empty-termlist? (term-list p)) (make-polynomial new-var (the-empty-poly-termlist)))
		; If the polynomial is empty, then return zero
		; ((empty-termlist? (term-list p)) 0)
		; if new-var is not really new, return the polynomial as it is
		((equal? (variable p) new-var) p)
		; If the polynomial has only one term
		((empty-termlist? (rest-terms (term-list p)))
			(let ((the-term (first-term (term-list p))))
				(cond
					; Coefficient is a non-empty polynomial
					((and (is-poly? (coeff the-term)) (not (empty-termlist? (term-list (coeff the-term)))))
	 					; coeff in the term is a polynomial by itself
						; First, convert the coefficient to be in the target variable.
						; Then do something like this (In this example, the converted coefficient is Cx^2):
						; Cx^2y4 needs to become Cy^4x^2
						; i.e.
						; ('polynomial y (4 (polynomial x (2 C)))) needs to become
						; ('polynomial x (2 (polynomial y (4 C))))
						(let ((converted-inner-poly (convert-polynomial (coeff the-term) new-var)))
							(let (
								  ; (cip-first-term-poly (first-term-part-poly converted-inner-poly))
								  (cip-rest-terms-poly (rest-terms-part-poly converted-inner-poly))
								  (cip-first-term-order (order (first-term (term-list converted-inner-poly))))
								  (cip-first-term-coeff (coeff (first-term (term-list converted-inner-poly))))
								 )
								(add
									(make-polynomial
										new-var
										(make-polynomial-sparse-terms
											(list
												(list
													; ORDER
													cip-first-term-order
													; COEFFICIENT
													(make-polynomial
														(variable p)
														(make-polynomial-sparse-terms
															(list
																(list
																	(order the-term)
																	cip-first-term-coeff
																)
															)
														)
													)
												)
											)
										)
									)
									; LOGIC PROBLEM HERE
									(convert-polynomial
										; cip-rest-terms-poly
										(make-polynomial
											(variable p)
											(make-polynomial-sparse-terms
												(list
													(list
														; ORDER
														(order the-term)
														; COEFFICIENT
														cip-rest-terms-poly
													)
												)
											)
										)
										new-var
									)
								)
							)
						)
	 					; coeff in the term is not a polynomial
	 					; Example: 7y^4 becomes (7y^4)x^0
	 				)
					; Coefficient is an empty polynomial, so return an empty polynomial in the new variable
					((and (is-poly? (coeff the-term)) (empty-termlist? (term-list (coeff the-term))))
						(make-polynomial new-var (the-empty-poly-termlist))
					)
					(else
	 					(make-polynomial new-var (make-polynomial-sparse-terms (list (list 0 p))))
					)
				)
			)
		)
		; If the polynomial has more than one term
		; This is the high level driving logic
		((not (equal? (variable p) new-var))
			(add
				(convert-polynomial
					(make-polynomial (variable p) (make-polynomial-sparse-terms (list (contents (first-term (term-list p))))))
					new-var
				)
				(convert-polynomial
					(make-polynomial (variable p) (rest-terms (term-list p)))
					new-var
				)
			)
		)
		(else
			(error "Unable to convert " (list p new-var))
		)
	)
)

; Term-list manipulation for dense polynomials
(define (adjoin-term-dense term term-list)
	; Structure of term-list
	; [OrderOfPolynomial, (list of coefficients)]
	; Maintaining the order of the polynomial as the first element allows us to avoid repeated
	; expensive 'length' calls on the coefficient list
	; Reminder: We assume that term lists are represented as lists of terms,
	; arranged from highest-order to lowest-order term.
	(if (=zero? (coeff-poly-term term))
		term-list
		(cond
			((= (order-poly-term term) (+ (polynomial-order-dense term-list) 1))
				; no need to insert a zero in the coefficient list
				(cons (+ 1 (polynomial-order-dense term-list)) (cons (coeff-poly-term term) (coefficients-dense term-list)))
			)
			((and (empty-termlist? term-list) (= 0 (order-poly-term term)))
				(list 0 (coeff-poly-term term))
			)
			((and (empty-termlist? term-list) (> (order-poly-term term) 0))
				(adjoin-term-dense term (list 0 0))
			)
			(else
				; we need to supply zero(s) if there are gaps
				(adjoin-term-dense term (cons (+ 1 (polynomial-order-dense term-list)) (cons 0 (coefficients-dense term-list))))
			)
		)
	)
)

(define (first-term-dense term-list)
	; Since we are using the term-list representation that is appropriate for
	; dense polynomials (see SICP text), we need to do some extra processing
	; to retrieve both the order and coefficient
	(if (pair? term-list)
		(if (pair? (coefficients-dense term-list))
			(list (polynomial-order-dense term-list) (car (coefficients-dense term-list)))	
			null
		)
		null
	)
)

(define (rest-terms-dense term-list)
	; Since we are using the term-list representation that is appropriate for
	; dense polynomials (see SICP text), we need to do some extra processing
	; to retrieve both the order and coefficient
	(if (> (polynomial-order-dense term-list) 0)
		(cons (- (polynomial-order-dense term-list) 1) (cdr (coefficients-dense term-list)))
		(the-empty-poly-termlist)
	)
)

(define (polynomial-order-dense term-list)
	(if (pair? term-list)
		(car term-list)
		0
	)
)

(define (coefficients-dense term-list)
	(if (pair? term-list)
		(cdr term-list)
		(list 0)
	)
)

; Term-list manipulation for sparse polynomials
(define (adjoin-term-sparse term term-list)
	(if (=zero? (coeff-poly-term term))
		term-list
		(cons term term-list)
	)
)

(define (first-term-sparse term-list)
	(cond
		((null? term-list) null)
		((not (pair? term-list)) (error "Procedure first-term-sparse: term-list is not a pair!"))
		(else
			(car term-list)
		)
	)
)

(define (rest-terms-sparse term-list)
	(cond
		((null? term-list) (the-empty-poly-termlist))
		((not (pair? term-list)) (error "Procedure rest-terms-sparse: term-list is not a pair!"))
		(else
			(cdr term-list)
		)
	)
)

; Term manipulation (used in both dense and sparse polynomial representations)
(define (make-poly-term order coeff) (list order coeff))

(define (order-poly-term term)
	(if (pair? term)
		(car term)
		0
	)
)

(define (coeff-poly-term term)
	(if (pair? term)
		(cadr term)
		0
	)
)

(define (higher-in-hierarchy? v1 v2)
	; returns true if v1 is higher than v2, otherwise false
	; Items in the below list are arranged from low to high
	(define var-hierarchy (list 'p 'q 'r 's 't 'u 'v 'w 'z 'y 'x))

	(define (find-position item items current-position)
		(cond
			((pair? items)
				(if (equal? item (car items))
					current-position
					(find-position item (cdr items) (+ current-position 1))
				)
			)
			(else
				-1
			)
		)
	)

	(let ((v1-pos (find-position v1 var-hierarchy 1)) (v2-pos (find-position v2 var-hierarchy 1)))
		(cond
			((< v1-pos v2-pos) false)
			((> v1-pos v2-pos) true)
			(else
				; default treatment
				false
			)
		)
	)
)

; Polynomial Operations
(define (add-poly p1 p2)
	(if (same-variable? (variable-poly p1) (variable-poly p2))
		(make-poly
			(variable-poly p1)
			(add-terms (term-list-poly p1) (term-list-poly p2))
		)
		(error "Polys not in same var -- ADD-POLY"
			(list p1 p2)
		)
	)
)

(define (sub-poly p1 p2)
	(add-poly p1 (negate-poly p2))
)

(define (mul-poly p1 p2)
	(if (same-variable? (variable-poly p1) (variable-poly p2))
		(make-poly
			(variable-poly p2)
			(mul-terms (term-list-poly p1) (term-list-poly p2))
		)
		(error "Polys not in same var -- MUL-POLY"
			(list p1 p2)
		)
	)
)

(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
					((greater? (order t1) (order t2))
						(adjoin-term t1 (add-terms (rest-terms L1) L2))
					)
					((lesser? (order t1) (order t2))
						(adjoin-term t2 (add-terms L1 (rest-terms L2)))
					)
					(else
						(adjoin-term
							(make-term (order t1) (add (coeff t1) (coeff t2)))
							(add-terms (rest-terms L1) (rest-terms L2))
						)
					)
				)
			)
		)
	)
)

(define (mul-terms L1 L2)
	(if (or (empty-termlist? L1) (empty-termlist? L2))
		(the-empty-poly-termlist)
		(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 (or (empty-termlist? L) (empty-term? t1))
		; Returning an empty list with an explicit tag so that the generic procedure
		; 'adjoin-term' below does not fail. Generic procedures expect a data tag for each of their arguments
		(list 'polynomial-sparse-terms (list 0 0))
		(let ((t2 (first-term L)))
			(adjoin-term
				(make-term (add (order t1) (order t2)) (mul (coeff t1) (coeff t2)))
				(mul-term-by-all-terms t1 (rest-terms L))
			)
		)
	)
)

(define (div-poly p-dividend p-divisor)
	(if (same-variable? (variable-poly p-dividend) (variable-poly p-divisor))
		(let ((result (div-terms (term-list-poly p-dividend) (term-list-poly p-divisor))))
			(list
				(make-poly (variable-poly p-dividend) (car result))
				(make-poly (variable-poly p-dividend) (cadr result))
			)
		)
		(error "Polys not in same var -- DIV-POLY"
			(list p-dividend p-divisor)
		)
	)
)

(define (div-terms L-dividend L-divisor)
	(if (empty-termlist? L-dividend)
		(list (the-empty-poly-termlist) (the-empty-poly-termlist))
		(let ((ft-dividend (first-term L-dividend)) (ft-divisor (first-term L-divisor)))
			(if (greater? (order ft-divisor) (order ft-dividend))
				(list (the-empty-poly-termlist) L-dividend)
				(let ((new-c (div (coeff ft-dividend) (coeff ft-divisor)))
					  (new-o (sub (order ft-dividend) (order ft-divisor))))
					(let ((rest-of-result
							(div-terms
								(term-list-poly
									(sub-poly
										(make-poly 'x L-dividend)
										(mul-poly
											(make-poly 'x L-divisor)
											(make-poly 'x (make-polynomial-sparse-terms (list (list new-o new-c))))
										)
									)
								)
								L-divisor
							)
						 ))
						; <form complete result>
						(list
							(add-terms
								(make-polynomial-sparse-terms (list (list new-o new-c)))
								(car rest-of-result)
							)
							(cadr rest-of-result)
						)
					)
				)
			)
		)
	)
)

(define (=zero-polynomial? p)
	; A polynomial is zero if its term-list is empty
	(empty-termlist? (term-list-poly p))
)

(define (lesser-polynomial-real? p r)
	; returns true if the polynomial is just a constant which is less than r
	(and (=zero? (order (first-term (term-list-poly p)))) (lesser? (coeff (first-term (term-list-poly p))) r))
)

(define (negate-poly p)
	(make-poly
		(variable-poly p)
		(negate-term-list (term-list-poly p))
	)
)

(define (negate-term-list t)
	(if (empty-termlist? t)
		t
		(adjoin-term (negate-term (first-term t)) (negate-term-list (rest-terms t)))
	)
)

(define (negate-poly-term t)
	(list (order-poly-term t) (mul -1 (coeff-poly-term t)))
)

(define (empty-termlist? t)
	; consider both with tag and without tag
	; This procedure will return true for the following data:
	; '()
	; ('polynomial-sparse-terms)
	; ('polynomial-dense-terms)
	; ('polynomial-sparse-terms (0 0))
	; ('polynomial-dense-terms 0 0)
	(or
		(null? t)
		(and (symbol? (car t)) (null? (cdr t)))
		(and (eq? (car t) 'polynomial-sparse-terms) (eq? (caadr t) 0) (eq? (cadadr t) 0))
		(and (eq? (car t) 'polynomial-dense-terms) (eq? (cadr t) 0) (eq? (caddr t) 0))
	)
)

(define (equal-polynomial-real? p r)
	; returns true if the polynomial is just a constant which is less than r
	(and (=zero? (order (first-term (term-list-poly p)))) (equ? (coeff (first-term (term-list-poly p))) r))
)

(define (empty-poly-term? t)
	(cond
		((null? t) true)
		((not (pair? t)) (error "Procedure empty-term: t not a pair!"))
		(else
			(eq? (cadr t) 0)
		)
	)
)

(define (same-variable? v1 v2)
	(and (variable? v1) (variable? v2) (eq? v1 v2))
)

(define (variable? x) (symbol? x))

; COMPLEX NUMBER PROCEDURES
(define (magnitude z) (apply-generic 'magnitude z))
(define (angle z) (apply-generic 'angle z))
(define (REAL-PART z) (apply-generic 'REAL-PART z))
(define (IMAG-PART z) (apply-generic 'IMAG-PART z))

(define (add-complex z1 z2)
	(make-from-real-imag
		(add (REAL-PART z1) (REAL-PART z2))
		(add (IMAG-PART z1) (IMAG-PART z2))
	)
)

(define (sub-complex z1 z2)
	(make-from-real-imag
		(sub (REAL-PART z1) (REAL-PART z2))
		(sub (IMAG-PART z1) (IMAG-PART z2))
	)
)

(define (mul-complex z1 z2)
	(make-from-mag-ang
		(mul (magnitude z1) (magnitude z2))
		(add (angle z1) (angle z2))
	)
)

(define (div-complex z1 z2)
	(make-from-mag-ang
		(div (magnitude z1) (magnitude z2))
		(sub (angle z1) (angle z2))
	)
)

(define (mul-and-scale-complex z1 z2 factor)
	(display "Entered proc mul-and-scale-complex")
	(newline)
	(let ((prod (mul-complex z1 z2)))
		(make-complex-from-real-imag (mul (REAL-PART prod) factor) (mul (IMAG-PART prod) factor))
	)
)

(define (add-four-complex-numbers z1 z2 z3 z4)
	(display "Entered proc add-four-complex-numbers")
	(newline)
	(make-from-real-imag
		(add (REAL-PART z1) (REAL-PART z2) (REAL-PART z3) (REAL-PART z4))
		(add (IMAG-PART z1) (IMAG-PART z2) (IMAG-PART z3) (IMAG-PART z4))
	)
)

(define (equal-complex? c1 c2)
	(and (equ? (REAL-PART c1) (REAL-PART c2)) (equ? (IMAG-PART c1) (IMAG-PART c2)))
)

(define (=zero-complex? c) (equ? 0 (magnitude c)))

(define (print-complex c)
	(if (=zero? (IMAG-PART c))
		(print (REAL-PART c))
		(begin
			(print (REAL-PART c))
			(display '+)
			(print (IMAG-PART c))
			(display 'i)
		)
	)
)

; (define (project-complex c) (make-real (REAL-PART c)))
(define (project-complex c) (REAL-PART c))

(define (square-complex c) 
	(mul-complex c c)
)

; Rectangular (Complex) Number procedures
(define (make-from-real-imag-rectangular x y) (cons x y))
(define (make-from-mag-ang-rectangular r a) (cons (mul r (cosine a)) (mul r (sine a))))

(define (magnitude-rectangular z)
	(square-root (add (square (real-part-rectangular z)) (square (imag-part-rectangular z))))
)

(define (angle-rectangular z) (tan-inverse (imag-part-rectangular z) (real-part-rectangular z)))

(define (real-part-rectangular z) (car z))
(define (imag-part-rectangular z) (cdr z))

(define (raise-rectangular r)
	(make-complex-from-real-imag (real-part-rectangular r) (imag-part-rectangular r))
)

; Polar (Complex) Number procedures
(define (make-from-real-imag-polar x y) (cons (square-root (add (square x) (square y))) (tan-inverse y x)))
(define (make-from-mag-ang-polar r a) (cons r a))
(define (magnitude-polar z) (car z))
(define (angle-polar z) (cdr z))
(define (real-part-polar z) (mul (magnitude-polar z) (cosine (angle-polar z))))
(define (imag-part-polar z) (mul (magnitude-polar z) (sine (angle-polar z))))

(define (raise-polar p)
	(make-complex-from-mag-ang (magnitude-polar p) (angle-polar p))
)

; REAL NUMBER PROCEDURES
(define (make-real-specific r)
	(if (real? r)
		r
		(error "Cannot make real with: " r)
	)
)

(define (abs-real r)
	(abs r)
)

(define (=zero-real? r)
	(= 0 r)
)

(define (print-real r)
	(display r)
)

(define (raise-real r)
	(make-complex-from-real-imag r 0)
)

(define (project-real r)
	(make-rational r 1)
)

(define (square-root-real r)
	(if (>= r 0)
		(sqrt r)
		(make-complex-from-real-imag 0 (sqrt (abs r)))
	)
)

(define (square-real r)
	(* r r)
)

(define (cosine-real r)
	(cos r)
)

(define (tan-inverse-real r1 r2)
	(atan r1 r2)
)

(define (sine-real r)
	(sin r)
)

; RATIONAL NUMBER PROCEDURES
(define (numer x) (car x))
(define (denom x) (cdr x))

(define (make-rational-specific n d)
	(define (construct-rational n d)
		(let ((g (gcd n d)))
			(cond
				((= d 0) (error "Denominator in a rational number cannot be zero"))
				((= n 0) (cons n d))
				((and (< n 0) (< d 0)) (cons (/ (abs n) g) (/ (abs d) g)))
				((and (< n 0) (> d 0)) (cons (/ n g) (/ d g)))
				((and (> n 0) (< d 0)) (cons (/ (* -1 n) g) (/ (abs d) g)))
				((and (> n 0) (> d 0)) (cons (/ n g) (/ d g)))
			)
		)
	)

	(cond
		((and (integer? n) (integer? d))
			(construct-rational (exact-round n) (exact-round d))
		)
		; if both supplied arguments are not integers, try combining them and try to make a rational
		; Example: 4 / 0.5 gives us 8 which is rational
		((integer? (/ n d))
			(construct-rational (exact-round (/ n d)) 1)
		)
		(else
			(error "Rational number cannot be made with: " n d)
		)
	)
)

