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Journal of Applied Mathematics
Volume 2014, Article ID 201214, 10 pages
Research Article

Formalization of Function Matrix Theory in HOL

1Beijing Key Laboratory of Electronic System Reliability Technology, Capital Normal University, Beijing 100048, China
2Guangxi Key Laboratory of Trusted Software, Guilin University of Electronic Technology, Guilin 541004, China
3College of Information Science and Engineering, Graduate University of Chinese Academy of Sciences, Beijing 100049, China
4College of Information Science and Technology, Beijing University of Chemical Technology, Beijing 100029, China
5School of Mechanical Engineering and Automation, Beijing University of Aeronautics and Astronautics, Beijing 100191, China

Received 12 January 2014; Accepted 22 April 2014; Published 24 July 2014

Academic Editor: Guiming Luo

Copyright © 2014 Zhiping Shi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Function matrices, in which elements are functions rather than numbers, are widely used in model analysis of dynamic systems such as control systems and robotics. In safety-critical applications, the dynamic systems are required to be analyzed formally and accurately to ensure their correctness and safeness. Higher-order logic (HOL) theorem proving is a promise technique to match the requirement. This paper proposes a higher-order logic formalization of the function vector and the function matrix theories using the HOL theorem prover, including data types, operations, and their properties, and further presents formalization of the differential and integral of function vectors and function matrices. The formalization is implemented as a library in the HOL system. A case study, a formal analysis of differential of quadratic functions, is presented to show the usefulness of the proposed formalization.