#lang racket

; returns true if v is a variable (not a number) not in vars
(define (is-new-var? v vars)
  (and (not (number? v)) (not (member v vars))))

; extracts out a list of all the variables in E.  Each variable appears
; exactly once in the result list
(define (exp-vars E) 
  
  ; given an current expression Ec, and a list vars of variables seen
  ; so far, add and new variables of Ec
  (define (exp-vars-r Ec vars)
    
    (cond
      ; atoms are themselves, so if not seen already, add a non-number
      ; to the list of variables
      [ (not (list? Ec)) 
        (if (is-new-var? Ec vars) (cons Ec vars) vars)  ]
      
      ; single element lists are the value of their first element, so
      ; get variables in the expression given by the first element.
      [ (null? (rest Ec)) (exp-vars-r (first Ec) vars) ] 
      
      ; binary expressions, update the variables with the ones on the
      ; left, and then update those with the ones on the right.  Note 
      ; the threading of vars through the recursion.
      [ else (exp-vars-r (third Ec) (exp-vars-r (first Ec) vars) ) ]
      
      ))

  (exp-vars-r E '())
  )