(define (add-rational x y) (make-rational-specific (+ (* (numer x) (denom y)) (* (numer y) (denom x))) (* (denom x) (denom y))))
(define (sub-rational x y) (make-rational-specific (- (* (numer x) (denom y)) (* (numer y) (denom x))) (* (denom x) (denom y))))
(define (mul-rational x y) (make-rational-specific (* (numer x) (numer y)) (* (denom x) (denom y))))
(define (div-rational x y) (make-rational-specific (* (numer x) (denom y)) (* (denom x) (numer y))))

(define (equal-rational? x y)
	; (display "Entered equal-rational?")
	; (newline)
	(and (= (numer x) (numer y)) (= (denom x) (denom y)))
)

(define (greater-rational? x y)
	(> (* (numer x) (denom y)) (* (denom x) (numer y)))
)

(define (lesser-rational? x y)
	(< (* (numer x) (denom y)) (* (denom x) (numer y)))
)

(define (=zero-rational? x)
	(= 0 (numer x))
)

(define (square-root-rational rat)
	(if (>= (numer rat) 0)
		(sqrt (/ (numer rat) (denom rat)))
		(make-complex-from-real-imag 0 (sqrt (abs (/ (numer rat) (denom rat)))))
	)
)

(define (square-rational rat)
	(cons (* (numer rat) (numer rat)) (* (denom rat) (denom rat)))
)

(define (cosine-rational rat)
	(cos (/ (numer rat) (denom rat)))
)

(define (tan-inverse-rational rat1 rat2)
	(atan (/ (numer rat1) (denom rat1)) (/ (numer rat2) (denom rat2)))
)

(define (sine-rational rat)
	(sin (/ (numer rat) (denom rat)))
)

(define (raise-rational r)
	(make-real (* 1.0 (/ (numer r) (denom r))))
)

(define (project-rational r)
	; tries to push this object down one step in the tower
	(make-integer (numer r))
)

(define (mul-five-rationals v w x y z)
	(mul-rational (mul-rational (mul-rational (mul-rational v w) x) y) z)
)

; INTEGER PROCEDURES
(define (abs-integer i)
	(abs i)
)

(define (make-integer-specific n)
	(if (integer? n)
		n
		(error "Cannot make integer with: " n)
	)
)

(define (print-int i)
	(display i)
)

(define (raise-int n)
	(make-rational n 1)
)

(define (project-int n)
	; tries to push this object down one step in the tower
	(make-natural n)
)

(define (square-root-integer i)
	(if (>= i 0)
		(sqrt i)
		(make-complex-from-real-imag 0 (sqrt (abs i)))
	)
)

(define (square-integer n)
	(* n n)
)

(define (cosine-integer i)
	(cos i)
)

(define (tan-inverse-integer i1 i2)
	(atan i1 i2)
)

(define (sine-integer i)
	(sin i)
)

; NATURAL NUMBER PROCEDURES
(define (make-natural-specific n)
	(if (and (>= n 0) (or (natural? n) (= n (exact-round n))))
		n
		(error "Cannot make natural with: " n)
	)
)

(define (abs-natural n)
	(abs n)
)

(define (print-natural n)
	(display n)
)

(define (raise-natural n)
	(make-integer n)
)

(define (square-root-natural n)
	(sqrt n)
)

(define (square-natural n)
	(* n n)
)

(define (cosine-natural n)
	(cos n)
)

(define (tan-inverse-natural n1 n2)
	(atan n1 n2)
)

(define (sine-natural n)
	(sin n)
)

(define (project-natural n)
	; Natural numbers are the lowest in the tower, we will simply return n itself (in effect getting rid of
	; the type-tag)
	n
)

; GENERIC PROCEDURE FRAMEWORK

(define (apply-generic op . args)
	
	(define (apply-generic-internal op position args)
		(let ((type-tags (map type-tag args)))
			(let ((proc (get op type-tags)))
				(if proc
					(apply proc (map contents args))
					; proc not found so we need to try coercion (provided the args are not all
					; of the same type)
					(if (dissimilar? type-tags)
						(if (<= position (length args))
							; Coerce all arguments to the type of the argument that is in 'position'
							; position in the list. If all the coercions obtained are valid,
							; then apply the operation on the coerced arguments.
							(let ((target-type (find-element type-tags position)))
								; type-tags will be something like:
								; ('complex 'rational 'real)
								; from this we want to generate a list of procedures like:
								; (complex->complex rational->complex 'real->complex)
								; If all the above procedures are valid, then we will apply them to
								; the respective arguments to do the coercion
								(let ((coercion-procs (build-coercion-proc-list type-tags target-type)))
									(if (allValid? coercion-procs)
										; coerce all the arguments
										(let ((coerced-args (coerce args coercion-procs)))
											(let ((coerced-type-tags (map type-tag coerced-args)))
												(let ((new-proc (get op coerced-type-tags)))
													(if new-proc
														; Found a valid procedure for the coerced args
														; So we are done
														(apply new-proc (map contents coerced-args))
														; Could not find a valid procedure for the coerced args so try the next coercion
														(apply-generic-internal op (+ position 1) args)
													)
												)
											)
										)
										; At least one coercion is not supported so try coercing
										; with the type of the next argument
										(apply-generic-internal op (+ position 1) args)
									)
								)
							)
							; tried all coercions so now try "raising", provided the op itself is not "raise"
							; Also, it does not make sense to raise up when we are trying to project down
							(if (and (not (equal? op 'raise)) (not (equal? op 'project)))
								(apply-raise op args)
								(error "Tried all coercions. No procedure was found for these types" (list op type-tags))
							)
						)
						; args are all of the same type so try raising, provided the op itself is not "raise"
						; Also, it does not make sense to raise up when we are trying to project down
						(if (and (not (equal? op 'raise)) (not (equal? op 'project)))
							(apply-raise op args)
							(error "(All arguments are of the same type and) no procedure was found for these types" (list op type-tags))
						)
					)
				)
			)
		)
	)

	; (display "Starting with args: ")
	; (display args)
	; (newline)
	(if (and (not (equal? op 'raise)) (not (equal? op 'project)))
		(drop (apply-generic-internal op 1 args))
		(apply-generic-internal op 1 args)
	)
)

(define (drop x)
	(with-handlers ([exn:fail? (lambda (exn)
									; (display "Could not drop beyond: ")
									; (display x)
									; (newline)
									x)])
		(if (not (pair? x))
			x
			(let ((projected-x (project x)))
				(cond
					((equ? x (raise projected-x)) (drop projected-x))
					(else
						; can't drop any further
						x
					)
				)
			)
		)
	)
)

(define (apply-raise op args)
	(let ((type-tags (map type-tag args)))
		(let ((new-args (raise-one-step args)))
			; (display "New args: ")
			; (display new-args)
			; (newline)
			(let ((new-type-tags (map type-tag new-args)))
				(if (not (equal? new-type-tags type-tags))
					; raise worked
					(let ((proc (get op new-type-tags)))
						(if proc
							; valid procedure found, so apply it
							(apply proc (map contents new-args))
							; valid procedure not found, so raise again
							(apply-raise op new-args)
						)
					)
					; we could not raise any more, so give up
					(error "Tried all raise options and failed to find a valid procedure for this operation" (list op args))
				)
			)
		)
	)
)

(define (raise-one-step args)
	; Logic of this procedure:
	; If the arguments are dissimilar, find the 'lowest' argument in the list and raise it one step
	; If the arguments are all similar (i.e. of the same type), then raise all of them one step
	; return the new arguments
	(let ((type-tags (map type-tag args)))
		(if (dissimilar? type-tags)
				(raise-lowest args)
				(raise-all args)
		)
	)
)

(define (raise-lowest args)

	(define (raise-lowest-internal arg-list position-of-lowest-type)
		(if (= position-of-lowest-type 1)
			(cons (try-raise (car arg-list)) (cdr arg-list))
			(cons
				(car arg-list)
				(raise-lowest-internal (cdr arg-list) (- position-of-lowest-type 1))
			)
		)
	)

	(let ((position-of-lowest-type (get-position-of-lowest-type args)))
		; (display "Raising the argument in position: ")
		; (display position-of-lowest-type)
		; (newline)
		(raise-lowest-internal args position-of-lowest-type)
	)
)

(define (get-position-of-lowest-type args)

	(define (get-position-of-lowest-type-internal
			arg-list
			current-candidate
			position-of-current-candidate
			current-position)
		(cond
			((null? arg-list) position-of-current-candidate)
			(else
				(if (lower-in-tower? (car arg-list) current-candidate)
					; we found a "lower" candidate
					(get-position-of-lowest-type-internal (cdr arg-list) (car arg-list) (+ current-position 1) (+ current-position 1))
					; we did not find a "lower" candidate, continue looking with the same current candidate
					(get-position-of-lowest-type-internal (cdr arg-list) current-candidate position-of-current-candidate (+ current-position 1))
				)
			)
		)
	)

	(cond
		((null? args) 0)
		((null? (cdr args)) 1)
		(else
			(get-position-of-lowest-type-internal (cdr args) (car args) 1 1)
		)
	)
)

(define (lower-in-tower? a b)
	; tests whether a is lower in the tower than b
	; returns true if a is lower, false if not
	; Logic: keep raising a till you find b or cannot raise any further
	(let ((new-a (try-raise a)))
		(if (same-type? new-a a)
			; raise failed
			false
			; raise worked and we got a new type
			(if (same-type? new-a b)
				; we successfully raised a to b
				true
				; a has not become b yet so keep looking
				(lower-in-tower? new-a b)
			)
		)
	)
)

(define (same-type? x y)
	(equal? (type-tag x) (type-tag y))
)

(define (raise-all arg-list)
	(cond
		((null? arg-list) (list))
		(else
			(cons
				(try-raise (car arg-list))
				(raise-all (cdr arg-list))
			)
		)
	)
)

(define (try-raise x)
	; Tries to raise the object. If successful, it returns the raised object
	; Otherwise it returns the same object
	(let ((proc (get 'raise (list (type-tag x)))))
		(if proc
			(proc (contents x))
			x
		)
	)
)

(define (dissimilar? items)
	; evaluates to true if the list contains dissimilar items
	; false if all the items are the same
	(cond
		((not (pair? items)) false)
		((null? items) false)
		; reached the last item in the list
		((null? (cdr items)) false)
		((equal? (car items) (cadr items))
			; continue looking
			(dissimilar? (cdr items))
		)
		(else
			true
		)
	)
)

(define (find-element items index)
	; finds and returns the item in the 'index' position of the list
	(cond
		((and (> index 0) (<= index (length items)))
			(if (= index 1)
				(car items)
				(find-element (cdr items) (- index 1))
			)
		)
		(else
			null
		)
	)
)

(define (build-coercion-proc-list type-tags target-type)
	(cond
		((null? type-tags) (list))
		((not (pair? type-tags)) (error "type-tags not a pair: " type-tags))
		(else
			(cons
				(get-coercion (car type-tags) target-type)
				(build-coercion-proc-list (cdr type-tags) target-type)
			)
		)
	)
)

(define (allValid? procs)
	(cond
		((null? procs) true)
		((not (pair? procs)) (error "procs not a pair: " procs))
		((car procs) (allValid? (cdr procs)))
		(else
			false
		)
	)
)

(define (coerce args coercion-procs)
	(cond
		((or (null? args) (null? coercion-procs)) (list))
		((not (pair? args)) (error "args not a pair: " args))
		((not (pair? coercion-procs)) (error "coercion-procs not a pair: " coercion-procs))
		((not (= (length args) (length coercion-procs))) (error "The number of coercion procs needs to be equal to the number of arguments"))
		(else
			(cons ((car coercion-procs) (car args)) (coerce (cdr args) (cdr coercion-procs)))
		)
	)
)

(define (type-tag datum)
	(cond
		((pair? datum) (car datum))
		((number? datum) (determine-number-type datum))
		((boolean? datum) 'boolean)
		(else
			(error "Bad tagged datum -- TYPE-TAG" datum)
		)
	)
)

(define (contents datum)
	(cond
		((pair? datum) (cdr datum))
		((number? datum) datum)
		((boolean? datum) datum)
		(else
			(error "Bad tagged datum -- CONTENTS" datum)
		)
	)
)

(define (determine-number-type x)
	(if (not (= (imag-part x) 0))
		; number has a non-zero imaginary part
		'complex
		; number is real
		'real
	)
)

; Coercion procedures

(define (complex->complex z) z)

(define (rational->complex x)
	(let ((rat (contents x)))
		(make-complex-from-real-imag (* 1.0 (/ (numer rat) (denom rat)))  0)
	)
)

(define (get-coercion type1 type2)
	(define coercion-table
		(list
			; Since I have re-written "apply-generic", there should be no harm in 
			; keeping coercion procedures that convert from one type to the same type. 
			; These should not cause infinite loops any more
			(cons
				'complex
				(list
					(cons 'complex complex->complex)
				)
			)
			(cons
				'rational
				(list
					(cons 'complex rational->complex)
				)
			)
		)
	)

	(define (find-row type-name table)
		(cond
			((not (pair? table)) false)
			((null? table) (error "Type row not found for " type-name))
			(else
				(if (equal? type-name (car (car table)))
					(car table)
					(find-row type-name (cdr table))
				)
			)
		)
	)

	(define (find-type-in-row type-list type-name)
		(cond
			((not (pair? type-list)) false)
			((null? type-list) (error "type not found: " type-name))
			(else
				(if (equal? type-name (car (car type-list)))
					(cdr (car type-list))
					(find-type-in-row (cdr type-list) type-name)
				)
			)
		)
	)

	(if (find-row type1 coercion-table)
		(find-type-in-row (cdr (find-row type1 coercion-table)) type2)
		false
	)
)

; Implementation of get on the operation table.
; The table is hard-coded here so no 'put's are needed

(define (get operation type)

	; (display "Searching op-table for ")
	; (display operation)
	; (display ", ")
	; (display type)
	; (newline)

