Automated Certification of Implicit Induction Proofs

Stratulat, Sorin; Demange, Vincent

doi:10.1007/978-3-642-25379-9_5

Cited by 5 publications

(11 citation statements)

References 15 publications

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“…Stratulat and Demange (Stratulat, 2010;Stratulat and Demange, 2011) succeeded to formalize the implicit induction reasoning directly into Coq scripts. Their goal was to automatically certify implicit induction proofs generated by native implicit induction provers as SPIKE.…”

Section: Related Workmentioning

confidence: 99%

Mechanically certifying formula-based Noetherian induction reasoning

Stratulat

2017

Journal of Symbolic Computation

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In first-order logic, the formula-based instances of the Noetherian induction principle allow to perform effectively simultaneous, mutual and lazy induction reasoning. Compared to the termbased Noetherian induction instances, they are not directly supported by the current proof assistants. We provide general formal tools for certifying formula-based Noetherian induction proofs by the Coq proof assistant, then show how to apply them to certify proofs of conjectures about conditional specifications, built with: i) a reductive rewrite-based induction system, and ii) a reductive-free cyclic induction system. The generation of reductive proofs and their certification process can be easily automatised, without requiring additional definitions or proof transformations, but may involve many ordering constraints to be checked during the certification process. On the other hand, the reductive-free proofs generate fewer ordering constraints, may involve more general specifications and the certification process is more effective. However, their proof generation is less automatic and the generated proofs need to be normalised before being certified. The methodology for certifying reductive-free cyclic induction proofs related to conditional specifications extends a previous approach used for implicit induction proofs and it can be easily adapted to certify any formula-based Noetherian induction reasoning. In practice, the methodology has been implemented to automatically certify implicit induction proofs generated by the SPIKE theorem prover as well as reductive-free cyclic proofs built by the same system but in a less automatic way.

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Section: Related Workmentioning

confidence: 99%

Mechanically certifying formula-based Noetherian induction reasoning

Stratulat

2017

Journal of Symbolic Computation

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“…In order to ease the translation from Spike proofs to Coq scripts process, the generated Spike specifications may contain inline Coq scripts, for example declaring the signature of the function symbols using Parameter. For our example, the Spike specification starts with: Translating the Spike proof into Coq script The Spike prover is automatically executed on the generated specification using a mode that can produce Coq script from a proof, as shown in [20,22]. In the following, we only recall the key steps of the translation process.…”

Section: The Spike Tactic Comes Into Four Variantsmentioning

confidence: 99%

“…We have also used the Spike tactic and its extensions for other examples treated in previous works [20,22]. Table 1 illustrates the number of lemmas (i.e., previously proved conjectures), hypotheses (i.e., not yet proved conjectures), the employed tactic and whether it has used parameters or not, as well as the execution time for some successfully proved conjectures involved in the validation of a telecommunications protocol [17].…”

Section: Statisticsmentioning

confidence: 99%

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Performing Implicit Induction Reasoning with Certifying Proof Environments

Henaien

Stratulat

2013

Electron. Proc. Theor. Comput. Sci.

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Largely adopted by proof assistants, the conventional induction methods based on explicit induction schemas are non-reductive and local, at schema level. On the other hand, the implicit induction methods used by automated theorem provers allow for lazy and mutual induction reasoning. In this paper, we present a new tactic for the Coq proof assistant able to perform automatically implicit induction reasoning. By using an automatic black-box approach, conjectures intended to be manually proved by the certifying proof environment that integrates Coq are proved instead by the Spike implicit induction theorem prover. The resulting proofs are translated afterwards into certified Coq scripts

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“…φ 1 is rewritten by the axioms to the tautology φ 1 : T rue = T rue, then deleted by DedNat. φ 2 1 is rewritten by the lemmas In the following, we will show how the certification process of implicit induction proofs [53] can be improved by representing them as D-proofs. Even if the implicit induction proofs are generated automatically by inference systems that implicitly check the ordering constraints, the certification process should explicitly validate every single proof step.…”

Section: Other Examplesmentioning

confidence: 99%

A Unified View of Induction Reasoning for First-Order Logic

Stratulat¹

EPiC Series in Computing

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Induction is a powerful proof technique adapted to reason on setswith an unbounded number of elements. In a first-order setting, twodifferent methods are distinguished: the conventional induction,based on explicit induction schemas, and the implicit induction,based on reductive procedures. We propose a new cycle-basedinduction method that keeps their best features, i.e. i) performslazy induction, ii) naturally fits for mutual induction, and iii) isfree of reductive constraints. The heart of the method is a proofstrategy that identifies in the proof script the subset of formulascontributing to validate the application of induction hypotheses.The conventional and implicit induction are particular cases of ourmethod.

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Automated Certification of Implicit Induction Proofs

Cited by 5 publications

References 15 publications

Mechanically certifying formula-based Noetherian induction reasoning

Mechanically certifying formula-based Noetherian induction reasoning

Performing Implicit Induction Reasoning with Certifying Proof Environments

A Unified View of Induction Reasoning for First-Order Logic

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