6 research outputs found
Normalization of IZF with Replacement
ZF 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
\iizfr. We define a typed lambda calculus \li corresponding to proofs in
\iizfr according to the Curry-Howard isomorphism principle. Using realizability
for \iizfr, we show weak normalization of \li. We use normalization to prove
the disjunction, numerical existence and term existence properties. An inner
extensional model is used to show these properties, along with the set
existence property, for full, extensional \izfr
Extracting Programs from Constructive HOL Proofs via IZF Set-Theoretic<br> Semantics
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.Comment: 17 page
A Normalizing Intuitionistic Set Theory with Inaccessible Sets
We propose a set theory strong enough to interpret powerful type theories
underlying proof assistants such as LEGO and also possibly Coq, which at the
same time enables program extraction from its constructive proofs. For this
purpose, we axiomatize an impredicative constructive version of
Zermelo-Fraenkel set theory IZF with Replacement and -many
inaccessibles, which we call \izfio. Our axiomatization utilizes set terms, an
inductive definition of inaccessible sets and the mutually recursive nature of
equality and membership relations. It allows us to define a weakly-normalizing
typed lambda calculus corresponding to proofs in \izfio according to the
Curry-Howard isomorphism principle. We use realizability to prove the
normalization theorem, which provides a basis for program extraction
capability.Comment: To be published in Logical Methods in Computer Scienc
Normalization of IZF with Replacement
ZF 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
\iizfr. We define a typed lambda calculus \li corresponding to proofs in
\iizfr according to the Curry-Howard isomorphism principle. Using realizability
for \iizfr, we show weak normalization of \li. We use normalization to prove
the disjunction, numerical existence and term existence properties. An inner
extensional model is used to show these properties, along with the set
existence property, for full, extensional \izfr
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 IZF_R, along with its intensional counterpart IZF_R^-. We define a typed lambda calculus corresponding to proofs in IZF_R^- according to the Curry-Howard isomorphism principle. Using realizability for IZF_R^-, we show weak normalization of the calculus 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 IZF_R