A short proof of the strong normalization of classical natural deduction with disjunction

David, René; Nour, Karim

doi:10.2178/jsl/1067620187

Cited by 42 publications

(24 citation statements)

References 18 publications

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“…It has disjunction, first-order and second-order existential quantification and their permutative conversions. We will give the definition of the system NJ 2 NJ 2 NJ . Below we will give the list of axioms and inference rules for NJ 2 NJ 2 NJ together with a standard assignment of the second order l-terms by Curry-Howard isomorphism.…”

Section: Second Order Natural Deduction Njmentioning

confidence: 99%

“…We will give the definition of the system NJ 2 NJ 2 NJ . Below we will give the list of axioms and inference rules for NJ 2 NJ 2 NJ together with a standard assignment of the second order l-terms by Curry-Howard isomorphism. The system of reductions is also standard and includes permutative conversions for  , $, $ 2 .…”

Section: Second Order Natural Deduction Njmentioning

confidence: 99%

“…Permutative Conversions Techniques and references second order  , $, $ 2 strong validity [15] inductive definitions [12] second order  , $ saturated sets [19] second order no reducibility [5,6] first order  , $ strong validity [10,20] type theory ITT 0 ITT 0 ITT S (weak permutative conversions) (ad hoc) [18] first order  CPS translation [3] inductive definitions [2,9] .., x n := t n ] are also defined in a standard way. They will be sometimes written using the vector notation such as s [x := t  t  t].…”

Section: Logical Systemsmentioning

confidence: 99%

“…Strong normalization of a type theory with P types, S types, and their weak permutative conversions is proved in [18]. Strong normalization of propositional natural deduction with disjunction and commutative conversions is proved in [3,2,9].We will prove strong normalization of second order intuitionistic natural deduction with permutative conversions by using Prawitz's strong validity. This paper completes Prawitz's original proof given in [15].…”

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Second order permutative conversions with Prawitz's strong validity

Tatsuta¹

2005

Prog. Inform.

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Permutative conversions transform a proof with a disjunction or existential quantification elimination rule followed by an elimination rule into a proof with the second rule in the minor deduction of the first rule [14,21]. Permutative conversions are indispensable for normalizing a proof in a natural deduction system with disjunction or existential quantification. Without permutative conversions, a normal proof fails to have the subformula property, because there may exist an introduction rule of a logical symbol in a minor deduction of a disjunction or existential quantification elimination rule followed by the elimination rule of the same logical symbol, which may break the subformula property.Strong normalization property is important. First it implies weak normalization property, which proves the subformula property and consistency [14,21]. In particular, weak normalization of a second order system has been often proved through showing its strong normalization [5,6]. Secondly, when we consider proof normalization as computation by the proofs-as-programs paradigm [8], strong normalization guarantees termination of programs. Thirdly, it is itself an interesting combinatorial problem in mathematics.Several papers have studied strong normalization of systems with or without permutative conversions. We summarize them in Table 1. Strong normalization of second order natural deduction with disjunction, existential quantification, and their commutative conversions is proved in [15,12]. Strong normalization of second order natural deduction with disjunction, first-order existential quantification, and their commutative conversions is proved in [19]. Strong normalization of second order natural deduction with disjunction and existential quantification without their commutative conversions is proved in [5,6]. Strong normalization of the negative fragment of second order natural deduction is discussed in [4,7,11,20,21]. Strong normalization of first order natural deduction with disjunction, existential quantification, and their commutative conversions is proved in [10,20]. Strong normalization of a type theory with P types, S types, and their weak permutative conversions is proved in [18]. Strong normalization of propositional natural deduction with disjunction and commutative conversions is proved in [3,2,9].We will prove strong normalization of second order intuitionistic natural deduction with permutative conversions by using Prawitz's strong validity. This paper completes Prawitz's original proof given in [15].By Curry-Howard isomorphism [8], a second order logical system corresponds to a functional programming language with polymorphic types and abstract data types. Second order universal quantification gives polymorphic Second order permutative conversions with Prawitz's strong validity Makoto TATSUTA National Institute of InformaticsThe Graduate University for Advanced Studies ABSTRACT A clear and complete proof of strong normalization of second order natural deduction with permutative conversions is given by us...

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Section: Second Order Natural Deduction Njmentioning

confidence: 99%

Section: Second Order Natural Deduction Njmentioning

confidence: 99%

Section: Logical Systemsmentioning

confidence: 99%

mentioning

confidence: 96%

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Second order permutative conversions with Prawitz's strong validity

Tatsuta¹

2005

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Arithmetical Proofs of Strong Normalization Results for the Symmetric λμ-Calculus

David

Nour

2005

Lecture Notes in Computer Science

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Abstract. The symmetric λµ-calculus is the λµ-calculus introduced by Parigot in which the reduction rule µ ′ , which is the symmetric of µ, is added. We give arithmetical proofs of some strong normalization results for this calculus. We show (this is a new result) that the µµ ′ -reduction is strongly normalizing for the un-typed calculus. We also show the strong normalization of the βµµ ′ -reduction for the typed calculus: this was already known but the previous proofs use candidates of reducibility where the interpretation of a type was defined as the fix point of some increasing operator and thus, were highly non arithmetical.

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Simple Saturated Sets for Disjunction and Second-Order Existential Quantification

Tatsuta

2007

Lecture Notes in Computer Science

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A short proof of the strong normalization of classical natural deduction with disjunction

Abstract: We give a direct, purely arithmetical and elementary proof of the strong normalization of the cut-elimination procedure for full (i.e. in presence of all the usual connectives) classical natural deduction.

Cited by 42 publications

References 18 publications

Second order permutative conversions with Prawitz's strong validity

Second order permutative conversions with Prawitz's strong validity

Arithmetical Proofs of Strong Normalization Results for the Symmetric λμ-Calculus

Simple Saturated Sets for Disjunction and Second-Order Existential Quantification

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