Wave equation numerical resolution: A comprehensive mechanized proof of a C program

Journal of Automated Reasoning

Volumn 50, Issue 4, 2013, Pages 423-456

  Boldo, Sylvie ^a,b   Clément, François ^c   Filliâtre, Jean Christophe ^b   Mayero, Micaela ^d,e   Melquiond, Guillaume ^a,b   Weis, Pierre ^c  

^a INRIA SACLAY   (France)

^b Laboratoire Interdisciplinaire des Sciences du Numérique   (France)

^c INRIA   (France)

^d CNRS   (France)

^e UNIVERSITÉ LYON 1   (France)

Author keywords

Acoustic wave equation; Convergence of numerical scheme; Coq formal proof; Formal proof of numerical program; Partial differential equation; Proof of C program; Rounding error analysis

Indexed keywords

C PROGRAMS; FORMAL PROOFS; NUMERICAL PROGRAMS; NUMERICAL SCHEME; ROUNDING ERROR ANALYSIS;

ACOUSTIC WAVE PROPAGATION; ACOUSTIC WAVES; C (PROGRAMMING LANGUAGE); ERROR ANALYSIS; PARTIAL DIFFERENTIAL EQUATIONS;

NUMERICAL METHODS;

EID: 84876490356     PISSN: 01687433     EISSN: 15730670     Source Type: Journal    
DOI: 10.1007/s10817-012-9255-4     Document Type: Article

References (58)

