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**11 - 19**of**19**### A Bargain for Intersection Types: A Simple Strong Normalization Proof

"... This pearl gives a discount proof of the folklore theorem that every strongly #-normalizing #-term is typable with an intersection type. (We consider typings that do not use the empty intersection # which can type any term.) The proof uses the perpetual reduction strategy which finds a longest path. ..."

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This pearl gives a discount proof of the folklore theorem that every strongly #-normalizing #-term is typable with an intersection type. (We consider typings that do not use the empty intersection # which can type any term.) The proof uses the perpetual reduction strategy which finds a longest path. This is a simplification over existing proofs that consider any longest reduction path. The choice of reduction strategy avoids the need for weakening or strengthening of type derivations. The proof becomes a bargain because it works for more intersection type systems, while being simpler than existing proofs.

### Characterizing Lambda-Terms With Equal Reduction Behavior

"... We define an equivalence relation on lambda-terms called shuffle-equivalence which attempts to capture the notion of reductional equivalence on strongly normalizing terms. The aim of reductional equivalence is to characterize the evaluation behavior of programs. The shuffle-equivalence classes are s ..."

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We define an equivalence relation on lambda-terms called shuffle-equivalence which attempts to capture the notion of reductional equivalence on strongly normalizing terms. The aim of reductional equivalence is to characterize the evaluation behavior of programs. The shuffle-equivalence classes are shown to divide the classes of beta-equal strongly normalising terms (programs which lead to the same final value/output) into smaller ones consisting of terms with similar evaluation behavior. We refine beta-reduction from a relation on terms to a relation on shuffle-equivalence classes, called shuffle-reduction, and show that this refinement captures existing generalisations of lambda-reduction. Shuffle-reduction allows one to make more redexes visible and to contract these newly visible redexes. Moreover, it allows more freedom in choosing the reduction path of a term, which can result in smaller terms along the reduction path if a clever reduction strategy is used. This can benefit both programming language...

### Generalised-reduction and explicit substitutions 8th international conference on Programming Languages: Implementations, Logics and

"... Abstract. Extending the -calculus with either explicit substitution or generalised reduction has been the subject of extensive research recently which still has many open problems. Due to this reason, the properties of a calculus combining both generalised reduction and explicit substitu-tions have ..."

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Abstract. Extending the -calculus with either explicit substitution or generalised reduction has been the subject of extensive research recently which still has many open problems. Due to this reason, the properties of a calculus combining both generalised reduction and explicit substitu-tions have never been studied. This paper presents such a calculus sg and shows that it is a desirable extension of the -calculus. In partic-ular, we show that sg preserves strong normalisation, is sound and it simulates classical -reduction. Furthermore, we study the simply typed -calculus extended with both generalised reduction and explicit substi-tution and show that well-typed terms are strongly normalising and that other properties such as subtyping and subject reduction hold.

### An Approximation of Reductional Equivalence

"... We define an equivalence relation on -terms called shuffle-equivalence which attempts to capture the notion of reductional equivalence on strongly normalizing terms. The shuffleequivalence classes are shown to divide the classes of fi-equal terms into smaller ones consisting of terms with similar re ..."

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We define an equivalence relation on -terms called shuffle-equivalence which attempts to capture the notion of reductional equivalence on strongly normalizing terms. The shuffleequivalence classes are shown to divide the classes of fi-equal terms into smaller ones consisting of terms with similar reduction behaviour. We refine fi-reduction from a relation on terms to a relation on shuffle-equivalence classes, called shuffle-reduction, and show that this refinement captures existing generalisations of fi-reduction. Shuffle-reduction moreover, apart from allowing one to make more redexes visible and to contract these newly visible redexes, enables one to have more freedom in choosing the reduction path of a term, which can result in smaller terms along the reduction path if a clever reduction strategy is used. This can benefit both programming languages and theorem provers since this flexibility and freedom in chosing reduction paths can be exploited to produce the shortest program evalu...

### Weak Orthogonality Implies Confluence: the Higher-Order Case

- In: Proc. of the 3 rd International Conference on Logical Foundations of Computer Science, LFCS'94, A. Nerode and Yu.V. Matiyasevich, eds., Springer LNCS
, 1994

"... In this paper we prove confluence for weakly orthogonal Higher-Order Rewriting Systems. This generalises all the known `confluence by orthogonality' results. AMS Subject Classification (1991): 68Q50 CR Subject Classification (1991): F.4.1, F.3.3 Keywords & Phrases: higher-order rewriting ..."

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In this paper we prove confluence for weakly orthogonal Higher-Order Rewriting Systems. This generalises all the known `confluence by orthogonality' results. AMS Subject Classification (1991): 68Q50 CR Subject Classification (1991): F.4.1, F.3.3 Keywords & Phrases: higher-order rewriting, weak orthogonality, confluence. Note: Most of the research of the first author has been carried out during his employment at the Vrije Universiteit, Amsterdam, The Netherlands. The research of the second author is supported by NWO/SION project 612-316-606. 1. Introduction This paper deals with higher-order term rewriting. Since our approach of higher-order term rewriting is different from the usual one, both in respect to the concept of `higher-order' and to the notion of `term rewriting', we first comment on our approach and the terminology used, before stating the general confluence result. term rewriting. In term rewriting as usually defined (see e.g. [DJ89, Klo92, Klo80, Nip91]) rewrite steps...

### De Bruijn's syntax and reductional equivalence of λ-terms

, 2001

"... In this paper, a notation inuenced by de Bruijn's syntax of the λ-calculus is used to describe canonical forms of terms and an equivalence relation which divides terms into classes according to their reductional behaviour. We show that this notation helps describe canonical forms more elegantly ..."

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In this paper, a notation inuenced by de Bruijn's syntax of the λ-calculus is used to describe canonical forms of terms and an equivalence relation which divides terms into classes according to their reductional behaviour. We show that this notation helps describe canonical forms more elegantly than the classical notation and we establish the desirable properties of our reduction modulo equivalence classes rather than single terms. Finally, we extend the cube consisting of eight type systems with class reduction and show that this extension satis es all the desirable properties of type systems.

### De Bruijn's syntax and reductional behaviour of λ-terms

"... In this paper, a notation influenced by de Bruijn's syntax of the λ-calculus is used to describe canonical forms of terms and an equivalence relation which divides terms into classes according to their reductional behaviour. We show that this notation helps describe canonical forms more eleg ..."

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In this paper, a notation influenced by de Bruijn's syntax of the λ-calculus is used to describe canonical forms of terms and an equivalence relation which divides terms into classes according to their reductional behaviour. We show that this notation helps describe canonical forms more elegantly than the classical notation. We de ne reduction modulo equivalence classes of terms up to the permutation of redexes in canonical forms and show that this reduction contains other notions of reductions in the literature including the -reduction of Regnier. We establish all the desirable properties of our reduction modulo equivalence classes. Then, we give two extensions of the Barendregt cube, one with the - reduction of Regnier and the other with our class reduction and show that the subject reduction property fails in each of these extensions. We then show that adding de nitions in the contexts of type derivations, enables each of these extensions to satisfy all the desirable properties of type systems, including subject reduction and strong normalisation.