Productive use of failure in inductive proof

Proof by mathematical induction gives rise to various kinds of eureka steps, e.g., missing lemmata and generalization. Most inductive theorem provers rely upon user intervention in supplying the required eureka steps. In contrast, we present a novel theorem-proving architecture for supporting the automatic discovery of eureka steps. We build upon rippling, a search control heuristic designed for inductive reasoning. We show how the failure if rippling can be used in bridging gaps in the search for inductive proofs.

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

[2]  Andrew Ireland,et al.  The Use of Planning Critics in Mechanizing Inductive Proofs , 1992, LPAR.

[3]  Alan Bundy,et al.  Using Meta-Level Inference for Selective Application of Multiple Rewrite Rule Sets in Algebraic Manipulation , 1980, Artif. Intell..

[4]  Dieter Hutter,et al.  Guiding Induction Proofs , 1990, CADE.

[5]  Frank van Harmelen,et al.  Extensions to the Rippling-Out Tactic for Guiding Inductive Proofs , 1990, CADE.

[6]  G. H. Hardy,et al.  I.—MATHEMATICAL PROOF , 1929 .

[7]  Putnam,et al.  The Collected Papers. , 1988 .

[8]  Jane Thurmann Hesketh,et al.  Using middle-out reasoning to guide inductive theorem proving , 1992 .

[9]  Zohar Manna,et al.  The logical basis for computer programming , 1985 .

[10]  Alan Bundy,et al.  Proof Plans for the Correction of False Conjectures , 1994, LPAR.

[11]  Toby Walsh,et al.  Difference Matching , 1992, CADE.

[12]  Toby Walsh,et al.  A Divergence Critic , 1994, CADE.

[13]  Michael J. C. Gordon,et al.  Edinburgh LCF: A mechanised logic of computation , 1979 .

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

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

[16]  Dale A. Miller,et al.  AN OVERVIEW OF PROLOG , 1988 .

[17]  Alan Bundy,et al.  Using Meta-Level Inference for Selective Application of Multiple Rewrite Rules in Algebraic Manipulation , 1980, CADE.

[18]  G A Miller,et al.  A MATHEMATICAL PROOF. , 1931, Science.

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

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

[21]  Dieter Hutter,et al.  A colored version of the λ-calculus , 1997 .

[22]  Muffy Calder,et al.  Inductive Inference for Solving Divergence in Knuth-Bendix Completion , 1989, AII.