Abstract
We describe a line of work that started in 2011 towards enriching Isabelle/HOL’s language with coinductive datatypes, which allow infinite values, and with a more expressive notion of inductive datatype than previously supported by any system based on higher-order logic. These (co)datatypes are complemented by definitional principles for (co)recursive functions and reasoning principles for (co)induction. In contrast with other systems offering codatatypes, no additional axioms or logic extensions are necessary with our approach.
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Bartels, F.: Generalised coinduction. Math. Struct. Comput. Sci. 13(2), 321–348 (2003)
Becker, H., Blanchette, J.C., Waldmann, U., Wand, D.: Formalization of Knuth–Bendix orders for lambda-free higher-order terms. Archive of Formal Proofs (2016). Formal proof development. http://isa-afp.org/entries/Lambda_Free_KBOs.shtml
Becker, H., Blanchette, J.C., Waldmann, U., Wand, D.: A transfinite Knuth–Bendix order for lambda-free higher-order terms. In: de Moura, L. (ed.) CADE-26. LNCS, vol. 10395, pp. 432–453. Springer, Cham (2017). doi:10.1007/978-3-319-63046-5_27
Berghofer, S., Wenzel, M.: Inductive datatypes in HOL—lessons learned in formal-logic engineering. In: Bertot, Y., Dowek, G., Théry, L., Hirschowitz, A., Paulin, C. (eds.) TPHOLs 1999. LNCS, vol. 1690, pp. 19–36. Springer, Heidelberg (1999). doi:10.1007/3-540-48256-3_3
Bertot, Y., Casteran, P.: Interactive Theorem Proving and Program Development–Coq’Art: The Calculus of Inductive Constructions. Texts in Theoretical Computer Science. Springer, Heidelberg (2004). doi:10.1007/978-3-662-07964-5
Blanchette, J.C.: Relational analysis of (co)inductive predicates, (co)inductive datatypes, and (co)recursive functions. Softw. Qual. J. 21(1), 101–126 (2013)
Blanchette, J.C., Fleury, M., Traytel, D.: Nested multisets, hereditary multisets, and syntactic ordinals in Isabelle/HOL. In: Miller, D. (ed.) FSCD 2017. LIPIcs, vol. 84, pp. 11:1–11:17 (2017). Schloss Dagstuhl—Leibniz-Zentrum für Informatik
Blanchette, J.C., Nipkow, T.: Nitpick: a counterexample generator for higher-order logic based on a relational model finder. In: Kaufmann, M., Paulson, L.C. (eds.) ITP 2010. LNCS, vol. 6172, pp. 131–146. Springer, Heidelberg (2010). doi:10.1007/978-3-642-14052-5_11
Blanchette, J.C., Popescu, A., Traytel, D.: Soundness and completeness proofs by coinductive methods. J. Autom. Reason. 58(1), 149–179 (2017)
Blanchette, J.C., Bouzy, A., Lochbihler, A., Popescu, A., Traytel, D.: Friends with benefits. In: Yang, H. (ed.) ESOP 2017. LNCS, vol. 10201, pp. 111–140. Springer, Heidelberg (2017). doi:10.1007/978-3-662-54434-1_5
Blanchette, J.C., Fleury, M., Traytel, D.: Formalization of nested multisets, hereditary multisets, and syntactic ordinals. Archive of Formal Proofs (2016). Formal proof development. http://isa-afp.org/entries/Nested_Multisets_Ordinals.shtml
Blanchette, J.C., Hölzl, J., Lochbihler, A., Panny, L., Popescu, A., Traytel, D.: Truly modular (co)datatypes for Isabelle/HOL. In: Klein, G., Gamboa, R. (eds.) ITP 2014. LNCS, vol. 8558, pp. 93–110. Springer, Cham (2014). doi:10.1007/978-3-319-08970-6_7
