left up

6. Bibliography

[aiello-meta] Aiello, L., Cecchi, C., and Sartini, D. (1986) Representation and use of metaknowledge.
Proceedings of the IEEE, 74, 1304--1321.

[andrews-tps] Andrews, P. B., Bishop, M., Issar, S., Nesmith, D., Pfenning, F., and Xi, H. (1994) TPS: A theorem proving system for classical type theory.
Research report 94-166, Department of Mathematics, Carnegie-Mellon University.

[anonymous-qed] Anonymous (1994) The QED Manifesto.
See [cade12], pp. 238--251.

[appel-4] Appel, K. and Haken, W. (1976) Every planar map is four colorable.
Bulletin of the American Mathematical Society, 82, 711--712.

[anl-new-results] {Argonne National Laboratories} (1995) New results with ANL's ATP software.
Unpublished; available on the Web as http://www.mcs.anl.gov/home/mccune/ar/new_results/index.html.

[armando-building] Armando, A., Cimatti, A., and Viganò, L. (1993) Building and executing proof strategies in a formal metatheory.
In Torasso, P. (ed.), Advances in Artifical Intelligence: Proceedings of the Third Congress of the Italian Association for Artificial Intelligence, IA*AI'93, Volume 728 of Lecture Notes in Computer Science, Torino, Italy, pp. 11--22. Springer-Verlag.

[arnold-newtonhooke] Arnol'd, V. I. (1990) Huygens and Barrow, Newton and Hooke: pioneers in mathematical analysis and catastrophe theory from evolents to quasicrystals.
Birkhauser.
Translated from the Russian by Eric J.F. Primrose.

[beckert-posegga] Beckert, B. and Posegga, J. (1994) leanT AP: lean, tableau-based theorem proving.
See [cade12], pp. 793--797.
Extended version available on the Web from http://i12www.ira.uka.de/ posegga/LeanTaP.ps.Z.

[benacerraf] Benacerraf, P. (1965) What numbers could not be.
Philosophical Review, 74, 47--73.
Reprinted in [benacerraf-putnam], pp. 272--294.

[benacerraf-putnam] Benacerraf, P. and Putnam, H. (1983) Philosophy of mathematics: selected readings (2nd ed.).
Cambridge University Press.

[biggs-graph] Biggs, N. L., Lloyd, E. K., and Wilson, R. J. (1976) Graph Theory 1736--1936.
Clarendon Press.

[bishop-bridges] Bishop, E. and Bridges, D. (1985) Constructive analysis, Volume 279 of Grundlehren der mathematischen Wissenschaften.
Springer-Verlag.

[boole-article] Boole, G. (1848) The calculus of logic.
The Cambridge and Dublin Mathematical Journal, 3, 183--198.

[boulton-thesis] Boulton, R. J. (1993) Efficiency in a fully-expansive theorem prover.
Technical Report 337, University of Cambridge Computer Laboratory, New Museums Site, Pembroke Street, Cambridge, CB2 3QG, UK.
Author's PhD thesis.

[bourbaki-found] Bourbaki, N. (1948) Foundations of mathematics for the working mathematician.
Journal of Symbolic Logic, 14, 1--14.

[bourbaki-topology1] Bourbaki, N. (1966) General topology, Volume 1 of Elements of mathematics.
Addison-Wesley.
Translated from French {'Topologie Genérale'} in the series {'Eléments de mathématique'}, originally published by Hermann in 1966.

[bourbaki-sets] Bourbaki, N. (1968) Theory of sets.
Elements of mathematics. Addison-Wesley.
Translated from French {'Théorie des ensembles'} in the series {'Eléments de mathématique'}, originally published by Hermann in 1968.

[boyer-acl] Boyer, R. S. and Moore, J. S. (1979) A Computational Logic.
ACM Monograph Series. Academic Press.

[boyer-moore-meta] Boyer, R. S. and Moore, J. S. (1981) Metafunctions: proving them correct and using them efficiently as new proof procedures.
In Boyer, R. S. and Moore, J. S. (eds.), The Correctness Problem in Computer Science, pp. 103--184. Academic Press.

