Formalization of the Complex Number Theory in HOL4

In this paper, the theory of complex numbers is formalized and the theorem library of complex numbers is embedded in HOL4, the theorem prover of High Order Logics. The theorem library introduces a data type C by an R × R type abbreviation, and defines arithmetic operations of complex numbers in terms of group and field theories. Moreover, the polar and exponential forms are provided for simplifying the applications in control theory and signal analysis. We define the scalar multiplication of complex numbers and prove some properties about R-module of complex numbers. The theorem library extends the scope of application of HOL4. The developed complex number theory has been released in HOL4 Kananaskis-7.