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On Girard’s “Candidats de Réductibilité
 Logic and Computer Science
, 1990
"... Abstract: We attempt to elucidate the conditions required on Girard’s candidates of reducibility (in French, “candidats de reductibilité”) in order to establish certain properties of various typed lambda calculi, such as strong normalization and ChurchRosser property. We present two generalizations ..."
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Abstract: We attempt to elucidate the conditions required on Girard’s candidates of reducibility (in French, “candidats de reductibilité”) in order to establish certain properties of various typed lambda calculi, such as strong normalization and ChurchRosser property. We present two generalizations of the candidates of reducibility, an untyped version in the line of Tait and Mitchell, and a typed version which is an adaptation of Girard’s original method. As an application of this general result, we give two proofs of strong normalization for the secondorder polymorphic lambda calculus under ⌘reduction (and thus underreduction). We present two sets of conditions for the typed version of the candidates. The first set consists of conditions similar to those used by Stenlund (basically the typed version of Tait’s conditions), and the second set consists of Girard’s original conditions. We also compare these conditions, and prove that Girard’s conditions are stronger than Tait’s conditions. We give a new proof of the ChurchRosser theorem for bothreduction and ⌘reduction, using the modified version of Girard’s method. We also compare various proofs that have appeared in the literature (see section 11). We conclude by sketching the extension of the above results to Girard’s higherorder polymorphic calculus F!, and in appendix 1, to F! with product types. i 1
A short and flexible proof of Strong Normalization for the Calculus of Constructions
, 1994
"... this paper can still go through (with slightly more technical effort) in case one can distinguish cases according to whether a specific subterm is a type or kind in a fixed context. The other property of type systems that is really actually required for the constructions in this paper to go through ..."
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this paper can still go through (with slightly more technical effort) in case one can distinguish cases according to whether a specific subterm is a type or kind in a fixed context. The other property of type systems that is really actually required for the constructions in this paper to go through is a slight strengthening of the Stripping property (also called Generation). This property says, for example, that if \Gamma ` v:T:M : U has a derivation D, then one can find a subderivation of
Modified Realizability Toposes and Strong Normalization Proofs (Extended Abstract)
 Typed Lambda Calculi and Applications, LNCS 664
, 1993
"... ) 1 J. M. E. Hyland 2 C.H. L. Ong 3 University of Cambridge, England Abstract This paper is motivated by the discovery that an appropriate quotient SN 3 of the strongly normalising untyped 3terms (where 3 is just a formal constant) forms a partial applicative structure with the inherent appl ..."
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) 1 J. M. E. Hyland 2 C.H. L. Ong 3 University of Cambridge, England Abstract This paper is motivated by the discovery that an appropriate quotient SN 3 of the strongly normalising untyped 3terms (where 3 is just a formal constant) forms a partial applicative structure with the inherent application operation. The quotient structure satisfies all but one of the axioms of a partial combinatory algebra (pca). We call such partial applicative structures conditionally partial combinatory algebras (cpca). Remarkably, an arbitrary rightabsorptive cpca gives rise to a tripos provided the underlying intuitionistic predicate logic is given an interpretation in the style of Kreisel's modified realizability, as opposed to the standard Kleenestyle realizability. Starting from an arbitrary rightabsorptive cpca U , the tripostotopos construction due to Hyland et al. can then be carried out to build a modified realizability topos TOPm (U ) of nonstandard sets equipped with an equali...
Lectures on the curryhoward isomorphism
, 1998
"... The CurryHoward isomorphism states an amazing correspondence between systems of formal logic as encountered in proof theory and computational calculi as found in type theory. For instance, minimal propositional logic corresponds to simply typed λcalculus, firstorder logic corresponds to dependent ..."