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

	(define op-table
		(list
			(cons
				'make-from-real-imag
				(list
					(cons 'complex (lambda (x y) (attach-tag 'complex (make-from-real-imag x y))))
					(cons 'rectangular (lambda (x y) (attach-tag 'rectangular (make-from-real-imag-rectangular x y))))
					(cons 'polar (lambda (x y) (attach-tag 'polar (make-from-real-imag-polar x y))))
				)
			)
			(cons
				'make-from-mag-ang
				(list
					(cons 'complex (lambda (r a) (attach-tag 'complex (make-from-mag-ang r a))))
					(cons 'rectangular (lambda (r a) (attach-tag 'rectangular (make-from-mag-ang-rectangular r a))))
					(cons 'polar (lambda (r a) (attach-tag 'polar (make-from-mag-ang-polar r a))))
				)
			)
			(cons
				'make
				(list
					(cons 'scheme-number (lambda (x) (attach-tag 'scheme-number x)))
					(cons 'natural (lambda (n) (attach-tag 'natural (make-natural-specific n))))
					(cons 'integer (lambda (n) (attach-tag 'integer (make-integer-specific n))))
					(cons 'rational (lambda (n d) (attach-tag 'rational (make-rational-specific n d))))
					(cons 'real (lambda (r) (attach-tag 'real (make-real-specific r))))
					(cons 'polynomial (lambda (var terms) (attach-tag 'polynomial (make-poly var terms))))
					(cons 'polynomial-dense-terms (lambda (terms) (attach-tag 'polynomial-dense-terms (make-dense-terms terms))))
					(cons 'polynomial-sparse-terms (lambda (terms) (attach-tag 'polynomial-sparse-terms (make-sparse-terms terms))))
					(cons 'polynomial-term (lambda (order coeff) (attach-tag 'polynomial-term (make-poly-term order coeff))))
				)
			)
			(cons
				'raise
				(list
					(cons '(natural) raise-natural)
					(cons '(integer) raise-int)
					(cons '(rational) raise-rational)
					(cons '(real) raise-real)
					(cons '(rectangular) raise-rectangular)
					(cons '(polar) raise-polar)
				)
			)
			(cons
				'project
				(list
					(cons '(complex) project-complex)
					(cons '(real) project-real)
					(cons '(rational) project-rational)
					(cons '(integer) project-int)
					(cons '(natural) project-natural)
				)
			)
			(cons
				'add
				(list
					(cons '(scheme-number scheme-number) (lambda (x y) (attach-tag 'scheme-number (+ x y))))
					(cons '(rational rational) (lambda (x y) (attach-tag 'rational (add-rational x y))))
					(cons '(real real) (lambda (x y) (attach-tag 'real (+ x y))))
					(cons '(complex complex) (lambda (z1 z2) (attach-tag 'complex (add-complex z1 z2))))
					(cons
						'(polynomial polynomial)
						(lambda (p1 p2)
							(attach-tag 'polynomial
								(if (same-variable? (variable-poly p1) (variable-poly p2))
									(add-poly p1 p2)
									; Note: The procedure convert-polynomial expects the type-tag in the object
									; we supply to it
									(let ((pg1 (make-polynomial (variable-poly p1) (term-list-poly p1)))
										  (pg2 (make-polynomial (variable-poly p2) (term-list-poly p2))))
										(if (higher-in-hierarchy? (variable-poly p1) (variable-poly p2))
											(let ((cpg2 (convert-polynomial pg2 (variable-poly p1))))
												(add-poly p1 (contents cpg2))
											)
											(let ((cpg1 (convert-polynomial pg1 (variable-poly p2))))
												(add-poly p2 (contents cpg1))
											)
										)
									)
								)
							)
						)
					)
					(cons
						'(polynomial complex)
						(lambda (p c)
							(attach-tag
								'polynomial
								; 'Raise' the complex object to a polynomial and then do ordinary addition
								(let ((pg (make-polynomial (variable-poly p) (make-polynomial-sparse-terms (list (list 0 c))))))
									(add-poly p (contents pg))
								)
							)
						)
					)
					(cons
						'(complex polynomial)
						(lambda (c p)
							(attach-tag
								'polynomial
								; 'Raise' the complex object to a polynomial and then do ordinary addition
								(let ((pg (make-polynomial (variable-poly p) (make-polynomial-sparse-terms (list (list 0 c))))))
									(add-poly p (contents pg))
								)
							)
						)
					)
				)
			)
			(cons
				'add-four-quantities
				(list
					(cons '(complex complex complex complex) (lambda (z1 z2 z3 z4) (attach-tag 'complex (add-four-complex-numbers z1 z2 z3 z4))))
				)
			)
			(cons
				'sub
				(list
					(cons '(scheme-number scheme-number) (lambda (x y) (attach-tag 'scheme-number (- x y))))
					(cons '(rational rational) (lambda (x y) (attach-tag 'rational (sub-rational x y))))
					(cons '(real real) (lambda (x y) (attach-tag 'real (- x y))))
					(cons '(complex complex) (lambda (z1 z2) (attach-tag 'complex (sub-complex z1 z2))))
					(cons
						'(polynomial polynomial)
						(lambda (p1 p2)
							(let ((pg1 (make-polynomial (variable-poly p1) (term-list-poly p1)))
							  	  (pg2 (make-polynomial (variable-poly p2) (term-list-poly p2))))
								  (add pg1 (NEGATE pg2))
							)
						)
						; (lambda (p1 p2)
; 							; (attach-tag 'polynomial (sub-poly p1 p2))
; 							(attach-tag 'polynomial
; 								(if (same-variable? (variable-poly p1) (variable-poly p2))
; 									(sub-poly p1 p2)
; 									; Note: The procedure convert-polynomial expects the type-tag in the object
; 									; we supply to it
; 									(let ((pg1 (make-polynomial (variable-poly p1) (term-list-poly p1)))
; 										  (pg2 (make-polynomial (variable-poly p2) (term-list-poly p2))))
; 										(if (higher-in-hierarchy? (variable-poly p1) (variable-poly p2))
; 											(let ((cpg2 (convert-polynomial pg2 (variable-poly p1))))
; 												(sub-poly p1 (contents cpg2))
; 											)
; 											(let ((cpg1 (convert-polynomial pg1 (variable-poly p2))))
; 												(sub-poly (contents cpg1) p2)
; 											)
; 										)
; 									)
; 								)
;							)
;						)
					)
				)
			)
			(cons
				'mul
				(list
					(cons '(scheme-number scheme-number) (lambda (x y) (attach-tag 'scheme-number (* x y))))
					(cons '(rational rational) (lambda (x y) (attach-tag 'rational (mul-rational x y))))
					(cons '(real real) (lambda (x y) (attach-tag 'real (* x y))))
					(cons '(complex complex) (lambda (z1 z2) (attach-tag 'complex (mul-complex z1 z2))))
					(cons
						'(polynomial polynomial)
						; (lambda (p1 p2) (attach-tag 'polynomial (mul-poly p1 p2)))
						(lambda (p1 p2)
							(attach-tag 'polynomial
								(if (same-variable? (variable-poly p1) (variable-poly p2))
									(mul-poly p1 p2)
									; Note: The procedure convert-polynomial expects the type-tag in the object
									; we supply to it
									(let ((pg1 (make-polynomial (variable-poly p1) (term-list-poly p1)))
										  (pg2 (make-polynomial (variable-poly p2) (term-list-poly p2))))
										(if (higher-in-hierarchy? (variable-poly p1) (variable-poly p2))
											(let ((cpg2 (convert-polynomial pg2 (variable-poly p1))))
												(mul-poly p1 (contents cpg2))
											)
											(let ((cpg1 (convert-polynomial pg1 (variable-poly p2))))
												(mul-poly p2 (contents cpg1))
											)
										)
									)
								)
							)
						)
					)
					(cons
						'(polynomial real)
						(lambda (p r)
							(attach-tag
								'polynomial
								; 'Raise' the real object to a polynomial and then do ordinary multiplication
								(let ((pg (make-polynomial (variable-poly p) (make-polynomial-sparse-terms (list (list 0 r))))))
									(mul-poly p (contents pg))
								)
							)
						)
					)
					(cons
						'(real polynomial)
						(lambda (r p)
							(attach-tag
								'polynomial
								; 'Raise' the real object to a polynomial and then do ordinary multiplication
								(let ((pg (make-polynomial (variable-poly p) (make-polynomial-sparse-terms (list (list 0 r))))))
									(mul-poly p (contents pg))
								)
							)
						)
					)
					(cons
						'(polynomial complex)
						(lambda (p c)
							(attach-tag
								'polynomial
								; 'Raise' the complex object to a polynomial and then do ordinary multiplication
								(let ((pg (make-polynomial (variable-poly p) (make-polynomial-sparse-terms (list (list 0 c))))))
									(mul-poly p (contents pg))
								)
							)
						)
					)
					(cons
						'(complex polynomial)
						(lambda (c p)
							(attach-tag
								'polynomial
								; 'Raise' the complex object to a polynomial and then do ordinary multiplication
								(let ((pg (make-polynomial (variable-poly p) (make-polynomial-sparse-terms (list (list 0 c))))))
									(mul-poly p (contents pg))
								)
							)
						)
					)
				)
			)
			(cons
				'div
				(list
					(cons '(scheme-number scheme-number) (lambda (x y) (attach-tag 'scheme-number (/ x y))))
					(cons '(rational rational) (lambda (x y) (attach-tag 'rational (div-rational x y))))
					(cons '(real real) (lambda (x y) (attach-tag 'real (/ x y))))
					(cons '(complex complex) (lambda (z1 z2) (attach-tag 'complex (div-complex z1 z2))))
					(cons '(polynomial polynomial) (lambda (p1 p2) (attach-tag 'polynomial-list (div-poly p1 p2))))
				)
			)
			(cons
				'exp
				(list
					(cons '(scheme-number scheme-number) (lambda (x y) (attach-tag 'scheme-number (expt x y))))
					(cons '(real real) (lambda (x y) (attach-tag 'real (expt x y))))
				)
			)
			(cons
				'square-root
				(list
					(cons '(real) square-root-real)
					(cons '(rational) square-root-rational)
					(cons '(integer) square-root-integer)
					(cons '(natural) square-root-natural)
				)
			)
			(cons
				'square
				(list
					(cons '(complex) (lambda (z) (attach-tag 'complex (square-complex z))))
					(cons '(real) (lambda (r) (attach-tag 'real (square-real r))))
					(cons '(rational) (lambda (r) (attach-tag 'rational (square-rational r))))
					(cons '(integer) (lambda (i) (attach-tag 'integer (square-integer i))))
					(cons '(natural) (lambda (n) (attach-tag 'natural (square-natural n))))
				)
			)
			(cons
				'negate
				(list
					(cons '(polynomial) (lambda (p) (attach-tag 'polynomial (negate-poly p))))
				)
			)
			(cons
				'sine
				(list
					(cons '(real) (lambda (x) (attach-tag 'real (sine-real x))))
					(cons '(rational) (lambda (x) (attach-tag 'real (sine-rational x))))
					(cons '(integer) (lambda (x) (attach-tag 'real (sine-integer x))))
					(cons '(natural) (lambda (x) (attach-tag 'real (sine-natural x))))
					(cons '(scheme-number) (lambda (x) (attach-tag 'real (sin x))))
				)
			)
			(cons
				'cosine
				(list
					(cons '(real) (lambda (x) (attach-tag 'real (cosine-real x))))
					(cons '(rational) (lambda (x) (attach-tag 'real (cosine-rational x))))
					(cons '(integer) (lambda (x) (attach-tag 'real (cosine-integer x))))
					(cons '(natural) (lambda (x) (attach-tag 'real (cosine-natural x))))
					(cons '(scheme-number) (lambda (x) (attach-tag 'real (cos x))))
				)
			)
			(cons
				'tan-inverse
				(list
					(cons '(real real) tan-inverse-real)
					(cons '(rational rational) tan-inverse-rational)
					(cons '(integer integer) tan-inverse-integer)
					(cons '(natural natural) tan-inverse-natural)
				)
			)
			(cons
				'mul-and-scale
				(list
					(cons '(complex complex scheme-number) mul-and-scale-complex)
				)
			)
			(cons
				'mul-five-quantities
				(list
					(cons '(rational rational rational rational rational) (lambda (r1 r2 r3 r4 r5) (attach-tag 'rational (mul-five-rationals r1 r2 r3 r4 r5))))
				)
			)
			(cons
				'equal?
				(list
					(cons '(scheme-number scheme-number) =)
					(cons '(natural natural) =)
					(cons '(integer integer) =)
					(cons '(rational rational) equal-rational?)
					(cons '(real real) =)
					(cons '(complex complex) equal-complex?)
					(cons '(polynomial real) equal-polynomial-real?)
				)
			)
			(cons
				'greater?
				(list
					(cons '(scheme-number scheme-number) >)
					(cons '(natural natural) >)
					(cons '(integer integer) >)
					(cons '(rational rational) greater-rational?)
					(cons '(real real) >)
				)
			)
			(cons
				'lesser?
				(list
					(cons '(scheme-number scheme-number) <)
					(cons '(natural natural) <)
					(cons '(integer integer) <)
					(cons '(rational rational) lesser-rational?)
					(cons '(real real) <)
					(cons '(polynomial real) lesser-polynomial-real?)
				)
			)
			(cons
				'=zero?
				(list
					(cons '(scheme-number) (lambda (x) (= 0 x)))
					(cons '(rational) =zero-rational?)
					(cons '(real) =zero-real?)
					(cons '(complex) =zero-complex?)
					(cons '(polynomial) =zero-polynomial?)
				)
			)
			(cons
				'empty-term?
				(list
					(cons '(polynomial-term) empty-poly-term?)
				)
			)
			(cons
				'REAL-PART
				(list
					(cons '(complex) REAL-PART)
					(cons '(rectangular) real-part-rectangular)
					(cons '(polar) real-part-polar)
				)
			)
			(cons
				'IMAG-PART
				(list
					(cons '(complex) IMAG-PART)
					(cons '(rectangular) imag-part-rectangular)
					(cons '(polar) imag-part-polar)
				)
			)
			(cons
				'magnitude
				(list
					(cons '(complex) magnitude)
					(cons '(rectangular) magnitude-rectangular)
					(cons '(polar) magnitude-polar)
				)
			)
			(cons
				'angle
				(list
					(cons '(complex) angle)
					(cons '(rectangular) angle-rectangular)
					(cons '(polar) angle-polar)
				)
			)
			(cons
				'first-term
				(list
					(cons '(polynomial-dense-terms) (lambda (x) (attach-tag 'polynomial-term (first-term-dense x))))
					(cons '(polynomial-sparse-terms) (lambda (x) (attach-tag 'polynomial-term (first-term-sparse x))))
				)
			)
			(cons
				'rest-terms
				(list
					(cons '(polynomial-dense-terms) (lambda (x) (attach-tag 'polynomial-dense-terms (rest-terms-dense x))))
					(cons '(polynomial-sparse-terms) (lambda (x) (attach-tag 'polynomial-sparse-terms (rest-terms-sparse x))))
				)
			)
			(cons
				'adjoin-term
				(list
					(cons '(polynomial-term polynomial-dense-terms) (lambda (term terms) (attach-tag 'polynomial-dense-terms (adjoin-term-dense term terms))))
					(cons '(polynomial-term polynomial-sparse-terms) (lambda (term terms) (attach-tag 'polynomial-sparse-terms (adjoin-term-sparse term terms))))
				)
			)
			(cons
				'order
				(list
					(cons '(polynomial-term) order-poly-term)
				)
			)
			(cons
				'coeff
				(list
					(cons '(polynomial-term) coeff-poly-term)
				)
			)
			(cons
				'negate-term
				(list
					(cons '(polynomial-term) (lambda (term) (attach-tag 'polynomial-term (negate-poly-term term))))
				)
			)
			(cons
				'variable
				(list
					(cons '(polynomial) variable-poly)
				)
			)
			(cons
				'term-list
				(list
					(cons '(polynomial) term-list-poly)
				)
			)
			(cons
				'print
				(list
					(cons '(polynomial) print-poly)
					(cons '(complex) print-complex)
					(cons '(real) print-real)
					(cons '(integer) print-int)
					(cons '(natural) print-natural)
					(cons '(boolean) (lambda (x) (if x (display 'True) (display 'False))))
				)
			)
			(cons
				'abs
				(list
					(cons '(polynomial) (lambda (x) (attach-tag 'polynomial x)))
					(cons '(complex) (lambda (x) (attach-tag 'complex x)))
					(cons '(real) (lambda (x) (attach-tag 'real (abs-real x))))
					(cons '(integer) (lambda (x) (attach-tag 'integer (abs-integer x))))
					(cons '(natural) (lambda (x) (attach-tag 'natural (abs-natural x))))
				)
			)
		)
	)

	(define (find-operation-row op table)
		(cond
			((not (pair? table)) (error "op-table Not a pair!"))
			((null? table) (error "Operation not found for " op))
			(else
				(if (equal? op (car (car table)))
					(car table)
					(find-operation-row op (cdr table))
				)
			)
		)
	)

	(define (find-type-in-op-row type-list t)
		(cond
			((not (pair? type-list)) false)
			((null? type-list) (error "type not found: " t))
			(else
				(if (equal? t (car (car type-list)))
					(cdr (car type-list))
					(find-type-in-op-row (cdr type-list) t)
				)
			)
		)
	)

	(find-type-in-op-row (cdr (find-operation-row operation op-table)) type)
)

; Test Driver

(define (run-test proc . args)

	(define (print-item-list items first-time?)
		(cond
			((not (pair? items)) (void))
			(else
				(if (not first-time?)
					(display ", ")
					(void)
				)
				(print (car items))
				(print-item-list (cdr items) false)
			)
		)
	)

	(display "Running Test: ") (display (cons proc args)) (display " ")
	(newline)
	(display "Applying ")
	(display proc)
	(display " on: ")
	(print-item-list args true)
	(newline)
	(let ((result (apply proc args)))
		(display "Result: ")
		(display result)
		(newline)
		(print result)
		(newline)
	)
	(newline)
)

; Tests

(define t1-sparse (make-polynomial-sparse-terms (list (list 0 0))))
(define t2-sparse (make-polynomial-sparse-terms (list (list 0 5))))
(define t3-sparse (make-polynomial-sparse-terms (list (list 1 7) (list 0 5))))
(define t4-sparse (make-polynomial-sparse-terms (list (list 1 9) (list 0 -25))))
(define t5-sparse (make-polynomial-sparse-terms (list (list 5 1) (list 4 2) (list 2 3) (list 1 -2) (list 0 -5))))
(define t6-sparse (make-polynomial-sparse-terms (list (list 5 2) (list 4 4) (list 2 5) (list 1 -7) (list 0 -15))))
(define t7-sparse (make-polynomial-sparse-terms (list (list 10 3) (list 7 4) (list 5 -11) (list 1 -7) (list 0 -15))))
(define t8-sparse (make-polynomial-sparse-terms (list (list 12 4) (list 6 13) (list 5 -12) (list 1 -7) (list 0 -15))))
(define t9-sparse (make-polynomial-sparse-terms (list (list 12 -4) (list 6 13) (list 5 -12) (list 1 -7) (list 0 -15))))
(define t10-sparse (make-polynomial-sparse-terms (list (list 12 0) (list 6 -13) (list 5 -12) (list 1 -7) (list 0 -15))))

(define t1-dense (make-polynomial-dense-terms (list 0 0)))
(define t2-dense (make-polynomial-dense-terms (list 0 5)))
(define t3-dense (make-polynomial-dense-terms (list 1 3 5)))
(define t4-dense (make-polynomial-dense-terms (list 1 4 9)))
(define t5-dense (make-polynomial-dense-terms (list 2 3 5 23)))
(define t6-dense (make-polynomial-dense-terms (list 2 4 9 19)))
(define t7-dense (make-polynomial-dense-terms (list 5 1 2 0 3 -2 -5)))
(define t8-dense (make-polynomial-dense-terms (list 5 2 4 0 5 -7 -15)))
(define t9-dense (make-polynomial-dense-terms (list 5 -2 4 0 5 -7 -15)))
(define t10-dense (make-polynomial-dense-terms (list 5 0 -4 0 5 -7 -15)))

(define sp1 (make-polynomial 'p t1-sparse))
(define sp2 (make-polynomial 'q t2-sparse))
(define sp3 (make-polynomial 'r t3-sparse))
(define sp4 (make-polynomial 's t4-sparse))
(define sp5 (make-polynomial 't t5-sparse))
(define sp6 (make-polynomial 'u t6-sparse))
(define sp7 (make-polynomial 'v t7-sparse))
(define sp8 (make-polynomial 'z t8-sparse))
(define sp9 (make-polynomial 'y t9-sparse))
(define sp10 (make-polynomial 'x t10-sparse))

(define dp1 (make-polynomial 'q t1-dense))
(define dp2 (make-polynomial 'r t2-dense))
(define dp3 (make-polynomial 's t3-dense))
(define dp4 (make-polynomial 't t4-dense))
(define dp5 (make-polynomial 'u t5-dense))
(define dp6 (make-polynomial 'v t6-dense))
(define dp7 (make-polynomial 'w t7-dense))
(define dp8 (make-polynomial 'z t8-dense))
(define dp9 (make-polynomial 'y t9-dense))
(define dp10 (make-polynomial 'x t10-dense))

(define p-1-divisor-terms (make-polynomial-sparse-terms (list (list 1 1))))
(define p-1-dividend-terms (make-polynomial-sparse-terms (list (list 1 1))))

(define p-1-divisor (make-polynomial 'y p-1-divisor-terms))
(define p-1-dividend (make-polynomial 'y p-1-dividend-terms))

(define p0-divisor-terms (make-polynomial-sparse-terms (list (list 4 1))))
(define p0-dividend-terms (make-polynomial-sparse-terms (list (list 6 1))))

(define p0-divisor (make-polynomial 'w p0-divisor-terms))
(define p0-dividend (make-polynomial 'w p0-dividend-terms))

(define p1-divisor-terms (make-polynomial-sparse-terms (list (list 4 1) (list 0 -1))))
(define p1-dividend-terms (make-polynomial-sparse-terms (list (list 5 1) (list 0 -1))))

