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Induction using term orderings

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Automated Deduction — CADE-12 (CADE 1994)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 814))

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Abstract

We present a procedure for proving inductive theorems which is based on explicit induction, yet supports mutual induction. Mutual induction allows the postulation of lemmas whose proofs use the theorems ex hypothesi while the theorems themselves use the lemmas. This feature has always been supported by induction procedures based on Knuth-Bendix completion, but these procedures are limited by the use of rewriting (or rewriting-like) inferences. Our procedure avoids this limitation by making explicit the implicit induction realized by these procedures. As a result, arbitrary deduction mechanisms can be used while still allowing mutual induction.

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References

  1. L. Bachmair. Proof by consistency in equational theories. In Proc. 3rd LICS Symp., Edinburgh (UK), pages 228–233, 1988.

    Google Scholar 

  2. E. Bevers and J. Lewi. Proof by consistency in conditional equational theories. In S. Kaplan and M. Okada, editors, 2nd CTRS Workshop, LNCS, vol. 516, pages 195–205. Springer-Verlag, 1991.

    Google Scholar 

  3. S. Biundo, B. Hummel, D. Hutter, and C. Walther. The Karlsruhe induction theorem proving system. In 8th CADE Conf., LNCS, vol. 230. Springer-Verlag, 1986.

    Google Scholar 

  4. Adel Bouhoula and MichaËl Rusinowitch. Automatic case analysis in proof by induction. In Ruzena Bajcsy, editor, Proc. 13th IJCAI Conf., Chambéry (France), volume 1, pages 88–94. Morgan Kaufmann, August 1993

    Google Scholar 

  5. R. S. Boyer and J. S. Moore. A Computational Logic. Academic Press, New York, 1979.

    Google Scholar 

  6. R. S. Boyer and J. S. Moore. A theorem prover for a computational logic. In M. E. Stickel, editor, Proc. 10th CADE Conf., Kaiserslautern (Germany), LNCS, vol. 449, pages 1–15. Springer-Verlag, 1990

    Google Scholar 

  7. F. Bronsard and U. S. Reddy. Conditional rewriting in Focus. In S. Kaplan and M. Okada, editors, 2nd CTRS Workshop, LNCS, vol. 516, pages 2-13. Springer-Verlag, 1991.

    Google Scholar 

  8. F. Bronsard and U. S. Reddy. Reduction techniques for first-order reasoning. In M. Rusinowitch and J. L. Rémy, editors, 3rd CTRS Workshop, LNCS, vol. 656, pages 242–256. Springer-Verlag, 1992.

    Google Scholar 

  9. A. Bundy. A rational reconstruction and extension of recursion analysis. In IJCAI, 1989.

    Google Scholar 

  10. R. M. Burstall. Proving properties of programs by structural induction. Computer Journal, 12:41–48, 1969.

    Google Scholar 

  11. N. Dershowitz. Completion and its applications. In Resolution of Equations in Algebraic Structures, volume 2: Rewriting Techniques, pages 31–86. Academic Press, San Diego, 1989.

    Google Scholar 

  12. N. Dershowitz and J.-P. Jouannaud. Rewrite systems. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science B: Formal Methods and Semantics, chapter 6, pages 243–320. North-Holland, Amsterdam, 1990.

    Google Scholar 

  13. N. Dershowitz and Z. Manna. Proving termination with multiset orderings. Comm. ACM, 22(8):465–476, August 1979.

    Google Scholar 

  14. S. J. Garland and J. V Guttag. Inductive methods for reasoning about abstract data types. In ACM POPL Symp., pages 219–228. ACM, 1988.

    Google Scholar 

  15. J. A. Goguen. How to prove inductive hypotheses without induction. In 5th CADE Conf., LNCS, vol. 87, pages 356–372. Springer Verlag, Jul 1980.

    Google Scholar 

  16. B. Gramlich. Induction theorem proving using refined unfailing completion techniques. In ECAI, 1989. (also vailable as Technical Report SR-89-14, Universität Kaiserslautern, Germany.).

