Extracting programs from constructive HOL proofs via IZF set-theoretic semantics

Authors:
Robert Constable;Wojciech Moczydłowski
Affiliations:
Department of Computer Science, Cornell University, Ithaca, NY;Department of Computer Science, Cornell University, Ithaca, NY
Venue:
IJCAR'06 Proceedings of the Third international joint conference on Automated Reasoning
Year:
2006

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Abstract

Church's Higher Order Logic is a basis for 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 modifying Rathjen's realizability for CZF and the other using a new direct weak normalization result for intensional IZF by Moczydłowski. The latter can also be used for the term existence property.