Verified completeness in Henkin-style for intuitionistic propositional logic

TL;DR

This paper formalizes and mechanizes a Henkin-style completeness proof for intuitionistic propositional logic using the Lean theorem prover, providing the first verified proof of its kind.

Contribution

It introduces a verified Henkin-style completeness proof for intuitionistic propositional logic, implemented in Lean, based on Troelstra and van Dalen's method.

Findings

First verified Henkin-style proof in Lean

Implementation details of prime extension lemma and canonical model

Accessible source code for the proof

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. The full source code can be found online at https://github.com/bbentzen/ipl.