Formalization of Asymptotic Notations in HOL4

Asymptotic notations characterize the limiting behavior of a function. They are extensively used in many branches of mathematics and computer science particularly in analytical number theory, combinatorics and computational complexity while analyzing algorithms. Traditionally, the mathematical analysis involving these notations has been done by paper-and-pencil proof methods or simulation. In order to introduce formal verification in this domain, this paper provides the higher-order-logic formalizations of $\mathrm {O}, \Theta , \Omega , 0$ and (${j}$) notations and the formal verification of most of their classical properties of interest. The formalization is based on the theory of sets, real and natural numbers and has been done using the HOL4 theorem prover.