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DOI: 10.4230/LIPIcs.ITP.2024.18

Mechanized HOL Reasoning in Set Theory

Authors: Simon Guilloud, Sankalp Gambhir, Andrea Gilot, and Viktor Kunčak

Published in: LIPIcs, Volume 309, 15th International Conference on Interactive Theorem Proving (ITP 2024)

Abstract

We present a mechanized embedding of higher-order logic (HOL) and algebraic data types (ADTs) into first-order logic with ZFC axioms. Our approach interprets types as sets, with function (arrow) types coinciding with set-theoretic function spaces. We assume traditional FOL syntax without notation for term-level binders. To embed λ-terms, we define a notion of context, defining the closure of all abstractions occuring inside a term. We implement the embedding in the Lisa proof assistant for schematic first-order logic and its library based on axiomatic set theory (presented at ITP 2023). We show how to implement type checking and the proof steps of HOL Light as proof-producing tactics in Lisa. The embedded HOL theorems and proofs are interoperable with the existing Lisa library. This yields a form of soft type system supporting top-level polymorphism and ADTs within set theory. The approach offers tools for Lisa users to carry HOL-style proofs within set theory. It also enables the import of HOL Light theorem statements into Lisa, as well as the replay of small HOL Light kernel proofs.

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Simon Guilloud, Sankalp Gambhir, Andrea Gilot, and Viktor Kunčak. Mechanized HOL Reasoning in Set Theory. In 15th International Conference on Interactive Theorem Proving (ITP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 309, pp. 18:1-18:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{guilloud_et_al:LIPIcs.ITP.2024.18,
  author =	{Guilloud, Simon and Gambhir, Sankalp and Gilot, Andrea and Kun\v{c}ak, Viktor},
  title =	{{Mechanized HOL Reasoning in Set Theory}},
  booktitle =	{15th International Conference on Interactive Theorem Proving (ITP 2024)},
  pages =	{18:1--18:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-337-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{309},
  editor =	{Bertot, Yves and Kutsia, Temur and Norrish, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2024.18},
  URN =		{urn:nbn:de:0030-drops-207464},
  doi =		{10.4230/LIPIcs.ITP.2024.18},
  annote =	{Keywords: Proof assistant, First Order Logic, Set Theory, Higher Order Logic}
}

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Mechanized HOL Reasoning in Set Theory

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