The Gauge Integral Theory in HOL4

Zhiping Shi, Capital Normal University
Weiqing Gu, Harvey Mudd College
Xiaojuan Li, Capital Normal University
Yong Guan, Capital Normal University
Shiwei Ye, Chinese Academy of Science
Jie Zhang, Beijing University of Chemical Technology
Hongxing Wei, Beijing University of Aeronautics and Astronautics


The integral is one of the most important foundations for modeling dynamical systems. The gauge integral is a generalization of the Riemann integral and the Lebesgue integral and applies to a much wider class of functions. In this paper, we formalize the operational properties which contain the linearity, monotonicity, integration by parts, the Cauchy-type integrability criterion, and other important theorems of the gauge integral in higher-order logic 4 (HOL4) and then use them to verify an inverting integrator. The formalized theorem library has been accepted by the HOL4 authority and will appear in HOL4 Kananaskis-9.