Antiderivatives and Integration

IF 1 Q1 MATHEMATICS
Noboru Endou
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引用次数: 0

Abstract

Summary In this paper, we introduce indefinite integrals [8] (antiderivatives) and proof integration by substitution in the Mizar system [2], [3]. In our previous article [15], we have introduced an indefinite-like integral, but it is inadequate because it must be an integral over the whole set of real numbers and in some sense it causes some duplication in the Mizar Mathematical Library [13]. For this reason, to define the antiderivative for a function, we use the derivative of an arbitrary interval as defined recently in [7]. Furthermore, antiderivatives are also used to modify the integration by substitution and integration by parts. In the first section, we summarize the basic theorems on continuity and derivativity (for interesting survey of formalizations of real analysis in another proof-assistants like ACL2 [12], Isabelle/HOL [11], Coq [4], see [5]). In the second section, we generalize some theorems that were noticed during the formalization process. In the last section, we define the antiderivatives and formalize the integration by substitution and the integration by parts. We referred to [1] and [6] in our development.
不定积分与积分
本文介绍了Mizar系统[2],[3]中的不定积分[8](不定积分)和代换证明积分。在我们之前的文章[15]中,我们引入了一个不定积分,但它是不充分的,因为它必须是整个实数集合上的积分,并且在某种意义上它在Mizar数学库[13]中造成了一些重复。因此,为了定义函数的不定积分,我们使用最近在[7]中定义的任意区间的导数。此外,还利用不定积分法修正了换元积分法和分部积分法。在第一部分中,我们总结了关于连续性和导数性的基本定理(对于另一个证明辅助工具如ACL2 [12], Isabelle/HOL [11], Coq[4]的形式化的有趣调查,参见[5])。在第二部分中,我们推广了在形式化过程中注意到的一些定理。在最后一节中,我们定义了不定积分,并形式化了代换积分和分部积分。我们在开发过程中参考了[1]和[6]。
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来源期刊
Formalized Mathematics
Formalized Mathematics MATHEMATICS-
自引率
0.00%
发文量
0
审稿时长
10 weeks
期刊介绍: Formalized Mathematics is to be issued quarterly and publishes papers which are abstracts of Mizar articles contributed to the Mizar Mathematical Library (MML) - the basis of a knowledge management system for mathematics.
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