HOCore in Coq

Maksimovic, Petar; Schmitt, Alan

doi:10.1007/978-3-319-22102-1_19

Cited by 14 publications

(14 citation statements)

References 26 publications

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“…Even worse, in logical frameworks with object-level constructors, so-called exotic terms can be derived." Similar problems claimed about HOAS can also be found in [72,85].…”

Section: Substitution Lemmas For Freesupporting

confidence: 77%

Mechanized Metatheory Revisited

Miller

2018

J Autom Reasoning

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HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Mechanized metatheory revisited Dale Miller To cite this version:Dale Miller. Mechanized metatheory revisited.Abstract When proof assistants and theorem provers implement the metatheory of logical systems, they must deal with a range of syntactic expressions (e.g., types, formulas, and proofs) that involve variable bindings. Since most mature proof assistants do not have built-in methods to treat bindings, they have been extended with various packages and libraries that allow them to encode such syntax using, for example, de Bruijn numerals. We put forward the argument that bindings are such an intimate aspect of the structure of expressions that they should be accounted for directly in the underlying programming language support for proof assistants and not via packages and libraries. We present an approach to designing programming languages and proof assistants that directly supports bindings in syntax. The roots of this approach can be found in the mobility of binders between term-level bindings, formula-level bindings (quantifiers), and proof-level bindings (eigenvariables). In particular, the combination of Church's approach to terms and formulas (found in his Simple Theory of Types) and Gentzen's approach to proofs (found in his sequent calculus) yields a framework for the interaction of bindings with a full range of logical connectives and quantifiers. We will also illustrate how that framework provides a direct and semantically clean treatment of computation and reasoning with syntax containing bindings. Some implemented systems, which support this intimate and built-in treatment of bindings, will be briefly described.Keywords mechanized metatheory, λ-tree syntax, mobility of binders 1 Metatheory and its mechanization Theorem proving-in both its interactive and automatic forms-has been applied in a wide range of domains. A frequent use of theorem provers is to formally establish correctness properties for specific programs: e.g., prove that a given program always terminates and correctly sorts a list or prove that a given loop satisfies a given invariant.A more niche domain to which theorem proving is applied is that of the metatheory of programming languages. In this domain, one takes a formal definition of a particular

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“…Even worse, in logical frameworks with object-level constructors, so-called exotic terms can be derived." Similar problems claimed about HOAS can also be found in [72,85].…”

Section: Substitution Lemmas For Freesupporting

confidence: 77%

Mechanized Metatheory Revisited

Miller

2018

J Autom Reasoning

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“…Even though bisimilarity can be defined using a coinductive datatype, we prefer to use the set theoretic approach, where two terms are bisimilar if there exists a bisimulation relating them. Not only it corresponds to Definition 2.2, but it is also more tractable in Coq [15]. In the following, test_proc, test_abs, and test_conc are notations representing the testing conditions of Definition 2.2 for each kind of agent ( ).…”

Section: Bisimilaritymentioning

confidence: 99%

“…To our knowledge, only two prior works [15,17] propose formalizations of higher-order process calculi. The calculus studied by Maksimović and Schmitt [15] is a sub-calculus of HOπ as it does not feature name restriction. Parrow et al [17] extend an existing formalization of the psicalculus to accommodate for higher-order communication.…”

Section: Related Work and Conclusionmentioning

confidence: 99%

“…If the first-order π -calculus has been formalized in various proof assistants, using different representations for binders (Gay [9] and Perera and Cheney [18] list some of them), only a few recent works propose a formalization of a higher-order calculus [15,17]. The semantics of the calculus of Parrow et al [17] is based on triggers and clauses to enable the execution of transmitted processes.…”

Section: Introductionmentioning

confidence: 99%

“…The semantics of the calculus of Parrow et al [17] is based on triggers and clauses to enable the execution of transmitted processes. Maksimović and Schmitt [15] have formalized a minimal higher-order calculus which lacks name restriction νa.P, an operator widely used in process calculi to restrict the scope of channel names.…”

Section: Introductionmentioning

confidence: 99%

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HOπ in Coq

Lenglet

Schmitt

2018

Proceedings of the 7th ACM SIGPLAN International Conference on Certified Programs and Proofs

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International audienceWe propose a formalization in Coq of HOπ , a process calculus where messages carry processes. Such a higher-order calculus features two very different kinds of binder: process input, similar to λ-abstraction, and name restriction, whose scope can be expanded by communication. We formalize strong context bisimilarity and prove it is compatible using Howe's method, based on several proof schemes we developed in a previous paper

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Trakhtenbrot’s Theorem in Coq

Kirst¹,

Larchey-Wendling²

2020

Automated Reasoning

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We study finite first-order satisfiability (FSAT) in the constructive setting of dependent type theory. Employing synthetic accounts of enumerability and decidability, we give a full classification of FSAT depending on the first-order signature of non-logical symbols. On the one hand, our development focuses on Trakhtenbrot's theorem, stating that FSAT is undecidable as soon as the signature contains an at least binary relation symbol. Our proof proceeds by a many-one reduction chain starting from the Post correspondence problem. On the other hand, we establish the decidability of FSAT for monadic first-order logic, i.e. where the signature only contains at most unary function and relation symbols, as well as the enumerability of FSAT for arbitrary enumerable signatures. All our results are mechanised in the framework of a growing Coq library of synthetic undecidability proofs.

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HOCore in Coq

Cited by 14 publications

References 26 publications

Mechanized Metatheory Revisited

Mechanized Metatheory Revisited

HOπ in Coq

Trakhtenbrot’s Theorem in Coq

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