1. Achenbach, J.D.: Wave Propagation in Elastic Solids. North Holland, Amsterdam (1973)
2. Andrews, G.E.; Askey, R.; Roy, R.: Special Functions. Cambridge University Press, Cambridge (1999)
3. Askey, R.; Gasper, G.: Certain rational functions whose power series have positive coefficients. Am. Math. Mon. 79, 327-341 (1972)
4. Avigad, J.; Donnelly, K.: A decision procedure for linear "Big O" equations. J. Autom. Reason. 38(4), 353-373 (2007)
5. Barrett, C.; Tinelli, C.: CVC3. In: 19th International Conference on Computer Aided Verification (CAV '07), LNCS, vol. 4590, pp. 298-302. Springer, Berlin (2007)
6. Baudin, P.; Cuoq, P.; Filliâtre, J.C.; Marché, C.; Monate, B.; Moy, Y.; Prevosto, V.: ACSL: ANSI/ISO C specification language, version 1.5 (2009). URL http://frama-c.cea.fr/acsl.html
7. Bécache, E.: Étude de schémas numériques pour la résolution de l'équation des ondes. Master Modélisation et simulation, Cours ENSTA (2009). URL http://www-rocq.inria.fr/~becache/COURS- ONDES/Poly-num-0209.pdf
8. Bertot, Y.; Castéran, P.: Interactive theorem proving and program development. Coq'Art: the calculus of inductive constructions. In: Texts in Theoretical Computer Science. Springer (2004)
9. Bertot, Y.; Gonthier, G.; Ould Biha, S.; Pasca, I.: Canonical big operators. In: 21st International Conference on Theorem Proving in Higher Order Logics (TPHOLs'08), LNCS, vol. 5170, pp. 86-101. Springer, Montreal (2008)
10. Bobot, F.; Conchon, S.; Contejean, E.; Iguernelala, M.; Lescuyer, S.; Mebsout, A.: The Alt-Ergo automated theorem prover (2008). URL http://alt-ergo.lri.fr/
11. Boldo, S.: Preuves formelles en arithmétiques à virgule flottante. Ph.D. thesis, École Normale Supérieure de Lyon (2004)
12. Boldo, S.: Floats & Ropes: a case study for formal numerical program verification. In: 36th International Colloquium on Automata, Languages and Programming, LNCS - ARCoSS, vol. 5556, pp. 91-102. Springer, Rhodos (2009)
13. Boldo, S.; Filliâtre, J.C.: Formal verification of floating-point programs. In: 18th IEEE International Symposium on Computer Arithmetic, pp. 187-194. Montpellier, France (2007)
14. Boldo, S.; Nguyen, T.M.T.: Proofs of numerical programs when the compiler optimizes. Innovations Syst. Softw. Eng. 7(2), 151-160 (2011)
15. Boldo, S.; Filliâtre, J.C.; Melquiond, G.: Combining Coq and Gappa for certifying floating-point programs. In: Carette, J.; Dixon, L.; Coen, C.S.; Watt, S.M. (eds.) 16th Calculemus Symposium, Lecture Notes in Artificial Intelligence, vol. 5625, pp. 59-74. Grand Bend, ON, Canada (2009)
16. Boldo, S.; Clément, F.; Filliâtre, J.C.; Mayero, M.; Melquiond, G.; Weis, P.: Formal proof of a wave equation resolution scheme: the method error. In: Kaufmann, M.; Paulson, L.C. (eds.) 1st Interactive Theorem Proving Conference (ITP), LNCS, vol. 6172, pp. 147-162. Springer, Edinburgh (2010)
17. Boldo, S.; Melquiond, G.: Flocq: A unified library for proving floating-point algorithms in Coq. In: Antelo, E.; Hough, D.; Ienne, P. (eds.) 20th IEEE Symposium on Computer Arithmetic, pp. 243-252. Tübingen, Germany (2011)
18. Brekhovskikh, L.M.; Goncharov, V.: Mechanics of Continua and Wave Dynamics. Springer (1994)
19. Conchon, S.; Contejean, E.; Kanig, J.; Lescuyer, S.: CC(X): semantical combination of congruence closure with solvable theories. In: Post-Proceedings of the 5th International Workshop on Satisfiability Modulo Theories (SMT 2007), Electronic Notes in Computer Science, vol. 198-2, pp. 51-69. Elsevier (2008)
20. Coquand, T.; Paulin-Mohring, C.: Inductively defined types. In: Martin-Löf, P.; Mints, G. (eds.) Colog'88, LNCS, vol. 417. Springer (1990)
21. Courant, R.; Friedrichs, K.; Lewy, H.: On the partial difference equations of mathematical physics. IBM J. Res. Develop. 11(2), 215-234 (1967)
22. Cousot, P.; Cousot, R.; Feret, J.; Mauborgne, L.; Miné, A.; Monniaux, D.; Rival, X.: The ASTRÉE analyzer. In: ESOP, no. 3444 in LNCS, pp. 21-30 (2005)
23. Cruz-Filipe, L.: A constructive formalization of the fundamental theorem of calculus. In: Geuvers, H.; Wiedijk, F. (eds.) 2nd International Workshop on Types for Proofs and Programs (TYPES 2002), LNCS, vol. 2646. Springer, Berg en Dal (2002)
24. Daumas, M.; Melquiond, G.: Certification of bounds on expressions involving rounded operators. Trans. Math. Softw. 37(1), 1-20 (2010)
25. Daumas, M.; Rideau, L.; Théry, L.: A generic library for floating-point numbers and its application to exact computing. In: TPHOLs, pp. 169-184 (2001)
26. Delmas, D.; Goubault, E.; Putot, S.; Souyris, J.; Tekkal, K.; Védrine, F.: Towards an industrial use of FLUCTUAT on safety-critical avionics software. In: FMICS, LNCS, vol. 5825, pp. 53-69. Springer (2009)
27. de Dinechin, F.; Lauter, C.; Melquiond, G.: Certifying the floating-point implementation of an elementary function using Gappa. IEEE Trans. Comput. 60(2), 242-253 (2011)
28. de Moura, L.; Bjørner, N.: Z3, an efficient SMT solver. In: TACAS, Lecture Notes in Computer Science, vol. 4963, pp. 337-340. Springer (2008)