[bridges-richmann] Bridges, D. and Richman, F. (1987) Varieties of Constructive Mathematics, Volume 97 of London Mathematical Society Lecture Note Series.
Cambridge University Press.

[debruijn-sad] de Bruijn, N. G. (1970) The mathematical language AUTOMATH, its usage and some of its extensions.
See [sadem], pp. 29--61.

[debruijn-Automat h] de Bruijn, N. G. (1980) A survey of the project AUTOMATH.
In Seldin, J. P. and Hindley, J. R. (eds.), To H. B. Curry: Essays in Combinatory Logic, Lambda Calculus, and Formalism, pp.\ 589--606. Academic Press.

[bryant] Bryant, R. E. (1986) Graph-based algorithms for Boolean function manipulation.
IEEE Transactions on Computers, C-35, 677--691.

[cade12] Bundy, A. (ed.) (1994) 12th International Conference on Automated Deduction, Volume 814 of Lecture Notes in Computer Science, Nancy, France. Springer-Verlag.

[church-number] Church, A. (1936) An unsolvable problem of elementary number-theory.
American Journal of Mathematics, 58, 345--363.

[church-types] Church, A. (1940) A formulation of the Simple Theory of Types.
Journal of Symbolic Logic, 5, 56--68.

[chwistek] Chwistek, L. (1922) Über die Antinomien de Principien der Mathematik.
Mathematische Annalen, 14, 236--243.

[constable-programs] Constable, R. L., Knoblock, T. B., and Bates, J. L. (1985) Writing programs that construct proofs.
Journal of Automated Reasoning, 1, 285--326.

[corella-thesis] Corella, F. (1990) Mechanizing set theory.
Technical Report 232, University of Cambridge Computer Laboratory, New Museums Site, Pembroke Street, Cambridge, CB2 3QG, UK.
Author's PhD thesis.

[curzon-changes] Curzon, P. (1995) Tracking design changes with formal machine-checked proof.
The Computer Journal, 38, 91--100.

[davis-undecidable] Davis, M. (ed.) (1965) The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions. Raven Press, NY.

[davis-schwartz] Davis, M. and Schwartz, J. T. (1979) Metatheoretic extensibility for theorem verifiers and proof-checkers.
Computers and Mathematics with Applications, 5, 217--230.

[dedekind-was] Dedekind, R. (1888) Was sind und was sollen die Zahlen.
Vieweg, Braunschweig.
English translation in [dedekind-essays].

[dedekind-essays] Dedekind, R. (ed.) (1901) Essays on the theory of numbers. Open Court, NY.
Reprinted by Dover, 1963.

[dijkstra-invarianc e] Dijkstra, E. W. (1985) Invariance and non-determinacy.
In Hoare, C. A. R. and Shepherdson, J. C. (eds.), Mathematical Logic and Programming Languages, Prentice-Hall International Series in Computer Science, pp. 157--165. Prentice-Hall.
The papers in this volume were first published in the Philosophical Transactions of the Royal Society, Series A, vol. 312, 1984.

[farmer-imps] Farmer, W., Guttman, J., and Thayer, J. (1990) IMPS: an interactive mathematical proof system.
In Stickel, M. E. (ed.), 10th International Conference on Automated Deduction, Volume 449 of Lecture Notes in Computer Science, Kaiserslautern, Federal Republic of Germany, pp. 653--654. Springer-Verlag.

[feferman-transfini te] Feferman, S. (1962) Transfinite recursive progressions of axiomatic theories.
Journal of Symbolic Logic, 27, 259--316.

[fitch-book] Fitch, F. B. (1952) Symbolic Logic: an introduction.
The Ronald Press Company, New York.

[fowler] Fowler, H. W. (1965) A Dictionary of Modern English Usage (2nd, revised ed.).
Oxford University Press.
Revised by Sir Ernest Gowers.

[frege-beg] Frege, G. (1879) Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens.
Halle.
English translation, 'Begriffsschrift, a formula language, modeled upon that of arithmetic, for pure thought' in [vanh], pp. 1--82.

