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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 13 (3 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 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
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 11 (1 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.
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 11 (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
Coq Modulo Theory
, 2010
"... Abstract. Coq Modulo Theory (CoqMT) is an extension of the Coq proof assistant incorporating, in its computational mechanism, validity entailment for userdefined firstorder equational theories. Such a mechanism strictly enriches the system (more terms are typable), eases the use of dependent types ..."
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Cited by 5 (1 self)
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Abstract. Coq Modulo Theory (CoqMT) is an extension of the Coq proof assistant incorporating, in its computational mechanism, validity entailment for userdefined firstorder equational theories. Such a mechanism strictly enriches the system (more terms are typable), eases the use of dependent types and provides more automation during the development of proofs. CoqMT improves over the Calculus of Congruent Inductive Constructions by getting rid of various restrictions and simplifying the typechecking algorithm and the integration of firstorder decision procedures. We present here CoqMT, and outline its metatheoretical study. We also give a brief description of our CoqMT implementation. 1
Higherorder termination: From kruskal to computability
 In 13th International Conf. on Logic for Programming, Artificial Intelligence, and Reasoning. Lecture Notes in Computer Science
"... Termination is a major question in both logic and computer science. In logic, termination is at the heart of proof theory where it is usually called strong normalization (of cut elimination). In computer science, termination has always been an important issue for showing programs correct. ..."
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Cited by 4 (2 self)
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Termination is a major question in both logic and computer science. In logic, termination is at the heart of proof theory where it is usually called strong normalization (of cut elimination). In computer science, termination has always been an important issue for showing programs correct.
A Semantic Normalization Proof for Inductive Types
, 2008
"... Abstract. Semantics methods have been used to prove cut elimination theorems for a long time. It is only recently that they have been extended to prove strong normalization results. For instance using the notion of superconsistency that is a semantic criterion for theories expressed in deduction mo ..."
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Abstract. Semantics methods have been used to prove cut elimination theorems for a long time. It is only recently that they have been extended to prove strong normalization results. For instance using the notion of superconsistency that is a semantic criterion for theories expressed in deduction modulo implying strong normalization. However, the strong normalization of System T has always been reluctant to such semantic methods. In this paper we give a semantic normalization proof of system T using the super consistency of some theory. We then extend the result to every strictly positive inductive type and discuss the extension to predicate logic. 1