Higher-Order Tarski Grothendieck as a Foundation for Formal Proof

Authors Chad E. Brown, Cezary Kaliszyk , Karol Pąk

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Author Details

Chad E. Brown
  • Czech Technical University in Prague, Czech Republic
Cezary Kaliszyk
  • University of Innsbruck, Austria
  • University of Warsaw, Poland
Karol Pąk
  • University of Białystok, Poland

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Chad E. Brown, Cezary Kaliszyk, and Karol Pąk. Higher-Order Tarski Grothendieck as a Foundation for Formal Proof. In 10th International Conference on Interactive Theorem Proving (ITP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 141, pp. 9:1-9:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Interactive proof systems
  • Theory of computation → Logic and verification
  • model
  • higher-order
  • Tarski Grothendieck
  • proof foundation


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