eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-09-05
9:1
9:16
10.4230/LIPIcs.ITP.2019.9
article
Higher-Order Tarski Grothendieck as a Foundation for Formal Proof
Brown, Chad E.
1
Kaliszyk, Cezary
2
3
https://orcid.org/0000-0002-8273-6059
Pąk, Karol
4
https://orcid.org/0000-0002-7099-1669
Czech Technical University in Prague, Czech Republic
University of Innsbruck, Austria
University of Warsaw, Poland
University of Białystok, Poland
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol141-itp2019/LIPIcs.ITP.2019.9/LIPIcs.ITP.2019.9.pdf
model
higher-order
Tarski Grothendieck
proof foundation