A case study in programming coinductive proofs: Howe’s method

Momigliano, Alberto; Pientka, Brigitte; Thibodeau, David

doi:10.1017/s0960129518000415

Cited by 5 publications

(4 citation statements)

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“…Finally, another area that deserves new benchmarks is coinductive reasoning. While this has been a staple in proof assistants since the late 90s, most case studies regarded properties of bisimlarity in process and λ-calculi, to name just a few recent papers (Tiu & Miller, 2010;Bengtson et al, 2016;Lenglet & Schmitt, 2018;Momigliano et al, 2019). These turned (a posteriori) not so challenging, since those coinductive proofs can be carried out more or less with the limited technology of guarded induction.…”

Section: Discussionmentioning

confidence: 99%

POPLMark reloaded: Mechanizing proofs by logical relations

et al. 2019

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We propose a new collection of benchmark problems in mechanizing the metatheory of programming languages, in order to compare and push the state of the art of proof assistants. In particular, we focus on proofs using logical relations (LRs) and propose establishing strong normalization of a simply typed calculus with a proof by Kripke-style LRs as a benchmark. We give a modern view of this well-understood problem by formulating our LR on well-typed terms. Using this case study, we share some of the lessons learned tackling this problem in different dependently typed proof environments. In particular, we consider the mechanization in Beluga, a proof environment that supports higher-order abstract syntax encodings and contrast it to the development and strategies used in general-purpose proof assistants such as Coq and Agda. The goal of this paper is to engage the community in discussions on what support in proof environments is needed to truly bring mechanized metatheory to the masses and engage said community in the crafting of future benchmarks.

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Section: Discussionmentioning

confidence: 99%

POPLMark reloaded: Mechanizing proofs by logical relations

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“…For this, Beluga is a strong candidate: in fact the type safety challenge is already in the bag, thanks to the techniques developed in [20]. The coinduction part is more challenging, but we have a good track record in a similar benchmark [14]. Solving the rest of CCFB in Beluga may also shed some light on the role of the ∇ quantifier [13] as a meta-reasoning tool, compared to Beluga's use of contextual LF as a specification language.…”

Section: Evaluation and Conclusionmentioning

confidence: 99%

A Beluga Formalization of the Harmony Lemma in the π-Calculus

Cecilia,

Momigliano

2024

Electron. Proc. Theor. Comput. Sci.

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The "Harmony Lemma", as formulated by Sangiorgi & Walker, establishes the equivalence between the labelled transition semantics and the reduction semantics in the π-calculus. Despite being a widely known and accepted result for the standard π-calculus, this assertion has never been rigorously proven, formally or informally. Hence, its validity may not be immediately apparent when considering extensions of the π-calculus. Contributing to the second challenge of the Concurrent Calculi Formalization Benchmark -a set of challenges tackling the main issues related to the mechanization of concurrent systems -we present a formalization of this result for the fragment of the π-calculus examined in the Benchmark. Our formalization is implemented in Beluga and draws inspiration from the HOAS formalization of the LTS semantics popularized by Honsell et al. In passing, we introduce a couple of useful encoding techniques for handling telescopes and lexicographic induction.

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“…Besides the Abella and Twelf system, a number of other implemented systems support some or all aspects of λ-tree syntax: these include Beluga [129], Hybrid [44], Isabelle [123], Minlog [147], and the Teyjus [115] and ELPI [40] implementations of λProlog. Some of these systems have been explicitly compared and contrasted in recent papers [45,46,78,112]. Additionally, benchmark problems that are unique to metaprogramming problems have been proposed to test the ability of mechanized metatheory provers [12,47].…”

Section: Related Workmentioning

confidence: 99%

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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A case study in programming coinductive proofs: Howe’s method

Cited by 5 publications

References 61 publications

POPLMark reloaded: Mechanizing proofs by logical relations

POPLMark reloaded: Mechanizing proofs by logical relations

A Beluga Formalization of the Harmony Lemma in the π-Calculus

Mechanized Metatheory Revisited

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