Abstract
We port Dawson and Goré’s general framework of deep embeddings of derivability from Isabelle to Coq. By using lists instead of multisets to encode sequents, we enable the encoding of genuinely substructural logics in which some combination of exchange, weakening and contraction are not admissible. We then show how to extend the framework to encode the linear nested sequent calculus \(\mathsf {LNS}_\mathsf {Kt}\) of Goré and Lellmann for the tense logic \(\mathsf {Kt}\) and prove cut-elimination and all required proof-theoretic theorems in Coq, based on their pen-and-paper proofs. Finally, we extract the proof of the cut-elimination theorem to obtain a formally verified Haskell program that produces cut-free derivations from those with cut. We believe it is the first published formally verified computer program for eliminating cuts in any proof calculus.
C. D’Abrera—Supported by an Australian Government Research Training Program Scholarship.
R. Goré—Work supported by the FWF projects I 2982 and P 33548.
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References
Belnap, N.: Display logic. J. Philos. Log. 11, 375–417 (1982)
Chaudhuri, K., Lima, L., Reis, G.: Formalized meta-theory of sequent calculi for linear logics. TCS 781, 24–38 (2019)
Dawson, J.E., Brotherston, J., Goré, R.: Machine-checked interpolation theorems for substructural logics using display calculi. In: Olivetti, N., Tiwari, A. (eds.) IJCAR 2016. LNCS (LNAI), vol. 9706, pp. 452–468. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-40229-1_31
Dawson, J.E., Clouston, R., Goré, R., Tiu, A.: From display calculi to deep nested sequent calculi: formalised for full intuitionistic linear logic. In: Diaz, J., Lanese, I., Sangiorgi, D. (eds.) TCS 2014. LNCS, vol. 8705, pp. 250–264. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-44602-7_20
Dawson, J.E., Goré, R.: Formalised cut admissibility for display logic. In: Carreño, V.A., Muñoz, C.A., Tahar, S. (eds.) TPHOLs 2002. LNCS, vol. 2410, pp. 131–147. Springer, Heidelberg (2002). https://doi.org/10.1007/3-540-45685-6_10
Dawson, J.E., Goré, R.: Generic methods for formalising sequent calculi applied to provability logic. In: Fermüller, C.G., Voronkov, A. (eds.) LPAR 2010. LNCS, vol. 6397, pp. 263–277. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-16242-8_19
Goré, R.: Tableau methods for modal and temporal logics. In: D’Agostino, M., Gabbay, D., Haehnle, R., Posegga, J. (eds.) Handbook of Tableau Methods, Kluwer, pp. 297–396 (1999)
Goré, R., Lellmann, B.: Syntactic cut-elimination and backward proof-search for tense logic via linear nested sequents. In: Cerrito, S., Popescu, A. (eds.) TABLEAUX 2019. LNCS (LNAI), vol. 11714, pp. 185–202. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-29026-9_11
Graham-Lengrand, S.: Polarities & focussing: a journey from realisability to automated reasoning, Habilitation Thesis, Université Paris-Sud (2014)
Kashima, R.: Cut-free sequent calculi for some tense logics. Stud. Log. 53(1), 119–136 (1994)
Larchey-Wendling, D.: Semantic cut elimination. https://github.com/. DmxLarchey/Coq-Phase-Semantics/blob/master/coq.type/cut_elim.v
Larchey-Wendling, D.: Constructive decision via redundancy-free proof-search. J. Autom. Reason. 64(7), 1197–1219 (2020)
Miller, D., Pimentel, E.: A formal framework for specifying sequent calculus proof systems. Theor. Comput. Sci. 474, 98–116 (2013)
Negri, S.: Proof analysis in modal logic. J. Philos. Logic 34(5–6), 507–544 (2005)
Olarte, C., Pimentel, E., Xavier, B.: A fresh view of linear logic as a logical framework. In: LSFA 2020. ENTCS, vol. 351, pp. 143–165. Elsevier (2020)
Pfenning, F.: Structural cut elimination. In: LICS 1995, pp. 156–166. IEEE Computer Society (1995)
Simmons, R.J.: Structural focalization. ACM Trans. Comput. Log. 15(3), 21:1–21:33 (2014)
Tews, H.: Formalizing cut elimination of coalgebraic logics in Coq. In: Galmiche, D., Larchey-Wendling, D. (eds.) TABLEAUX 2013. LNCS (LNAI), vol. 8123, pp. 257–272. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40537-2_22
Troelstra, A., Schwichtenberg, H.: Basic Proof Theory. Number 43 in Cambridge Tracts in Theoretical Computer Science. Cambridge University Press (1996)
Urban, C., Zhu, B.: Revisiting cut-elimination: one difficult proof is really a proof. In: Voronkov, A. (ed.) RTA 2008. LNCS, vol. 5117, pp. 409–424. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-70590-1_28
Xavier, B., Olarte, C., Reis, G., Nigam, V.: Mechanizing focused linear logic in Coq. In: LSFA 2017. ENTCS, vol. 338, pp. 219–236. Elsevier (2017)
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D’Abrera, C., Dawson, J., Goré, R. (2021). A Formally Verified Cut-Elimination Procedure for Linear Nested Sequents for Tense Logic. In: Das, A., Negri, S. (eds) Automated Reasoning with Analytic Tableaux and Related Methods. TABLEAUX 2021. Lecture Notes in Computer Science(), vol 12842. Springer, Cham. https://doi.org/10.1007/978-3-030-86059-2_17
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