Translating HOL to Dedukti

Assaf, Ali; Burel, Guillaume

doi:10.4204/eptcs.186.8

Cited by 14 publications

(21 citation statements)

References 19 publications

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“…Through the Curry-Howard correspondence, Dedukti can be used as a proof-checker for a wide variety of logics [8,2,1]. It is commonly used to check proofs coming from the Deduction modulo provers Iprover Modulo [4] and Zenon Modulo [6].…”

Section: Deduktimentioning

confidence: 99%

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ML Pattern-Matching, Recursion, and Rewriting: From FoCaLiZe to Dedukti

Cauderlier

Dubois

2016

Theoretical Aspects of Computing – ICTAC 2016

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Abstract. The programming environment FoCaLiZe allows the user to specify, implement, and prove programs with the help of the theorem prover Zenon. In the actual version, those proofs are verified by Coq. In this paper we propose to extend the FoCaLiZe compiler by a backend to the Dedukti language in order to benefit from Zenon Modulo, an extension of Zenon for Deduction modulo. By doing so, FoCaLiZe can benefit from a technique for finding and verifying proofs more quickly. The paper focuses mainly on the process that overcomes the lack of local pattern-matching and recursive definitions in Dedukti.

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Section: Deduktimentioning

confidence: 99%

“…This work is also a first step in the direction of interoperability between FoCaLiZe and other proof languages translated to Dedukti [8,1,2]. This new compilation backend to Dedukti is based on the existing backend to Coq.…”

Section: Introductionmentioning

confidence: 99%

ML Pattern-Matching, Recursion, and Rewriting: From FoCaLiZe to Dedukti

Cauderlier

Dubois

2016

Theoretical Aspects of Computing – ICTAC 2016

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“…By doing so, we would be able to orient the theory used in the provers of the HOL family without using axioms, thus improving the performance of the translator Holide [1].…”

Section: Conclusion and Further Workmentioning

confidence: 99%

“…In particular, this is the case of one of the axioms of the theory used in the provers of the HOL family (HOL4, HOL Light, or even Isabelle/HOL). In the translation of proofs in the OpenTheory format [20] into proofs that can be checked by Dedukti [1], this axiom could not be easily presented as a rewriting rule, and should therefore remain as an axiom, losing partially the benefit of working modulo the theory. As we will see in Example 8, this axiom can be naturally presented as a conditional rewriting rule.…”

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confidence: 99%

Cut Admissibility by Saturation

Burel

2014

Lecture Notes in Computer Science

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Abstract. Deduction modulo is a framework in which theories are integrated into proof systems such as natural deduction or sequent calculus by presenting them using rewriting rules. When only terms are rewritten, cut admissibility in those systems is equivalent to the confluence of the rewriting system, as shown by Dowek, RTA 2003, LNCS 2706. This is no longer true when considering rewriting rules involving propositions. In this paper, we show that, in the same way that it is possible to recover confluence using Knuth-Bendix completion, one can regain cut admissibility in the general case using standard saturation techniques. This work relies on a view of proposition rewriting rules as oriented clauses, like term rewriting rules can be seen as oriented equations. This also leads us to introduce an extension of deduction modulo with conditional term rewriting rules.Whatever their origin, proofs rarely need to be search for without context: Program verification requires arithmetic, theories of lists or arrays, etc. Mathematical theorems are in general not proved in pure predicate logic. Consequently, even if (automated and interactive) proof systems have achieved a high degree of maturity, they need to be able to deal with theories in a efficient way. This explains the particular interest focused on the SMT (Satisfiability Modulo Theory) provers in the latter years. However, one of the drawbacks of the SMT approach is that the way theories are integrated is not completely generic, in the sense that each theory needs a special treatment.A more generic approach to integrate theories into a proof system was proposed by Dowek, Hardin and Kirchner [15]. In Deduction Modulo 1 , a theory is represented by a congruence over formulae, and proofs are search for modulo this congruence. In practice, this congruence is most often described as a rewriting system. However, using only term rewriting rules would not be enough to capture interesting theories. For instance, Vorobyov [24] showed that even quantifier-free Presburger arithmetic cannot be presented as a convergent term rewriting system. To overcome this, Deduction Modulo also deals with proposition rewriting rules, that rewrite atomic formulae into formulae. Thanks to this, it was possible to present many theories in Deduction Modulo : simple type theory (also known as higher-order logic) [14], arithmetic [17], set theory [16], B set theory [21], any pure type system, including the calculus of construction which is the 1 Although it may sound rather strange, the absence of subsequent to the term "modulo" follows the original works about this field.

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“…The LFSC system is an extension of the dependently typed λ-calculus with side-conditions and an implementation of it has successfully been used to check proofs coming from the SMT solvers CLSAT and CVC4 [48]. Deduction modulo [18] is another extension to dependently typed λ-terms in which rewriting is available: the Dedukti checker, based on that extension, has been successfully used to check proofs from such systems as Coq [9] and HOL [4]. In the domain of higher-order classical logic, the GAPT system [22] can check proofs given by sequent calculus, resolution, and expansion trees and allows for checking and transforming among proofs in those formats.…”

Section: Introductionmentioning

confidence: 99%

Translating Between Implicit and Explicit Versions of Proof

Blanco

Chihani

Miller

2017

Automated Deduction – CADE 26

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Abstract. The Foundational Proof Certificate (FPC) framework can be used to define the semantics of a wide range of proof evidence. For example, such definitions exist for a number of textbook proof systems as well as for the proof evidence output from some existing theorem proving systems. An important decision in designing a proof certificate format is the choice of how many details are to be placed within certificates. Formats with fewer details are smaller and easier for theorem provers to output but they require more sophistication from checkers since checking will involve some proof reconstruction. Conversely, certificate formats containing many details are larger but are checkable by less sophisticated checkers. Since the FPC framework is based on well-established proof theory principles, proof certificates can be manipulated in meaningful ways. In this paper, we illustrate how it is possible to automate moving from implicit to explicit (elaboration) and from explicit to implicit (distillation) proof evidence via the proof checking of a pair of proof certificates. Performing elaboration makes it possible to transform a proof certificate with details missing into a certificate packed with enough details so that a simple kernel (without support for proof reconstruction) can check the elaborated certificate. We illustrate how trust in only a single, simple checker of explicitly described proofs can be used to provide trust in a range of theorem provers employing a range of proof structures.

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Translating HOL to Dedukti

Cited by 14 publications

References 19 publications

ML Pattern-Matching, Recursion, and Rewriting: From FoCaLiZe to Dedukti

ML Pattern-Matching, Recursion, and Rewriting: From FoCaLiZe to Dedukti

Cut Admissibility by Saturation

Translating Between Implicit and Explicit Versions of Proof

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