Wojciech Moczydlowski

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

Churchs Higher Order Logic is a basis for proof assistants — HOL and PVS. Churchs 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 Rathjens 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.

Normalization of IZF with replacement

IZF is a well investigated impredicative constructive version of Zermelo-Fraenkel set theory. Using set terms, we axiomatize IZF with Replacement, which we call IZFR, along with its intensional counterpart IZF

Extracting the resolution algorithm from a completeness proof for the propositional calculus

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Termination of single-threaded one-rule semi-thue systems

. We define a typed lambda calculus λZ corresponding to proofs in IZF

A Normalizing Intuitionistic Set Theory with Inaccessible Sets

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A Dependent Set Theory

according to the Curry-Howard isomorphism principle. Using realizability for IZF

Extracting the Resolution Algorithm from a Completeness Proof for the Propositional Calculus

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

, we show weak normalization of λZ by employing a reduction-preserving erasure map from lambda terms to realizers. We use normalization to prove disjunction, numerical existence, set existence and term existence properties. An inner extensional model is used to show the properties for full, extensional IZFR.

Investigations on sets and types

Abstract We prove constructively that for any propositional formula ϕ in Conjunctive Normal Form, we can either find a satisfying assignment of true and false to its variables, or a refutation of ϕ showing that it is unsatisfiable. This refutation is a resolution proof of ¬ ϕ . From the formalization of our proof in Coq, we extract Robinson’s famous resolution algorithm as a Haskell program correct by construction. The account is an example of the genre of highly readable formalized mathematics.

Normalization of IZF with Replacement

This paper is a contribution to the long standing open problem of uniform termination of Semi-Thue Systems that consist of one rule s→ t. McNaughton previously showed that rules incapable of (1) deleting t completely from both sides, (2) deleting t completely from the left, and (3) deleting t completely from the right, have a decidable uniform termination problem. We use a novel approach to show that Premise (2) or, symmetrically, Premise (3), is inessential. Our approach is based on derivations in which every pair of successive steps has an overlap. We call such derivations single-threaded.