Blanchette, J.C., Meier, F., Popescu, A., Traytel, D.: Foundational nonuniform (co)datatypes for higher-order logic. In: Ouaknine, J. (ed.) LICS 2017. IEEE Computer Society (2017)
Blanchette, J.C., Popescu, A., Traytel, D.: Abstract completeness. Archive of Formal Proofs (2014). Formal proof development. http://isa-afp.org/entries/Abstract_Completeness.shtml
Blanchette, J.C., Popescu, A., Traytel, D.: Cardinals in Isabelle/HOL. In: Klein, G., Gamboa, R. (eds.) ITP 2014. LNCS, vol. 8558, pp. 111–127. Springer, Cham (2014). doi:10.1007/978-3-319-08970-6_8
Blanchette, J.C., Popescu, A., Traytel, D.: Unified classical logic completeness. In: Demri, S., Kapur, D., Weidenbach, C. (eds.) IJCAR 2014. LNCS (LNAI), vol. 8562, pp. 46–60. Springer, Cham (2014). doi:10.1007/978-3-319-08587-6_4
Blanchette, J.C., Popescu, A., Traytel, D.: Foundational extensible corecursion–a proof assistant perspective. In: Fisher, K., Reppy, J.H. (eds.) ICFP 2015, pp. 192–204. ACM (2015)
Blanchette, J.C., Popescu, A., Traytel, D.: Witnessing (co)datatypes. In: Vitek, J. (ed.) ESOP 2015. LNCS, vol. 9032, pp. 359–382. Springer, Heidelberg (2015). doi:10.1007/978-3-662-46669-8_15
Bulwahn, L., Krauss, A., Nipkow, T.: Finding lexicographic orders for termination proofs in Isabelle/HOL. In: Schneider, K., Brandt, J. (eds.) TPHOLs 2007. LNCS, vol. 4732, pp. 38–53. Springer, Heidelberg (2007). doi:10.1007/978-3-540-74591-4_5
Dershowitz, N., Manna, Z.: Proving termination with multiset orderings. In: Maurer, H.A. (ed.) ICALP 1979. LNCS, vol. 71, pp. 188–202. Springer, Heidelberg (1979). doi:10.1007/3-540-09510-1_15
Gödel, K.: Über die Vollständigkeit des Logikkalküls. Ph.D. thesis, Universität Wien (1929)
Gunter, E.L.: Why we can’t have SML-style datatype declarations in HOL. In: TPHOLs 1992. IFIP Transactions, vol. A-20, pp. 561–568. North-Holland/Elsevier (1993)
Hinze, R., Paterson, R.: Finger trees: a simple general-purpose data structure. J. Funct. Program. 16(2), 197–217 (2006)
Hölzl, J.: Markov chains and Markov decision processes in Isabelle/HOL. J. Autom. Reason. doi:10.1007/s10817-016-9401-5
Hölzl, J.: Markov processes in Isabelle/HOL. In: Bertot, Y., Vafeiadis, V. (eds.) CPP 2017, pp. 100–111. ACM (2017)
Huffman, B., Kunčar, O.: Lifting and transfer: a modular design for quotients in Isabelle/HOL. In: Gonthier, G., Norrish, M. (eds.) CPP 2013. LNCS, vol. 8307, pp. 131–146. Springer, Cham (2013). doi:10.1007/978-3-319-03545-1_9
Kleene, S.C.: Mathematical Logic. Wiley, New York (1967)
Kovács, L., Robillard, S., Voronkov, A.: Coming to terms with quantified reasoning. In: Castagna, G., Gordon, A.D. (eds.) POPL 2017, pp. 260–270. ACM (2017)
Krauss, A.: Partial recursive functions in higher-order logic. In: Furbach, U., Shankar, N. (eds.) IJCAR 2006. LNCS (LNAI), vol. 4130, pp. 589–603. Springer, Heidelberg (2006). doi:10.1007/11814771_48
Lochbihler, A.: Jinja with threads. Archive of Formal Proofs (2007). Formal proof development. http://isa-afp.org/entries/JinjaThreads.shtml
Lochbihler, A.: Coinductive. Archive of Formal Proofs (2010). Formal proof development. http://afp.sf.net/entries/Coinductive.shtml
