Extracting Programs from Constructive HOL Proofs via IZF Set-Theoretic<br> Semantics

TL;DR

This paper develops a set-theoretic semantics for HOL and ZFC that enables program extraction from constructive proofs by leveraging properties of IZF, with implementations based on realizability and normalization techniques.

Contribution

It introduces a novel semantics mapping HOL's constructive core into IZF, facilitating program extraction from proofs using new realizability and normalization methods.

Findings

Program extraction is feasible from HOL proofs via IZF semantics.

Two methods for implementing properties: realizability and normalization.

The approach enhances proof assistant capabilities with constructive semantics.

Abstract

Church's Higher Order Logic is a basis for influential proof assistants -- HOL and PVS. Church's logic has a simple set-theoretic semantics, making it trustworthy and extensible. We factor HOL into a constructive core plus axioms of excluded middle and choice. We similarly factor standard set theory, ZFC, into a constructive core, IZF, and axioms of excluded middle and choice. Then we provide the standard set-theoretic semantics in such a way that the constructive core of HOL is mapped into IZF. We use the disjunction, numerical existence and term existence properties of IZF to provide a program extraction capability from proofs in the constructive core. We can implement the disjunction and numerical existence properties in two different ways: one using Rathjen's realizability for IZF and the other using a new direct weak normalization result for IZF by Moczydlowski. The latter can also be used for the term existence property.