[freyd-allegories] Freyd, P. J. and Scedrov, A. (1990) Categories, allegories.
North-Holland.

[vangasteren-shape] van Gasteren, A. J. M. (1990) On the shape of mathematical arguments, Volume 445 of Lecture Notes in Computer Science.
Springer-Verlag.
Foreword by E. W. Dijkstra.

[gentzen-investigatio ns] Gentzen, G. (1935) Über das logische Schliessen.
Mathematische Zeitschrift, 39, 176--210.
This was Gentzen's Inaugural Dissertation at Göttingen. English translation, 'Investigations into Logical Deduction', in [gentzen], p. 68--131.

[godel-undecidabl e] Gödel, K. (1931) Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I.
Monatshefte für Mathematik und Physik, 38, 173--198.
English translation, 'On Formally Undecidable Propositions of Principia Mathematica and Related Systems, I', in [vanh], pp. 592--618 or [davis-undecidable], pp. 4--38.

[gordon-st] Gordon, M. (1994) Merging HOL with set theory: preliminary experiments.
Technical Report 353, University of Cambridge Computer Laboratory, New Museums Site, Pembroke Street, Cambridge, CB2 3QG, UK.

[gordon-holbook] Gordon, M. J. C. and Melham, T. F. (1993) Introduction to {HOL: a theorem proving environment for higher order logic}.
Cambridge University Press.

[gordon-lcfbook] Gordon, M. J. C., Milner, R., and Wadsworth, C. P. (1979) Edinburgh {LCF: A Mechanised Logic of Computation}, Volume 78 of Lecture Notes in Computer Science.
Springer-Verlag.

[halmos-ba] Halmos, P. R. (1963) Lectures on Boolean algebras, Volume 1 of Van Nostrand mathematical studies.
Van Nostrand.

[harrison-hol93] Harrison, J. (1993) A HOL decision procedure for elementary real algebra.
See [hol93], pp. 426--436.

[harrison-fmsd94] Harrison, J. (1994) Constructing the real numbers in HOL.
Formal Methods in System Design, 5, 35--59.

[harrison-cj95] Harrison, J. (1995) Binary decision diagrams as a HOL derived rule.
The Computer Journal, 38, 162--170.

[harrison-thery2] Harrison, J. and Théry, L. (1993) Extending the HOL theorem prover with a computer algebra system to reason about the reals.
See [hol93], pp. 174--184.

[heaywood] Heawood, P. J. (1890) Map-colour theorem.
Quarterly Journal of Pure and Applied Mathematics, 24, 332--338.
Reprinted in [biggs-graph].

[vanh] van Heijenoort, J. (ed.) (1967) From Frege to Gödel: A Source Book in Mathematical Logic 1879--1931. Harvard University Press.

[henkin-ptypes] Henkin, L. (1963) A theory of propositional types.
Fundamenta Mathematicae, 52, 323--344.

[hilbert-fgeom] Hilbert, D. (1899) Grundlagen der Geometrie.
Teubner.
English translation 'Foundations of Geometry' published in 1902 by Open Court, Chicago.

[holmes-naive] Holmes, R. M. (1995) Naive set theory with a universal set.
Unpublished; available on the Web as http://math.idbsu.edu/faculty/holmes/naive1.ps.

[howe-computational] Howe, D. J. (1988) Computational metatheory in Nuprl.
In Lusk, E. and Overbeek, R. (eds.), 9th International Conference on Automated Deduction, Volume 310 of Lecture Notes in Computer Science, Argonne, Illinois, USA, pp. 238--257. Springer-Verlag.

[hol93] Joyce, J. J. and Seger, C. (eds.) (1993) Proceedings of the 1993 International Workshop on the {HOL theorem proving system and its applications}, Volume 780 of Lecture Notes in Computer Science, UBC, Vancouver, Canada. Springer-Verlag.

[jutting-thesis] van Bentham Jutting, L. S. (1977) Checking Landau's ''Grundlagen'' in the {AUTOMATH System}.
Ph. D. thesis, Eindhoven University of Technology.
Useful summary in [nederpelt-automath], pp. 701--732.

