Categorical combinators [12, 21, 43] and more recently oe-calculus [1, 23], have been introduced to provide an explicit treatment of substitutions in the -calculus. We reintroduce here the ingredients of these calculi in a self-contained and stepwise way, with a special emphasis on confluence properties. The main new results of the paper w.r.t. [12, 21, 1, 23] are the following: 1. We present a confluent weak calculus of substitutions, where no variable clashes can be feared. 2. We solve a conjecture raised in [1]: oe-calculus is not confluent (it is confluent on ground terms only). This unfortunate result is "repaired" by presenting a confluent version of oe-calculus, named the Env-calculus in [23], called here the confluent oe-calculus.

Combinatory Reduction Systems, or CRSs for short, were designed to combine the usual first-order format of term rewriting with the presence of bound variables as in pure -calculus and various typed -calculi. Bound variables are also present in many other rewrite systems, such as systems with simplification rules for proof normalization. The original idea of CRSs is due to Aczel, who introduced a restricted class of CRSs and, under the assumption of orthogonality, proved confluence. Orthogonality means that the rules are non-ambiguous (no overlap leading to a critical pair) and left-linear (no global comparison of terms necessary). We introduce the class of orthogonal CRSs, illustrated with many examples, discuss its expressive power, and give an outline of a short proof of confluence. This proof is a direct generalization of Aczel's original proof, which is close to the well-known confluence proof for -calculus by Tait and Martin-Lof. There is a well-known connection between the para...

In a previous work (“Abstract Data Type Systems”, TCS 173(2), 1997), the last two authors presented a combined language made of a (strongly normalizing) algebraic rewrite system and a typed λ-calculus enriched by pattern-matching definitions following a certain format, called the “General Schema”, which generalizes the usual recursor definitions for natural numbers and similar “basic inductive types”. This combined language was shown to be strongly normalizing. The purpose of this paper is to reformulate and extend the General Schema in order to make it easily extensible, to capture a more general class of inductive types, called “strictly positive”, and to ease the strong normalization proof of the resulting system. This result provides a computation model for the combination of an algebraic specification language based on abstract data types and of a strongly typed functional language with strictly positive inductive types.

www.loria.fr/{~cirstea,~ckirchne,~lliquori} Abstract. In this paper we give a simple and uniform presentation of the rewriting calculus, also called Rho Calculus. In addition to its simplicity, this formulation explicitly allows us to encode complex structures such as lists, sets, and objects. We provide extensive examples of the calculus, and we focus on its ability to represent some object oriented calculi, namely the Lambda Calculus of Objects of Fisher, Honsell, and Mitchell, and the Object Calculus of Abadi and Cardelli. Furthermore, the calculus allows us to get object oriented constructions unreachable in other calculi. In summa, we intend to show that because of its matching ability, the Rho Calculus represents a lingua franca to naturally encode many paradigms of computations. This enlightens the capabilities of the rewriting calculus based language ELAN to be used as a logical as well as powerful semantical framework. 1

In this paper we present the algebraic--cube, an extension of Barendregt's -cube with first- and higherorder algebraic rewriting. We show that strong normalization is a modular property of all systems in the algebraic--cube, provided that the first-order rewrite rules are non-duplicating and the higher-order rules satisfy the general schema of Jouannaud and Okada. This result is proven for the algebraic extension of the Calculus of Constructions, which contains all the systems of the algebraic--cube. We also prove that local confluence is a modular property of all the systems in the algebraic--cube, provided that the higher-order rules do not introduce critical pairs. This property and the strong normalization result imply the modularity of confluence. 1 Introduction Many different computational models have been developed and studied by theoretical computer scientists. One of the main motivations for the development This research was partially supported by ESPRIT Basic Research Act...

Abstract. In a previous work, we proved that an important part of the Calculus of Inductive Constructions (CIC), the basis of the Coq proof assistant, can be seen as a Calculus of Algebraic Constructions (CAC), an extension of the Calculus of Constructions with functions and predicates defined by higher-order rewrite rules. In this paper, we prove that almost all CIC can be seen as a CAC, and that it can be further extended with non-strictly positive types and inductive-recursive types together with non-free constructors and pattern-matching on defined symbols. 1.

We investigate the system obtained by adding an algebraic rewriting system R to an untyped lambda calculus in which terms are formed using the function symbols from R as constants. On certain classes of terms, called here "stable", we prove that the resulting calculus is confluent if R is confluent, and terminating if R is terminating. The termination result has the corresponding theorems for several typed calculi as corollaries. The proof of the confluence result suggests a general method for proving confluence of typed β reduction plus rewriting; we sketch the application to the polymorphic lambda calculus.

The ρ-calculus permits to express in a uniform and simple way firstorder rewriting, λ-calculus and non-deterministic computations as well as their combination. In this paper, we present the main components of the ρ-calculus and we give a full first-order presentation of this rewriting calculus using an explicit substitution setting, called ρσ, that generalizes the λσ-calculus. The basic properties of the non-explicit and explicit substitution versions are presented. We then detail how to use the ρ-calculus to give an operational semantics to the rewrite rules of the ELAN language. 1

The last few years have seen the development of a new calculus which can be considered as an outcome of the last decade of various researches on (higher order) term rewriting systems, and lambda calculi. In the Rewriting Calculus (or Rho Calculus, ρCal), algebraic rules are considered as sophisticated forms of “lambda terms with patterns”, and rule applications as lambda applications with pattern matching facilities. The calculus can be customized to work modulo sophisticated theories, like commutativity, associativity, associativity-commutativity, etc. This allows us to encode complex structures such as list, sets, and more generally objects. The calculus can either be presented “à la Curry ” or “à la Church ” without sacrificing readability and without complicating too much the metatheory. Many static type systems can be easily plugged-in on top of the calculus in the spirit of the rich type-oriented literature. The Rewriting Calculus could represent a lingua franca to encode many paradigms of computations together with a formal basis used to build powerful theorem provers based on lambda calculus and efficient rewriting, and a step towards new proof engines based on the Curry-Howard isomorphism. 1

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