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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
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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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
Author manuscript, published in "Logic In Computer Science (LICS 2010) (2010)" Coq Modulo Theory
, 2010
"... Theorem provers like COQ [3] based on the CurryHoward isomorphism enjoy a mechanism which incorporates computations within deductions. This allows replacing the proof of a proposition by the proof of an equivalent proposition obtained from the former thanks to possibly complex computations. Adding ..."
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Theorem provers like COQ [3] based on the CurryHoward isomorphism enjoy a mechanism which incorporates computations within deductions. This allows replacing the proof of a proposition by the proof of an equivalent proposition obtained from the former thanks to possibly complex computations. Adding more power to this mechanism leads to a calculus which is more expressive (more terms are typable), which provides more automation (more deduction steps are hidden in computations) and, most importantly, eases the use of dependent data types in proof development. COQ was initially based on the Calculus of Constructions (CC) of Coquand and Huet [4], which is an impredicative type theory incorporating polymorphism, dependent types and type constructors. At that time, computations