Journal of Automated Reasoning

, Volume 45, Issue 3, pp 301–325 | Cite as

A Formalization of the Knuth–Bendix(–Huet) Critical Pair Theorem

  • André L. Galdino
  • Mauricio Ayala-Rincón


A mechanical proof of the Knuth–Bendix Critical Pair Theorem in the higher-order language of the theorem prover PVS is described. This well-known theorem states that a Term Rewriting System is locally confluent if and only if all its critical pairs are joinable. The formalization of this theorem follows Huet’s well-known structure of proof in which the restriction on strong normalization or Noetherian was dropped and the result presented as a lemma. In order to formalize the Knuth–Bendix Critical Pair Theorem we rely on previously developed PVS theories for abstract reduction systems, named ars, and term rewriting systems, named trs, which were built upon the PVS libraries for finite sequences and sets. On the one hand, the theory trs is composed of subtheories for dealing with the structure of terms, for replacements of subterms and substitutions and jointly with the theory ars it allows for adequate specifications of elaborate notions of term rewriting systems such as the one of critical pairs. On the other hand, ars specifies basic definitions and notions of abstract reduction systems such as reduction, termination, normal forms, and confluence as well as non basic concepts such as strong normalization.


Abstract reduction systems Term rewriting systems Formalization of theorems PVS Critical pair theorem 


