# * Extracting Programs from Constructive HOL Proofs via IZF Set-Theoretic Semantics *

## by Robert L. Constable, Wojciech Moczydlowski

2006

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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 Moczydlowski. The latter can also be used for the term existence property

**bibTex ref: CM06**

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