Conversion of HOL Light proofs into Metamath

TL;DR

This paper introduces an algorithm to convert HOL Light proofs into Metamath, enabling the transfer of proofs and consistency verification between these formal systems.

Contribution

The paper presents a novel two-step translation process from HOL Light proofs to Metamath, linking their foundations and automations.

Findings

Successful translation of HOL Light proofs into Metamath formalizations

Proofs of HOL Light theorems verified within Metamath's ZFC framework

Demonstration of HOL Light's consistency relative to Metamath's foundations

Abstract

We present an algorithm for converting proofs from the OpenTheory interchange format, which can be translated to and from any of the HOL family of proof languages (HOL4, HOL Light, ProofPower, and Isabelle), into the ZFC-based Metamath language. This task is divided into two steps: the translation of an OpenTheory proof into a Metamath HOL formalization, $\mathtt{\text{hol.mm}}$, followed by the embedding of the HOL formalization into the main ZFC foundations of the main Metamath library, $\mathtt{\text{set.mm}}$. This process provides a means to link the simplicity of the Metamath foundations to the intense automation efforts which have borne fruit in HOL Light, allowing the production of complete Metamath proofs of theorems in HOL Light, while also proving that HOL Light is consistent, relative to Metamath's ZFC axiomatization.