(define p1-divisor (make-polynomial 'p p1-divisor-terms))
(define p1-dividend (make-polynomial 'p p1-dividend-terms))

(define p2-divisor-terms (make-polynomial-sparse-terms (list (list 2 1) (list 0 -1))))
(define p2-dividend-terms (make-polynomial-sparse-terms (list (list 5 1) (list 0 -1))))

(define p2-divisor (make-polynomial 'x p2-divisor-terms))
(define p2-dividend (make-polynomial 'x p2-dividend-terms))

(define p3-divisor-terms (make-polynomial-sparse-terms (list (list 2 2) (list 1 3) (list 0 4))))
(define p3-dividend-terms (make-polynomial-sparse-terms (list (list 5 6) (list 4 9) (list 3 8) (list 2 -15) (list 1 -16) (list 0 5))))

(define p3-divisor (make-polynomial 'x p3-divisor-terms))
(define p3-dividend (make-polynomial 'x p3-dividend-terms))

(run-test =zero? sp1)
(run-test =zero? sp2)
(run-test =zero? sp3)
(run-test =zero? sp4)
(run-test =zero? dp1)
(run-test =zero? dp2)
(run-test =zero? dp3)
(run-test =zero? dp4)
(run-test =zero? dp5)
(run-test =zero? dp6)

(newline)

(run-test add sp1 sp1)
(run-test add sp1 sp2)
(run-test add sp2 sp3)
(run-test add sp3 sp4)
(run-test add sp4 sp5)
(run-test add sp5 sp6)
(run-test add sp6 sp7)
(run-test add sp7 sp8)

(newline)

(run-test add dp1 dp1)
(run-test add dp1 dp2)
(run-test add dp2 dp3)
(run-test add dp3 dp4)
(run-test add dp4 dp5)
(run-test add dp5 dp6)
(run-test add dp6 dp7)
(run-test add dp7 dp8)

(newline)

(run-test add sp1 dp2)
(run-test add dp2 sp1)
(run-test add sp2 dp3)
(run-test add sp3 dp4)
(run-test add sp4 dp5)
(run-test add sp5 dp6)
(run-test add sp6 dp7)
(run-test add sp7 dp8)

(newline)

(run-test add dp1 sp2)
(run-test add sp2 dp1)
(run-test add dp2 sp3)
(run-test add dp3 sp4)
(run-test add dp4 sp5)
(run-test add dp5 sp6)
(run-test add dp6 sp7)
(run-test add dp7 sp8)

(newline)

(run-test mul sp1 sp1)
(run-test mul sp1 sp2)
(run-test mul sp2 sp2)
(run-test mul sp2 sp3)
(run-test mul sp3 sp4)
(run-test mul sp4 sp4)
(run-test mul sp4 sp5)
(run-test mul sp5 sp6)
(run-test mul sp6 sp7)
(run-test mul sp7 sp8)

(newline)

(run-test mul dp1 dp1)
(run-test mul dp1 dp2)
(run-test mul dp2 dp2)
(run-test mul dp2 dp3)
(run-test mul dp3 dp4)
(run-test mul dp4 dp5)
(run-test mul dp5 dp6)
(run-test mul dp6 dp7)
(run-test mul dp7 dp8)

(newline)

(run-test mul sp1 dp2)
(run-test mul dp2 sp1)
(run-test mul sp2 dp3)
(run-test mul sp3 dp4)
(run-test mul sp4 dp5)
(run-test mul sp5 dp6)
(run-test mul sp6 dp7)
(run-test mul sp7 dp8)

(newline)

(run-test mul dp1 sp2)
(run-test mul sp2 dp1)
(run-test mul dp2 sp3)
(run-test mul dp3 sp4)
(run-test mul dp4 sp5)
(run-test mul dp5 sp6)
(run-test mul dp6 sp7)
(run-test mul dp7 sp8)

(newline)

(run-test NEGATE sp1)
(run-test NEGATE sp2)
(run-test NEGATE sp2)
(run-test NEGATE sp3)
(run-test NEGATE sp4)
(run-test NEGATE sp5)
(run-test NEGATE sp6)
(run-test NEGATE sp7)
(run-test NEGATE sp8)

(newline)

(run-test NEGATE dp1)
(run-test NEGATE dp2)
(run-test NEGATE dp2)
(run-test NEGATE dp3)
(run-test NEGATE dp4)
(run-test NEGATE dp5)
(run-test NEGATE dp6)
(run-test NEGATE dp7)
(run-test NEGATE dp8)

(run-test sub sp1 sp1)
(run-test sub sp1 sp2)
(run-test sub sp2 sp2)
(run-test sub sp2 sp3)
(run-test sub sp3 sp4)
(run-test sub sp4 sp5)
(run-test sub sp5 sp6)
(run-test sub sp6 sp7)
(run-test sub sp7 sp8)

(newline)

(run-test sub dp1 dp1)
(run-test sub dp1 dp2)
(run-test sub dp2 dp2)
(run-test sub dp2 dp3)
(run-test sub dp3 dp4)
(run-test sub dp4 dp5)
(run-test sub dp5 dp6)
(run-test sub dp6 dp7)
(run-test sub dp7 dp8)

(newline)

(run-test sub sp1 dp2)
(run-test sub dp2 sp1)
(run-test sub sp2 dp3)
(run-test sub sp3 dp4)
(run-test sub sp4 dp5)
(run-test sub sp5 dp6)
(run-test sub sp6 dp7)
(run-test sub sp7 dp8)

(newline)

(run-test sub dp1 sp2)
(run-test sub sp2 dp1)
(run-test sub dp2 sp3)
(run-test sub dp3 sp4)
(run-test sub dp4 sp5)
(run-test sub dp5 sp6)
(run-test sub dp6 sp7)
(run-test sub dp7 sp8)

(newline)

(run-test =zero? (sub dp6 dp6))
(run-test =zero? (sub sp4 sp4))

(newline)

(div p-1-dividend p-1-divisor)
(div p0-dividend p0-divisor)
(div p1-dividend p1-divisor)
(div p2-dividend p2-divisor)
(div p3-dividend p3-divisor)

(newline)

(define t11-sparse (make-polynomial-sparse-terms (list (list 1 3) (list 0 -2))))
(define t11-dense (make-polynomial-dense-terms (list 1 7 3)))
(define P1x (make-polynomial 'x t11-sparse))
(define P2x (make-polynomial 'x t11-dense))

(define t12-sparse (make-polynomial-sparse-terms (list (list 1 5) (list 0 -4))))
(define t12-dense (make-polynomial-dense-terms (list 1 9 -6)))
(define P3x (make-polynomial 'x t12-sparse))
(define P4x (make-polynomial 'x t12-dense))

(define t13-sparse (make-polynomial-sparse-terms (list (list 1 P1x) (list 0 P2x))))
(define t13-dense (make-polynomial-dense-terms (list 1 P3x P4x)))

(define P1y (make-polynomial 'y t13-sparse))
(define P2y (make-polynomial 'y t13-dense))

(define t14-sparse (make-polynomial-sparse-terms (list (list 1 P1y) (list 0 P2y))))
(define P1z (make-polynomial 'z t14-sparse))

; P(y) = y in terms of x
(define P3y (make-polynomial 'y (make-polynomial-sparse-terms (list (list 1 1)))))
(display "P3y: ")
P3y
(display "P3y: ")
(print P3y)
(newline)
(display "P3y in terms of x: ")
(convert-polynomial P3y 'x)
(display "P3y in terms of x: ")
(print (convert-polynomial P3y 'x))
(newline)

(newline)

; P(y) = xy in terms of x should become yx
(define X (make-polynomial 'x (make-polynomial-sparse-terms (list (list 1 1)))))
(define P4y (make-polynomial 'y (make-polynomial-sparse-terms (list (list 1 X)))))
(display "X: ")
X
(display "X: ")
(print X)
(newline)
(display "P4y: ")
P4y
(display "P4y: ")
(print P4y)
(newline)
(display "P4y in terms of x: ")
(convert-polynomial P4y 'x)
(display "P4y in terms of x: ")
(print (convert-polynomial P4y 'x))
(newline)

(newline)

; P(y) = x^2y in terms of x should become yx^2
(define Xsq (make-polynomial 'x (make-polynomial-sparse-terms (list (list 2 1)))))
(define P5y (make-polynomial 'y (make-polynomial-sparse-terms (list (list 1 Xsq)))))
(display "Xsq: ")
Xsq
(display "Xsq: ")
(print Xsq)
(newline)
(display "P5y: ")
P5y
(display "P5y: ")
(print P5y)
(newline)
(display "P5y in terms of x: ")
(convert-polynomial P5y 'x)
(display "P5y in terms of x: ")
(print (convert-polynomial P5y 'x))
(newline)

(newline)

; P(y) = 5x^2y in terms of x should become 5yx^2
(define 5Xsq (make-polynomial 'x (make-polynomial-sparse-terms (list (list 2 5)))))
(define P6y (make-polynomial 'y (make-polynomial-sparse-terms (list (list 1 5Xsq)))))
(display "5Xsq: ")
5Xsq
(display "5Xsq: ")
(print 5Xsq)
(newline)
(display "P6y: ")
P6y
(display "P6y: ")
(print P6y)
(newline)
(display "P6y in terms of x: ")
(convert-polynomial P6y 'x)
(display "P6y in terms of x: ")
(print (convert-polynomial P6y 'x))
(newline)

(newline)

; P(y) = 5x^3y^7 in terms of x should become 5y^7x^3
(define 5Xcb (make-polynomial 'x (make-polynomial-sparse-terms (list (list 3 5)))))
(define P7y (make-polynomial 'y (make-polynomial-sparse-terms (list (list 7 5Xcb)))))
(display "5Xcb: ")
5Xcb
(display "5Xcb: ")
(print 5Xcb)
(newline)
(display "P7y: ")
P7y
(display "P7y: ")
(print P7y)
(newline)
(display "P7y in terms of x: ")
(convert-polynomial P7y 'x)
(display "P7y in terms of x: ")
(print (convert-polynomial P7y 'x))
(newline)

(newline)

; Two terms
(define P8y (add P6y P7y))
(display "P8y: ")
P8y
(display "P8y: ")
(print P8y)
(newline)
(display "P8y in terms of x: ")
(convert-polynomial P8y 'x)
(display "P8y in terms of x: ")
(print (convert-polynomial P8y 'x))
(newline)

(newline)

; Two terms with same order of x but different orders of y
(define P9y (make-polynomial 'y (make-polynomial-sparse-terms (list (list 4 Xsq)))))
(display "P9y: ")
(print P9y)
(newline)
(display "P6y: ")
(print P6y)
(newline)
(define P10y (add P9y P6y))
(display "P10y = P9y + P6y: ")
P10y
(display "P10y = P9y + P6y: ")
(print P10y)
(newline)
(display "P10y in terms of x: ")
(convert-polynomial P10y 'x)
(display "P10y in terms of x: ")
(print (convert-polynomial P10y 'x))
(newline)

(newline)

; Two terms with same order of x and same orders of y
(define P61y (make-polynomial 'y (make-polynomial-sparse-terms (list (list 4 5Xsq)))))
(display "P9y: ")
(print P9y)
(newline)
(display "P61y: ")
(print P61y)
(newline)
(define P11y (add P9y P61y))
(display "P11y = P9y + P61y: ")
P11y
(display "P11y = P9y + P61y: ")
(print P11y)
(newline)
(display "P11y in terms of x: ")
(convert-polynomial P11y 'x)
(display "P11y in terms of x: ")
(print (convert-polynomial P11y 'x))
(newline)

(newline)
(display "P1x: ")
P1x
(display "P1x: ")
(print P1x)
(newline)
(display "P1x in terms of z: ")
(convert-polynomial P1x 'z)
(display "P1x in terms of z: ")
(print (convert-polynomial P1x 'z))
(newline)

(newline)
(display "P1y: ")
P1y
(display "P1y: ")
(print P1y)
(newline)
(display "P1y in terms of x: ")
(convert-polynomial P1y 'x)
(display "P1y in terms of x: ")
(print (convert-polynomial P1y 'x))
(newline)

(newline)
(display "P1z: ")
P1z
(display "P1z: ")
(print P1z)
(newline)
(display "P1z in terms of y: ")
(convert-polynomial P1z 'y)
(display "P1z in terms of y: ")
(print (convert-polynomial P1z 'y))
(newline)

(newline)
(display "P1z in terms of x: ")
(convert-polynomial P1z 'x)
(display "P1z in terms of x: ")
(print (convert-polynomial P1z 'x))
(newline)

; Addition of two similar polynomials in different variables
(newline)
(define P5x (make-polynomial 'x (make-polynomial-sparse-terms (list (list 1 11)))))
(define P12y (make-polynomial 'y (make-polynomial-sparse-terms (list (list 1 15)))))
(display "P5x: ")
P5x
(display "P5x: ")
(print P5x)
(newline)
(display "P12y: ")
P12y
(display "P12y: ")
(print P12y)
(newline)
(display "P5x + P12y: ")
(newline)
(run-test add P5x P12y)
(display "P5x + P12y: ")
(newline)
(print (add P5x P12y))
(newline)

; Addition of two similar polynomials in different variables
(newline)
(define P6x (make-polynomial 'x (make-polynomial-sparse-terms (list (list 1 11) (list 0 12)))))
(define P13y (make-polynomial 'y (make-polynomial-sparse-terms (list (list 1 15) (list 0 17)))))
(display "P6x: ")
P6x
(display "P6x: ")
(print P6x)
(newline)
(display "P13y: ")
P13y
(display "P13y: ")
(print P13y)
(newline)
(display "P6x + P13y: ")
(newline)
(run-test add P6x P13y)
(display "P6x + P13y: ")
(newline)
(print (add P6x P13y))
(newline)

; Addition of two similar polynomials in different variables
(newline)
(define P7x (make-polynomial 'x (make-polynomial-sparse-terms (list (list 2 10) (list 1 11) (list 0 12)))))
(define P14y (make-polynomial 'y (make-polynomial-sparse-terms (list (list 2 13) (list 1 15) (list 0 17)))))
(display "P7x: ")
P7x
(display "P7x: ")
(print P7x)
(newline)
(display "P14y: ")
P14y
(display "P14y: ")
(print P14y)
(newline)
(display "P7x + P14y: ")
(newline)
(run-test add P7x P14y)
(display "P7x + P14y: ")
(newline)
(print (add P7x P14y))
(newline)

Welcome to DrRacket, version 6.11 [3m].
Language: racket, with debugging; memory limit: 512 MB.
Running Test: (#<procedure:=zero?> (polynomial p polynomial-sparse-terms (0 0))) 
Applying #<procedure:=zero?> on: {}
Result: #t
True

Running Test: (#<procedure:=zero?> (polynomial q polynomial-sparse-terms (0 5))) 
Applying #<procedure:=zero?> on: {5}
Result: #f
False

Running Test: (#<procedure:=zero?> (polynomial r polynomial-sparse-terms (1 7) (0 5))) 
Applying #<procedure:=zero?> on: {7r + 5}
Result: #f
False

Running Test: (#<procedure:=zero?> (polynomial s polynomial-sparse-terms (1 9) (0 -25))) 
Applying #<procedure:=zero?> on: {9s - 25}
Result: #f
False

Running Test: (#<procedure:=zero?> (polynomial q polynomial-dense-terms 0 0)) 
Applying #<procedure:=zero?> on: {}
Result: #t
True

Running Test: (#<procedure:=zero?> (polynomial r polynomial-dense-terms 0 5)) 
Applying #<procedure:=zero?> on: {5}
Result: #f
False

Running Test: (#<procedure:=zero?> (polynomial s polynomial-dense-terms 1 3 5)) 
Applying #<procedure:=zero?> on: {3s + 5}
Result: #f
False

Running Test: (#<procedure:=zero?> (polynomial t polynomial-dense-terms 1 4 9)) 
Applying #<procedure:=zero?> on: {4t + 9}
Result: #f
False

Running Test: (#<procedure:=zero?> (polynomial u polynomial-dense-terms 2 3 5 23)) 
Applying #<procedure:=zero?> on: {3u^2 + 5u + 23}
Result: #f
False

Running Test: (#<procedure:=zero?> (polynomial v polynomial-dense-terms 2 4 9 19)) 
Applying #<procedure:=zero?> on: {4v^2 + 9v + 19}
Result: #f
False


Running Test: (#<procedure:add> (polynomial p polynomial-sparse-terms (0 0)) (polynomial p polynomial-sparse-terms (0 0))) 
Applying #<procedure:add> on: {}, {}
Result: (polynomial p polynomial-sparse-terms (0 0))
{}

Running Test: (#<procedure:add> (polynomial p polynomial-sparse-terms (0 0)) (polynomial q polynomial-sparse-terms (0 5))) 
Applying #<procedure:add> on: {}, {5}
Result: (polynomial q polynomial-sparse-terms (0 5))
{5}

Running Test: (#<procedure:add> (polynomial q polynomial-sparse-terms (0 5)) (polynomial r polynomial-sparse-terms (1 7) (0 5))) 
Applying #<procedure:add> on: {5}, {7r + 5}
Result: (polynomial r polynomial-sparse-terms (1 7) (0 (polynomial q polynomial-sparse-terms (0 10))))
{7r + {10}}

Running Test: (#<procedure:add> (polynomial r polynomial-sparse-terms (1 7) (0 5)) (polynomial s polynomial-sparse-terms (1 9) (0 -25))) 
Applying #<procedure:add> on: {7r + 5}, {9s - 25}
Result: (polynomial s polynomial-sparse-terms (1 9) (0 (polynomial r polynomial-sparse-terms (1 7) (0 (integer . -20)))))
{9s + {7r - 20}}

Running Test: (#<procedure:add> (polynomial s polynomial-sparse-terms (1 9) (0 -25)) (polynomial t polynomial-sparse-terms (5 1) (4 2) (2 3) (1 -2) (0 -5))) 
Applying #<procedure:add> on: {9s - 25}, {t^5 + 2t^4 + 3t^2 - 2t - 5}
Result: (polynomial t polynomial-sparse-terms (5 1) (4 2) (2 3) (1 -2) (0 (polynomial s polynomial-sparse-terms (1 9) (0 (integer . -30)))))
{t^5 + 2t^4 + 3t^2 - 2t + {9s - 30}}

