Formalizing O notation in Isabelle/HOL Jeremy Avigad and Kevin Donnelly Summary: Formalizing O notation in Isabelle/HOL Jeremy Avigad and Kevin Donnelly Carnegie Mellon University Abstract. We describe a formalization of asymptotic O notation using the Isabelle/HOL proof assistant. 1 Introduction Asymptotic notions are used to characterize the approximate long-term behavior of functions in a number of branches of mathematics and computer science, including analysis, combinatorics, and computational complexity. Our goal here is to describe an implementation of one important asymptotic notion -- "big O notation" -- using the Isabelle/HOL proof assistant. Developing a library to support such reasoning poses a number of interesting challenges. First of all, ordinary mathematical practice involving O notation relies on a number of conventions, some determinate and some ambiguous, so deciding on an appropriate formal representation requires some thought. Second, we will see that a natural way of handling the notation is inherently higher- order; thus the implementation is a way of putting the higher-order features of a theorem prover to the test. Finally, O notation is quite general, since many of the definitions and basic properties make sense for the analysis of any domain of functions A B where B has the structure of an ordered ring (or even, more Collections: Multidisciplinary Databases and Resources; Mathematics