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Embedding pure type systems in the lambdaPicalculus modulo
 TLCA
, 2007
"... The lambdaPicalculus allows to express proofs of minimal predicate logic. It can be extended, in a very simple way, by adding computation rules. This leads to the lambdaPicalculus modulo. We show in this paper that this simple extension is surprisingly expressive and, in particular, that all fu ..."
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Cited by 34 (9 self)
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The lambdaPicalculus allows to express proofs of minimal predicate logic. It can be extended, in a very simple way, by adding computation rules. This leads to the lambdaPicalculus modulo. We show in this paper that this simple extension is surprisingly expressive and, in particular, that all functional Pure Type Systems, such as the system F, or the Calculus of Constructions, can be embedded in it. And, moreover, that this embedding is conservative under termination hypothesis.
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
CoLoR: a Coq library on wellfounded rewrite relations and its application to the automated verification of termination certificates
, 2010
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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.
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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(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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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 Recursive Path Orderings à la carte
"... Introduction Rewrite rules are increasingly used in programming languages and logical systems, with two main goals: defining functions by pattern matching; describing rulebased decision procedures. Our ambition is to develop for the higherorder/type case the kind of semiautomated termination pro ..."
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Cited by 11 (2 self)
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Introduction Rewrite rules are increasingly used in programming languages and logical systems, with two main goals: defining functions by pattern matching; describing rulebased decision procedures. Our ambition is to develop for the higherorder/type case the kind of semiautomated termination proof techniques that are available for the firstorder case, of which the most popular one is the recursive path ordering [4]. At LICS'99, we contributed to this program with a reduction ordering for typed higherorder terms which conservatively extends Dershowitz's recursive path ordering for firstorder terms. In the latter, the precedence rule allows to decrease from the term s = f(s 1 ; : : : ; s n ) to the term g(t 1 ; : : : ; t n ), provided that (i) f is bigger than g in the given precedence on function symbols, and (ii) s is bigger than every t i . For typing reasons, in our ordering the latter condition becomes: (ii) for every t i , either s is bigger than t i or some s j is bigger t
On the strength of proofirrelevant type theories
 of Lecture Notes in Computer Science
, 2006
"... Vol. 4 (3:13) 2008, pp. 1–20 ..."
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Towards Rewriting in Coq
"... Equational reasoning in Coq is not straightforward. For a few years now there has been an ongoing research process towards adding rewriting to Coq. However, there are many research problems on this way. In this paper we give a coherent view of rewriting in Coq, we describe what is already done and w ..."
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Equational reasoning in Coq is not straightforward. For a few years now there has been an ongoing research process towards adding rewriting to Coq. However, there are many research problems on this way. In this paper we give a coherent view of rewriting in Coq, we describe what is already done and what remains to be done. We discuss such issues as strong normalization, confluence, logical consistency, completeness, modularity and extraction.
Typed Applicative Structures and Normalization by Evaluation for System F ω
"... Abstract. We present a normalizationbyevaluation (NbE) algorithm for System F ω with βηequality, the simplest impredicative type theory with computation on the type level. Values are kept abstract and requirements on values are kept to a minimum, allowing many different implementations of the alg ..."
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Abstract. We present a normalizationbyevaluation (NbE) algorithm for System F ω with βηequality, the simplest impredicative type theory with computation on the type level. Values are kept abstract and requirements on values are kept to a minimum, allowing many different implementations of the algorithm. The algorithm is verified through a general model construction using typed applicative structures, called type and object structures. Both soundness and completeness of NbE are conceived as an instance of a single fundamental theorem.