文章基本信息

标题：Higher-Order Tarski Grothendieck as a Foundation for Formal Proof
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作者：Chad E. Brown ; Cezary Kaliszyk ; Karol Pak 等
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2019
卷号：141
页码：1-16
DOI：10.4230/LIPIcs.ITP.2019.9
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：We formally introduce a foundation for computer verified proofs based on higher-order Tarski-Grothendieck set theory. We show that this theory has a model if a 2-inaccessible cardinal exists. This assumption is the same as the one needed for a model of plain Tarski-Grothendieck set theory. The foundation allows the co-existence of proofs based on two major competing foundations for formal proofs: higher-order logic and TG set theory. We align two co-existing Isabelle libraries, Isabelle/HOL and Isabelle/Mizar, in a single foundation in the Isabelle logical framework. We do this by defining isomorphisms between the basic concepts, including integers, functions, lists, and algebraic structures that preserve the important operations. With this we can transfer theorems proved in higher-order logic to TG set theory and vice versa. We practically show this by formally transferring Lagrange's four-square theorem, Fermat 3-4, and other theorems between the foundations in the Isabelle framework.
关键词：model; higher-order; Tarski Grothendieck; proof foundation