Lochbihler, A.: Verifying a compiler for Java threads. In: Gordon, A.D. (ed.) ESOP 2010. LNCS, vol. 6012, pp. 427–447. Springer, Heidelberg (2010). doi:10.1007/978-3-642-11957-6_23
Lochbihler, A.: Making the Java memory model safe. ACM Trans. Program. Lang. Syst. 35(4), 12:1–12:65 (2014)
Lochbihler, A.: Probabilistic functions and cryptographic oracles in higher order logic. In: Thiemann, P. (ed.) ESOP 2016. LNCS, vol. 9632, pp. 503–531. Springer, Heidelberg (2016). doi:10.1007/978-3-662-49498-1_20
Lochbihler, A., Hölzl, J.: Recursive functions on lazy lists via domains and topologies. In: Klein, G., Gamboa, R. (eds.) ITP 2014. LNCS, vol. 8558, pp. 341–357. Springer, Cham (2014). doi:10.1007/978-3-319-08970-6_22
Meier, F.: Non-uniform datatypes in Isabelle/HOL. M.Sc. thesis, ETH Zürich (2016)
Milius, S., Moss, L.S., Schwencke, D.: Abstract GSOS rules and a modular treatment of recursive definitions. Log. Methods Comput. Sci. 9(3), 1–52 (2013)
Moura, L., Bjørner, N.: Z3: an efficient SMT solver. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 337–340. Springer, Heidelberg (2008). doi:10.1007/978-3-540-78800-3_24
Nipkow, T., Wenzel, M., Paulson, L.C. (eds.): Isabelle/HOL: A Proof Assistant for Higher-Order Logic. LNCS, vol. 2283. Springer, Heidelberg (2002). doi:10.1007/3-540-45949-9
Okasaki, C.: Purely Functional Data Structures. Cambridge University Press, Cambridge (1999)
Panny, L.: Primitively (co)recursive function definitions for Isabelle/HOL. B.Sc. thesis, Technische Universität München (2014)
Reynolds, A., Blanchette, J.C.: A decision procedure for (co)datatypes in SMT solvers. J. Autom. Reason. 58(3), 341–362 (2017)
Reynolds, J.C.: Types, abstraction and parametric polymorphism. In: IFIP 1983, pp. 513–523 (1983)
Rutten, J.J.M.M.: Automata and coinduction (an exercise in coalgebra). In: Sangiorgi, D., Simone, R. (eds.) CONCUR 1998. LNCS, vol. 1466, pp. 194–218. Springer, Heidelberg (1998). doi:10.1007/BFb0055624
Schropp, A., Popescu, A.: Nonfree datatypes in Isabelle/HOL. In: Gonthier, G., Norrish, M. (eds.) CPP 2013. LNCS, vol. 8307, pp. 114–130. Springer, Cham (2013). doi:10.1007/978-3-319-03545-1_8
Sternagel, C., Thiemann, R.: Deriving comparators and show functions in Isabelle/HOL. In: Urban, C., Zhang, X. (eds.) ITP 2015. LNCS, vol. 9236, pp. 421–437. Springer, Cham (2015). doi:10.1007/978-3-319-22102-1_28
Sternagel, C., Thiemann, R.: Deriving class instances for datatypes. Archive of Formal Proofs (2015). Formal proof development. http://isa-afp.org/entries/Deriving.shtml
Thiemann, R., Sternagel, C.: Certification of termination proofs using CeTA. In: Berghofer, S., Nipkow, T., Urban, C., Wenzel, M. (eds.) TPHOLs 2009. LNCS, vol. 5674, pp. 452–468. Springer, Heidelberg (2009). doi:10.1007/978-3-642-03359-9_31
Torlak, E., Jackson, D.: Kodkod: a relational model finder. In: Grumberg, O., Huth, M. (eds.) TACAS 2007. LNCS, vol. 4424, pp. 632–647. Springer, Heidelberg (2007). doi:10.1007/978-3-540-71209-1_49
Traytel, D.: Formal languages, formally and coinductively. In: Kesner, D., Pientka, B. (eds.) FSCD 2016. LIPIcs, vol. 52, pp. 31:1–31:17 (2016). Schloss Dagstuhl—Leibniz-Zentrum für Informatik
