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Tr. Mat. Inst. Steklova, 2003, Volume 242, Pages 147–175 (Mi tm413)  

This article is cited in 35 scientific papers (total in 35 papers)

Proof Mining: A Systematic Way of Analyzing Proofs in Mathematics

U. Kohlenbach, P. Oliva

University of Aarhus

Abstract: Proof mining is the process of logically analyzing proofs in mathematics with the aim of obtaining new information. In this survey paper, we discuss, by means of examples from mathematics, some of the main techniques used in proof mining. We show that these techniques apply not only to proofs based on classical logic but also to proofs that involve noneffective principles such as the attainment of the infimum of $f\in C[0,1]$ and the convergence for bounded monotone sequences of reals. We also report on recent case studies in approximation theory and fixed point theory where new results were obtained.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2003, 242, 136–164

Bibliographic databases:
UDC: 510.6
Received in June 2002
Language:

Citation: U. Kohlenbach, P. Oliva, “Proof Mining: A Systematic Way of Analyzing Proofs in Mathematics”, Mathematical logic and algebra, Collected papers. Dedicated to the 100th birthday of academician Petr Sergeevich Novikov, Tr. Mat. Inst. Steklova, 242, Nauka, MAIK Nauka/Inteperiodika, M., 2003, 147–175; Proc. Steklov Inst. Math., 242 (2003), 136–164

Citation in format AMSBIB
\Bibitem{KohOli03}
\by U.~Kohlenbach, P.~Oliva
\paper Proof Mining: A~Systematic Way of Analyzing Proofs in Mathematics
\inbook Mathematical logic and algebra
\bookinfo Collected papers. Dedicated to the 100th birthday of academician Petr Sergeevich Novikov
\serial Tr. Mat. Inst. Steklova
\yr 2003
\vol 242
\pages 147--175
\publ Nauka, MAIK Nauka/Inteperiodika
\publaddr M.
\mathnet{http://mi.mathnet.ru/tm413}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2054493}
\zmath{https://zbmath.org/?q=an:1079.03045}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2003
\vol 242
\pages 136--164


