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Formalizing Calculus without Limit Theory in Coq

Beijing Key Laboratory of Space-Ground Interconnection and Convergence, School of Electronic Engineering, Beijing University of Posts and Telecommunications, Beijing 100876, China
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Academic Editor: Radi Romansky
Mathematics 2021, 9(12), 1377; https://doi.org/10.3390/math9121377
Received: 11 May 2021 / Revised: 3 June 2021 / Accepted: 11 June 2021 / Published: 14 June 2021
(This article belongs to the Section Mathematics and Computer Science)
Formal verification of mathematical theory has received widespread concern and grown rapidly. The formalization of the fundamental theory will contribute to the development of large projects. In this paper, we present the formalization in Coq of calculus without limit theory. The theory aims to found a new form of calculus more easily but rigorously. This theory as an innovation differs from traditional calculus but is equivalent and more comprehensible. First, the definition of the difference-quotient control function is given intuitively from the physical facts. Further, conditions are added to it to get the derivative, and define the integral by the axiomatization. Then some important conclusions in calculus such as the Newton–Leibniz formula and the Taylor formula can be formally verified. This shows that this theory can be independent of limit theory, and any proof does not involve real number completeness. This work can help learners to study calculus and lay the foundation for many applications. View Full-Text
Keywords: calculus; difference-quotient control function; Coq; formalization; limit theory calculus; difference-quotient control function; Coq; formalization; limit theory
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MDPI and ACS Style

Fu, Y.; Yu, W. Formalizing Calculus without Limit Theory in Coq. Mathematics 2021, 9, 1377. https://doi.org/10.3390/math9121377

AMA Style

Fu Y, Yu W. Formalizing Calculus without Limit Theory in Coq. Mathematics. 2021; 9(12):1377. https://doi.org/10.3390/math9121377

Chicago/Turabian Style

Fu, Yaoshun, and Wensheng Yu. 2021. "Formalizing Calculus without Limit Theory in Coq" Mathematics 9, no. 12: 1377. https://doi.org/10.3390/math9121377

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