Running Test: (#<procedure:add> (polynomial t polynomial-sparse-terms (5 1) (4 2) (2 3) (1 -2) (0 -5)) (polynomial u polynomial-sparse-terms (5 2) (4 4) (2 5) (1 -7) (0 -15))) 
Applying #<procedure:add> on: {t^5 + 2t^4 + 3t^2 - 2t - 5}, {2u^5 + 4u^4 + 5u^2 - 7u - 15}
Result: (polynomial u polynomial-sparse-terms (5 2) (4 4) (2 5) (1 -7) (0 (polynomial t polynomial-sparse-terms (5 1) (4 2) (2 3) (1 -2) (0 (integer . -20)))))
{2u^5 + 4u^4 + 5u^2 - 7u + {t^5 + 2t^4 + 3t^2 - 2t - 20}}

Running Test: (#<procedure:add> (polynomial u polynomial-sparse-terms (5 2) (4 4) (2 5) (1 -7) (0 -15)) (polynomial v polynomial-sparse-terms (10 3) (7 4) (5 -11) (1 -7) (0 -15))) 
Applying #<procedure:add> on: {2u^5 + 4u^4 + 5u^2 - 7u - 15}, {3v^10 + 4v^7 - 11v^5 - 7v - 15}
Result: (polynomial v polynomial-sparse-terms (10 3) (7 4) (5 -11) (1 -7) (0 (polynomial u polynomial-sparse-terms (5 2) (4 4) (2 5) (1 -7) (0 (integer . -30)))))
{3v^10 + 4v^7 - 11v^5 - 7v + {2u^5 + 4u^4 + 5u^2 - 7u - 30}}

Running Test: (#<procedure:add> (polynomial v polynomial-sparse-terms (10 3) (7 4) (5 -11) (1 -7) (0 -15)) (polynomial z polynomial-sparse-terms (12 4) (6 13) (5 -12) (1 -7) (0 -15))) 
Applying #<procedure:add> on: {3v^10 + 4v^7 - 11v^5 - 7v - 15}, {4z^12 + 13z^6 - 12z^5 - 7z - 15}
Result: (polynomial z polynomial-sparse-terms (12 4) (6 13) (5 -12) (1 -7) (0 (polynomial v polynomial-sparse-terms (10 3) (7 4) (5 -11) (1 -7) (0 (integer . -30)))))
{4z^12 + 13z^6 - 12z^5 - 7z + {3v^10 + 4v^7 - 11v^5 - 7v - 30}}


Running Test: (#<procedure:add> (polynomial q polynomial-dense-terms 0 0) (polynomial q polynomial-dense-terms 0 0)) 
Applying #<procedure:add> on: {}, {}
Result: (polynomial q polynomial-dense-terms 0 0)
{}

Running Test: (#<procedure:add> (polynomial q polynomial-dense-terms 0 0) (polynomial r polynomial-dense-terms 0 5)) 
Applying #<procedure:add> on: {}, {5}
Result: (polynomial r polynomial-dense-terms 0 5)
{5}

Running Test: (#<procedure:add> (polynomial r polynomial-dense-terms 0 5) (polynomial s polynomial-dense-terms 1 3 5)) 
Applying #<procedure:add> on: {5}, {3s + 5}
Result: (polynomial s polynomial-sparse-terms (1 3) (0 (polynomial r polynomial-sparse-terms (0 10))))
{3s + {10}}

Running Test: (#<procedure:add> (polynomial s polynomial-dense-terms 1 3 5) (polynomial t polynomial-dense-terms 1 4 9)) 
Applying #<procedure:add> on: {3s + 5}, {4t + 9}
Result: (polynomial t polynomial-sparse-terms (1 4) (0 (polynomial s polynomial-sparse-terms (1 3) (0 14))))
{4t + {3s + 14}}

Running Test: (#<procedure:add> (polynomial t polynomial-dense-terms 1 4 9) (polynomial u polynomial-dense-terms 2 3 5 23)) 
Applying #<procedure:add> on: {4t + 9}, {3u^2 + 5u + 23}
Result: (polynomial u polynomial-sparse-terms (2 3) (1 5) (0 (polynomial t polynomial-sparse-terms (1 4) (0 32))))
{3u^2 + 5u + {4t + 32}}

Running Test: (#<procedure:add> (polynomial u polynomial-dense-terms 2 3 5 23) (polynomial v polynomial-dense-terms 2 4 9 19)) 
Applying #<procedure:add> on: {3u^2 + 5u + 23}, {4v^2 + 9v + 19}
Result: (polynomial v polynomial-sparse-terms (2 4) (1 9) (0 (polynomial u polynomial-sparse-terms (2 3) (1 5) (0 42))))
{4v^2 + 9v + {3u^2 + 5u + 42}}

Running Test: (#<procedure:add> (polynomial v polynomial-dense-terms 2 4 9 19) (polynomial w polynomial-dense-terms 5 1 2 0 3 -2 -5)) 
Applying #<procedure:add> on: {4v^2 + 9v + 19}, {w^5 + 2w^4 + 3w^2 - 2w - 5}
Result: (polynomial w polynomial-sparse-terms (5 1) (4 2) (2 3) (1 -2) (0 (polynomial v polynomial-sparse-terms (2 4) (1 9) (0 14))))
{w^5 + 2w^4 + 3w^2 - 2w + {4v^2 + 9v + 14}}

Running Test: (#<procedure:add> (polynomial w polynomial-dense-terms 5 1 2 0 3 -2 -5) (polynomial z polynomial-dense-terms 5 2 4 0 5 -7 -15)) 
Applying #<procedure:add> on: {w^5 + 2w^4 + 3w^2 - 2w - 5}, {2z^5 + 4z^4 + 5z^2 - 7z - 15}
Result: (polynomial z polynomial-sparse-terms (5 2) (4 4) (2 5) (1 -7) (0 (polynomial w polynomial-sparse-terms (5 1) (4 2) (2 3) (1 -2) (0 (integer . -20)))))
{2z^5 + 4z^4 + 5z^2 - 7z + {w^5 + 2w^4 + 3w^2 - 2w - 20}}


Running Test: (#<procedure:add> (polynomial p polynomial-sparse-terms (0 0)) (polynomial r polynomial-dense-terms 0 5)) 
Applying #<procedure:add> on: {}, {5}
Result: (polynomial r polynomial-dense-terms 0 5)
{5}

Running Test: (#<procedure:add> (polynomial r polynomial-dense-terms 0 5) (polynomial p polynomial-sparse-terms (0 0))) 
Applying #<procedure:add> on: {5}, {}
Result: (polynomial r polynomial-dense-terms 0 5)
{5}

Running Test: (#<procedure:add> (polynomial q polynomial-sparse-terms (0 5)) (polynomial s polynomial-dense-terms 1 3 5)) 
Applying #<procedure:add> on: {5}, {3s + 5}
Result: (polynomial s polynomial-sparse-terms (1 3) (0 (polynomial q polynomial-sparse-terms (0 10))))
{3s + {10}}

Running Test: (#<procedure:add> (polynomial r polynomial-sparse-terms (1 7) (0 5)) (polynomial t polynomial-dense-terms 1 4 9)) 
Applying #<procedure:add> on: {7r + 5}, {4t + 9}
Result: (polynomial t polynomial-sparse-terms (1 4) (0 (polynomial r polynomial-sparse-terms (1 7) (0 14))))
{4t + {7r + 14}}

Running Test: (#<procedure:add> (polynomial s polynomial-sparse-terms (1 9) (0 -25)) (polynomial u polynomial-dense-terms 2 3 5 23)) 
Applying #<procedure:add> on: {9s - 25}, {3u^2 + 5u + 23}
Result: (polynomial u polynomial-sparse-terms (2 3) (1 5) (0 (polynomial s polynomial-sparse-terms (1 9) (0 (integer . -2)))))
{3u^2 + 5u + {9s - 2}}

Running Test: (#<procedure:add> (polynomial t polynomial-sparse-terms (5 1) (4 2) (2 3) (1 -2) (0 -5)) (polynomial v polynomial-dense-terms 2 4 9 19)) 
Applying #<procedure:add> on: {t^5 + 2t^4 + 3t^2 - 2t - 5}, {4v^2 + 9v + 19}
Result: (polynomial v polynomial-sparse-terms (2 4) (1 9) (0 (polynomial t polynomial-sparse-terms (5 1) (4 2) (2 3) (1 -2) (0 14))))
{4v^2 + 9v + {t^5 + 2t^4 + 3t^2 - 2t + 14}}

Running Test: (#<procedure:add> (polynomial u polynomial-sparse-terms (5 2) (4 4) (2 5) (1 -7) (0 -15)) (polynomial w polynomial-dense-terms 5 1 2 0 3 -2 -5)) 
Applying #<procedure:add> on: {2u^5 + 4u^4 + 5u^2 - 7u - 15}, {w^5 + 2w^4 + 3w^2 - 2w - 5}
Result: (polynomial w polynomial-sparse-terms (5 1) (4 2) (2 3) (1 -2) (0 (polynomial u polynomial-sparse-terms (5 2) (4 4) (2 5) (1 -7) (0 (integer . -20)))))
{w^5 + 2w^4 + 3w^2 - 2w + {2u^5 + 4u^4 + 5u^2 - 7u - 20}}

Running Test: (#<procedure:add> (polynomial v polynomial-sparse-terms (10 3) (7 4) (5 -11) (1 -7) (0 -15)) (polynomial z polynomial-dense-terms 5 2 4 0 5 -7 -15)) 
Applying #<procedure:add> on: {3v^10 + 4v^7 - 11v^5 - 7v - 15}, {2z^5 + 4z^4 + 5z^2 - 7z - 15}
Result: (polynomial z polynomial-sparse-terms (5 2) (4 4) (2 5) (1 -7) (0 (polynomial v polynomial-sparse-terms (10 3) (7 4) (5 -11) (1 -7) (0 (integer . -30)))))
{2z^5 + 4z^4 + 5z^2 - 7z + {3v^10 + 4v^7 - 11v^5 - 7v - 30}}


Running Test: (#<procedure:add> (polynomial q polynomial-dense-terms 0 0) (polynomial q polynomial-sparse-terms (0 5))) 
Applying #<procedure:add> on: {}, {5}
Result: (polynomial q polynomial-sparse-terms (0 5))
{5}

Running Test: (#<procedure:add> (polynomial q polynomial-sparse-terms (0 5)) (polynomial q polynomial-dense-terms 0 0)) 
Applying #<procedure:add> on: {5}, {}
Result: (polynomial q polynomial-sparse-terms (0 5))
{5}

Running Test: (#<procedure:add> (polynomial r polynomial-dense-terms 0 5) (polynomial r polynomial-sparse-terms (1 7) (0 5))) 
Applying #<procedure:add> on: {5}, {7r + 5}
Result: (polynomial r polynomial-sparse-terms (1 7) (0 10))
{7r + 10}

Running Test: (#<procedure:add> (polynomial s polynomial-dense-terms 1 3 5) (polynomial s polynomial-sparse-terms (1 9) (0 -25))) 
Applying #<procedure:add> on: {3s + 5}, {9s - 25}
Result: (polynomial s polynomial-sparse-terms (1 12) (0 (integer . -20)))
{12s - 20}

Running Test: (#<procedure:add> (polynomial t polynomial-dense-terms 1 4 9) (polynomial t polynomial-sparse-terms (5 1) (4 2) (2 3) (1 -2) (0 -5))) 
Applying #<procedure:add> on: {4t + 9}, {t^5 + 2t^4 + 3t^2 - 2t - 5}
Result: (polynomial t polynomial-sparse-terms (5 1) (4 2) (2 3) (1 2) (0 4))
{t^5 + 2t^4 + 3t^2 + 2t + 4}

Running Test: (#<procedure:add> (polynomial u polynomial-dense-terms 2 3 5 23) (polynomial u polynomial-sparse-terms (5 2) (4 4) (2 5) (1 -7) (0 -15))) 
Applying #<procedure:add> on: {3u^2 + 5u + 23}, {2u^5 + 4u^4 + 5u^2 - 7u - 15}
Result: (polynomial u polynomial-sparse-terms (5 2) (4 4) (2 8) (1 (integer . -2)) (0 8))
{2u^5 + 4u^4 + 8u^2 - 2u + 8}

Running Test: (#<procedure:add> (polynomial v polynomial-dense-terms 2 4 9 19) (polynomial v polynomial-sparse-terms (10 3) (7 4) (5 -11) (1 -7) (0 -15))) 
Applying #<procedure:add> on: {4v^2 + 9v + 19}, {3v^10 + 4v^7 - 11v^5 - 7v - 15}
Result: (polynomial v polynomial-sparse-terms (10 3) (7 4) (5 -11) (2 4) (1 2) (0 4))
{3v^10 + 4v^7 - 11v^5 + 4v^2 + 2v + 4}

Running Test: (#<procedure:add> (polynomial w polynomial-dense-terms 5 1 2 0 3 -2 -5) (polynomial z polynomial-sparse-terms (12 4) (6 13) (5 -12) (1 -7) (0 -15))) 
Applying #<procedure:add> on: {w^5 + 2w^4 + 3w^2 - 2w - 5}, {4z^12 + 13z^6 - 12z^5 - 7z - 15}
Result: (polynomial z polynomial-sparse-terms (12 4) (6 13) (5 -12) (1 -7) (0 (polynomial w polynomial-sparse-terms (5 1) (4 2) (2 3) (1 -2) (0 (integer . -20)))))
{4z^12 + 13z^6 - 12z^5 - 7z + {w^5 + 2w^4 + 3w^2 - 2w - 20}}


Running Test: (#<procedure:mul> (polynomial p polynomial-sparse-terms (0 0)) (polynomial p polynomial-sparse-terms (0 0))) 
Applying #<procedure:mul> on: {}, {}
Result: (polynomial p)
{}

Running Test: (#<procedure:mul> (polynomial p polynomial-sparse-terms (0 0)) (polynomial q polynomial-sparse-terms (0 5))) 
Applying #<procedure:mul> on: {}, {5}
Result: (polynomial q)
{}

Running Test: (#<procedure:mul> (polynomial q polynomial-sparse-terms (0 5)) (polynomial q polynomial-sparse-terms (0 5))) 
Applying #<procedure:mul> on: {5}, {5}
Result: (polynomial q polynomial-sparse-terms (0 25) (0 0))
{25}

Running Test: (#<procedure:mul> (polynomial q polynomial-sparse-terms (0 5)) (polynomial r polynomial-sparse-terms (1 7) (0 5))) 
Applying #<procedure:mul> on: {5}, {7r + 5}
Result: (polynomial r polynomial-sparse-terms (1 (polynomial q polynomial-sparse-terms (0 35) (0 0))) (0 (polynomial q polynomial-sparse-terms (0 25) (0 0))) (0 0))
{{35}r + {25}}

Running Test: (#<procedure:mul> (polynomial r polynomial-sparse-terms (1 7) (0 5)) (polynomial s polynomial-sparse-terms (1 9) (0 -25))) 
Applying #<procedure:mul> on: {7r + 5}, {9s - 25}
Result: (polynomial s polynomial-sparse-terms (1 (polynomial r polynomial-sparse-terms (1 63) (0 45) (0 0))) (0 (polynomial r polynomial-sparse-terms (1 (integer . -175)) (0 (integer . -125)) (0 0))) (0 0))
{{63r + 45}s + { - 175r - 125}}

Running Test: (#<procedure:mul> (polynomial s polynomial-sparse-terms (1 9) (0 -25)) (polynomial s polynomial-sparse-terms (1 9) (0 -25))) 
Applying #<procedure:mul> on: {9s - 25}, {9s - 25}
Result: (polynomial s polynomial-sparse-terms (2 81) (1 (integer . -450)) (0 625) (0 0))
{81s^2 - 450s + 625}

Running Test: (#<procedure:mul> (polynomial s polynomial-sparse-terms (1 9) (0 -25)) (polynomial t polynomial-sparse-terms (5 1) (4 2) (2 3) (1 -2) (0 -5))) 
Applying #<procedure:mul> on: {9s - 25}, {t^5 + 2t^4 + 3t^2 - 2t - 5}
Result: (polynomial t polynomial-sparse-terms (5 (polynomial s polynomial-sparse-terms (1 9) (0 (integer . -25)) (0 0))) (4 (polynomial s polynomial-sparse-terms (1 18) (0 (integer . -50)) (0 0))) (2 (polynomial s polynomial-sparse-terms (1 27) (0 (integer . -75)) (0 0))) (1 (polynomial s polynomial-sparse-terms (1 (integer . -18)) (0 50) (0 0))) (0 (polynomial s polynomial-sparse-terms (1 (integer . -45)) (0 125) (0 0))) (0 0))
{{9s - 25}t^5 + {18s - 50}t^4 + {27s - 75}t^2 + { - 18s + 50}t + { - 45s + 125}}

Running Test: (#<procedure:mul> (polynomial t polynomial-sparse-terms (5 1) (4 2) (2 3) (1 -2) (0 -5)) (polynomial u polynomial-sparse-terms (5 2) (4 4) (2 5) (1 -7) (0 -15))) 
Applying #<procedure:mul> on: {t^5 + 2t^4 + 3t^2 - 2t - 5}, {2u^5 + 4u^4 + 5u^2 - 7u - 15}
Result: (polynomial u polynomial-sparse-terms (5 (polynomial t polynomial-sparse-terms (5 2) (4 4) (2 6) (1 (integer . -4)) (0 (integer . -10)) (0 0))) (4 (polynomial t polynomial-sparse-terms (5 4) (4 8) (2 12) (1 (integer . -8)) (0 (integer . -20)) (0 0))) (2 (polynomial t polynomial-sparse-terms (5 5) (4 10) (2 15) (1 (integer . -10)) (0 (integer . -25)) (0 0))) (1 (polynomial t polynomial-sparse-terms (5 (integer . -7)) (4 (integer . -14)) (2 (integer . -21)) (1 14) (0 35) (0 0))) (0 (polynomial t polynomial-sparse-terms (5 (integer . -15)) (4 (integer . -30)) (2 (integer . -45)) (1 30) (0 75) (0 0))) (0 0))
{{2t^5 + 4t^4 + 6t^2 - 4t - 10}u^5 + {4t^5 + 8t^4 + 12t^2 - 8t - 20}u^4 + {5t^5 + 10t^4 + 15t^2 - 10t - 25}u^2 + { - 7t^5 - 14t^4 - 21t^2 + 14t + 35}u + { - 15t^5 - 30t^4 - 45t^2 + 30t + 75}}