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Kohlenbach U., “Some logical metatheorems with applications in functional analysis”, Trans. Amer. Math. Soc., 357:1 (2005), 89–128  crossref  mathscinet  zmath  isi  scopus
    2. Macintyre A., “The mathematical significance of proof theory”, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 363:1835 (2005), 2419–2435  crossref  mathscinet  zmath  adsnasa  isi  scopus
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    4. Ferreira F., Oliva P., “Bounded functional interpretation”, Ann. Pure Appl. Logic, 135:1-3 (2005), 73–112  crossref  mathscinet  zmath  isi  scopus
    5. Kohlenbach U., “Proof mining in functional analysis”, New computational paradigms, First conference on computability in Europe, CiE 2005 (Amsterdam, The Netherlands, June 8–12, 2005), Proceedings, Lecture Notes in Comput. Sci., 3526, Springer, Berlin, 2005, 233–234  crossref  zmath  isi
    6. Hayashi S., “Can proofs be animated by games?”, Typed lambda calculi and applications, Lecture Notes in Comput. Sci., 3461, Springer, Berlin, 2005, 11–22  crossref  mathscinet  zmath  isi
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    8. Schuster P., “Unique solutions”, MLQ Math. Log. Q., 52:6 (2006), 534–539  crossref  mathscinet  zmath  isi  scopus
    9. Gerhardy Ph., Kohlenbach U., “Strongly uniform bounds from semi-constructive proofs”, Ann. Pure Appl. Logic, 141:1-2 (2006), 89–107  crossref  mathscinet  zmath  isi  scopus
    10. Ferreira F., Nunes A., “Bounded modified realizability”, J. Symbolic Logic, 71:1 (2006), 329–346  crossref  mathscinet  zmath  isi  scopus
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    15. Schuster P., “Corrigendum to: “Unique solutions” [MLQ Math. Log. Q. 52 (2006), no. 6, 534–539]”, MLQ Math. Log. Q., 53:2 (2007), 214  crossref  mathscinet  zmath  adsnasa  isi  scopus
    16. Ishihara H., “Unique existence and computability in constructive reverse mathematics”, Computation and Logic in the Real World, Proceedings, Lecture Notes in Computer Science, 4497, 2007, 368–377  crossref  mathscinet  zmath  isi  scopus
    17. Schuster P., “Problems as solutions”, Computation and Logic in the Real World, Proceedings, Lecture Notes in Computer Science, 4497, 2007, 676–684  crossref  mathscinet  zmath  isi  scopus
    18. Briseid E.M., “Fixed points of generalized contractive mappings”, J. Nonlinear Convex Anal., 9:2 (2008), 181–204  mathscinet  zmath  isi
    19. Seisenberger M., “Programs from proofs using classical dependent choice”, Ann. Pure Appl. Logic, 153:1-3 (2008), 97–110  crossref  mathscinet  zmath  isi  scopus
    20. Gerhardy Ph., Kohlenbach U., “General logical metatheorems for functional analysis”, Trans. Amer. Math. Soc., 360:5 (2008), 2615–2660  crossref  mathscinet  zmath  isi  scopus
    21. Cantini A., “On formal proofs”, Deduction, Computation, Experiment: Exploring the Effectiveness of Proof, 2008, 29–48  crossref  mathscinet  zmath  isi
    22. Ferreira F., “A most artistic package of a jumble of ideas (Kurt Gödel)”, Dialectica, 62:2 (2008), 205–222  crossref  mathscinet  zmath  isi  scopus
    23. Kohlenbach U., “Gödel's functional interpretation and its use in current mathematics (Reprinted from 'Horizons of Truth - Logics, Foundations of Mathematics, and the Quest for Understanding the Nature of Knowledge', 2008)”, Dialectica, 62:2 (2008), 223–267  crossref  mathscinet  isi  elib  scopus
    24. Oliva P., “An analysis of Gödel's 'Dialectica' interpretation via linear logic”, Dialectica, 62:2 (2008), 269–290  crossref  mathscinet  isi  scopus
    25. Hernest M.-D., Oliva P., “Hybrid functional interpretations”, Logic and Theory of Algorithms, Lecture Notes in Computer Science, 5028, 2008, 251–260  crossref  mathscinet  zmath  isi  scopus
    26. Briseid E.M., “Logical aspects of rates of convergence in metric spaces”, J. Symbolic Logic, 74:4 (2009), 1401–1428  crossref  mathscinet  zmath  isi  scopus
    27. Hernest M.-D., “Light monotone Dialectica methods for proof mining”, MLQ Math. Log. Q., 55:5 (2009), 551–561  crossref  mathscinet  zmath  isi  scopus
    28. Schuster P., “Problems, solutions, and completions”, J. Log. Algebr. Program., 79:1 (2010), 84–91  crossref  mathscinet  zmath  isi  scopus
    29. Avigad J., Towsner H., “Metastability in the Furstenberg-Zimmer tower”, Fund Math, 210:3 (2010), 243–268  crossref  mathscinet  zmath  isi  scopus
    30. Safarik P., Kohlenbach U., “On the computational content of the Bolzano-WeierstraB Principle”, MLQ Math Log Q, 56:5 (2010), 508–532  crossref  mathscinet  zmath  isi
    31. Singh Sh.L., Mishra S.N., Pant R., “Fixed Points of Generalized Asymptotic Contractions”, Fixed Point Theory, 12:2 (2011), 475–484  mathscinet  zmath  isi
    32. Oliva P., “Hybrid Functional Interpretations of Linear and Intuitionistic Logic”, J. Logic Comput., 22:2, SI (2012), 305–328  crossref  mathscinet  zmath  isi  scopus
    33. Hadzihasanovic A., van den Berg B., “Nonstandard Functional Interpretations and Categorical Models”, Notre Dame J. Form. Log., 58:3 (2017), 343–380  crossref  mathscinet  zmath  isi  scopus
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