A Formalization of Properties of Continuous Functions on Closed Intervals

Fu, Yaoshun

doi:10.1007/978-3-030-52200-1_27

Cited by 3 publications

(4 citation statements)

References 8 publications

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“…In our previous conference paper [42], we had completed the proof of equivalence between Dedekind fundamental theorem and Supremum theorem. Furthermore, we present the properties of continuous functions on closed intervals including boundedness theorem, extreme value theorem, intermediate value theorem, and uniform continuity theorem.…”

Section: Related Workmentioning

confidence: 99%

“…Natural numbers are usually defined through Peano axioms, which can be formalized in Coq by variety of approaches. In Morse-Kelley axiomatic set theory [47,48], the Peano axioms can be deduced as theorems.On the other hand, the formalization of Peano axioms can be presented directly, and the details can be found in our previous conference paper [42].…”

Section: Natural Numbers Fractions Rational Numbersmentioning

confidence: 99%

“…Then, there exists exactly one real number Ξ such that each number smaller than Ξ belongs to the first class and each number greater than Ξ belongs to the second class [15,42].…”

Section: Dedekind's Fundamental Theoremmentioning

confidence: 99%

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Formalization of the Equivalence among Completeness Theorems of Real Number in Coq

2020

Mathematics

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The formalization of mathematics based on theorem prover becomes increasingly important in mathematics and computer science, and, particularly, formalizing fundamental mathematical theories becomes especially essential. In this paper, we describe the formalization in Coq of eight very representative completeness theorems of real numbers. These theorems include the Dedekind fundamental theorem, Supremum theorem, Monotone convergence theorem, Nested interval theorem, Finite cover theorem, Accumulation point theorem, Sequential compactness theorem, and Cauchy completeness theorem. We formalize the real number theory strictly following Landau’s Foundations of Analysis where the Dedekind fundamental theorem can be proved. We extend this system and complete the related notions and properties for finiteness and sequence. We prove these theorems in turn from Dedekind fundamental theorem, and finally prove the Dedekind fundamental theorem by the Cauchy completeness theorem. The full details of formal proof are checked by the proof assistant Coq, which embodies the characteristics of reliability and interactivity. This work can lay the foundation for many applications, especially in calculus and topology.

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Section: Related Workmentioning

confidence: 99%

Section: Natural Numbers Fractions Rational Numbersmentioning

confidence: 99%

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Formalization of the Equivalence among Completeness Theorems of Real Number in Coq

2020

Mathematics

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“…Starting from the Peano axioms, the whole theory of the number system from natural numbers to complex numbers is given in turn. As an application of our system, we formally completed the cyclic proof of eight completeness theorems and the proof of properties of a continuous function on closed intervals [35,36]. It should be noted that we only use the contents before Section 4.4 which do not involve completeness (Dedekind fundamental theorem) yet.…”

Section: Introductionmentioning

confidence: 99%

Formalizing Calculus without Limit Theory in Coq

2021

Mathematics

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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.

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Formalizing equivalence between real number completeness and intermediate value theorem

2021

2021 China Automation Congress (CAC)

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A Formalization of Properties of Continuous Functions on Closed Intervals

Cited by 3 publications

References 8 publications

Formalization of the Equivalence among Completeness Theorems of Real Number in Coq

Formalization of the Equivalence among Completeness Theorems of Real Number in Coq

Formalizing Calculus without Limit Theory in Coq

Formalizing equivalence between real number completeness and intermediate value theorem

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