Finite Set Theory based on Fully Ordered Lists

We present a new finite set theory implementation for ACL2 wherein sets are implemented as fully ordered lists. This order unifies the notions of set equality and element equality by creating a unique representation for each set, which in turn enables nested sets to be trivially supported and eliminates the need for congruence rules. We demonstrate that ordered sets can be reasoned about in the traditional style of membership arguments. Using this technique, we prove the classic properties of set operations in a natural and effortless manner. We then use the exciting new MBE feature of ACL2 to provide linear-time implementations of all basic set operations. These optimizations are made “behind the scenes” and do not adversely impact reasoning ability. We finally develop a framework for reasoning about quantification over set elements. We also begin to provide common higher-order patterns from functional programming. The net result is an efficient library that is easy to use and reason about.