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149
Definitions by Rewriting in the Calculus of Constructions
, 2001
"... The main novelty of this paper is to consider an extension of the Calculus of Constructions where predicates can be defined with a general form of rewrite rules. ..."
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Cited by 50 (7 self)
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The main novelty of this paper is to consider an extension of the Calculus of Constructions where predicates can be defined with a general form of rewrite rules.
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 33 (11 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.
Termination Checking with Types
, 1999
"... The paradigm of typebased termination is explored for functional programming with recursive data types. The article introduces , a lambdacalculus with recursion, inductive types, subtyping and bounded quanti cation. Decorated type variables representing approximations of inductive types ..."
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Cited by 36 (6 self)
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The paradigm of typebased termination is explored for functional programming with recursive data types. The article introduces , a lambdacalculus with recursion, inductive types, subtyping and bounded quanti cation. Decorated type variables representing approximations of inductive types are used to track the size of function arguments and return values. The system is shown to be type safe and strongly normalizing. The main novelty is a bidirectional type checking algorithm whose soundness is established formally.
Polymorphic higherorder recursive path orderings
 Journal of the ACM
, 2005
"... This paper extends the termination proof techniques based on reduction orderings to a higherorder setting, by defining a family of recursive path orderings for terms of a typed lambdacalculus generated by a signature of polymorphic higherorder function symbols. These relations can be generated fro ..."
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Cited by 25 (6 self)
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This paper extends the termination proof techniques based on reduction orderings to a higherorder setting, by defining a family of recursive path orderings for terms of a typed lambdacalculus generated by a signature of polymorphic higherorder function symbols. These relations can be generated from two given wellfounded orderings, on the function symbols and on the type constructors. The obtained orderings on terms are wellfounded, monotonic, stable under substitution and include βreductions. They can be used to prove the strong normalization property of higherorder calculi in which constants can be defined by higherorder rewrite rules using firstorder pattern matching. For example, the polymorphic version of Gödel’s recursor for the natural numbers is easily oriented. And indeed, our ordering is polymorphic, in the sense that a single comparison allows to prove the termination property of all monomorphic instances of a polymorphic rewrite rule. Many nontrivial examples are given which exemplify the expressive power of these orderings. All have been checked by our implementation. This paper is an extended and improved version of [Jouannaud and Rubio 1999]. Polymorphic algebras have been made more expressive than in our previous framework. The intuitive notion of a polymorphic higherorder ordering has now been made precise. The higherorder recursive
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 22 (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
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 19 (6 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
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 19 (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.
From formal proofs to mathematical proofs: A safe, incremental way for building in firstorder decision procedures
 In TCS 2008: 5th IFIP International Conference on Theoretical Computer Science
, 2008
"... (CIC) on which the proof assistant Coq is based: the Calculus of Congruent Inductive Constructions, which truly extends CIC by building in arbitrary firstorder decision procedures: deduction is still in charge of the CIC kernel, while computation is outsourced to dedicated firstorder decision proc ..."
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Cited by 14 (0 self)
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(CIC) on which the proof assistant Coq is based: the Calculus of Congruent Inductive Constructions, which truly extends CIC by building in arbitrary firstorder decision procedures: deduction is still in charge of the CIC kernel, while computation is outsourced to dedicated firstorder decision procedures that can be taken from the shelves provided they deliver a proof certificate. The soundness of the whole system becomes an incremental property following from the soundness of the certificate checkers and that of the kernel. A detailed example shows that the resulting style of proofs becomes closer to that of the working mathematician. 1
Canonical Abstract Syntax Trees
 WRLA 2006
, 2006
"... This paper presents GOM, a language for describing abstract syntax trees and generating a Java implementation for those trees. GOM includes features allowing the user to specify and modify the interface of the data structure. These features provide in particular the capability to maintain the intern ..."
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Cited by 17 (6 self)
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This paper presents GOM, a language for describing abstract syntax trees and generating a Java implementation for those trees. GOM includes features allowing the user to specify and modify the interface of the data structure. These features provide in particular the capability to maintain the internal representation of data in canonical form with respect to a rewrite system. This explicitly guarantees that the client program only manipulates normal forms for this rewrite system, a feature which is only implicitly used in many implementations.
Building decision procedures in the calculus of inductive constructions
 of Lecture Notes in Computer Science
, 2007
"... It is commonly agreed that the success of future proof assistants will rely on their ability to incorporate computations within deduction in order to mimic the mathematician when replacing the proof of a proposition P by the proof of an equivalent proposition P ’ obtained from P thanks to possibly c ..."
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Cited by 14 (3 self)
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It is commonly agreed that the success of future proof assistants will rely on their ability to incorporate computations within deduction in order to mimic the mathematician when replacing the proof of a proposition P by the proof of an equivalent proposition P ’ obtained from P thanks to possibly complex calculations. In this paper, we investigate a new version of the calculus of inductive constructions which incorporates arbitrary decision procedures into deduction via the conversion rule of the calculus. The novelty of the problem in the context of the calculus of inductive constructions lies in the fact that the computation mechanism varies along proofchecking: goals are sent to the decision procedure together with the set of user hypotheses available from the current context. Our main result shows that this extension of the calculus of constructions does not compromise its main properties: confluence, subject reduction, strong normalization and consistency are all preserved.
Results 1  10
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149