We report on an original formalization of measure and integration theory in the Coq proof assistant. We build the Lebesgue measure following a standard construction that has not yet been formalized in type theory-based proof assistants: by extension of a measure over a semiring of sets. We achieve this formalization by leveraging on existing techniques from the Mathematics Components project. We explain how we extend Mathematical Components' iterated operators and mathematical structures for analysis to provide support for infinite sums and extended real numbers. We introduce new mathematical structures for measure theory and incidentally provide an illustrative, concrete application of Hierarchy-Builder, a generic tool for the formalization of hierarchies of mathematical structures. This formalization of measure theory provides the basis for a new formalization of the Lebesgue integration compatible with the Mathematical Components project.
翻译:我们在科克校准助理中报告计量和集成理论的最初正式化。我们根据标准结构构建了Lebesgue措施,该标准结构尚未在基于理论的证明助理类型中正式化:扩展了一种衡量和集成的分数,我们利用数学构件项目的现有技术实现了这一正规化。我们解释了我们如何扩大数学构件的迭代操作员和用于分析的数学结构,为无限量和扩展的真数提供支持。我们引入了计量理论的新数学结构,并顺便提供了等级结构-建筑的示例和具体应用,这是数学结构等级正规化的通用工具。这种正式化计量理论为与数学构件项目相匹配的Lebesgue集成提供了新的正规化基础。