[kanger] Kanger, S. (1963) A simplified proof method for elementary logic.
In Braffort, P. and Hirschberg, D. (eds.), Computer programming and formal systems, pp. 89--95. North-Holland.

[kempe] Kempe, A. B. (1879) On the geographical problem of the four colours.
American Journal of Mathematics, 2, 193--200.
Reprinted in [biggs-graph].

[knuth-bendix] Knuth, D. and Bendix, P. (1970) Simple word problems in universal algebras.
In Leech, J. (ed.), Computational Problems in Abstract Algebra. Pergamon Press.

[knuth-ss] Knuth, D. E. (1973) The Art of Computer Programming; Volume 3: Sorting and Searching.
Addison-Wesley Series in Computer Science and Information processing. Addison-Wesley.

[kreisel-hilbert] Kreisel, G. (1958) Hilbert's programme.
Dialectica, 12, 346--372.
Revised version in [benacerraf-putnam].

[kreisel-sad] Kreisel, G. (1970) Hilbert's programme and the search for automatic proof procedures.
See [sadem], pp. 128--146.

[kreisel-distractions] Kreisel, G. (1990) Logical aspects of computation: Contributions and distractions.
In Logic and Computer Science, Volume 31 of APIC Studies in Data Processing, pp. 205--278. Academic Press.

[kreisel-krivine] Kreisel, G. and Krivine, J.-L. (1971) Elements of mathematical logic: model theory (Revised second ed.).
Studies in Logic and the Foundations of Mathematics. North-Holland.
First edition 1967. Translation of the French 'Eléments de logique mathématique, théorie des modeles' published by Dunod, Paris in 1964.

[kumar-faust] Kumar, R., Kropf, T., and Schneider, K. (1991) Integrating a first-order automatic prover in the HOL environment.
In Archer, M., Joyce, J. J., Levitt, K. N., and Windley, P. J. (eds.), Proceedings of the 1991 International Workshop on the {HOL theorem proving system and its Applications}, University of California at Davis, Davis CA, USA, pp. 170--176. IEEE Computer Society Press.

[lakatos-pr] Lakatos, I. (1976) Proofs and Refutations: the Logic of Mathematical Discovery.
Cambridge University Press.
Edited by John Worrall and Elie Zahar. Derived from Lakatos's Cambridge PhD thesis; an earlier version was published in the British Journal for the Philosophy of Science vol. 14.

[lakatos-proof] Lakatos, I. (1980) What does a mathematical proof prove?
In Worrall, J. and Currie, G. (eds.), Mathematics, science and epistemology. Imre Lakatos: Philosophical papers vol. 2, pp. 61--69. Cambridge University Press.

[lam-proof] Lam, C. W. H. (1990) How reliable is a computer-based proof?
The Mathematical Intelligencer, 12, 8--12.

[landau] Landau, E. (1930) Grundlagen der Analysis.
Leipzig.
English translation by F. Steinhardt: 'Foundations of analysis: the arithmetic of whole, rational, irrational, and complex numbers. A supplement to textbooks on the differential and integral calculus', published by Chelsea; 3rd edition 1966.

[lang-algebra] Lang, S. (1994) Algebra (3rd ed.).
Addison-Wesley.

[sadem] Laudet, M., Lacombe, D., Nolin, L., and Schützenberger, M. (eds.) (1970) Symposium on Automatic Demonstration, Volume 125 of Lecture Notes in Mathematics. Springer-Verlag.

[lecat-errors] Lecat, M. (1935) Erreurs de Mathématiciens.
Brussels.

[littlewood-mis cellany] Littlewood, J. E. (1986) Littlewood's Miscellany.
Cambridge University Press.
Edited by Bela Bollobas.

[lob-theorem] Löb, M. H. (1955) Solution of a problem of Leon Henkin.
Journal of Symbolic Logic, 20, 115--118.

[loveland-me1] Loveland, D. W. (1968) Mechanical theorem-proving by model elimination.
Journal of the ACM, 15, 236--251.

