Descente Infinie + Deduction

Inductive theorem proving in the form of descente infinie was known to the ancient Greeks and is the standard induction method of a working mathematician since it was reinvented in the middle of the 17th century. We present an integration of descente infinie into state-of-the-art free-variable sequent and tableau calculi. It is well-suited for an efficient interplay of human interaction and automation and combines raising, explicit representation of dependence between variables, the liberalized δ-rule, preservation of solutions, and unrestricted applicability of lemmas and induction hypotheses. The semantical requirements are satisfied for a variety of two-valued logics, such as clausal logic, classical first-order logic, and higher-order modal logic.

[1]  Charisma Lee.  A completeness theorem and a computer program for finding theorems derivable from given axioms , 1967 .

[2]  Karel Berka,et al.  Logik-Texte - kommentierte Auswahl zur Geschichte der modernen Logik (4. Aufl.) , 1971 .

[3]  Cora Diamond,et al.  Wittgenstein's Lectures on the Foundations of Mathematics. , 1977 .

[4]  Frank van Harmelen,et al.  The Oyster-Clam System , 1990, CADE.

[5]  Dieter Hutter,et al.  The Karlsruhe Induction Theorem Proving System , 1986, CADE.

[6]  L. M. Milne-Thomson,et al.  Grundlagen der Mathematik , 1935, Nature.

[7]  Robert S. Boyer,et al.  Computational Logic , 1990, ESPRIT Basic Research Series.

[8]  Martin Giese,et al.  Hilbert's epsilon-Terms in Automated Theorem Proving , 1999, TABLEAUX.

[9]  P. Howard,et al.  Consequences of the axiom of choice , 1998 .

[10]  Tobias Schmidt-Samoa,et al.  Flexible heuristics for simplification with conditional lemmas by marking formulas as forbidden, mandatory, obligatory, and generous , 2006, J. Appl. Non Class. Logics.

[11]  Claus-Peter Wirth,et al.  Abstract Notions and Inference Systems for Proofs by Mathematical Induction , 1994, CTRS.

[12]  Dov M. Gabbay,et al.  Handbook of logic in artificial intelligence and logic programming (vol. 1) , 1993 .

[13]  A. Leisenring.  Mathematical logic and Hilbert's ε-symbol , 1971 .

[14]  Peter B. Andrews.  General Models, Descriptions, and Choice in Type Theory , 1972, J. Symb. Log..

[15]  Balder ten Cate,et al.  Question Answering: From Partitions to Prolog , 2002, TABLEAUX.

[16]  Alan Bundy,et al.  The Automation of Proof by Mathematical Induction , 1999, Handbook of Automated Reasoning.

[17]  Domenico Cantone,et al.  A Further and Effective Liberalization of the delta-Rule in Free Variable Semantic Tableaux , 1998, FTP.

[18]  Dov M. Gabbay,et al.  Handbook of the history of logic , 2004 .

[19]  W. H. Bussey.  The Origin of Mathematical Induction , 1917 .

[20]  Claus-Peter Wirth,et al.  Conditional Equational Specifications of Data Types with Partial Operations for Inductive Theorem Proving , 1997, RTA.

[21]  Christoph Walther,et al.  Mathematical induction , 1994, Handbook of Logic in Artificial Intelligence and Logic Programming.

[22]  Robert S. Boyer,et al.  A computational logic handbook , 1979, Perspectives in computing.

[23]  Mike Paterson,et al.  Linear unification , 1976, STOC '76.

[24]  Graham Wrightson,et al.  Automation of Reasoning , 1983 .

[25]  Claus-Peter Wirth.  A New Indefinite Semantics for Hilbert's Epsilon , 2002, TABLEAUX.

[26]  Sloman,et al.  Automation of Reasoning , 1983, Symbolic Computation.

[27]  Stig Kanger,et al.  A Simplified Proof Method for Elementary Logic , 1959 .

[28]  Christian G. Fermüller,et al.  Non-elementary Speedups between Different Versions of Tableaux , 1995, TABLEAUX.

[29]  Claus-Peter Wirth.  Positive negative conditional equations: a constructor-based framework for specification and inductive theorem proving , 1997 .

[30]  Wilfried Sieg,et al.  Normal Natural Deduction Proofs (in classical logic) , 1998, Stud Logica.

[31]  P. Fermat,et al.  Oeuvres de Fermat , 1891 .

[32]  Michael S. Mahoney,et al.  The Mathematical Career of Pierre de Fermat, 1601-1665 , 1973 .

[33]  Serge Autexier,et al.  Hierarchical contextual reasoning , 2003 .

[34]  Melvin Fitting,et al.  First-Order Logic and Automated Theorem Proving , 1990, Graduate Texts in Computer Science.

