;;; Name:
;;; Login:
;;; TA:
;;; Section:

(define (assert-equal v1 v2)
  (if (equal? v1 v2)
    (print 'ok)
    (print (list v2 'does 'not 'equal v1))))

;;; Q1.

(define (test-q1)
  (define counts (list 1 2 3 4))
  (assert-equal 2 (cadr counts))
  (assert-equal 3 (caddr counts)))

(define (cddr s)
  (cdr (cdr s)))

(define (cadr s)
  'YourCodeHere)

(define (caddr s)
  'YourCodeHere)

;;; Q2.

(define (test-q2)
  (assert-equal -1 (sign -42))
  (assert-equal 0 (sign 0))
  (assert-equal 1 (sign 42)))

(define (sign x)
  'YourCodeHere)


; Derive returns the derivative of exp with respect to var.
(define (derive exp var)
  (cond ((number? exp) 0)
        ((variable? exp) (if (same-variable? exp var) 1 0))
        ((sum? exp) (derive-sum exp var))
        ((product? exp) (derive-product exp var))
        ((exponentiation? exp) (derive-exponentiation exp var))
        (else 'Error)))


; Variables are represented as symbols
(define (variable? x) (symbol? x))
(define (same-variable? v1 v2)
  (and (variable? v1) (variable? v2) (eq? v1 v2)))

; Numbers are compared with =
(define (=number? exp num)
  (and (number? exp) (= exp num)))

; Sums are represented as lists that start with +.
(define (make-sum a1 a2)
  (cond ((=number? a1 0) a2)
        ((=number? a2 0) a1)
        ((and (number? a1) (number? a2)) (+ a1 a2))
        (else (list '+ a1 a2))))
(define (sum? x)
  (and (pair? x) (eq? (car x) '+)))
(define (addend s) (cadr s))
(define (augend s) (caddr s))

; Products are represented as lists that start with *.
(define (make-product m1 m2)
  (cond ((or (=number? m1 0) (=number? m2 0)) 0)
        ((=number? m1 1) m2)
        ((=number? m2 1) m1)
        ((and (number? m1) (number? m2)) (* m1 m2))
        (else (list '* m1 m2))))
(define (product? x)
  (and (pair? x) (eq? (car x) '*)))
(define (multiplier p) (cadr p))
(define (multiplicand p) (caddr p))

(define (test-sum)
  (assert-equal '(+ a x) (make-sum 'a 'x))
  (assert-equal '(+ a (+ x 1)) (make-sum 'a (make-sum 'x 1)))
  (assert-equal 'x (make-sum 'x 0))
  (assert-equal 'x (make-sum 0 'x))
  (assert-equal 4 (make-sum 1 3)))

(define (test-product)
  (assert-equal '(* a x) (make-product 'a 'x))
  (assert-equal 0 (make-product 'x 0))
  (assert-equal 'x (make-product 1 'x))
  (assert-equal 6 (make-product 2 3)))

;;; Q3.

(define (test-q3)
  (assert-equal 1 (derive '(+ x 3) 'x)))

(define (derive-sum exp var)
  'YourCodeHere)

;;; Q4.

(define (test-q4)
  (assert-equal 'y (derive '(* x y) 'x))
  (assert-equal '(+ (* y (+ x 3)) (* x y))
                (derive '(* (* x y) (+ x 3)) 'x)))

(define (derive-product exp var)
  'YourCodeHere)

;;; Q5.

(define (test-q5)
  (assert-equal 1024 (pow 2 10))
  (assert-equal 243 (pow 3 5)))

(define (square x) (* x x))

(define (pow b n)
  'YourCodeHere)

;;; Q6.

(define (test-q6)
  (let ((x^2 (make-exponentiation 'x 2))
        (x^3 (make-exponentiation 'x 3))
        (xy+x^2 (make-exponentiation (make-sum (make-product 'x 'y) 'x) 2)))
    (assert-equal 'x (make-exponentiation 'x 1))
    (assert-equal 1 (make-exponentiation 'x 0))
    (assert-equal 16 (make-exponentiation 2 4))
    (assert-equal (make-product 3 x^2) (derive x^3 'x))
    (assert-equal '(* 2 x) (derive x^2 'x))
    (assert-equal '(* (* 2 (+ (* x y) x)) (+ y 1)) (derive xy+x^2 'x))
    (assert-equal '(* (* 2 (+ (* x y) x)) x) (derive xy+x^2 'y))))

(define (make-exponentiation base exponent)
  'YourCodeHere)

(define (base exponentiation)
  'YourCodeHere)

(define (exponent exponentiation)
  'YourCodeHere)

(define (exponentiation? exp)
  'YourCodeHere)

(define (derive-exponentiation exp var)
  'YourCodeHere)


(test-q1)
(test-q2)
(test-q3)
(test-q4)
(test-q5)
(test-q6)