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  1. 1.
    Altenkirch, T.: A formalization of the strong normalization proof for system F in LEGO. In: Bezem, M., Groote, J.F. (eds.) Proceedings of the International Conference on Typed Lambda Calculi and Applications, TLCA’93. Lecture Notes in Computer Science, vol. 664, pp. 13–28. Springer, Utrecht (1993)CrossRefGoogle Scholar
  2. 2.
    Altenkirch, T.: Proving strong normalization of CC by modifying realizability semantics. In: Barendregt, H.P., Nipkow, T. (eds.) Types for proofs and programs. Lecture Notes in Computer Science, vol. 806, pp. 3–18. Springer, New York (1994)Google Scholar
  3. 3.
    Altenkirch, T., Dybjer, P., Hofmann, M., Scott, P.: Normalization by evaluation for typed lambda calculus with coproducts. In: Halpern, J. (ed.) Proceedings of the Sixteenth Annual IEEE Symposium on Logic in Computer Science, pp. 303–310. IEEE Computer Society, Boston (2001)Google Scholar
  4. 4.
    Ayala-Rincón, M., Llanos, C.H., Jacobi, R.P., Hartenstein, R.W.: Prototyping time- and space-efficient computations of algebraic operations over dynamically reconfigurable systems modeled by rewriting-logic. ACM Transact. Des. Automat. Electron. Syst. 11(2), 251–281 (2006)CrossRefGoogle Scholar
  5. 5.
    Ayala-Rincón, M., Sant’Ana, T.M.: SAEPTUM: verification of ELAN hardware specifications using the proof assistant PVS. In: 19th Symp. on Integrated Circuits and System Design, pp. 125–130. ACM Press (2006)Google Scholar
  6. 6.
    Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, Cambridge (1998)Google Scholar
  7. 7.
    Barras, B.: Auto-validation d’un système de preuves avec familles inductives. Thèse de doctorat, Université Paris 7 (1999)Google Scholar
  8. 8.
    Bezem, M., Coquand, T.: Neman’s lemma—a case study in proof automation and geometric logic. Bull. Eur. Assoc. Theor. Comput. Sci. 79, 86–100 (2003)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Bezem, M., Klop, J.W., de Vrijer, R. (eds.): Term Rewriting Systems by TeReSe. Cambridge Tracts in Theoretical Computer Science, no. 55. Cambridge University Press, Cambridge (2003)Google Scholar
  10. 10.
    Blanqui, F., Coupet-Grimal, S., Delobel, W., Hinderer, S., Koprowski, A.: CoLoR, a Coq library on rewriting and termination. In: 8th International Workshop on Termination (WST ’06) (2006)Google Scholar
  11. 11.
    Boyer, R.S., Moore, J.S.: A Computational Logic Handbook. Academic Press Professional, San Diego (1988)zbMATHGoogle Scholar
  12. 12.
    Contejean, E., Courtieu, P., Forest, J., Pons, O., Urbain, X.: Certification of automated termination proofs. In: Konev, B., Wolter, F. (eds.) 6th International Symposium on Frontiers of Combining Systems (FroCos 07). Lecture Notes in Artificial Intelligence, vol. 4720, pp. 148–162. Springer, Liverpool (2007)CrossRefGoogle Scholar
  13. 13.
    Ford, J.M., Mason, I.A.: Operational techniques in PVS—a preliminary evaluation. In: Proceedings of the Australasian Theory Symposium, CATS’01 (2001)Google Scholar
  14. 14.
    Galdino, A.L., Ayala-Rincón, M.: A formalization of Newman’s and Yokouchi lemmas in a higher-order language. J. Formal. Reasoning 1(1), 39–50 (2008)zbMATHGoogle Scholar
  15. 15.
    Galdino, A.L., Ayala-Rincón, M.: A theory for abstract reduction systems in PVS. CLEI Electr. J. 11(2), 12 pp. (2008) (Special Issue of Best Papers preented at CLEI’07)Google Scholar
  16. 16.
    Galdino, A.L., Ayala-Rincón, M.: A PVS theory for term rewriting systems. In: Pimentel, E., Benevides, M. (eds.) Proceedings of the Third Workshop on Logical and Semantic Frameworks, with Applications—LSFA 2008. Electronic Notes in Theoretical Computer Science, vol. 247, pp. 67–83. Elsevier, Amsterdam (2009)Google Scholar
  17. 17.
    Huet, G.: Confluent reductions: abstract properties and applications to term rewriting systems. J. Assoc. Comput. Mach. 27(4), 797–821 (1980)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Huet, G.: Residual theory in λ-calculus: a formal development. J. Funct. Program. 4(3), 371–394 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Kapur, D., Zhang, H.: An overview of Rewrite Rule Laboratory (RRL). In: Dershowitz, N. (ed.) Proc. Third Int. Conf. on Rewriting techniques and Applications, Chapel-Hill, NC. Lecture Notes in Computer Science, vol. 355, pp. 559–563. Springer, New York (1989)Google Scholar
  20. 20.
    Knuth, D.E., Bendix, P.B.: Simple word problems in universal algebra. Computational problems in abstract algebra, pp. 263–297 (1970)Google Scholar
  21. 21.
    Koprowski, A.: A formalization of the simply typed lambda in Coq (2006).
  22. 22.
    McKinna, J., Pollack, R.: Some lambda calculus and type theory formalized. J. Autom. Reason. 23(3–4), 373–409 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Morra, C., Becker, J., Ayala-Rincón, M., Hartenstein, R.W.: FELIX: using rewriting-logic for generating functionally equivalent implementations. In: 15th Int. Conference on Field Programmable Logic and Applications—FPL 2005, pp. 25–30. IEEE CS (2005)Google Scholar
  24. 24.
    Nipkow, T.: More Church-Rosser proofs (in Isabelle/HOL). In: McRobbie, M., Slaney, J. (eds.) Proceedings of the 13th International Conference on Automated Deduction (CADE-13). Lecture Notes in Artificial Intelligence, vol. 1104, pp. 733–747. Springer, New Brunswick (1996)Google Scholar
  25. 25.
    Nipkow, T.: More Church-Rosser proofs. J. Autom. Reason. 26(1), 51–66 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    van Oostrom, V.: Development closed critical pairs. In: Selected Papers from the Second International Workshop on Higher-Order Algebra, Logic, and Term Rewriting—HOA’95. Lecture Notes in Computer Science, vol. 1074, pp. 185–200. Springer, London (1996)Google Scholar
  27. 27.
    Pfenning, F.: A proof of the Church–Rosser theorem and its representation in a logical framework. A preliminary version is available as Carnegie Mellon technical report CMU-CS-92-186. (1992)
  28. 28.
    Rasmussen, O.: The Church–Rosser theorem in Isabelle: a proof porting experiment. Tech. Rep. UCAM-CL-TR-364, Computer Laboratory, University of Cambridge, Cambridge (1995)Google Scholar
  29. 29.
    Ruiz-Reina, J.L., Alonso, J.A., Hidalgo, M.J., Martín-Mateos, F.J.: Formal proofs about rewriting using ACL2. Ann. Math. Artif. Intell. 36(3), 239–262 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Saïbi, A.: Formalization of a lamda-calculus with explicit substitutions in Coq. In: TYPES’94: Selected Papers from the International Workshop on Types for Proofs and Programs. Lecture Notes in Computer Science, vol. 996, pp. 183–202. Springer, London (1995)Google Scholar
  31. 31.
    Shankar, N.: A mechanical proof of the Church–Rosser theorem. J. Assoc. Comput. Mach. 35, 475–522 (1988)zbMATHMathSciNetGoogle Scholar

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© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidade Federal de GoiásCatalãoBrazil
  2. 2.Instituto de Ciências ExatasUniversidade de BrasíliaBrasíliaBrazil

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