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The CurryHoward isomorphism states an amazing correspondence between systems of formal logic as encountered in proof theory and computational calculi as found in type theory. For instance, minimal propositional logic corresponds to simply typed λcalculus, firstorder logic corresponds to dependent types, secondorder logic corresponds to polymorphic types, etc. The isomorphism has many aspects, even at the syntactic level: formulas correspond to types, proofs correspond to terms, provability corresponds to inhabitation, proof normalization corresponds to term reduction, etc. But there is much more to the isomorphism than this. For instance, it is an old idea—due to Brouwer, Kolmogorov, and Heyting, and later formalized by Kleene’s realizability interpretation—that a constructive proof of an implication is a procedure that transforms proofs of the antecedent into proofs of the succedent; the CurryHoward isomorphism gives syntactic representations of such procedures. These notes give an introduction to parts of proof theory and related
Typing untyped λterms, or Reducibility strikes again!
, 1995
"... It was observed by Curry that when (untyped) λterms can be assigned types, for example, simple types, these terms have nice properties (for example, they are strongly normalizing). Coppo, Dezani, and Veneri, introduced type systems using conjunctive types, and showed that several important classes ..."
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It was observed by Curry that when (untyped) λterms can be assigned types, for example, simple types, these terms have nice properties (for example, they are strongly normalizing). Coppo, Dezani, and Veneri, introduced type systems using conjunctive types, and showed that several important classes of (untyped) terms can be characterized according to the shape of the types that can be assigned to these terms. For example, the strongly normalizable terms, the normalizable terms, and the terms having headnormal forms, can be characterized in some systems D and D. The proofs use variants of the method of reducibility. In this paper, we presenta uniform approach for proving several metatheorems relating properties ofterms and their typability in the systems D and D. Our proofs use a new and more modular version of the reducibility method. As an application of our metatheorems, we show how the characterizations obtained by Coppo, Dezani, Veneri, and Pottinger, can be easily rederived. We alsocharacterize the terms that have weak headnormal forms, which appears to be new. We conclude by stating a number of challenging open problems regarding possible generalizations of the realizability method.
Normalization for Typed Lambda Calculi with Explicit Substitution
 University of Cambridge, Computer Laboratory, Technical Report
, 1994
"... This paper shows that the strong normalization property holds for a restricted class of reductions: those which push a substitution under a abstraction only if this is the outermost constructor. All standard implementations of the typed calculus, like those using a lazy or eager strategy, have th ..."
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This paper shows that the strong normalization property holds for a restricted class of reductions: those which push a substitution under a abstraction only if this is the outermost constructor. All standard implementations of the typed calculus, like those using a lazy or eager strategy, have this property, hence we can conclude that they terminate. Furthermore, this result means that an implementation of a typed 
A Lambda Model Characterizing Computational Behaviours of Terms
 PROCEEDINGS OF THE AND LIKAVEC INTERNATIONAL WORKSHOP REWRITING IN PROOF AND COMPUTATION
, 2001
"... We build a lambda model which characterizes completely (persistently) normalizing, (persistently) head normalizing, and (persistently) weak head normalizing terms. ..."
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Cited by 6 (4 self)
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We build a lambda model which characterizes completely (persistently) normalizing, (persistently) head normalizing, and (persistently) weak head normalizing terms.
On the Correspondence Between Proofs and λTerms
 Cahiers Du Centre de Logique
, 1995
"... Abstract. The correspondence between natural deduction proofs and λterms is presented and discussed. A variant of the reducibility method is presented, and a general theorem for establishing properties of typed (firstorder) λterms is proved. As a corollary, we obtain a simple proof of the Church ..."
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Abstract. The correspondence between natural deduction proofs and λterms is presented and discussed. A variant of the reducibility method is presented, and a general theorem for establishing properties of typed (firstorder) λterms is proved. As a corollary, we obtain a simple proof of the ChurchRosser property, and of the strong normalization property, for the typed λcalculus associated with the system of (intuitionistic) firstorder natural deduction, including all the connectors
Two behavioural lambda models
 Types for Proofs and Programs
, 2003
"... Abstract. We build a lambda model which characterizes completely (persistently) normalizing, (persistently) head normalizing, and (persistently) weak head normalizing terms. This is proved by using the finitary logical description of the model obtained by defining a suitable intersection type assign ..."
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Cited by 5 (4 self)
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Abstract. We build a lambda model which characterizes completely (persistently) normalizing, (persistently) head normalizing, and (persistently) weak head normalizing terms. This is proved by using the finitary logical description of the model obtained by defining a suitable intersection type assignment system.