Running Test: (#<procedure:mul> (polynomial u polynomial-sparse-terms (5 2) (4 4) (2 5) (1 -7) (0 -15)) (polynomial v polynomial-sparse-terms (10 3) (7 4) (5 -11) (1 -7) (0 -15))) 
Applying #<procedure:mul> on: {2u^5 + 4u^4 + 5u^2 - 7u - 15}, {3v^10 + 4v^7 - 11v^5 - 7v - 15}
Result: (polynomial v polynomial-sparse-terms (10 (polynomial u polynomial-sparse-terms (5 6) (4 12) (2 15) (1 (integer . -21)) (0 (integer . -45)) (0 0))) (7 (polynomial u polynomial-sparse-terms (5 8) (4 16) (2 20) (1 (integer . -28)) (0 (integer . -60)) (0 0))) (5 (polynomial u polynomial-sparse-terms (5 (integer . -22)) (4 (integer . -44)) (2 (integer . -55)) (1 77) (0 165) (0 0))) (1 (polynomial u polynomial-sparse-terms (5 (integer . -14)) (4 (integer . -28)) (2 (integer . -35)) (1 49) (0 105) (0 0))) (0 (polynomial u polynomial-sparse-terms (5 (integer . -30)) (4 (integer . -60)) (2 (integer . -75)) (1 105) (0 225) (0 0))) (0 0))
{{6u^5 + 12u^4 + 15u^2 - 21u - 45}v^10 + {8u^5 + 16u^4 + 20u^2 - 28u - 60}v^7 + { - 22u^5 - 44u^4 - 55u^2 + 77u + 165}v^5 + { - 14u^5 - 28u^4 - 35u^2 + 49u + 105}v + { - 30u^5 - 60u^4 - 75u^2 + 105u + 225}}

Running Test: (#<procedure:mul> (polynomial v polynomial-sparse-terms (10 3) (7 4) (5 -11) (1 -7) (0 -15)) (polynomial z polynomial-sparse-terms (12 4) (6 13) (5 -12) (1 -7) (0 -15))) 
Applying #<procedure:mul> on: {3v^10 + 4v^7 - 11v^5 - 7v - 15}, {4z^12 + 13z^6 - 12z^5 - 7z - 15}
Result: (polynomial z polynomial-sparse-terms (12 (polynomial v polynomial-sparse-terms (10 12) (7 16) (5 (integer . -44)) (1 (integer . -28)) (0 (integer . -60)) (0 0))) (6 (polynomial v polynomial-sparse-terms (10 39) (7 52) (5 (integer . -143)) (1 (integer . -91)) (0 (integer . -195)) (0 0))) (5 (polynomial v polynomial-sparse-terms (10 (integer . -36)) (7 (integer . -48)) (5 132) (1 84) (0 180) (0 0))) (1 (polynomial v polynomial-sparse-terms (10 (integer . -21)) (7 (integer . -28)) (5 77) (1 49) (0 105) (0 0))) (0 (polynomial v polynomial-sparse-terms (10 (integer . -45)) (7 (integer . -60)) (5 165) (1 105) (0 225) (0 0))) (0 0))
{{12v^10 + 16v^7 - 44v^5 - 28v - 60}z^12 + {39v^10 + 52v^7 - 143v^5 - 91v - 195}z^6 + { - 36v^10 - 48v^7 + 132v^5 + 84v + 180}z^5 + { - 21v^10 - 28v^7 + 77v^5 + 49v + 105}z + { - 45v^10 - 60v^7 + 165v^5 + 105v + 225}}


Running Test: (#<procedure:mul> (polynomial q polynomial-dense-terms 0 0) (polynomial q polynomial-dense-terms 0 0)) 
Applying #<procedure:mul> on: {}, {}
Result: (polynomial q)
{}

Running Test: (#<procedure:mul> (polynomial q polynomial-dense-terms 0 0) (polynomial r polynomial-dense-terms 0 5)) 
Applying #<procedure:mul> on: {}, {5}
Result: (polynomial r)
{}

Running Test: (#<procedure:mul> (polynomial r polynomial-dense-terms 0 5) (polynomial r polynomial-dense-terms 0 5)) 
Applying #<procedure:mul> on: {5}, {5}
Result: (polynomial r polynomial-sparse-terms (0 25) (0 0))
{25}

Running Test: (#<procedure:mul> (polynomial r polynomial-dense-terms 0 5) (polynomial s polynomial-dense-terms 1 3 5)) 
Applying #<procedure:mul> on: {5}, {3s + 5}
Result: (polynomial s polynomial-sparse-terms (1 (polynomial r polynomial-sparse-terms (0 15) (0 0))) (0 (polynomial r polynomial-sparse-terms (0 25) (0 0))) (0 0))
{{15}s + {25}}

Running Test: (#<procedure:mul> (polynomial s polynomial-dense-terms 1 3 5) (polynomial t polynomial-dense-terms 1 4 9)) 
Applying #<procedure:mul> on: {3s + 5}, {4t + 9}
Result: (polynomial t polynomial-sparse-terms (1 (polynomial s polynomial-sparse-terms (1 12) (0 20) (0 0))) (0 (polynomial s polynomial-sparse-terms (1 27) (0 45) (0 0))) (0 0))
{{12s + 20}t + {27s + 45}}

Running Test: (#<procedure:mul> (polynomial t polynomial-dense-terms 1 4 9) (polynomial u polynomial-dense-terms 2 3 5 23)) 
Applying #<procedure:mul> on: {4t + 9}, {3u^2 + 5u + 23}
Result: (polynomial u polynomial-sparse-terms (2 (polynomial t polynomial-sparse-terms (1 12) (0 27) (0 0))) (1 (polynomial t polynomial-sparse-terms (1 20) (0 45) (0 0))) (0 (polynomial t polynomial-sparse-terms (1 92) (0 207) (0 0))) (0 0))
{{12t + 27}u^2 + {20t + 45}u + {92t + 207}}

Running Test: (#<procedure:mul> (polynomial u polynomial-dense-terms 2 3 5 23) (polynomial v polynomial-dense-terms 2 4 9 19)) 
Applying #<procedure:mul> on: {3u^2 + 5u + 23}, {4v^2 + 9v + 19}
Result: (polynomial v polynomial-sparse-terms (2 (polynomial u polynomial-sparse-terms (2 12) (1 20) (0 92) (0 0))) (1 (polynomial u polynomial-sparse-terms (2 27) (1 45) (0 207) (0 0))) (0 (polynomial u polynomial-sparse-terms (2 57) (1 95) (0 437) (0 0))) (0 0))
{{12u^2 + 20u + 92}v^2 + {27u^2 + 45u + 207}v + {57u^2 + 95u + 437}}

Running Test: (#<procedure:mul> (polynomial v polynomial-dense-terms 2 4 9 19) (polynomial w polynomial-dense-terms 5 1 2 0 3 -2 -5)) 
Applying #<procedure:mul> on: {4v^2 + 9v + 19}, {w^5 + 2w^4 + 3w^2 - 2w - 5}
Result: (polynomial w polynomial-sparse-terms (5 (polynomial v polynomial-sparse-terms (2 4) (1 9) (0 19) (0 0))) (4 (polynomial v polynomial-sparse-terms (2 8) (1 18) (0 38) (0 0))) (2 (polynomial v polynomial-sparse-terms (2 12) (1 27) (0 57) (0 0))) (1 (polynomial v polynomial-sparse-terms (2 (integer . -8)) (1 (integer . -18)) (0 (integer . -38)) (0 0))) (0 (polynomial v polynomial-sparse-terms (2 (integer . -20)) (1 (integer . -45)) (0 (integer . -95)) (0 0))) (0 0))
{{4v^2 + 9v + 19}w^5 + {8v^2 + 18v + 38}w^4 + {12v^2 + 27v + 57}w^2 + { - 8v^2 - 18v - 38}w + { - 20v^2 - 45v - 95}}

Running Test: (#<procedure:mul> (polynomial w polynomial-dense-terms 5 1 2 0 3 -2 -5) (polynomial z polynomial-dense-terms 5 2 4 0 5 -7 -15)) 
Applying #<procedure:mul> on: {w^5 + 2w^4 + 3w^2 - 2w - 5}, {2z^5 + 4z^4 + 5z^2 - 7z - 15}
Result: (polynomial z polynomial-sparse-terms (5 (polynomial w polynomial-sparse-terms (5 2) (4 4) (2 6) (1 (integer . -4)) (0 (integer . -10)) (0 0))) (4 (polynomial w polynomial-sparse-terms (5 4) (4 8) (2 12) (1 (integer . -8)) (0 (integer . -20)) (0 0))) (2 (polynomial w polynomial-sparse-terms (5 5) (4 10) (2 15) (1 (integer . -10)) (0 (integer . -25)) (0 0))) (1 (polynomial w polynomial-sparse-terms (5 (integer . -7)) (4 (integer . -14)) (2 (integer . -21)) (1 14) (0 35) (0 0))) (0 (polynomial w polynomial-sparse-terms (5 (integer . -15)) (4 (integer . -30)) (2 (integer . -45)) (1 30) (0 75) (0 0))) (0 0))
{{2w^5 + 4w^4 + 6w^2 - 4w - 10}z^5 + {4w^5 + 8w^4 + 12w^2 - 8w - 20}z^4 + {5w^5 + 10w^4 + 15w^2 - 10w - 25}z^2 + { - 7w^5 - 14w^4 - 21w^2 + 14w + 35}z + { - 15w^5 - 30w^4 - 45w^2 + 30w + 75}}


Running Test: (#<procedure:mul> (polynomial p polynomial-sparse-terms (0 0)) (polynomial r polynomial-dense-terms 0 5)) 
Applying #<procedure:mul> on: {}, {5}
Result: (polynomial r)
{}

Running Test: (#<procedure:mul> (polynomial r polynomial-dense-terms 0 5) (polynomial p polynomial-sparse-terms (0 0))) 
Applying #<procedure:mul> on: {5}, {}
Result: (polynomial r)
{}

Running Test: (#<procedure:mul> (polynomial q polynomial-sparse-terms (0 5)) (polynomial s polynomial-dense-terms 1 3 5)) 
Applying #<procedure:mul> on: {5}, {3s + 5}
Result: (polynomial s polynomial-sparse-terms (1 (polynomial q polynomial-sparse-terms (0 15) (0 0))) (0 (polynomial q polynomial-sparse-terms (0 25) (0 0))) (0 0))
{{15}s + {25}}

Running Test: (#<procedure:mul> (polynomial r polynomial-sparse-terms (1 7) (0 5)) (polynomial t polynomial-dense-terms 1 4 9)) 
Applying #<procedure:mul> on: {7r + 5}, {4t + 9}
Result: (polynomial t polynomial-sparse-terms (1 (polynomial r polynomial-sparse-terms (1 28) (0 20) (0 0))) (0 (polynomial r polynomial-sparse-terms (1 63) (0 45) (0 0))) (0 0))
{{28r + 20}t + {63r + 45}}

Running Test: (#<procedure:mul> (polynomial s polynomial-sparse-terms (1 9) (0 -25)) (polynomial u polynomial-dense-terms 2 3 5 23)) 
Applying #<procedure:mul> on: {9s - 25}, {3u^2 + 5u + 23}
Result: (polynomial u polynomial-sparse-terms (2 (polynomial s polynomial-sparse-terms (1 27) (0 (integer . -75)) (0 0))) (1 (polynomial s polynomial-sparse-terms (1 45) (0 (integer . -125)) (0 0))) (0 (polynomial s polynomial-sparse-terms (1 207) (0 (integer . -575)) (0 0))) (0 0))
{{27s - 75}u^2 + {45s - 125}u + {207s - 575}}

Running Test: (#<procedure:mul> (polynomial t polynomial-sparse-terms (5 1) (4 2) (2 3) (1 -2) (0 -5)) (polynomial v polynomial-dense-terms 2 4 9 19)) 
Applying #<procedure:mul> on: {t^5 + 2t^4 + 3t^2 - 2t - 5}, {4v^2 + 9v + 19}
Result: (polynomial v polynomial-sparse-terms (2 (polynomial t polynomial-sparse-terms (5 4) (4 8) (2 12) (1 (integer . -8)) (0 (integer . -20)) (0 0))) (1 (polynomial t polynomial-sparse-terms (5 9) (4 18) (2 27) (1 (integer . -18)) (0 (integer . -45)) (0 0))) (0 (polynomial t polynomial-sparse-terms (5 19) (4 38) (2 57) (1 (integer . -38)) (0 (integer . -95)) (0 0))) (0 0))
{{4t^5 + 8t^4 + 12t^2 - 8t - 20}v^2 + {9t^5 + 18t^4 + 27t^2 - 18t - 45}v + {19t^5 + 38t^4 + 57t^2 - 38t - 95}}

Running Test: (#<procedure:mul> (polynomial u polynomial-sparse-terms (5 2) (4 4) (2 5) (1 -7) (0 -15)) (polynomial w polynomial-dense-terms 5 1 2 0 3 -2 -5)) 
Applying #<procedure:mul> on: {2u^5 + 4u^4 + 5u^2 - 7u - 15}, {w^5 + 2w^4 + 3w^2 - 2w - 5}
Result: (polynomial w polynomial-sparse-terms (5 (polynomial u polynomial-sparse-terms (5 2) (4 4) (2 5) (1 (integer . -7)) (0 (integer . -15)) (0 0))) (4 (polynomial u polynomial-sparse-terms (5 4) (4 8) (2 10) (1 (integer . -14)) (0 (integer . -30)) (0 0))) (2 (polynomial u polynomial-sparse-terms (5 6) (4 12) (2 15) (1 (integer . -21)) (0 (integer . -45)) (0 0))) (1 (polynomial u polynomial-sparse-terms (5 (integer . -4)) (4 (integer . -8)) (2 (integer . -10)) (1 14) (0 30) (0 0))) (0 (polynomial u polynomial-sparse-terms (5 (integer . -10)) (4 (integer . -20)) (2 (integer . -25)) (1 35) (0 75) (0 0))) (0 0))
{{2u^5 + 4u^4 + 5u^2 - 7u - 15}w^5 + {4u^5 + 8u^4 + 10u^2 - 14u - 30}w^4 + {6u^5 + 12u^4 + 15u^2 - 21u - 45}w^2 + { - 4u^5 - 8u^4 - 10u^2 + 14u + 30}w + { - 10u^5 - 20u^4 - 25u^2 + 35u + 75}}

Running Test: (#<procedure:mul> (polynomial v polynomial-sparse-terms (10 3) (7 4) (5 -11) (1 -7) (0 -15)) (polynomial z polynomial-dense-terms 5 2 4 0 5 -7 -15)) 
Applying #<procedure:mul> on: {3v^10 + 4v^7 - 11v^5 - 7v - 15}, {2z^5 + 4z^4 + 5z^2 - 7z - 15}
Result: (polynomial z polynomial-sparse-terms (5 (polynomial v polynomial-sparse-terms (10 6) (7 8) (5 (integer . -22)) (1 (integer . -14)) (0 (integer . -30)) (0 0))) (4 (polynomial v polynomial-sparse-terms (10 12) (7 16) (5 (integer . -44)) (1 (integer . -28)) (0 (integer . -60)) (0 0))) (2 (polynomial v polynomial-sparse-terms (10 15) (7 20) (5 (integer . -55)) (1 (integer . -35)) (0 (integer . -75)) (0 0))) (1 (polynomial v polynomial-sparse-terms (10 (integer . -21)) (7 (integer . -28)) (5 77) (1 49) (0 105) (0 0))) (0 (polynomial v polynomial-sparse-terms (10 (integer . -45)) (7 (integer . -60)) (5 165) (1 105) (0 225) (0 0))) (0 0))
{{6v^10 + 8v^7 - 22v^5 - 14v - 30}z^5 + {12v^10 + 16v^7 - 44v^5 - 28v - 60}z^4 + {15v^10 + 20v^7 - 55v^5 - 35v - 75}z^2 + { - 21v^10 - 28v^7 + 77v^5 + 49v + 105}z + { - 45v^10 - 60v^7 + 165v^5 + 105v + 225}}


Running Test: (#<procedure:mul> (polynomial q polynomial-dense-terms 0 0) (polynomial q polynomial-sparse-terms (0 5))) 
Applying #<procedure:mul> on: {}, {5}
Result: (polynomial q)
{}

Running Test: (#<procedure:mul> (polynomial q polynomial-sparse-terms (0 5)) (polynomial q polynomial-dense-terms 0 0)) 
Applying #<procedure:mul> on: {5}, {}
Result: (polynomial q)
{}

Running Test: (#<procedure:mul> (polynomial r polynomial-dense-terms 0 5) (polynomial r polynomial-sparse-terms (1 7) (0 5))) 
Applying #<procedure:mul> on: {5}, {7r + 5}
Result: (polynomial r polynomial-sparse-terms (1 35) (0 25) (0 0))
{35r + 25}

Running Test: (#<procedure:mul> (polynomial s polynomial-dense-terms 1 3 5) (polynomial s polynomial-sparse-terms (1 9) (0 -25))) 
Applying #<procedure:mul> on: {3s + 5}, {9s - 25}
Result: (polynomial s polynomial-sparse-terms (2 27) (1 (integer . -30)) (0 (integer . -125)) (0 0))
{27s^2 - 30s - 125}

