We systematically formalize calculus without limit theory based on the existing research results. Compared with this work, there are innovations in the following aspects that have been made. Our previous paper formalized all definitions and theorems of the paper, and the development completely corresponds to the structure of the paper. Together with them, Lin supplements the real number axioms and function continuity to enrich the theory. In recent years, Zhang and Tong proposed the difference-quotient control function based on the previous concept, and they gave a new concept of macro derivative. Thus the calculus system without limit can be established wholly and rigorously. This theory gives the intuitive definition of an integral system and reveals the necessary and sufficient conditions for uniform derivative and strong derivative. On the other hand, Zhang introduced the concept that the difference quotient of one function is the median of another function, which shows the relationship between a function and its derivative and the integral in essence. This work is also based on the concept of the differentiability of the Lipschitz function. After that, Livshits proposed a method to directly define differential and integral without depending on real number completeness, limit and continuity. This is an outstanding progression of calculus without limit theory however, he did not form a systematic and complete theory. In 2005, Sparks avoided the limit concept in his book “Calculus Without Limits”, where the basic concepts and calculus formulas are explained with simple examples. ![]() Further he explicitly put forward the uniform inequality as the primary definition of derivative, which can be used as a new guide to calculus without limit theory. Later, Lin pointed out that the fundamental theorem of calculus can be simplified by consistent derivative. At the same time, he admitted that the differentiability introduced by him was not rigorous compared to traditional calculus theory. In 1999, Dovermann introduced the concept of differentiability without limit based on the Lipschitz condition so that his students could quickly learn the calculus theory. They used this concept to simplify the calculus reasoning, but the process still depended on the limit. proposed the concept of uniformly derivable, which can be proved to be equivalent to the concept of continuously derivable. This work can help learners to study calculus and lay the foundation for many applications. This shows that this theory can be independent of limit theory, and any proof does not involve real number completeness. Then some important conclusions in calculus such as the Newton–Leibniz formula and the Taylor formula can be formally verified. Further, conditions are added to it to get the derivative, and define the integral by the axiomatization. First, the definition of the difference-quotient control function is given intuitively from the physical facts. This theory as an innovation differs from traditional calculus but is equivalent and more comprehensible. The theory aims to found a new form of calculus more easily but rigorously. In this paper, we present the formalization in Coq of calculus without limit theory. ![]() ![]() ![]() The formalization of the fundamental theory will contribute to the development of large projects. So completeness is basically handier than the supremum axiom in proofs from how I understood it but I still don't understand why we need it here or in the Banach fixpoint theorem.Formal verification of mathematical theory has received widespread concern and grown rapidly. Things unclear: I don't understand completeness why do I need it as a precondition or do I not for the real-valued contraction theorem? But this one professor from whom I read the script and proof of the fixpoint theorem keeps stressing that you can't prove that $\mathbb R$ is uncountable, the mean value theorem, and so on without completeness. Let $x_1,x_2\in$ be fixpoints of $\varphi$. Preconditions: $\varphi$ is a contraction, an arbitrary Cauchy series converges. $$\lvert \varphi(a) - \varphi(b)\rvert \le \delta \lvert a-b\rvert$$ If $\varphi$ is continous differentiable, then $\varphi'$ is bounded on the closed intervall $$, by a constant $\delta$ therefor Show that c is unique with the mean value theorem Show with the mean value theorem that $$|x_ x_x$ sloves the fixed-point equation $$\varphi(c)=c \ \ \ (2)$$ Let $\varphi:\to \mathbb R$ be continuous differentiable and $|\varphi'(x)|1,$ and let $x_n\in$ for all $n\in \mathbb N$
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