[35]  Dieter Hutter,et al.  Coloring Terms to Control Equational Reasoning , 1997, Journal of Automated Reasoning.

[36]  M. Fitting.  Types, Tableaus, and Gödel's God , 2002 .

[37]  Gopalan Nadathur,et al.  Handbook of Logic in Artificial Intelligence and Logic Programming, Volume2, Deduction Methodologies , 1994, Handbook of Logic in Artificial Intelligence and Logic Programming, Volume 2.

[38]  Peter Aczel,et al.  Non-well-founded sets , 1988, CSLI lecture notes series.

[39]  Alan Bundy,et al.  The Use of Explicit Plans to Guide Inductive Proofs , 1988, CADE.

[40]  Michael S. Mahoney.  The mathematical career of Pierre de Fermat, 1601-1665 , 1996 .

[41]  Claus-Peter Wirth.  Full First-Order Free Variable Sequents and Tableaux in Implicit Induction , 1999, TABLEAUX.

[42]  David Thomas,et al.  The Art in Computer Programming , 2001 .

[43]  M. Newman.  On Theories with a Combinatorial Definition of "Equivalence" , 1942 .

[44]  Kurt Von Fritz,et al.  The Discovry of Incommensurability by Hippasus of Metapontum , 1945 .

[45]  Manfred Kerber,et al.  Proof Planning: A Practical Approach to Mechanized Reasoning in Mathematics , 1998 .

[46]  Claus-Peter Wirth.  Full First-Order Sequent and Tableau Calculi with Preservation of Solutions and the Liberalized delta-Rule but without Skolemization , 1998, FTP.

[47]  Leonard Gillman,et al.  Writing mathematics well , 1987 .

[48]  J. Hintikka.  The Principles of Mathematics Revisited: Introduction , 1996 .

[49]  V. P. Orevkov.  The Calculi of symbolic logic, 1 , 1971 .

[50]  Victor J. Katz,et al.  A History of Mathematics: An Introduction , 1998 .

[51]  Tobias Schmidt-Samoa.  Flexible heuristic control for combining automation and user-interaction in inductive theorem proving , 2005 .

[52]  Frank van Harmelen,et al.  Rippling: A Heuristic for Guiding Inductive Proofs , 1993, Artif. Intell..

[53]  J. Bell,et al.  Lectures on the Foundations of Mathematics , 2003 .

[54]  G. Gibson.  The Thirteen Books of Euclid's Elements , 1927, Nature.

[55]  Serge Autexier.  On the Dynamic Increase of Multiplicities in Matrix Proof Methods for Classical Higher-Order Logic , 2005, TABLEAUX.

[56]  Bernhard Gramlich,et al.  On Notions of Inductive Validity for First-Oder Equational Clauses , 1994, CADE.

[57]  Christoph Walther,et al.  Computing Induction Axioms , 1992, LPAR.

[58]  Michaël Rusinowitch,et al.  Implicit induction in conditional theories , 2004, Journal of Automated Reasoning.

[59]  Peter Baumgartner,et al.  Computing Answers with Model Elimination , 1997, Artif. Intell..

[60]  J. A. Robinson,et al.  A Machine-Oriented Logic Based on the Resolution Principle , 1965, JACM.

[61]  Wayne Snyder,et al.  Higher-Order Unification Revisited: Complete Sets of Transformations , 1989, J. Symb. Comput..

[62]  Peter B. Andrews.  Theorem Proving via General Matings , 1981, JACM.

[63]  Bernhard Beckert,et al.  The Even More Liberalized delta-Rule in Free Variable Semantic Tableaux , 1993, Kurt Gödel Colloquium.

[64]  Claus-Peter Wirth,et al.  History and Future of Implicit and Inductionless Induction: Beware the Old Jade and the Zombie! , 2005, Mechanizing Mathematical Reasoning.

[65]  K. Barner.  Das Leben Fermats , 2001 .

[66]  Dieter Hutter,et al.  Mechanizing Mathematical Reasoning, Essays in Honor of Jörg H. Siekmann on the Occasion of His 60th Birthday , 2005, Mechanizing Mathematical Reasoning.

[67]  Martin Protzen.  Lazy generation of induction hypotheses and patching faulty conjectures , 1995, DISKI.

[68]  H. Hornich.  Die gegenwärtige Lage in der mathematischen Grundlagenforschung. — Neue Fassung des Widerspruchsfreiheitsbeweises für die reine Zahlentheorie , 1939 .

[69]  Christian G. Fermüller,et al.  Lean Induction Principles for Tableaux , 1997, TABLEAUX.

[70]  Alan Robinson,et al.  Handbook of automated reasoning , 2001 .

[71]  Peter Padawitz.  Inductive Theorem Proving for Design Specification , 1996, J. Symb. Comput..