Running Test: (#<procedure:mul> (polynomial t polynomial-dense-terms 1 4 9) (polynomial t polynomial-sparse-terms (5 1) (4 2) (2 3) (1 -2) (0 -5))) 
Applying #<procedure:mul> on: {4t + 9}, {t^5 + 2t^4 + 3t^2 - 2t - 5}
Result: (polynomial t polynomial-sparse-terms (6 4) (5 17) (4 18) (3 12) (2 19) (1 (integer . -38)) (0 (integer . -45)) (0 0))
{4t^6 + 17t^5 + 18t^4 + 12t^3 + 19t^2 - 38t - 45}

Running Test: (#<procedure:mul> (polynomial u polynomial-dense-terms 2 3 5 23) (polynomial u polynomial-sparse-terms (5 2) (4 4) (2 5) (1 -7) (0 -15))) 
Applying #<procedure:mul> on: {3u^2 + 5u + 23}, {2u^5 + 4u^4 + 5u^2 - 7u - 15}
Result: (polynomial u polynomial-sparse-terms (7 6) (6 22) (5 66) (4 107) (3 4) (2 35) (1 (integer . -236)) (0 (integer . -345)) (0 0))
{6u^7 + 22u^6 + 66u^5 + 107u^4 + 4u^3 + 35u^2 - 236u - 345}

Running Test: (#<procedure:mul> (polynomial v polynomial-dense-terms 2 4 9 19) (polynomial v polynomial-sparse-terms (10 3) (7 4) (5 -11) (1 -7) (0 -15))) 
Applying #<procedure:mul> on: {4v^2 + 9v + 19}, {3v^10 + 4v^7 - 11v^5 - 7v - 15}
Result: (polynomial v polynomial-sparse-terms (12 12) (11 27) (10 57) (9 16) (8 36) (7 32) (6 (integer . -99)) (5 (integer . -209)) (3 (integer . -28)) (2 (integer . -123)) (1 (integer . -268)) (0 (integer . -285)) (0 0))
{12v^12 + 27v^11 + 57v^10 + 16v^9 + 36v^8 + 32v^7 - 99v^6 - 209v^5 - 28v^3 - 123v^2 - 268v - 285}

Running Test: (#<procedure:mul> (polynomial w polynomial-dense-terms 5 1 2 0 3 -2 -5) (polynomial z polynomial-sparse-terms (12 4) (6 13) (5 -12) (1 -7) (0 -15))) 
Applying #<procedure:mul> on: {w^5 + 2w^4 + 3w^2 - 2w - 5}, {4z^12 + 13z^6 - 12z^5 - 7z - 15}
Result: (polynomial z polynomial-sparse-terms (12 (polynomial w polynomial-sparse-terms (5 4) (4 8) (2 12) (1 (integer . -8)) (0 (integer . -20)) (0 0))) (6 (polynomial w polynomial-sparse-terms (5 13) (4 26) (2 39) (1 (integer . -26)) (0 (integer . -65)) (0 0))) (5 (polynomial w polynomial-sparse-terms (5 (integer . -12)) (4 (integer . -24)) (2 (integer . -36)) (1 24) (0 60) (0 0))) (1 (polynomial w polynomial-sparse-terms (5 (integer . -7)) (4 (integer . -14)) (2 (integer . -21)) (1 14) (0 35) (0 0))) (0 (polynomial w polynomial-sparse-terms (5 (integer . -15)) (4 (integer . -30)) (2 (integer . -45)) (1 30) (0 75) (0 0))) (0 0))
{{4w^5 + 8w^4 + 12w^2 - 8w - 20}z^12 + {13w^5 + 26w^4 + 39w^2 - 26w - 65}z^6 + { - 12w^5 - 24w^4 - 36w^2 + 24w + 60}z^5 + { - 7w^5 - 14w^4 - 21w^2 + 14w + 35}z + { - 15w^5 - 30w^4 - 45w^2 + 30w + 75}}


Running Test: (#<procedure:NEGATE> (polynomial p polynomial-sparse-terms (0 0))) 
Applying #<procedure:NEGATE> on: {}
Result: (polynomial p polynomial-sparse-terms (0 0))
{}

Running Test: (#<procedure:NEGATE> (polynomial q polynomial-sparse-terms (0 5))) 
Applying #<procedure:NEGATE> on: {5}
Result: (polynomial q polynomial-sparse-terms (0 (integer . -5)))
{ - 5}

Running Test: (#<procedure:NEGATE> (polynomial q polynomial-sparse-terms (0 5))) 
Applying #<procedure:NEGATE> on: {5}
Result: (polynomial q polynomial-sparse-terms (0 (integer . -5)))
{ - 5}

Running Test: (#<procedure:NEGATE> (polynomial r polynomial-sparse-terms (1 7) (0 5))) 
Applying #<procedure:NEGATE> on: {7r + 5}
Result: (polynomial r polynomial-sparse-terms (1 (integer . -7)) (0 (integer . -5)))
{ - 7r - 5}

Running Test: (#<procedure:NEGATE> (polynomial s polynomial-sparse-terms (1 9) (0 -25))) 
Applying #<procedure:NEGATE> on: {9s - 25}
Result: (polynomial s polynomial-sparse-terms (1 (integer . -9)) (0 25))
{ - 9s + 25}

Running Test: (#<procedure:NEGATE> (polynomial t polynomial-sparse-terms (5 1) (4 2) (2 3) (1 -2) (0 -5))) 
Applying #<procedure:NEGATE> on: {t^5 + 2t^4 + 3t^2 - 2t - 5}
Result: (polynomial t polynomial-sparse-terms (5 (integer . -1)) (4 (integer . -2)) (2 (integer . -3)) (1 2) (0 5))
{ - 1t^5 - 2t^4 - 3t^2 + 2t + 5}

Running Test: (#<procedure:NEGATE> (polynomial u polynomial-sparse-terms (5 2) (4 4) (2 5) (1 -7) (0 -15))) 
Applying #<procedure:NEGATE> on: {2u^5 + 4u^4 + 5u^2 - 7u - 15}
Result: (polynomial u polynomial-sparse-terms (5 (integer . -2)) (4 (integer . -4)) (2 (integer . -5)) (1 7) (0 15))
{ - 2u^5 - 4u^4 - 5u^2 + 7u + 15}

Running Test: (#<procedure:NEGATE> (polynomial v polynomial-sparse-terms (10 3) (7 4) (5 -11) (1 -7) (0 -15))) 
Applying #<procedure:NEGATE> on: {3v^10 + 4v^7 - 11v^5 - 7v - 15}
Result: (polynomial v polynomial-sparse-terms (10 (integer . -3)) (7 (integer . -4)) (5 11) (1 7) (0 15))
{ - 3v^10 - 4v^7 + 11v^5 + 7v + 15}

Running Test: (#<procedure:NEGATE> (polynomial z polynomial-sparse-terms (12 4) (6 13) (5 -12) (1 -7) (0 -15))) 
Applying #<procedure:NEGATE> on: {4z^12 + 13z^6 - 12z^5 - 7z - 15}
Result: (polynomial z polynomial-sparse-terms (12 (integer . -4)) (6 (integer . -13)) (5 12) (1 7) (0 15))
{ - 4z^12 - 13z^6 + 12z^5 + 7z + 15}


Running Test: (#<procedure:NEGATE> (polynomial q polynomial-dense-terms 0 0)) 
Applying #<procedure:NEGATE> on: {}
Result: (polynomial q polynomial-dense-terms 0 0)
{}

Running Test: (#<procedure:NEGATE> (polynomial r polynomial-dense-terms 0 5)) 
Applying #<procedure:NEGATE> on: {5}
Result: (polynomial r polynomial-dense-terms 0 (integer . -5))
{ - 5}

Running Test: (#<procedure:NEGATE> (polynomial r polynomial-dense-terms 0 5)) 
Applying #<procedure:NEGATE> on: {5}
Result: (polynomial r polynomial-dense-terms 0 (integer . -5))
{ - 5}

Running Test: (#<procedure:NEGATE> (polynomial s polynomial-dense-terms 1 3 5)) 
Applying #<procedure:NEGATE> on: {3s + 5}
Result: (polynomial s polynomial-dense-terms 1 (integer . -3) (integer . -5))
{ - 3s - 5}

Running Test: (#<procedure:NEGATE> (polynomial t polynomial-dense-terms 1 4 9)) 
Applying #<procedure:NEGATE> on: {4t + 9}
Result: (polynomial t polynomial-dense-terms 1 (integer . -4) (integer . -9))
{ - 4t - 9}

Running Test: (#<procedure:NEGATE> (polynomial u polynomial-dense-terms 2 3 5 23)) 
Applying #<procedure:NEGATE> on: {3u^2 + 5u + 23}
Result: (polynomial u polynomial-dense-terms 2 (integer . -3) (integer . -5) (integer . -23))
{ - 3u^2 - 5u - 23}

Running Test: (#<procedure:NEGATE> (polynomial v polynomial-dense-terms 2 4 9 19)) 
Applying #<procedure:NEGATE> on: {4v^2 + 9v + 19}
Result: (polynomial v polynomial-dense-terms 2 (integer . -4) (integer . -9) (integer . -19))
{ - 4v^2 - 9v - 19}

Running Test: (#<procedure:NEGATE> (polynomial w polynomial-dense-terms 5 1 2 0 3 -2 -5)) 
Applying #<procedure:NEGATE> on: {w^5 + 2w^4 + 3w^2 - 2w - 5}
Result: (polynomial w polynomial-dense-terms 5 (integer . -1) (integer . -2) 0 (integer . -3) 2 5)
{ - 1w^5 - 2w^4 - 3w^2 + 2w + 5}

Running Test: (#<procedure:NEGATE> (polynomial z polynomial-dense-terms 5 2 4 0 5 -7 -15)) 
Applying #<procedure:NEGATE> on: {2z^5 + 4z^4 + 5z^2 - 7z - 15}
Result: (polynomial z polynomial-dense-terms 5 (integer . -2) (integer . -4) 0 (integer . -5) 7 15)
{ - 2z^5 - 4z^4 - 5z^2 + 7z + 15}

Running Test: (#<procedure:sub> (polynomial p polynomial-sparse-terms (0 0)) (polynomial p polynomial-sparse-terms (0 0))) 
Applying #<procedure:sub> on: {}, {}
Result: (polynomial p polynomial-sparse-terms (0 0))
{}

Running Test: (#<procedure:sub> (polynomial p polynomial-sparse-terms (0 0)) (polynomial q polynomial-sparse-terms (0 5))) 
Applying #<procedure:sub> on: {}, {5}
Result: (polynomial q polynomial-sparse-terms (0 (integer . -5)))
{ - 5}

Running Test: (#<procedure:sub> (polynomial q polynomial-sparse-terms (0 5)) (polynomial q polynomial-sparse-terms (0 5))) 
Applying #<procedure:sub> on: {5}, {5}
Result: (polynomial q polynomial-sparse-terms)
{}

Running Test: (#<procedure:sub> (polynomial q polynomial-sparse-terms (0 5)) (polynomial r polynomial-sparse-terms (1 7) (0 5))) 
Applying #<procedure:sub> on: {5}, {7r + 5}
Result: (polynomial r polynomial-sparse-terms (1 (integer . -7)))
{ - 7r}

Running Test: (#<procedure:sub> (polynomial r polynomial-sparse-terms (1 7) (0 5)) (polynomial s polynomial-sparse-terms (1 9) (0 -25))) 
Applying #<procedure:sub> on: {7r + 5}, {9s - 25}
Result: (polynomial s polynomial-sparse-terms (1 (integer . -9)) (0 (polynomial r polynomial-sparse-terms (1 7) (0 30))))
{ - 9s + {7r + 30}}

Running Test: (#<procedure:sub> (polynomial s polynomial-sparse-terms (1 9) (0 -25)) (polynomial t polynomial-sparse-terms (5 1) (4 2) (2 3) (1 -2) (0 -5))) 
Applying #<procedure:sub> on: {9s - 25}, {t^5 + 2t^4 + 3t^2 - 2t - 5}
Result: (polynomial t polynomial-sparse-terms (5 (integer . -1)) (4 (integer . -2)) (2 (integer . -3)) (1 2) (0 (polynomial s polynomial-sparse-terms (1 9) (0 (integer . -20)))))
{ - 1t^5 - 2t^4 - 3t^2 + 2t + {9s - 20}}

Running Test: (#<procedure:sub> (polynomial t polynomial-sparse-terms (5 1) (4 2) (2 3) (1 -2) (0 -5)) (polynomial u polynomial-sparse-terms (5 2) (4 4) (2 5) (1 -7) (0 -15))) 
Applying #<procedure:sub> on: {t^5 + 2t^4 + 3t^2 - 2t - 5}, {2u^5 + 4u^4 + 5u^2 - 7u - 15}
Result: (polynomial u polynomial-sparse-terms (5 (integer . -2)) (4 (integer . -4)) (2 (integer . -5)) (1 7) (0 (polynomial t polynomial-sparse-terms (5 1) (4 2) (2 3) (1 -2) (0 10))))
{ - 2u^5 - 4u^4 - 5u^2 + 7u + {t^5 + 2t^4 + 3t^2 - 2t + 10}}

Running Test: (#<procedure:sub> (polynomial u polynomial-sparse-terms (5 2) (4 4) (2 5) (1 -7) (0 -15)) (polynomial v polynomial-sparse-terms (10 3) (7 4) (5 -11) (1 -7) (0 -15))) 
Applying #<procedure:sub> on: {2u^5 + 4u^4 + 5u^2 - 7u - 15}, {3v^10 + 4v^7 - 11v^5 - 7v - 15}
Result: (polynomial v polynomial-sparse-terms (10 (integer . -3)) (7 (integer . -4)) (5 11) (1 7) (0 (polynomial u polynomial-sparse-terms (5 2) (4 4) (2 5) (1 -7))))
{ - 3v^10 - 4v^7 + 11v^5 + 7v + {2u^5 + 4u^4 + 5u^2 - 7u}}

Running Test: (#<procedure:sub> (polynomial v polynomial-sparse-terms (10 3) (7 4) (5 -11) (1 -7) (0 -15)) (polynomial z polynomial-sparse-terms (12 4) (6 13) (5 -12) (1 -7) (0 -15))) 
Applying #<procedure:sub> on: {3v^10 + 4v^7 - 11v^5 - 7v - 15}, {4z^12 + 13z^6 - 12z^5 - 7z - 15}
Result: (polynomial z polynomial-sparse-terms (12 (integer . -4)) (6 (integer . -13)) (5 12) (1 7) (0 (polynomial v polynomial-sparse-terms (10 3) (7 4) (5 -11) (1 -7))))
{ - 4z^12 - 13z^6 + 12z^5 + 7z + {3v^10 + 4v^7 - 11v^5 - 7v}}


Running Test: (#<procedure:sub> (polynomial q polynomial-dense-terms 0 0) (polynomial q polynomial-dense-terms 0 0)) 
Applying #<procedure:sub> on: {}, {}
Result: (polynomial q polynomial-dense-terms 0 0)
{}

Running Test: (#<procedure:sub> (polynomial q polynomial-dense-terms 0 0) (polynomial r polynomial-dense-terms 0 5)) 
Applying #<procedure:sub> on: {}, {5}
Result: (polynomial r polynomial-dense-terms 0 (integer . -5))
{ - 5}

Running Test: (#<procedure:sub> (polynomial r polynomial-dense-terms 0 5) (polynomial r polynomial-dense-terms 0 5)) 
Applying #<procedure:sub> on: {5}, {5}
Result: (polynomial r polynomial-dense-terms)
{}

Running Test: (#<procedure:sub> (polynomial r polynomial-dense-terms 0 5) (polynomial s polynomial-dense-terms 1 3 5)) 
Applying #<procedure:sub> on: {5}, {3s + 5}
Result: (polynomial s polynomial-sparse-terms (1 (integer . -3)))
{ - 3s}

Running Test: (#<procedure:sub> (polynomial s polynomial-dense-terms 1 3 5) (polynomial t polynomial-dense-terms 1 4 9)) 
Applying #<procedure:sub> on: {3s + 5}, {4t + 9}
Result: (polynomial t polynomial-sparse-terms (1 (integer . -4)) (0 (polynomial s polynomial-sparse-terms (1 3) (0 (integer . -4)))))
{ - 4t + {3s - 4}}

Running Test: (#<procedure:sub> (polynomial t polynomial-dense-terms 1 4 9) (polynomial u polynomial-dense-terms 2 3 5 23)) 
Applying #<procedure:sub> on: {4t + 9}, {3u^2 + 5u + 23}
Result: (polynomial u polynomial-sparse-terms (2 (integer . -3)) (1 (integer . -5)) (0 (polynomial t polynomial-sparse-terms (1 4) (0 (integer . -14)))))
{ - 3u^2 - 5u + {4t - 14}}

Running Test: (#<procedure:sub> (polynomial u polynomial-dense-terms 2 3 5 23) (polynomial v polynomial-dense-terms 2 4 9 19)) 
Applying #<procedure:sub> on: {3u^2 + 5u + 23}, {4v^2 + 9v + 19}
Result: (polynomial v polynomial-sparse-terms (2 (integer . -4)) (1 (integer . -9)) (0 (polynomial u polynomial-sparse-terms (2 3) (1 5) (0 4))))
{ - 4v^2 - 9v + {3u^2 + 5u + 4}}

Running Test: (#<procedure:sub> (polynomial v polynomial-dense-terms 2 4 9 19) (polynomial w polynomial-dense-terms 5 1 2 0 3 -2 -5)) 
Applying #<procedure:sub> on: {4v^2 + 9v + 19}, {w^5 + 2w^4 + 3w^2 - 2w - 5}
Result: (polynomial w polynomial-sparse-terms (5 (integer . -1)) (4 (integer . -2)) (2 (integer . -3)) (1 2) (0 (polynomial v polynomial-sparse-terms (2 4) (1 9) (0 24))))
{ - 1w^5 - 2w^4 - 3w^2 + 2w + {4v^2 + 9v + 24}}