Traytel, D., Popescu, A., Blanchette, J.C.: Foundational, compositional (co)datatypes for higher-order logic—category theory applied to theorem proving. In: LICS 2012, pp. 596–605. IEEE Computer Society (2012)
Traytel, D.: A category theory based (co)datatype package for Isabelle/HOL. M.Sc. thesis, Technische Universität München (2012)
Wenzel, M.: Isabelle/Isar—a generic framework for human-readable proof documents. From Insight to Proof: Festschrift in Honour of Andrzej Trybulec, Studies in Logic, Grammar, and Rhetoric 10(23), 277–298 (2007). Uniwersytet w Białymstoku
Wenzel, M.: Re: [isabelle] “Unfolding” the sum-of-products encoding of datatypes (2015). https://lists.cam.ac.uk/pipermail/cl-isabelle-users/2015-November/msg00082.html
Acknowledgments
We first want to acknowledge the support and encouragement of past and current bosses: David Basin, Wan Fokkink, Stephan Merz, Aart Middeldorp, Tobias Nipkow, and Christoph Weidenbach. We are grateful to the FroCoS 2017 program chairs, Clare Dixon and Marcelo Finger, and to the program committee for giving us this opportunity to present our research. We are also indebted to Andreas Abel, Stefan Berghofer, Sascha Böhme, Lukas Bulwahn, Elsa Gunter, Florian Haftmann, Martin Hofmann, Brian Huffman, Lars Hupel, Alexander Krauss, Peter Lammich, Rustan Leino, Stefan Milius, Lutz Schröder, Mark Summerfield, Christian Urban, Daniel Wand, and Makarius Wenzel, and to dozens of anonymous reviewers (including those who rejected our manuscript “Witnessing (co)datatypes” [18] six times).
Blanchette was supported by the Deutsche Forschungsgemeinschaft (DFG) projects “Quis Custodiet” (NI 491/11-2) and “Den Hammer härten” (NI 491/14-1). He also received funding from the European Research Council under the European Union’s Horizon 2020 research and innovation program (grant agreement No. 713999, Matryoshka). Hölzl was supported by the DFG project “Verifikation probabilistischer Modelle in interaktiven Theorembeweisern” (NI 491/15-1). Kunčar and Popescu were supported by the DFG project “Security Type Systems and Deduction” (NI 491/13-2 and NI 491/13-3) as part of the program Reliably Secure Software Systems (RS\(^3\), priority program 1496). Kunčar was also supported by the DFG project “Integration der Logik HOL mit den Programmiersprachen ML und Haskell” (NI 491/10-2). Lochbihler was supported by the Swiss National Science Foundation (SNSF) grant “Formalising Computational Soundness for Protocol Implementations” (153217). Popescu was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) starting grant “VOWS: Verification of Web-based Systems” (EP/N019547/1). Sternagel and Thiemann were supported by the Austrian Science Fund (FWF): P27502 and Y757. Traytel was supported by the DFG program “Programm- und Modell-Analyse” (PUMA, doctorate program 1480). The authors are listed alphabetically.
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Biendarra, J. et al. (2017). Foundational (Co)datatypes and (Co)recursion for Higher-Order Logic. In: Dixon, C., Finger, M. (eds) Frontiers of Combining Systems. FroCoS 2017. Lecture Notes in Computer Science(), vol 10483. Springer, Cham. https://doi.org/10.1007/978-3-319-66167-4_1
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