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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.
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...
Rank 2 Intersection Type Assignment in Term Rewriting Systems
, 1996
"... A notion of type assignment on Curryfied Term Rewriting Systems is introduced that uses Intersection Types of Rank 2, and in which all function symbols are assumed to have a type. Type assignment will consist of specifying derivation rules that describe how types can be assigned to terms, using the ..."
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Cited by 23 (15 self)
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A notion of type assignment on Curryfied Term Rewriting Systems is introduced that uses Intersection Types of Rank 2, and in which all function symbols are assumed to have a type. Type assignment will consist of specifying derivation rules that describe how types can be assigned to terms, using the types of function symbols. Using a modified unification procedure, for each term the principal pair (of basis and type) will be defined in the following sense: from these all admissible pairs can be generated by chains of operations on pairs, consisting of the operations substitution, copying, and weakening. In general, given an arbitrary typeable CuTRS, the subject reduction property does not hold. Using the principal type for the lefthand side of a rewrite rule, a sufficient and decidable condition will be formulated that typeable rewrite rules should satisfy in order to obtain this property. Introduction In the recent years, several paradigms have been investigated for the implementatio...
The Computability Path Ordering: the End of a Quest
"... Abstract. In this paper, we first briefly survey automated termination proof methods for higherorder calculi. We then concentrate on the higherorder recursive path ordering, for which we provide an improved definition, the Computability Path Ordering. This new definition appears indeed to capture ..."
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Cited by 13 (2 self)
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Abstract. In this paper, we first briefly survey automated termination proof methods for higherorder calculi. We then concentrate on the higherorder recursive path ordering, for which we provide an improved definition, the Computability Path Ordering. This new definition appears indeed to capture the essence of computability arguments à la Tait and Girard, therefore explaining the name of the improved ordering. 1
Strong Normalization of Typeable Rewrite Systems
, 1994
"... This paper studies termination properties of rewrite systems that are typeable using intersection types. It introduces a notion of partial type assignment on Curryfied Term Rewrite Systems, that consists of assigning intersection types to function symbols, and specifying the way in which types can b ..."
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Cited by 12 (11 self)
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This paper studies termination properties of rewrite systems that are typeable using intersection types. It introduces a notion of partial type assignment on Curryfied Term Rewrite Systems, that consists of assigning intersection types to function symbols, and specifying the way in which types can be assigned to nodes and edges between nodes in the tree representation of terms. Two operations on types are specified that are used to define type assignment on terms and rewrite rules, and are proven to be sound on both terms and rewrite rules. Using a more liberal approach to recursion, a general scheme for recursive definitions is presented, that generalizes primitive recursion, but has full Turingmachine computational power. It will be proved that, for all systems that satisfy this scheme, every typeable term is strongly normalizable. Introduction Most functional programming languages, like Miranda [23] or ML [19] for instance, although implemented through an extended Lambda Calculus ...
Modularity of Strong Normalization in the Algebraicλcube
, 1996
"... In this paper we present the algebraicλcube, an extension of Barendregt's λcube with first and higherorder algebraic rewriting. We show that strong normalization is a modular property of all systems in the algebraicλcube, provided that the firstorder rewrite rules are nonduplicating and the ..."
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Cited by 8 (2 self)
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In this paper we present the algebraicλcube, an extension of Barendregt's λcube with first and higherorder algebraic rewriting. We show that strong normalization is a modular property of all systems in the algebraicλcube, 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 algebraicλcube. We also prove that local confluence is a modular property of all the systems in the algebraicλcube, provided that the higherorder rules do not introduce critical pairs. This property and the strong normalization result imply the modularity of confluence.
Expanding the Cube
 Proc. nd FOSSACS 1999 (Amsterdam), volume 1578 of lncs
, 1999
"... . We prove strong normalization of fireduction+jexpansion for the Calculus of Constructions, thus providing the first strong normalization result for fireduction+jexpansion in calculi of dependent types and answering in the affirmative a conjecture by Di Cosmo and Ghani. In addition, we prove st ..."
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Cited by 4 (2 self)
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. We prove strong normalization of fireduction+jexpansion for the Calculus of Constructions, thus providing the first strong normalization result for fireduction+jexpansion in calculi of dependent types and answering in the affirmative a conjecture by Di Cosmo and Ghani. In addition, we prove strong normalization of fireduction+jexpansion+algebraic reduction for the Algebraic Calculus of Constructions, which extends the Calculus of Constructions with firstorder termrewriting systems. The latter result, which requires the termrewriting system to be nonduplicating, partially answers in the affirmative another conjecture by Di Cosmo and Ghani. 1 Introduction Extensionality, as embodied in jconversion x:A: M x = M if x 62 FV(M) is a basic notion in calculus and type theory. Traditionally, jconversion has been oriented from left to right, thus leading to jreduction. Recently, several authors have advocated a different computational interpretation, in which jconversion is or...
Termination of algebraic type systems: the syntactic approach
 Proceedings of ALP '97  HOA '97, volume 1298 of Lecture Notes in Computer Science
, 1997
"... Combinations of type theory and rewriting are of obvious interest for the study of higherorder programming and program transformation with algebraic data types specifications; more recently, they also found applications in proofchecking. ..."
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Cited by 1 (1 self)
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Combinations of type theory and rewriting are of obvious interest for the study of higherorder programming and program transformation with algebraic data types specifications; more recently, they also found applications in proofchecking.
Inductive Data Type Systems: Strong Normalization
, 1997
"... : This paper is concerned with the foundations of Inductive Data Type Systems, an extension of pure type systems by inductive data types. IDTS generalize (inductive) types equipped with primitive recursion of highertype, by providing definitions of functions by pattern matching of a form which is g ..."
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Cited by 1 (0 self)
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: This paper is concerned with the foundations of Inductive Data Type Systems, an extension of pure type systems by inductive data types. IDTS generalize (inductive) types equipped with primitive recursion of highertype, by providing definitions of functions by pattern matching of a form which is general enough to capture recursor definitions for strictly positive inductive types. IDTS also generalize the firstorder framework of abstract data types by providing function types and higherorder rewrite rules. The main result of the paper is the strong normalization property of inductive data type systems, in case of a simple type discipline. 1 Introduction The recent years have seen a proliferation of formalisms for programming and proof development. The present paper is a contribution towards their unification in the lines of [6, 17]. Our goal is to argue in favor of a language which borrows from algebraic languages like OBJ their structuring mechanisms as well as functional definit...
Variants of the Basic Calculus of Constructions
, 2004
"... In this paper, a number of different versions of the basic calculus of constructions that have appeared in the literature are compared and the exact relationships between them are determined. The biggest differences between versions are those between the original version of Coquand and the version i ..."
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Cited by 1 (0 self)
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In this paper, a number of different versions of the basic calculus of constructions that have appeared in the literature are compared and the exact relationships between them are determined. The biggest differences between versions are those between the original version of Coquand and the version in early papers on the subject by Seldin. None of these results is very deep, but it seems useful to collect them in one place.