[maclane-mff] Mac Lane, S. (1986) Mathematics: Form and Function.
Springer-Verlag.

[mackenzie-proof] MacKenzie, D. (1995) The automation of proof: A historical and sociological exploration.
IEEE Annals of the History of Computing, 17(3), 7--29.

[maslov-inverse] Maslov, S. J. (1964) An inverse method of establishing deducibility in classical predicate calculus.
Doklady Akademii Nauk, 159, 17--20.

[mathias-bourbaki] Mathias, A. R. D. (1991) The ignorance of Bourbaki.
Physis; rivista internazionale di storia della scienza (nuova serie), 28, 887--904.
Also in 'The Mathematical Intelligencer' vol. 14, nr. 3, pp. 4--13, 1992.

[matsumura] Matsumura, H. (1986) Commutative Ring Theory, Volume 8 of Cambridge Studies in Advanced Mathematics.
Cambridge University Press.
Translated from Japanese 'Kakan kan ron' by Miles Reid.

[matthews-fs0-theor y] Matthews, S. (1994) A theory and its metatheory in {FS0}.
Publication unknown.

[melham-tydef] Melham, T. F. (1989) Automating recursive type definitions in higher order logic.
In Birtwistle, G. and Subrahmanyam, P. A. (eds.), Current Trends in Hardware Verification and Automated Theorem Proving, pp. 341--386. Springer-Verlag.

[morse-sets] Morse, A. P. (1965) A theory of sets.
New York.

[naur-pvf] Naur, P. (1994) Proof versus formalization.
BIT, 34, 148--164.

[nederpelt-automath] Nederpelt, R. P., Geuvers, J. H., and de Vrijer, R. C. (eds.) (1994) Selected Papers on Automath, Volume 133 of Studies in Logic and the Foundations of Mathematics. North-Holland.

[newell-simon] Newell, A. and Simon, H. A. (1956) The logic theory machine.
IRE Transactions on Information Theory, 2, 61--79.

[paris-harrington] Paris, J. and Harrington, L. (1991) A mathematical incompleteness in Peano Arithmetic.
In Barwise, J. and Keisler, H. (eds.), Handbook of mathematical logic, Volume 90 of Studies in Logic and the Foundations of Mathematics, pp. 1133--1142. North-Holland.

[paulson-coalgebra] Paulson, L. C. (1994) A concrete final coalgebra theorem for ZF set theory.
Technical Report 334, University of Cambridge Computer Laboratory, New Museums Site, Pembroke Street, Cambridge, CB2 3QG, UK.

[pollack-extensibilit y] Pollack, R. (1995) On extensibility of proof checkers.
In Dybjer, P., Nordström, B., and Smith, J. (eds.), Types for Proofs and Programs: selected papers from TYPES'94, Volume 996 of Lecture Notes in Computer Science, B\aastad, pp. 140--161. Springer-Verlag.

[prawitz-book] Prawitz, D. (1965) Natural deduction; a proof-theoretical study, Volume 3 of Stockholm Studies in Philosophy.
Almqvist and Wiksells.

[prawitz-pp] Prawitz, D., Prawitz, H., and Voghera, N. (1960) A mechanical proof procedure and its realization in an electronic computer.
Journal of the ACM, 7, 102--128.

[ramsey-fm] Ramsey, F. P. (1926) The foundations of mathematics.
Proceedings of the London Mathematical Society (2), 25, 338--384.

[rasiowa-sikorski] Rasiowa, H. and Sikorski, R. (1970) The Mathematics of Metamathematics (3rd ed.), Volume 41 of Monografie Matematyczne, Instytut Matematyczny Polskiej Akademii Nauk.
Polish Scientific Publishers.

[robinson-resolutio n] Robinson, J. A. (1965) A machine-oriented logic based on the resolution principle.
Journal of the ACM, 12, 23--41.

[rudnicki-obvious] Rudnicki, P. (1987) Obvious inferences.
Journal of Automated Reasoning, 3, 383--393.

[shostak-presburger] Shostak, R. (1979) A practical decision procedure for arithmetic with function symbols.
Journal of the ACM, 26, 351--360.

