Verified completeness in Henkin-style for intuitionistic propositional logic

In Bruno Bentzen, Beishui Liao, Davide Liga, Reka Markovich, Bin Wei, Minghui Xiong & Tianwen Xu (eds.), Logics for AI and Law: Joint Proceedings of the Third International Workshop on Logics for New-Generation Artificial Intelligence and the International Workshop on Logic, AI and Law, September 8-9 and 11-12, 2023, Hangzhou. College Publications. pp. 36-48 (2023)
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Abstract

This paper presents a formalization of the classical proof of completeness in Henkin-style developed by Troelstra and van Dalen for intuitionistic logic with respect to Kripke models. The completeness proof incorporates their insights in a fresh and elegant manner that is better suited for mechanization. We discuss details of our implementation in the Lean theorem prover with emphasis on the prime extension lemma and construction of the canonical model. Our implementation is restricted to a system of intuitionistic propositional logic with implication, conjunction, disjunction, and falsity given in terms of a Hilbert-style axiomatization. As far as we know, our implementation is the first verified Henkin-style proof of completeness for intuitionistic logic following Troelstra and van Dalen's method in the literature.

Author Profiles

Huayu Guo
Zhejiang University
Bruno Bentzen
Zhejiang University

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