Abstract. The correspondence between natural deduction proofs and λ-terms is presented and discussed. A variant of the reducibility method is presented, and a general theorem for establishing properties of typed (first-order) λ-terms is proved. As a corollary, we obtain a simple proof of the Church-Rosser property, and of the strong normalization property, for the typed λ-calculus associated with the system of (intuitionistic) first-order natural deduction, including all the connectors

. It was observed by Curry that when (untyped) -terms 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 head-normal forms, can be characterized in some systems D and D \Omega\Gamma The proofs use variants of the method of reducibility. In this paper, we present a uniform approach for proving several meta-theorems relating properties of -terms 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 easi...

. The correspondence between natural deduction proofs and -terms is presented and discussed. A variant of the reducibility method is presented, and a general theorem for establishing properties of typed (first-order) -terms is proved. As a corollary, we obtain a simple proof of the Church-Rosser property, and of the strong normalization property, for the typed -calculus associated with the system of (intuitionistic) first-order natural deduction, including all the connectors !, \Theta, +, 8, 9, and ? (falsity) (with or without j-like rules). This research was partially supported by ONR Grant NOOO14-88-K-0593. Contents 1 Introduction 3 2 Natural Deduction, Simply-Typed -Calculus 5 3 Adding Conjunction, Negation, and Disjunction 11 4 First-Order Quantifiers 14 5 P-Candidates for the Arrow Type Constructor ! 18 6 Adding Product and Sum Types \Theta and + 23 7 Adding the Absurdity Type ? 28 8 Adding First-Order Quantifiers 8 and 9 35 9 Adding j-like Reduction Rules 52 1 Introductio...

. The main purpose of this paper is to take apart the reducibility method in order to understand how its pieces fit together, and in particular, to recast the conditions on candidates of reducibility as sheaf conditions. There has been a feeling among experts on this subject that it should be possible to present the reducibility method using more semantic means, and that a deeper understanding would then be gained. This paper gives mathematical substance to this feeling, by presenting a generalization of the reducibility method based on a semantic notion of realizability which uses the notion of a cover algebra (as in abstract sheaf theory). A key technical ingredient is the introduction a new class of semantic structures equipped with preorders, called pre-applicative structures. These structures need not be extensional. In this framework, a general realizability theorem can be shown. Kleene's recursive realizability and a variant of Kreisel's modified realizability both fit into this...

By using an infinity of extra variables every λ-term with indexed redexes is interpreted into a term in the simply typed lambda calculus à la Curry. A development becomes a usual β-reduction in the simply typed lambda calculus and the corresponding properties of developments come out from the corresponding properties (strong normalization and Church-Rosser) holding in this system. In this way we obtain a complete simulation of the notion of development into the system of simply typed lambda calculus. Keywords: developments, strong normalization, Church-Rosser property, simple types 1

Conclusion George Koletsos, Yiorgos Stavrinos: Embedding developments into simply typed λ-calculus 2/14Developments Simply typed λ-calculus Embedding Finiteness of developments Conclusion • V = {x, y, z,...} infinite set of variables Λ: set of λ-terms M:: = x | λx.M | MM β-reduction: (λx.M)N −→β M[N/x] • Development F a set of redex occurrences in M

proofs in the λ-calculus

3.1 Presentation of the λ-Calculus..................... 5

Reducibility has been used to prove a number of properties in the λ-calculus and is well known to offer on one hand very general proofs which can be applied to a number of instantiations, and on the other hand, to be quite mysterious and inflexible. It has, amongst other things, been used along with the so called method of parallel reductions to prove the Church-Rosser property. In this paper, we concentrate on using the methods of reducibility and of parallel reductions for proving Church-Rosser for both β- and βη-reduction. Our contributions are two fold: We give a simple proof of CR for β-reduction which unlike the common proofs in the literature, avoids any type machinery and is solely carried out in a completely untyped setting. We give a new proof of CR for βη-reduction which is a generalisation of our simple proof for β-reduction. Keywords:

Reducibility has been used to prove a number of properties in the λ-calculus and is well known to offer on one hand very general proofs which can be applied to a number of instantiations, and on the other hand, to be quite mysterious and inflexible. In this paper, we look at two related but different results in λ-calculi with intersection types. We show that one such result (which aims at giving reducibility proofs of Church-Rosser, standardisation and weak normalisation for the untyped λ-calculus) faces serious problems which break the reducibility method and then we provide a proposal to partially repair the method. Then, we consider a second result whose purpose is to use reducibility for typed terms to show Church-Rosser of β-developments for untyped terms (without needing to use strong normalisation), from which Church-Rosser of β-reduction easily follows. We extend the second result to encompass both βI- and βη-reduction rather than simply β-reduction. 1