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Type theories
 In STACS ’02: Proceedings of the 19th Annual Symposium on Theoretical Aspects of Computer Science
, 1995
"... Abstract. Deduction modulo is a way to express a theory using computation rules instead of axioms. We present in this paper an extension of deduction modulo, called Polarized deduction modulo, where some rules can only be used at positive occurrences, while others can only be used at negative ones. ..."
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Abstract. Deduction modulo is a way to express a theory using computation rules instead of axioms. We present in this paper an extension of deduction modulo, called Polarized deduction modulo, where some rules can only be used at positive occurrences, while others can only be used at negative ones. We show that all theories in propositional calculus can be expressed in this framework and that cuts can always be eliminated with such theories. Mathematical proofs are almost never built in pure logic, but besides the deduction rules and the logical axioms that express the meaning of the connectors and quantifiers, they use something else a theory that expresses the meaning of the other symbols of the language. Examples of theories are equational theories, arithmetic, type theory, set theory,... The usual definition of a theory, as a set of axioms, is sufficient when one is interested in the provability relation, but, as wellknown, it is not when one is interested in the structure of proofs and in the theorem proving process. For
Cut elimination for Zermelo’s set theory
, 2006
"... We show how to express intuitionistic Zermelo set theory in deduction modulo (i.e. by replacing its axioms by rewrite rules) in such a way that the corresponding notion of proof enjoys the normalization property. To do so, we first rephrase set theory as a theory of pointed graphs (following a para ..."
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We show how to express intuitionistic Zermelo set theory in deduction modulo (i.e. by replacing its axioms by rewrite rules) in such a way that the corresponding notion of proof enjoys the normalization property. To do so, we first rephrase set theory as a theory of pointed graphs (following a paradigm due to P. Aczel) by interpreting settheoretic equality as bisimilarity, and show that in this setting, Zermelo’s axioms can be decomposed into graphtheoretic primitives that can be turned into rewrite rules. We then show that the theory we obtain in deduction modulo is a conservative extension of (a minor extension of) Zermelo set theory. Finally, we prove the normalization of the intuitionistic fragment of the theory.
The Stratified Foundations as a theory modulo
"... The Stratified Foundations are a restriction of naive set theory where the comprehension scheme is restricted to stratifiable propositions. It is known that this theory is consistent and that proofs strongly normalize in this theory. Deduction modulo is a formulation of firstorder logic with a ..."
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The Stratified Foundations are a restriction of naive set theory where the comprehension scheme is restricted to stratifiable propositions. It is known that this theory is consistent and that proofs strongly normalize in this theory. Deduction modulo is a formulation of firstorder logic with a general notion of cut. It is known that proofs normalize in a theory modulo if it has some kind of manyvalued model called a premodel. We show in this note that the Stratified Foundations can be presented in deduction modulo and that the method used in the original normalization proof can be adapted to construct a premodel for this theory. The Stratified Foundations are a restriction of naive set theory where the comprehension scheme is restricted to stratifiable propositions. This theory is consistent [8] and proofs in this theory strongly normalize [2], while naive set theory is contradictory and the consistency of the Stratified Foundations together with the extensionality axiom...
Deduction modulo theory
"... 1.1 Weaker vs. stronger systems Contemporary proof theory goes into several directions at the same time. One of them aims at analysing proofs, propositions, connectives, etc., that is at decomposing them into more atomic objects. This often leads to design systems ..."
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1.1 Weaker vs. stronger systems Contemporary proof theory goes into several directions at the same time. One of them aims at analysing proofs, propositions, connectives, etc., that is at decomposing them into more atomic objects. This often leads to design systems