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21
Inductive Data Type Systems
 THEORETICAL COMPUTER SCIENCE
, 1997
"... 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 λcalculus enriched by patternmatching definitions following a certain format, called the “General Schema”, w ..."
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Cited by 44 (10 self)
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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 λcalculus enriched by patternmatching 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.
Matching Power
 Proceedings of RTA’2001, Lecture Notes in Computer Science, Utrecht (The Netherlands
, 2001
"... 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 pr ..."
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Cited by 30 (20 self)
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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
The Calculus of Algebraic Constructions
 In Proc. of the 10th Int. Conf. on Rewriting Techniques and Applications, LNCS 1631
, 1999
"... 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 hi ..."
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Cited by 27 (10 self)
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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 higherorder 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 nonstrictly positive types and inductiverecursive types together with nonfree constructors and patternmatching on defined symbols. 1.
Modularity of Strong Normalization and Confluence in the algebraiclambdacube
, 1994
"... In this paper we present the algebraiccube, 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 algebraiccube, provided that the firstorder rewrite rules are nonduplicating and the hig ..."
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Cited by 25 (7 self)
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In this paper we present the algebraiccube, 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 algebraiccube, provided that the firstorder rewrite rules are nonduplicating and the higherorder 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 algebraiccube. We also prove that local confluence is a modular property of all the systems in the algebraiccube, provided that the higherorder 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...
Rewriting calculus with(out) types
 Proceedings of the fourth workshop on rewriting logic and applications
, 2002
"... 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 sophisticat ..."
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Cited by 22 (13 self)
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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, associativitycommutativity, 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 pluggedin on top of the calculus in the spirit of the rich typeoriented 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 CurryHoward isomorphism. 1
Combining HigherOrder and FirstOrder Computation Using ρcalculus: Towards a Semantics of ELAN
 In Frontiers of Combining Systems 2
, 1999
"... The ρcalculus permits to express in a uniform and simple way firstorder rewriting, λcalculus and nondeterministic computations as well as their combination. In this paper, we present the main components of the ρcalculus and we give a full firstorder presentation of this rewriting calculus using ..."
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Cited by 20 (9 self)
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The ρcalculus permits to express in a uniform and simple way firstorder rewriting, λcalculus and nondeterministic computations as well as their combination. In this paper, we present the main components of the ρcalculus and we give a full firstorder presentation of this rewriting calculus using an explicit substitution setting, called ρσ, that generalizes the λσcalculus. The basic properties of the nonexplicit and explicit substitution versions are presented. We then detail how to use the ρcalculus to give an operational semantics to the rewrite rules of the ELAN language. 1
Inductive types in the calculus of algebraic constructions
 FUNDAMENTA INFORMATICAE 65(12) (2005) 61–86 JOURNAL VERSION OF TLCA’03
, 2005
"... In a previous work, we proved that almost all of the Calculus of Inductive Constructions (CIC), the basis of the proof assistant Coq, can be seen as a Calculus of Algebraic Constructions (CAC), an extension of the Calculus of Constructions with functions and predicates defined by higherorder rewrit ..."
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Cited by 15 (4 self)
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In a previous work, we proved that almost all of the Calculus of Inductive Constructions (CIC), the basis of the proof assistant Coq, can be seen as a Calculus of Algebraic Constructions (CAC), an extension of the Calculus of Constructions with functions and predicates defined by higherorder rewrite rules. In this paper, we prove that CIC as a whole can be seen as a CAC, and that it can be extended with nonstrictly positive types and inductiverecursive types together with nonfree constructors and patternmatching on defined symbols.
Termination and confluence of higherorder rewrite systems
 In Proc. RTA ’00, volume 1833 of LNCS
, 2000
"... Abstract: In the last twenty years, several approaches to higherorder rewriting have been proposed, among which Klop’s Combinatory Rewrite Systems (CRSs), Nipkow’s Higherorder Rewrite Systems (HRSs) and Jouannaud and Okada’s higherorder algebraic specification languages, of which only the last on ..."
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Cited by 14 (8 self)
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Abstract: In the last twenty years, several approaches to higherorder rewriting have been proposed, among which Klop’s Combinatory Rewrite Systems (CRSs), Nipkow’s Higherorder Rewrite Systems (HRSs) and Jouannaud and Okada’s higherorder algebraic specification languages, of which only the last one considers typed terms. The later approach has been extended by Jouannaud, Okada and the present author into Inductive Data Type Systems (IDTSs). In this paper, we extend IDTSs with the CRS higherorder patternmatching mechanism, resulting in simplytyped CRSs. Then, we show how the termination criterion developed for IDTSs with firstorder patternmatching, called the General Schema, can be extended so as to prove the strong normalization of IDTSs with higherorder patternmatching. Next, we compare the unified approach with HRSs. We first prove that the extended General Schema can also be applied to HRSs. Second, we show how Nipkow’s higherorder critical pair analysis technique for proving local confluence can be applied to IDTSs. 1
A Combinatory Logic Approach to Higherorder Eunification
 in Proceedings of the Eleventh International Conference on Automated Deduction, SpringerVerlag LNAI 607
, 1992
"... Let E be a firstorder equational theory. A translation of typed higherorder Eunification problems into a typed combinatory logic framework is presented and justified. The case in which E admits presentation as a convergent term rewriting system is treated in detail: in this situation, a modifi ..."
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Cited by 9 (3 self)
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Let E be a firstorder equational theory. A translation of typed higherorder Eunification problems into a typed combinatory logic framework is presented and justified. The case in which E admits presentation as a convergent term rewriting system is treated in detail: in this situation, a modification of ordinary narrowing is shown to be a complete method for enumerating higherorder Eunifiers. In fact, we treat a more general problem, in which the types of terms contain type variables. 1 Introduction Investigation of the interaction between firstorder and higherorder equational reasoning has emerged as an active line of research. The collective import of a recent series of papers, originating with [Bre88] and including (among others) [Bar90], [BG91a], [BG91b], [Dou92], [JO91] and [Oka89], is that when various typed calculi are enriched by firstorder equational theories, the validity problem is wellbehaved, and furthermore that the respective computational approaches to ...
On the Power of Simple Diagrams
, 1996
"... . In this paper we focus on a set of abstract lemmas that are easy to apply and turn out to be quite valuable in order to establish confluence and/or normalization modularly, especially when adding rewriting rules for extensional equalities to various calculi. We show the usefulness of the lemmas by ..."
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Cited by 9 (3 self)
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. In this paper we focus on a set of abstract lemmas that are easy to apply and turn out to be quite valuable in order to establish confluence and/or normalization modularly, especially when adding rewriting rules for extensional equalities to various calculi. We show the usefulness of the lemmas by applying them to various systems, ranging from simply typed lambda calculus to higher order lambda calculi, for which we can establish systematically confluence and/or normalization (or decidability of equality) in a simple way. Many result are new, but we also discuss systems for which our technique allows to provide a much simpler proof than what can be found in the literature. 1 Introduction During a recent investigation of confluence and normalization properties of polymorphic lambda calculus with an expansive version of the # rule, we came across a nice lemma that gives a simple but quite powerful sufficient condition to check the Church Rosser property for a compound rewriting system...