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The Mechanisation of BarendregtStyle Equational Proofs (the Residual Perspective)
, 2001
"... We show how to mechanise equational proofs about higherorder languages by using the primitive proof principles of firstorder abstract syntax over onesorted variable names. We illustrate the method here by proving (in Isabelle/HOL) a technical property which makes the method widely applicable for ..."
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We show how to mechanise equational proofs about higherorder languages by using the primitive proof principles of firstorder abstract syntax over onesorted variable names. We illustrate the method here by proving (in Isabelle/HOL) a technical property which makes the method widely applicable for the λcalculus: the residual theory of β is renamingfree upto an initiality condition akin to the socalled Barendregt Variable Convention. We use our results to give a new diagrambased proof of the development part of the strong finite development property for the λcalculus. The proof has the same equational implications (e.g., confluence) as the proof of the full property but without the need to prove SN. We account for two other uses of the proof method, as presented elsewhere. One has been mechanised in full in Isabelle/HOL.
ContinuationPassing Style and Strong Normalisation for Intuitionistic Sequent Calculi
"... Abstract. The intuitionistic fragment of the callbyname version of Curien and Herbelin’s λµ˜µcalculus is isolated and proved strongly normalising by means of an embedding into the simplytyped λcalculus. Our embedding is a continuationandgarbagepassing style translation, the inspiring idea co ..."
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Cited by 3 (2 self)
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Abstract. The intuitionistic fragment of the callbyname version of Curien and Herbelin’s λµ˜µcalculus is isolated and proved strongly normalising by means of an embedding into the simplytyped λcalculus. Our embedding is a continuationandgarbagepassing style translation, the inspiring idea coming from Ikeda and Nakazawa’s translation of Parigot’s λµcalculus. The embedding simulates reductions while usual continuationpassingstyle transformations erase permutative reduction steps. For our intuitionistic sequent calculus, we even only need “units of garbage ” to be passed. We apply the same method to other calculi, namely successive extensions of the simplytyped λcalculus leading to our intuitionistic system, and already for the simplest extension we consider (λcalculus with generalised application), this yields the first proof of strong normalisation through a reductionpreserving embedding. 1
Characterizing Strongly Normalizing Terms of a lambdaCalculus with Generalized Applications via Intersection Types
"... An intersection type assignment system for the extension LJ of the untyped lcalculus, introduced by Joachimski and Matthes, is given and proven to characterize the strongly normalizing terms of LJ. Since LJ's generalized applications naturally allow permutative/commuting conversions, this is th ..."
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An intersection type assignment system for the extension LJ of the untyped lcalculus, introduced by Joachimski and Matthes, is given and proven to characterize the strongly normalizing terms of LJ. Since LJ's generalized applications naturally allow permutative/commuting conversions, this is the first analysis of a term rewrite system with permutative conversions by help of intersection types. Two proofs are given for the fact that the typable terms are strongly normalizing: One by the computability predicates method a la Tait and one showing directly that strongly normalizing typable terms are closed under (generalized) application and substitution. It is also shown that a straightforward extension of the type assignment for lcalculus fails to capture the strongly normalizing terms. Keywords Intersection Types, Strong Normalization, Permutative Conversions, Saturated Sets. 1 Introduction In [5] an extension LJ of lcalculus with generalized applications inspired by vo...
On Zucker's isomorphism for LJ and its extension to Pure Type Systems
, 2003
"... It is shown how the sequent calculus LJ can be embedded into a simple extension of the calculus by generalized applications, called J. The reduction rules of cut elimination and normalization can be precisely correlated, if explicit substitutions are added to J. The resulting system J2 is prove ..."
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It is shown how the sequent calculus LJ can be embedded into a simple extension of the calculus by generalized applications, called J. The reduction rules of cut elimination and normalization can be precisely correlated, if explicit substitutions are added to J. The resulting system J2 is proved strongly normalizing, thus showing strong normalization for Gentzen's cut elimination steps. This re nes previous results by Zucker, Pottinger and Herbelin on the isomorphism between natural deduction and sequent calculus.
Issues in a calculus of multiary sequent terms
, 2006
"... In this talk we overview our study on an extension of the λcalculus introduced in [1], exhibiting the features of multiarity and generality. The former feature is the ability of applying a term to a list of arguments. The latter is the ability of specifying a future use, or “continuation”, for a (p ..."
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In this talk we overview our study on an extension of the λcalculus introduced in [1], exhibiting the features of multiarity and generality. The former feature is the ability of applying a term to a list of arguments. The latter is the ability of specifying a future use, or “continuation”, for a (possibly multiary) application. The calculus was named the generalised multiary λcalculus, or the λJ mcalculus for short. In its simply typed version, the calculus corresponds to a sequent calculus, and the novelty of the mentioned features relates to the novelty of the left introduction rule relatively to a natural deduction format. The calculus was introduced in [1] as a calculus of multiary sequent terms (in the sense of [4]) for a study of permutative conversions in sequent calculus. The main lesson of this study is the existence of a relationship between permutative conversions, subsystems of λJ m and the features of generality and multiarity. Moreover, some subsystems turn out to be isomorphic to fragments of natural deduction, taking natural deduction in the extended sense of von Plato [5]. Later work [2] observes an overlap between the features of generality and multiarity. In [2] the overlap is mainly used to transfer results about reduction from ΛJ to λJ m, where ΛJ is the λcalculus with generalised application introduced in [3]. In addition this overlap suggests a refinement of the original view of λJ m as obtained from λ by modularly adding the two new features. This refinement is ongoing work. Another issue we are considering is that of obtaining a λterm in βnormal form out of a λJ mterm. This requires the combination of reduction and permutative conversion and, in particular, raises questions of termination. These questions relate to the problem of preservation of strong normalisation in calculi of explicit substitutions.