    Google Scholar 

  17. D. Hofbauer and R. D. Kutsche. Proving inductive theorems based on term rewriting systems. In J. Grabowski, P. Lescanne, and W. Wechler, editors, Proc. 1st ALP Workshop, pages 180–190. Akademie Verlag, 1988.

    Google Scholar 

  18. G. Huet and J.-M. Hullot. Proofs by induction in equational theories with constructors. J. Comp. and System Sciences, 25:239–266, 1982.

    Google Scholar 

  19. J.-P. Jouannaud and E. Kounalis. Automatic proofs by induction in equational theories without constructors. Information and Computation, 82:1–33, 1989.

    Google Scholar 

  20. D. Kapur and D. R. Musser. Proof by consistency. Artificial Intelligence, 31(2):125–157, February 1987.

    Google Scholar 

  21. D. Kapur, P. Narendran, and H. Zhang. Automating inductionless induction using test sets. J. Symbolic Computation, 11:83–112, 1991.

    Google Scholar 

  22. D. Knuth and P. Bendix. Simple word problems in universal algebras. In J. Leech, editor, Computational Problems in Abstract Algebra, pages 263–297. Pergamon Press, Oxford, 1970.

    Google Scholar 

  23. E. Kounalis and M. Rusinowitch. Mechanizing inductive reasoning. In Proc. AAAI Conf., pages 240–245. AAAI Press and MIT Press, July 1990.

    Google Scholar 

  24. R. Kowalski. Studies in the completeness and efficiency of theorem-proving by resolution. PhD thesis, University of Edinburgh, 1970.

    Google Scholar 

  25. D. McAllester. Term rewriting induction. theorem-provers@ai.mit.edu electronic bulletin board, 1990.

    Google Scholar 

  26. D. R. Musser. On proving inductive properties of abstract data types. In ACM POPL Symp., pages 154–162. ACM, 1980.

    Google Scholar 

  27. U. S. Reddy. Term rewriting induction. In M. Stickel, editor, 10th CADE Conf., volume 449 of LNAI, pages 162–177. Springer-Verlag, 1990.

    Google Scholar 

  28. Jean-Luc Rémy. Etude des systèmes de Réécriture Conditionnels et Applications aux Types Abstraits Algébriques. Th. Etat, INPL, Nancy (France), 1982.

    Google Scholar 

  29. G. A. Robinson and L. T. Wos. Paramodulation and first-order theorem proving. In B. Meltzer and D. Mitchie, editors, Machine Intelligence 4, pages 135–150. Edinburgh University Press, 1969.

    Google Scholar 

  30. J. A. Robinson. A machine-oriented logic based on the resolution principle. J. ACM, 12:23–41, 1965.

    Google Scholar 

  31. C.-P. Wirth and B. Gramlich. On notions of inductive validity for first-order equational clauses. In 12th CADE Conf., 1994.

    Google Scholar 

  32. H. Zhang, D. Kapur, and M. S. Krishnamoorthy. A mechanizable induction principle for equational specifications. In E. Lusk and R. Overbeek, editors, 9th CADE Conf., LNCS, vol. 310, pages 162–181. Springer-Verlag, 1988.

    Google Scholar 

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Authors and Affiliations

  1. The University of Illinois at Urbana-Champaign, USA

    Francois Bronsard, Uday S. Reddy & Robert W. Hasker

Authors
  1. Francois Bronsard
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  2. Uday S. Reddy
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  3. Robert W. Hasker
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Corresponding author

Correspondence to Francois Bronsard .

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Alan Bundy

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© 1994 Springer-Verlag Berlin Heidelberg

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Bronsard, F., Reddy, U.S., Hasker, R.W. (1994). Induction using term orderings. In: Bundy, A. (eds) Automated Deduction — CADE-12. CADE 1994. Lecture Notes in Computer Science, vol 814. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-58156-1_8

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  • DOI: https://doi.org/10.1007/3-540-58156-1_8

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-58156-7

  • Online ISBN: 978-3-540-48467-7

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