Running Test: (#<procedure:sub> (polynomial w polynomial-dense-terms 5 1 2 0 3 -2 -5) (polynomial z polynomial-dense-terms 5 2 4 0 5 -7 -15)) 
Applying #<procedure:sub> on: {w^5 + 2w^4 + 3w^2 - 2w - 5}, {2z^5 + 4z^4 + 5z^2 - 7z - 15}
Result: (polynomial z polynomial-sparse-terms (5 (integer . -2)) (4 (integer . -4)) (2 (integer . -5)) (1 7) (0 (polynomial w polynomial-sparse-terms (5 1) (4 2) (2 3) (1 -2) (0 10))))
{ - 2z^5 - 4z^4 - 5z^2 + 7z + {w^5 + 2w^4 + 3w^2 - 2w + 10}}


Running Test: (#<procedure:sub> (polynomial p polynomial-sparse-terms (0 0)) (polynomial r polynomial-dense-terms 0 5)) 
Applying #<procedure:sub> on: {}, {5}
Result: (polynomial r polynomial-dense-terms 0 (integer . -5))
{ - 5}

Running Test: (#<procedure:sub> (polynomial r polynomial-dense-terms 0 5) (polynomial p polynomial-sparse-terms (0 0))) 
Applying #<procedure:sub> on: {5}, {}
Result: (polynomial r polynomial-dense-terms 0 5)
{5}

Running Test: (#<procedure:sub> (polynomial q polynomial-sparse-terms (0 5)) (polynomial s polynomial-dense-terms 1 3 5)) 
Applying #<procedure:sub> on: {5}, {3s + 5}
Result: (polynomial s polynomial-sparse-terms (1 (integer . -3)))
{ - 3s}

Running Test: (#<procedure:sub> (polynomial r polynomial-sparse-terms (1 7) (0 5)) (polynomial t polynomial-dense-terms 1 4 9)) 
Applying #<procedure:sub> on: {7r + 5}, {4t + 9}
Result: (polynomial t polynomial-sparse-terms (1 (integer . -4)) (0 (polynomial r polynomial-sparse-terms (1 7) (0 (integer . -4)))))
{ - 4t + {7r - 4}}

Running Test: (#<procedure:sub> (polynomial s polynomial-sparse-terms (1 9) (0 -25)) (polynomial u polynomial-dense-terms 2 3 5 23)) 
Applying #<procedure:sub> on: {9s - 25}, {3u^2 + 5u + 23}
Result: (polynomial u polynomial-sparse-terms (2 (integer . -3)) (1 (integer . -5)) (0 (polynomial s polynomial-sparse-terms (1 9) (0 (integer . -48)))))
{ - 3u^2 - 5u + {9s - 48}}

Running Test: (#<procedure:sub> (polynomial t polynomial-sparse-terms (5 1) (4 2) (2 3) (1 -2) (0 -5)) (polynomial v polynomial-dense-terms 2 4 9 19)) 
Applying #<procedure:sub> on: {t^5 + 2t^4 + 3t^2 - 2t - 5}, {4v^2 + 9v + 19}
Result: (polynomial v polynomial-sparse-terms (2 (integer . -4)) (1 (integer . -9)) (0 (polynomial t polynomial-sparse-terms (5 1) (4 2) (2 3) (1 -2) (0 (integer . -24)))))
{ - 4v^2 - 9v + {t^5 + 2t^4 + 3t^2 - 2t - 24}}

Running Test: (#<procedure:sub> (polynomial u polynomial-sparse-terms (5 2) (4 4) (2 5) (1 -7) (0 -15)) (polynomial w polynomial-dense-terms 5 1 2 0 3 -2 -5)) 
Applying #<procedure:sub> on: {2u^5 + 4u^4 + 5u^2 - 7u - 15}, {w^5 + 2w^4 + 3w^2 - 2w - 5}
Result: (polynomial w polynomial-sparse-terms (5 (integer . -1)) (4 (integer . -2)) (2 (integer . -3)) (1 2) (0 (polynomial u polynomial-sparse-terms (5 2) (4 4) (2 5) (1 -7) (0 (integer . -10)))))
{ - 1w^5 - 2w^4 - 3w^2 + 2w + {2u^5 + 4u^4 + 5u^2 - 7u - 10}}

Running Test: (#<procedure:sub> (polynomial v polynomial-sparse-terms (10 3) (7 4) (5 -11) (1 -7) (0 -15)) (polynomial z polynomial-dense-terms 5 2 4 0 5 -7 -15)) 
Applying #<procedure:sub> on: {3v^10 + 4v^7 - 11v^5 - 7v - 15}, {2z^5 + 4z^4 + 5z^2 - 7z - 15}
Result: (polynomial z polynomial-sparse-terms (5 (integer . -2)) (4 (integer . -4)) (2 (integer . -5)) (1 7) (0 (polynomial v polynomial-sparse-terms (10 3) (7 4) (5 -11) (1 -7))))
{ - 2z^5 - 4z^4 - 5z^2 + 7z + {3v^10 + 4v^7 - 11v^5 - 7v}}


Running Test: (#<procedure:sub> (polynomial q polynomial-dense-terms 0 0) (polynomial q polynomial-sparse-terms (0 5))) 
Applying #<procedure:sub> on: {}, {5}
Result: (polynomial q polynomial-sparse-terms (0 (integer . -5)))
{ - 5}

Running Test: (#<procedure:sub> (polynomial q polynomial-sparse-terms (0 5)) (polynomial q polynomial-dense-terms 0 0)) 
Applying #<procedure:sub> on: {5}, {}
Result: (polynomial q polynomial-sparse-terms (0 5))
{5}

Running Test: (#<procedure:sub> (polynomial r polynomial-dense-terms 0 5) (polynomial r polynomial-sparse-terms (1 7) (0 5))) 
Applying #<procedure:sub> on: {5}, {7r + 5}
Result: (polynomial r polynomial-sparse-terms (1 (integer . -7)))
{ - 7r}

Running Test: (#<procedure:sub> (polynomial s polynomial-dense-terms 1 3 5) (polynomial s polynomial-sparse-terms (1 9) (0 -25))) 
Applying #<procedure:sub> on: {3s + 5}, {9s - 25}
Result: (polynomial s polynomial-sparse-terms (1 (integer . -6)) (0 30))
{ - 6s + 30}

Running Test: (#<procedure:sub> (polynomial t polynomial-dense-terms 1 4 9) (polynomial t polynomial-sparse-terms (5 1) (4 2) (2 3) (1 -2) (0 -5))) 
Applying #<procedure:sub> on: {4t + 9}, {t^5 + 2t^4 + 3t^2 - 2t - 5}
Result: (polynomial t polynomial-sparse-terms (5 (integer . -1)) (4 (integer . -2)) (2 (integer . -3)) (1 6) (0 14))
{ - 1t^5 - 2t^4 - 3t^2 + 6t + 14}

Running Test: (#<procedure:sub> (polynomial u polynomial-dense-terms 2 3 5 23) (polynomial u polynomial-sparse-terms (5 2) (4 4) (2 5) (1 -7) (0 -15))) 
Applying #<procedure:sub> on: {3u^2 + 5u + 23}, {2u^5 + 4u^4 + 5u^2 - 7u - 15}
Result: (polynomial u polynomial-sparse-terms (5 (integer . -2)) (4 (integer . -4)) (2 (integer . -2)) (1 12) (0 38))
{ - 2u^5 - 4u^4 - 2u^2 + 12u + 38}

Running Test: (#<procedure:sub> (polynomial v polynomial-dense-terms 2 4 9 19) (polynomial v polynomial-sparse-terms (10 3) (7 4) (5 -11) (1 -7) (0 -15))) 
Applying #<procedure:sub> on: {4v^2 + 9v + 19}, {3v^10 + 4v^7 - 11v^5 - 7v - 15}
Result: (polynomial v polynomial-sparse-terms (10 (integer . -3)) (7 (integer . -4)) (5 11) (2 4) (1 16) (0 34))
{ - 3v^10 - 4v^7 + 11v^5 + 4v^2 + 16v + 34}

Running Test: (#<procedure:sub> (polynomial w polynomial-dense-terms 5 1 2 0 3 -2 -5) (polynomial z polynomial-sparse-terms (12 4) (6 13) (5 -12) (1 -7) (0 -15))) 
Applying #<procedure:sub> on: {w^5 + 2w^4 + 3w^2 - 2w - 5}, {4z^12 + 13z^6 - 12z^5 - 7z - 15}
Result: (polynomial z polynomial-sparse-terms (12 (integer . -4)) (6 (integer . -13)) (5 12) (1 7) (0 (polynomial w polynomial-sparse-terms (5 1) (4 2) (2 3) (1 -2) (0 10))))
{ - 4z^12 - 13z^6 + 12z^5 + 7z + {w^5 + 2w^4 + 3w^2 - 2w + 10}}


Running Test: (#<procedure:=zero?> (polynomial v polynomial-dense-terms)) 
Applying #<procedure:=zero?> on: {}
Result: #t
True

Running Test: (#<procedure:=zero?> (polynomial s polynomial-sparse-terms)) 
Applying #<procedure:=zero?> on: {}
Result: #t
True


'(polynomial-list (y polynomial-sparse-terms (0 1)) (y))
'(polynomial-list (w polynomial-sparse-terms (2 1)) (w))
'(polynomial-list (p polynomial-sparse-terms (1 1)) (p polynomial-sparse-terms (1 1) (0 -1)))
'(polynomial-list (x polynomial-sparse-terms (3 1) (1 1)) (x polynomial-sparse-terms (1 1) (0 -1)))
'(polynomial-list (x polynomial-sparse-terms (3 3) (1 (integer . -2)) (0 (real . -4.5))) (x polynomial-sparse-terms (1 (real . 5.5)) (0 23) (0 0)))

P3y: '(polynomial y polynomial-sparse-terms (1 1))
P3y: {y}
P3y in terms of x: '(polynomial x polynomial-sparse-terms (0 (polynomial y polynomial-sparse-terms (1 1))))
P3y in terms of x: {{y}}

X: '(polynomial x polynomial-sparse-terms (1 1))
X: {x}
P4y: '(polynomial y polynomial-sparse-terms (1 (polynomial x polynomial-sparse-terms (1 1))))
P4y: {{x}y}
P4y in terms of x: '(polynomial x polynomial-sparse-terms (1 (polynomial y polynomial-sparse-terms (1 1))))
P4y in terms of x: {{y}x}

Xsq: '(polynomial x polynomial-sparse-terms (2 1))
Xsq: {x^2}
P5y: '(polynomial y polynomial-sparse-terms (1 (polynomial x polynomial-sparse-terms (2 1))))
P5y: {{x^2}y}
P5y in terms of x: '(polynomial x polynomial-sparse-terms (2 (polynomial y polynomial-sparse-terms (1 1))))
P5y in terms of x: {{y}x^2}

5Xsq: '(polynomial x polynomial-sparse-terms (2 5))
5Xsq: {5x^2}
P6y: '(polynomial y polynomial-sparse-terms (1 (polynomial x polynomial-sparse-terms (2 5))))
P6y: {{5x^2}y}
P6y in terms of x: '(polynomial x polynomial-sparse-terms (2 (polynomial y polynomial-sparse-terms (1 5))))
P6y in terms of x: {{5y}x^2}

5Xcb: '(polynomial x polynomial-sparse-terms (3 5))
5Xcb: {5x^3}
P7y: '(polynomial y polynomial-sparse-terms (7 (polynomial x polynomial-sparse-terms (3 5))))
P7y: {{5x^3}y^7}
P7y in terms of x: '(polynomial x polynomial-sparse-terms (3 (polynomial y polynomial-sparse-terms (7 5))))
P7y in terms of x: {{5y^7}x^3}

P8y: '(polynomial y polynomial-sparse-terms (7 (polynomial x polynomial-sparse-terms (3 5))) (1 (polynomial x polynomial-sparse-terms (2 5))))
P8y: {{5x^3}y^7 + {5x^2}y}
P8y in terms of x: '(polynomial x polynomial-sparse-terms (3 (polynomial y polynomial-sparse-terms (7 5))) (2 (polynomial y polynomial-sparse-terms (1 5))))
P8y in terms of x: {{5y^7}x^3 + {5y}x^2}

P9y: {{x^2}y^4}
P6y: {{5x^2}y}
P10y = P9y + P6y: '(polynomial y polynomial-sparse-terms (4 (polynomial x polynomial-sparse-terms (2 1))) (1 (polynomial x polynomial-sparse-terms (2 5))))
P10y = P9y + P6y: {{x^2}y^4 + {5x^2}y}
P10y in terms of x: '(polynomial x polynomial-sparse-terms (2 (polynomial y polynomial-sparse-terms (4 1) (1 5))))
P10y in terms of x: {{y^4 + 5y}x^2}

P9y: {{x^2}y^4}
P61y: {{5x^2}y^4}
P11y = P9y + P61y: '(polynomial y polynomial-sparse-terms (4 (polynomial x polynomial-sparse-terms (2 6))))
P11y = P9y + P61y: {{6x^2}y^4}
P11y in terms of x: '(polynomial x polynomial-sparse-terms (2 (polynomial y polynomial-sparse-terms (4 6))))
P11y in terms of x: {{6y^4}x^2}

P1x: '(polynomial x polynomial-sparse-terms (1 3) (0 -2))
P1x: {3x - 2}
P1x in terms of z: '(polynomial z polynomial-sparse-terms (0 (polynomial x polynomial-sparse-terms (1 3) (0 -2))))
P1x in terms of z: {{3x - 2}}

P1y: '(polynomial y polynomial-sparse-terms (1 (polynomial x polynomial-sparse-terms (1 3) (0 -2))) (0 (polynomial x polynomial-dense-terms 1 7 3)))
P1y: {{3x - 2}y + {7x + 3}}
P1y in terms of x: '(polynomial x polynomial-sparse-terms (1 (polynomial y polynomial-sparse-terms (1 3) (0 7))) (0 (polynomial y polynomial-sparse-terms (1 -2) (0 3))))
P1y in terms of x: {{3y + 7}x + { - 2y + 3}}

P1z: '(polynomial
  z
  polynomial-sparse-terms
  (1 (polynomial y polynomial-sparse-terms (1 (polynomial x polynomial-sparse-terms (1 3) (0 -2))) (0 (polynomial x polynomial-dense-terms 1 7 3))))
  (0 (polynomial y polynomial-dense-terms 1 (polynomial x polynomial-sparse-terms (1 5) (0 -4)) (polynomial x polynomial-dense-terms 1 9 -6))))
P1z: {{{3x - 2}y + {7x + 3}}z + {{5x - 4}y + {9x - 6}}}
P1z in terms of y: '(polynomial
  y
  polynomial-sparse-terms
  (1 (polynomial z polynomial-sparse-terms (1 (polynomial x polynomial-sparse-terms (1 3) (0 -2))) (0 (polynomial x polynomial-sparse-terms (1 5) (0 -4)))))
  (0 (polynomial z polynomial-sparse-terms (1 (polynomial x polynomial-dense-terms 1 7 3)) (0 (polynomial x polynomial-dense-terms 1 9 -6)))))
P1z in terms of y: {{{3x - 2}z + {5x - 4}}y + {{7x + 3}z + {9x - 6}}}

P1z in terms of x: '(polynomial
  x
  polynomial-sparse-terms
  (1 (polynomial z polynomial-sparse-terms (1 (polynomial y polynomial-sparse-terms (1 3) (0 7))) (0 (polynomial y polynomial-sparse-terms (1 5) (0 9)))))
  (0 (polynomial z polynomial-sparse-terms (1 (polynomial y polynomial-sparse-terms (1 -2) (0 3))) (0 (polynomial y polynomial-sparse-terms (1 -4) (0 -6))))))
P1z in terms of x: {{{3y + 7}z + {5y + 9}}x + {{ - 2y + 3}z + { - 4y - 6}}}

P5x: '(polynomial x polynomial-sparse-terms (1 11))
P5x: {11x}
P12y: '(polynomial y polynomial-sparse-terms (1 15))
P12y: {15y}
P5x + P12y: 
Running Test: (#<procedure:add> (polynomial x polynomial-sparse-terms (1 11)) (polynomial y polynomial-sparse-terms (1 15))) 
Applying #<procedure:add> on: {11x}, {15y}
Result: (polynomial x polynomial-sparse-terms (1 11) (0 (polynomial y polynomial-sparse-terms (1 15))))
{11x + {15y}}

P5x + P12y: 
{11x + {15y}}

P6x: '(polynomial x polynomial-sparse-terms (1 11) (0 12))
P6x: {11x + 12}
P13y: '(polynomial y polynomial-sparse-terms (1 15) (0 17))
P13y: {15y + 17}
P6x + P13y: 
Running Test: (#<procedure:add> (polynomial x polynomial-sparse-terms (1 11) (0 12)) (polynomial y polynomial-sparse-terms (1 15) (0 17))) 
Applying #<procedure:add> on: {11x + 12}, {15y + 17}
Result: (polynomial x polynomial-sparse-terms (1 11) (0 (polynomial y polynomial-sparse-terms (1 15) (0 29))))
{11x + {15y + 29}}

P6x + P13y: 
{11x + {15y + 29}}

P7x: '(polynomial x polynomial-sparse-terms (2 10) (1 11) (0 12))
P7x: {10x^2 + 11x + 12}
P14y: '(polynomial y polynomial-sparse-terms (2 13) (1 15) (0 17))
P14y: {13y^2 + 15y + 17}
P7x + P14y: 
Running Test: (#<procedure:add> (polynomial x polynomial-sparse-terms (2 10) (1 11) (0 12)) (polynomial y polynomial-sparse-terms (2 13) (1 15) (0 17))) 
Applying #<procedure:add> on: {10x^2 + 11x + 12}, {13y^2 + 15y + 17}
Result: (polynomial x polynomial-sparse-terms (2 10) (1 11) (0 (polynomial y polynomial-sparse-terms (2 13) (1 15) (0 29))))
{10x^2 + 11x + {13y^2 + 15y + 29}}

P7x + P14y: 
{10x^2 + 11x + {13y^2 + 15y + 29}}
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