Structural Induction and the λCalculus
"... Abstract. We consider formal provability with structural induction and related proof principles in the λcalculus presented with firstorder abstract syntax over onesorted variable names. As well as summarising and elaborating on earlier, formally verified proofs (in Isabelle/HOL) of the relative re ..."
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Abstract. We consider formal provability with structural induction and related proof principles in the λcalculus presented with firstorder abstract syntax over onesorted variable names. As well as summarising and elaborating on earlier, formally verified proofs (in Isabelle/HOL) of the relative renamingfreeness of βresidual theory and βconfluence, we also present proofs of ηconfluence, βηconfluence, the strong weaklyfinite βdevelopment (aka residualcompletion) property, residual βconfluence, ηoverβpostponement, and notably βstandardisation. In the latter case, the known proofs fail in instructive ways. Interestingly, our uniform proof methodology, which has relevance beyond the λcalculus, properly contains penandpaper proof practices in a precise sense. The proof methodology also makes precise what is the full algebraic proof burden of the considered results, which we, moreover, appear to be the first to resolve. 1
Some properties of the λµ ∧ ∨calculus Karim NOUR & Khelifa SABER
"... In this paper, we present the λµ ∧ ∨calculus which at the typed level corresponds to the full classical propositional natural deduction system. ChurchRosser property of this system is proved using the standardization and the finiteness developments theorem. We define also the leftmost reduction an ..."
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In this paper, we present the λµ ∧ ∨calculus which at the typed level corresponds to the full classical propositional natural deduction system. ChurchRosser property of this system is proved using the standardization and the finiteness developments theorem. We define also the leftmost reduction and prove that it is a winning strategy. 1
Issues in a calculus of multiary sequent terms J. Espírito Santo ∗ , M. J. Frade +, L. Pinto ∗ ∗Departamento de Matemática
"... In this talk we overview our study on an extension of the λcalculus introduced in [1], exhibiting the features of multiarity and generality. The former feature is the ability of applying a term to a list of arguments. The latter is the ability of specifying a future use, or “continuation”, for a (p ..."
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In this talk we overview our study on an extension of the λcalculus introduced in [1], exhibiting the features of multiarity and generality. The former feature is the ability of applying a term to a list of arguments. The latter is the ability of specifying a future use, or “continuation”, for a (possibly multiary) application. The calculus was named the generalised multiary λcalculus, or the λJ mcalculus for short. In its simply typed version, the calculus corresponds to a sequent calculus, and the novelty of the mentioned features relates to the novelty of the left introduction rule relatively to a natural deduction format. The calculus was introduced in [1] as a calculus of multiary sequent terms (in the sense of [4]) for a study of permutative conversions in sequent calculus. The main lesson of this study is the existence of a relationship between permutative conversions, subsystems of λJ m and the features of generality and multiarity. Moreover, some subsystems turn out to be isomorphic to fragments of natural deduction, taking natural deduction in the extended sense of von Plato [5]. Later work [2] observes an overlap between the features of generality and multiarity. In [2] the overlap is mainly used to transfer results about reduction from ΛJ to λJ m, where ΛJ is the λcalculus with generalised application introduced in [3]. In addition this overlap suggests a refinement of the original view of λJ m as obtained from λ by modularly adding the two new features. This refinement is ongoing work. Another issue we are considering is that of obtaining a λterm in βnormal form out of a λJ mterm. This requires the combination of reduction and permutative conversion and, in particular, raises questions of termination. These questions relate to the problem of preservation of strong normalisation in calculi of explicit substitutions.
Strong normalization of classical natural deduction with disjunctions
"... This paper proves strong normalization of classical natural deduction with disjunction and permutative conversions, by using CPStranslation and augmentations. By them, this paper also proves strong normalization of classical natural deduction with general elimination rules for implication and conju ..."
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This paper proves strong normalization of classical natural deduction with disjunction and permutative conversions, by using CPStranslation and augmentations. By them, this paper also proves strong normalization of classical natural deduction with general elimination rules for implication and conjunction, and their permutative conversions. This paper also proves natural deduction can be embedded into natural deduction with general elimination rules, strictly preserving proof normalization.
Proving Strong Normalisation via Nondeterministic Translations into Klop’s Extended λCalculus
"... In this paper we present strong normalisation proofs using a technique of nondeterministic translations into Klop’s extended λcalculus. We first illustrate the technique by showing strong normalisation of a typed calculus that corresponds to natural deduction with general elimination rules. Then w ..."
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In this paper we present strong normalisation proofs using a technique of nondeterministic translations into Klop’s extended λcalculus. We first illustrate the technique by showing strong normalisation of a typed calculus that corresponds to natural deduction with general elimination rules. Then we study its explicit substitution version, the typefree calculus of which does not satisfy PSN with respect to reduction of the original calculus; nevertheless it is shown that typed terms are strongly normalising with respect to reduction of the explicit substitution calculus. In the same framework we prove strong normalisation of Sørensen and Urzyczyn’s cutelimination system in intuitionistic sequent calculus.