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A focused linear logical framework and its application to metatheory of object logics

Published online by Cambridge University Press:  15 November 2021

Amy Felty ,
Carlos Olarte Open the ORCID record for Carlos Olarte [Opens in a new window]  and
Bruno Xavier
Amy Felty
Affiliation:
University of Ottawa, Ottawa, Canada
Carlos Olarte*
Affiliation:
LIPN, CNRS UMR 7030, Université Sorbonne Paris Nord, Villetaneuse, France and ECT, Universidade Federal do Rio Grande do Norte, Natal, Brazil
Bruno Xavier
Affiliation:
DIMAp, Universidade Federal do Rio Grande do Norte, Natal, Brazil
*
*Corresponding author. Email: olarte@lipn.univ-paris13.fr
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Abstract

Linear logic (LL) has been used as a foundation (and inspiration) for the development of programming languages, logical frameworks, and models for concurrency. LL’s cut-elimination and the completeness of focusing are two of its fundamental properties that have been exploited in such applications. This paper formalizes the proof of cut-elimination for focused LL. For that, we propose a set of five cut-rules that allows us to prove cut-elimination directly on the focused system. We also encode the inference rules of other logics as LL theories and formalize the necessary conditions for those logics to have cut-elimination. We then obtain, for free, cut-elimination for first-order classical, intuitionistic, and variants of LL. We also use the LL metatheory to formalize the relative completeness of natural deduction and sequent calculus in first-order minimal logic. Hence, we propose a framework that can be used to formalize fundamental properties of logical systems specified as LL theories.

Keywords

Linear logiccut-eliminationfocusingCoq
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 31 , Special Issue 3: Logical and Semantic Frameworks with Applications , March 2021 , pp. 312 - 340
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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