{"title":"区间微分","authors":"Noboru Endou","doi":"10.2478/forma-2023-0002","DOIUrl":null,"url":null,"abstract":"Summary This article generalizes the differential method on intervals, using the Mizar system [2], [3], [12]. Differentiation of real one-variable functions is introduced in Mizar [13], along standard lines (for interesting survey of formalizations of real analysis in various proof-assistants like ACL2 [11], Isabelle/HOL [10], Coq [4], see [5]), but the differentiable interval is restricted to open intervals. However, when considering the relationship with integration [9], since integration is an operation on a closed interval, it would be convenient for differentiation to be able to handle derivates on a closed interval as well. Regarding differentiability on a closed interval, the right and left differentiability have already been formalized [6], but they are the derivatives at the endpoints of an interval and not demonstrated as a differentiation over intervals. Therefore, in this paper, based on these results, although it is limited to real one-variable functions, we formalize the differentiation on arbitrary intervals and summarize them as various basic propositions. In particular, the chain rule [1] is an important formula in relation to differentiation and integration, extending recent formalized results [7], [8] in the latter field of research.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Differentiation on Interval\",\"authors\":\"Noboru Endou\",\"doi\":\"10.2478/forma-2023-0002\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Summary This article generalizes the differential method on intervals, using the Mizar system [2], [3], [12]. Differentiation of real one-variable functions is introduced in Mizar [13], along standard lines (for interesting survey of formalizations of real analysis in various proof-assistants like ACL2 [11], Isabelle/HOL [10], Coq [4], see [5]), but the differentiable interval is restricted to open intervals. However, when considering the relationship with integration [9], since integration is an operation on a closed interval, it would be convenient for differentiation to be able to handle derivates on a closed interval as well. Regarding differentiability on a closed interval, the right and left differentiability have already been formalized [6], but they are the derivatives at the endpoints of an interval and not demonstrated as a differentiation over intervals. Therefore, in this paper, based on these results, although it is limited to real one-variable functions, we formalize the differentiation on arbitrary intervals and summarize them as various basic propositions. In particular, the chain rule [1] is an important formula in relation to differentiation and integration, extending recent formalized results [7], [8] in the latter field of research.\",\"PeriodicalId\":42667,\"journal\":{\"name\":\"Formalized Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2023-09-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Formalized Mathematics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.2478/forma-2023-0002\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Formalized Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.2478/forma-2023-0002","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Summary This article generalizes the differential method on intervals, using the Mizar system [2], [3], [12]. Differentiation of real one-variable functions is introduced in Mizar [13], along standard lines (for interesting survey of formalizations of real analysis in various proof-assistants like ACL2 [11], Isabelle/HOL [10], Coq [4], see [5]), but the differentiable interval is restricted to open intervals. However, when considering the relationship with integration [9], since integration is an operation on a closed interval, it would be convenient for differentiation to be able to handle derivates on a closed interval as well. Regarding differentiability on a closed interval, the right and left differentiability have already been formalized [6], but they are the derivatives at the endpoints of an interval and not demonstrated as a differentiation over intervals. Therefore, in this paper, based on these results, although it is limited to real one-variable functions, we formalize the differentiation on arbitrary intervals and summarize them as various basic propositions. In particular, the chain rule [1] is an important formula in relation to differentiation and integration, extending recent formalized results [7], [8] in the latter field of research.
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.