[72]  Bernhard Gramlich,et al.  A Guide to UNICOM, an Inductive Theorem Prover Based on Rewriting and Completion Techniques , 1999 .

[73]  Tobias Schmidt-Samoa,et al.  An Even Closer Integration of Linear Arithmetic into Inductive Theorem Proving , 2006, Calculemus.

[74]  Victor J. Katz.  The history of mathematics , 1992 .

[75]  Joseph R. Shoenfield,et al.  Mathematical logic , 1967 .

[76]  Gernot Salzer,et al.  Automated Deduction in Classical and Non-Classical Logics , 2002, Lecture Notes in Computer Science.

[77]  Peter B. Andrews.  An introduction to mathematical logic and type theory - to truth through proof , 1986, Computer science and applied mathematics.

[78]  Andrei Voronkov,et al.  The Inverse Method , 2001, Handbook of Automated Reasoning.

[79]  Alexander Leitsch,et al.  The Resolution Calculus , 1997, Texts in Theoretical Computer Science An EATCS Series.

[80]  Melvin Fitting,et al.  On Quantified Modal Logic , 1999, Fundam. Informaticae.

[81]  Alan Bundy,et al.  The Design of the CADE-16 Inductive Theorem Prover Contest , 1999, CADE.

[82]  Deepak Kapur,et al.  An Overview of Rewrite Rule Laboratory (RRL) , 1989, RTA.

[83]  G. Gentzen.  Untersuchungen über das logische Schließen. I , 1935 .

[84]  A. O. Slisenko,et al.  An Algorithm for a Machine Search of a Natural Logical Deduction in a Propositional Calculus , 1983 .

[85]  F. Acerbi,et al.  Plato: Parmenides 149a7-c3. A Proof by Complete Induction? , 2000 .

[86]  Serge Autexier,et al.  The CoRe Calculus , 2005, CADE.

[87]  W. Bibel,et al.  Automated deduction : a basis for applications , 1998 .

[88]  Bernhard Gramlich,et al.  A Constructor-Based Approach for Positive/Negative-Conditional Equational Specifications , 1992, CTRS.

[89]  Andreas. Nonnengart.  Strong Skolemization , 1996 .

[90]  Nachum Dershowitz,et al.  In handbook of automated reasoning , 2001 .

[91]  Kurt von Fritz.  The discovery of incommensurability by Hippasus of Metapontum , 2004 .

[92]  Claus-Peter Wirth,et al.  How to Prove Inductive Theorems? QUODLIBET! , 2003, CADE.

[93]  Christoph Walther,et al.  About eriFun , 2003 .

[94]  W. V. Quine,et al.  Natural deduction , 2021, An Introduction to Proof Theory.

[95]  Ulrich Kühler.  A tactic-based inductive theorem prover for data types with partial operations , 2000, DISKI.

[96]  Wilhelm Ackermann,et al.  Zur Widerspruchsfreiheit der Zahlentheorie , 1940 .

[97]  K. Gödel.  Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I , 1931 .

[98]  Bernhard Gramlich,et al.  A Constructor-Based Approach to Positive/Negative-Conditional Equational Specifications , 1994, J. Symb. Comput..

[99]  Hantao Zhang,et al.  An overview of Rewrite Rule Laboratory (RRL) , 1995 .

[100]  Peter H. Schmitt,et al.  The liberalized δ-rule in free variable semantic tableaux , 2004, Journal of Automated Reasoning.

[101]  K. Gödel.  Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I , 1931 .

[102]  Dale Miller,et al.  Unification Under a Mixed Prefix , 1992, J. Symb. Comput..

[103]  Martin Protzen,et al.  Lazy Generation of Induction Hypotheses , 1994, CADE.

[104]  Lincoln A. Wallen,et al.  Automated proof search in non-classical logics - efficient matrix proof methods for modal and intuitionistic logics , 1990, MIT Press series in artificial intelligence.

[105]  Christoph Kreitz,et al.  Matrix-Based Inductive Theorem Proving , 2000, TABLEAUX.

[106]  Krzysztof Gra̧bczewski,et al.  Equivalents of the axiom of choice , 2001 .

[107]  Marta Cialdea Mayer,et al.  Free-Variable Tableaux for Constant-Domain Quantified Modal Logics with Rigid and Non-rigid Designation , 2001, IJCAR.

[108]  M. E. Szabo,et al.  The collected papers of Gerhard Gentzen , 1969 .

[109]  Leo Bachmair,et al.  Proof by consistency in equational theories , 1988, [1988] Proceedings. Third Annual Information Symposium on Logic in Computer Science.

[110]  Christoph Benzmüller,et al.  Higher-order semantics and extensionality , 2004, Journal of Symbolic Logic.

[111]  Paolo Bussotti,et al.  From Fermat to Gauss: indefinite descent and methods of reduction in number theory , 2006 .