In his 1948 paper" A Decision Method for Elementary Algebra and Geometry," Alfred Tarski gave a decision procedure for the first-order theory of the real numbers with+,", and=, commonly known as the theory of real closed fields. His decision procedure was nonelementary (in the complexity-theoretic sense). An elementary decision procedure was given by Leonard Monk in his Berkeley PhD thesis, and in 1974, double-exponential-time decision procedures were given independently by Collins and by Monk/Solovay. Various improvements and heuristics notwithstanding, that worst-case bound has stood since that time.