;;1.  (+ 10 -10)
(define express0 (make-add 10 -10))
;;2.  (+ (* 20 3) 33)
(define express1 (make-add (make-mul 20 3) 33))
;;3.  (* 3.14 (* r r))
(define express2 (make-mul 3.14 (make-mul 'r 'r)))
;;4.  (+ (* 9/5 c) 32)
(define express3 (make-add (make-mul 9/5 'c) 32))
;;5  (+ (* 3.14 (* o o)) (* 3.14 (* i i)))
(define express4 
  (make-add (make-mul 3.14 (make-mul 'o 'o)) (make-mul 3.14 (make-mul 'i 'i))))

;; numeric?:  Expression --> Boolean
;; returns true if there are no symbols in express.
(define (numeric? express)
  (cond
    [(add? express)
          (add-numeric? express)]
    [(mul? express)
          (mul-numeric? express)]
    [else false]))

;; add-numeric?:  (make-add Expression Expression) --> Boolean
;; determines if all pieces of express are numeric
(define (add-numeric? express)
  (cond
    [(and (add? (add-left express))
          (add? (add-right express)))
          (and (add-numeric? (add-left express))
               (add-numeric? (add-right express)))]
    [(and (number? (add-left express))
          (add? (add-right express)))
          (and (number? (add-left express))
               (add-numeric? (add-right express)))]
    [(and (add? (add-left express))
          (number? (add-right express)))
          (and (add-numeric? (add-left express))
               (number? (add-right express)))]
    [(and (number? (add-left express))
          (number? (add-right express)))
     true]
    [(and (mul? (add-left express))
          (mul? (add-right express)))
     (and (mul-numeric? (add-left express))
          (mul-numeric? (add-right express)))]
    [(and (mul? (add-left express))
          (add? (add-right express)))
     (and (mul-numeric? (add-left express))
          (add-numeric? (add-right express)))]
    [(and (add? (add-left express))
          (mul? (add-right express)))
     (and (add-numeric? (add-left express))
          (mul-numeric? (add-right express)))]
    [(and (mul? (add-left express))
          (number? (add-right express)))
     (and (mul-numeric? (add-left express))
          (number? (add-right express)))]
    [(and (number? (add-left express))
          (mul? (add-right express)))
     (and (number? (add-left express))
          (mul-numeric? (add-right express)))]
    [else false]
    )
  )

;;mul-numeric?:  (make-mul Expression Expression) --> Boolean
;;determines if all pieces of express are numeric
(define (mul-numeric? express)
  (cond
    [(and (mul? (mul-left express))
          (mul? (mul-right express)))
          (and (mul-numeric? (mul-left express))
               (mul-numeric? (mul-right express)))]
    [(and (number? (mul-left express))
          (mul? (mul-right express)))
          (and (number? (mul-left express))
               (mul-numeric? (mul-right express)))]
    [(and (mul? (mul-left express))
          (number? (mul-right express)))
          (and (mul-numeric? (mul-left express))
               (number? (mul-right express)))]
    [(and (number? (mul-left express))
          (number? (mul-right express)))
     true]
    [(and (add? (mul-left express))
          (add? (mul-right express)))
     (and (add-numeric? (mul-left express))
          (mul-numeric? (mul-right express)))]
    [(and (mul? (mul-left express))
          (add? (mul-right express)))
     (and (mul-numeric? (mul-left express))
          (add-numeric? (mul-right express)))]
    [(and (add? (mul-left express))
          (mul? (mul-right express)))
     (and (add-numeric? (mul-left express))
          (mul-numeric? (mul-right express)))]
    [(and (add? (mul-left express))
          (number? (mul-right express)))
     (and (add-numeric? (mul-left express))
          (number? (mul-right express)))]
    [(and (number? (mul-left express))
          (add? (mul-right express)))
     (and (number? (mul-left express))
          (add-numeric? (mul-right express)))]
    [else false]
    )
  )
;;examples/tests
(equal? (numeric? express0) ;;should be
        true)
(equal? (numeric? express1) ;;should be
        true)
(equal? (numeric? express2) ;;should be
        false)
(equal? (numeric? express3) ;;should be
        false)
(equal? (numeric? express4) ;;should be
        false)