[spivey-understanding] Spivey, J. M. (1988) Understanding Z: a specification language and its formal semantics, Volume 3 of Cambridge Tracts in Theoretical Computer Science.
Cambridge University Press.

[stalmarc k-patent] St\aalmarck, G. (1994) System for determining propositional logic theorems by applying values and rules to triplets that are generated from Boolean formula.
United States Patent number 5,276,897.

[stickel-pttp] Stickel, M. E. (1988) A Prolog Technology Theorem Prover: Implementation by an extended Prolog compiler.
Journal of Automated Reasoning, 4, 353--380.

[gentzen] Szabo, M. E. (ed.) (1969) The collected papers of Gerhard Gentzen, Studies in Logic and the Foundations of Mathematics. North-Holland.

[szczerba-mizar] Szczerba, L. W. (1989) The use of Mizar MSE in a course in foundations of geometry.
In Srzednicki, J. (ed.), Initiatives in logic, Volume 2 of Reason and argument series. M. Nijhoff.

[takeuti-uses] Takeuti, G. (1978) Two applications of logic to mathematics.
Number 13 in Publications of the Mathematical Society of Japan. Iwanami Shoten, Tokyo.
Number 3 in Kanô memorial lectures.

[tarski-truth] Tarski, A. (1936) Der Wahrheitsbegriff in den formalisierten Sprachen.
Studia Philosophica, 1, 261--405.
English translation, 'The Concept of Truth in Formalized Languages', in [tarski-lsm], pp. 152--278.

[tarski-decision] Tarski, A. (1954) A Decision Method for Elementary Algebra and Geometry.
University of California Press.

[tarski-lsm] Tarski, A. (ed.) (1956) Logic, Semantics and Metamathematics. Clarendon Press.

[trybulec-ilcc] Trybulec, A. (1978) The Mizar-QC/6000 logic information language.
ALCC Bulletin (Association for Literary and Linguistic Computing), 6, 136--140.

[trybulec-z] Trybulec, Z. and Swi\ceczkowska, H. (1991-1992) The language of mathematical texts.
Studies in Logic, Grammar and Rhetoric, 10/11, 103--124.

[turing] Turing, A. M. (1936) On computable numbers, with an application to the Entscheidungsproblem.
Proceedings of the London Mathematical Society (2), 42, 230--265.

[turing-ordinals] Turing, A. M. (1939) Systems of logic based on ordinals.
Proceedings of the London Mathematical Society (2), 45, 161--228.
Reprinted in [davis-undecidable], pp. 154--222.

[upsenskii-mechan ics] Upsenskii, V. A. (1961) Some Applications of Mechanics to Mathematics, Volume 3 of Popular lectures in mathematics.
Pergamon.
Translated from the Russian by Halina Moss; translation editor Ian N. Sneddon.

[whitehead-intro] Whitehead, A. N. (1919) An Introduction to Mathematics.
Williams and Norgate.

[whitehead-principia] Whitehead, A. N. and Russell, B. (1910) Principia Mathematica (3 vols).
Cambridge University Press.

[wiener-pair] Wiener, N. (1914) A simplification of the logic of relations.
Proceedings of the Cambridge Philosophical Society, 17, 387--390.

[wang-mm] Wang, H. (1960) Toward mechanical mathematics.
IBM Journal of research and development, 4, 2--22.

[yamamoto-graphs] Yamamoto, M., Nishizaha, S.-y., Hagiya, M., and Toda, Y. (1995) Formalization of planar graphs.
In Windley, P. J., Schubert, T., and Alves-Foss, J. (eds.), Higher Order Logic Theorem Proving and Its Applications: Proceedings of the 8th International Workshop, Volume 971 of Lecture Notes in Computer Science, Aspen Grove, Utah, pp. 369--384. Springer-Verlag.

[zermelo-new] Zermelo, E. (1908) Neuer Beweis für die Möglichkeit einer wohlordnung.
Mathematische Annalen, 65, 107--128.
English translation, 'A new proof of the possibility of a wellordering' in [vanh], pp. 183--198.


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