29. Dutertre, B.: Elements of mathematical analysis in PVS. In: von Wright, J.; Grundy, J.; Harrison, J. (eds.) 9th International Conference on Theorem Proving in Higher Order Logics (TPHOLs'96), LNCS, vol. 1125, pp. 141-156. Springer, Turku (1996)
30. Filliâtre, J.C.; Marché, C.: The Why/Krakatoa/Caduceus platform for deductive program verification. In: 19th International Conference on Computer Aided Verification, LNCS, vol. 4590, pp. 173-177. Springer, Berlin (2007)
31. Fleuriot, J.D.: On the mechanization of real analysis in Isabelle/HOL. In: Aagaard, M.; Harrison, J. (eds.) 13th International Conference on Theorem Proving and Higher-Order Logic (TPHOLs'00), LNCS, vol. 1869, pp. 145-161. Springer (2000)
32. Gamboa, R.; Kaufmann, M.: Nonstandard analysis in ACL2. J. Autom. Reason. 27(4), 323-351 (2001)
33. Geuvers, H.; Niqui, M.: Constructive reals in Coq: axioms and categoricity. In: Callaghan, P.; Luo, Z.; McKinna, J.; Pollack, R. (eds.) 1st International Workshop on Types for Proofs and Programs (TYPES 2000), LNCS, vol. 2277, pp. 79-95. Springer, Durham (2002)
34. Harrison, J.: Theorem Proving with the Real Numbers. Springer (1998)
35. Harrison, J.: A HOL theory of Euclidean space. In: Hurd, J.; Melham, T.F. (eds.) 18th International Conference on Theorem Proving and Higher-Order Logic (TPHOLs'05), LNCS, vol. 3603, pp. 114-129. Springer (2005)
36. John, F.: Partial Differential Equations. Springer (1986)
37. Krebbers, R.; Spitters, B.: Type classes for efficient exact real arithmetic in Coq. arXiv:1106.3448v1 (2011). URL http://arXiv.org/abs/1106.3448
38. le Rond D'Alembert, J.: Recherches sur la courbe que forme une corde tendue mise en vibrations. In: Histoire de l'Académie Royale des Sciences et Belles Lettres (Année 1747), vol. 3, pp. 214-249. Haude et Spener, Berlin (1749)
39. Lee, G.; Werner, B.: Proof-irrelevant model of CC with predicative induction and judgmental equality. Logical Methods in Computer Science 7(4) (2011)
40. Lelay, C.; Melquiond, G.: Différentiabilité et intégrabilité en Coq. Application à la formule de d'Alembert. In: 23èmes Journées Francophones des Langages Applicatifs, pp. 119-133. Carnac, France (2012)
41. Letouzey, P.: A new extraction for Coq. In: Geuvers, H.; Wiedijk, F. (eds.) 2nd International Workshop on Types for Proofs and Programs (TYPES 2002), LNCS, vol. 2646. Springer, Berg en Dal (2003)
42. Marché, C.: Jessie: an intermediate language for Java and C verification. In: Programming Languages meets Program Verification (PLPV), pp. 1-2. ACM, Freiburg (2007)
43. Mayero, M.: Formalisation et automatisation de preuves en analyses réelle et numérique. Ph.D. thesis, Université Paris VI (2001)
44. Mayero, M.: Using theorem proving for numerical analysis (correctness proof of an automatic differentiation algorithm). In: Carreño, V.; Muñoz, C.; Tahar, S. (eds.) 15th International Conference on Theorem Proving and Higher-Order Logic, LNCS, vol. 2410, pp. 246-262. Springer, Hampton (2002)
45. Microprocessor Standards Committee: IEEE Standard for Floating-Point Arithmetic. IEEE Std. 754-2008, pp. 1-58 (2008). doi: 10.1109/IEEESTD.2008. 4610935
46. Newton, I.: Axiomata, sive Leges Motus. In: Philosophiae Naturalis Principia Mathematica, vol. 1. London (1687)
47. O'Connor, R.: Certified exact transcendental real number computation in Coq. In: 21st International Conference on Theorem Proving in Higher Order Logics (TPHOLs'08), LNCS, vol. 5170, pp. 246-261. Springer (2008)
48. O'Connor, R.; Spitters, B.: A computer-verified monadic functional implementation of the integral. Theor. Comp. Sci. 411(37), 3386-3402 (2010)
49. Rosinger, E.E.: Propagation of round-off errors and the role of stability in numerical methods for linear and nonlinear PDEs. Appl. Math. Model. 9(5), 331-336 (1985)
50. Rosinger, E.E.: L-convergence paradox in numerical methods for PDEs. Appl. Math. Model. 15(3), 158-163 (1991)
51. Roy, C.J.; Oberkampf, W.L.: A comprehensive framework for verification, validation, and uncertainty quantification in scientific computing. Comput. Methods Appl. Mech. Eng. 200(25-28), 2131-2144 (2011)
52. Rudnicki, P.: An overview of the MIZAR project. In: Types for Proofs and Programs, pp. 311-332 (1992)
53. Szyszka, B.: An interval method for solving the one-dimensional wave equation. In: 7th EUROMECH Solid Mechanics Conference (ESMC2009). Lisbon, Portugal (2009)
54. The Coq reference manual. URL http://coq.inria.fr/refman/
55. The Frama-C platform for static analysis of C programs (2008). URL http://www.frama-c.cea.fr/
56. Thomas, J.W.: Numerical partial differential equations: finite difference methods. In: Texts in Applied Mathematics, no. 22. Springer (1995)
57. Zach, R.: Hilbert's "Verunglueckter Beweis," the first epsilon theorem, and consistency proofs. URL http://front.math.ucdavis.edu/math.LO/ 0204255
58. Zwillinger, D.: Handbook of Differential Equations. Academic Press (1998)