### Abstract

Speculating intermediate lemmas is one of the main reason of user interaction/guidance while mechanically attempting proofs by induction. An approach for generating intermediate lemmas is developed, and its effectiveness is demonstrated while proving properties of recursively defined functions. The approach is guided by the paradigm of attempting to generate a proof of the conclusion subgoal in an induction step by the application of an induction hypothesis (es). Generation of intermediate conjectures is motivated by attempts to find appropriate instantiations for non-induction variables in the main conjecture. In case, the main conjecture does not have any non-induction variables, such variables are introduced by attempting its generalization. A constraint based paradigm is proposed for guessing the missing side of an intermediate conjecture by identifying constraints on the term schemes introduced for the missing side. Definitions and properties of functions are judiciously used for generating instantiations and intermediate conjectures. Heuristics are identified for performing such analysis. The approach fails if appropriate instantiations of non-induction variables cannot be generated. Otherwise, proofs of intermediate conjectures are attempted and the proposed method is recursively applied. The method has proven to be surprisingly effective in speculating intermediate lemmas for tail-recursive programs.

Original language | English (US) |
---|---|

Title of host publication | Automated Deduction – Cade-13 - 13th International Conference on Automated Deduction, Proceedings |

Editors | John K. Slaney, Michael A. McRobbie |

Publisher | Springer Verlag |

Pages | 538-552 |

Number of pages | 15 |

ISBN (Print) | 3540615113, 9783540615118 |

DOIs | |

State | Published - Jan 1 1996 |

Event | 13th International Conference on Automated Deduction, CADE 1996 - New Brunswick, United States Duration: Jul 30 1996 → Aug 3 1996 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
---|---|

Volume | 1104 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 13th International Conference on Automated Deduction, CADE 1996 |
---|---|

Country | United States |

City | New Brunswick |

Period | 7/30/96 → 8/3/96 |

### Fingerprint

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Automated Deduction – Cade-13 - 13th International Conference on Automated Deduction, Proceedings*(pp. 538-552). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 1104). Springer Verlag. https://doi.org/10.1007/3-540-61511-3_112

**Lemma discovery in automating induction.** / Kapur, Deepak; Subramaniam, M.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Automated Deduction – Cade-13 - 13th International Conference on Automated Deduction, Proceedings.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 1104, Springer Verlag, pp. 538-552, 13th International Conference on Automated Deduction, CADE 1996, New Brunswick, United States, 7/30/96. https://doi.org/10.1007/3-540-61511-3_112

}

TY - GEN

T1 - Lemma discovery in automating induction

AU - Kapur, Deepak

AU - Subramaniam, M.

PY - 1996/1/1

Y1 - 1996/1/1

N2 - Speculating intermediate lemmas is one of the main reason of user interaction/guidance while mechanically attempting proofs by induction. An approach for generating intermediate lemmas is developed, and its effectiveness is demonstrated while proving properties of recursively defined functions. The approach is guided by the paradigm of attempting to generate a proof of the conclusion subgoal in an induction step by the application of an induction hypothesis (es). Generation of intermediate conjectures is motivated by attempts to find appropriate instantiations for non-induction variables in the main conjecture. In case, the main conjecture does not have any non-induction variables, such variables are introduced by attempting its generalization. A constraint based paradigm is proposed for guessing the missing side of an intermediate conjecture by identifying constraints on the term schemes introduced for the missing side. Definitions and properties of functions are judiciously used for generating instantiations and intermediate conjectures. Heuristics are identified for performing such analysis. The approach fails if appropriate instantiations of non-induction variables cannot be generated. Otherwise, proofs of intermediate conjectures are attempted and the proposed method is recursively applied. The method has proven to be surprisingly effective in speculating intermediate lemmas for tail-recursive programs.

AB - Speculating intermediate lemmas is one of the main reason of user interaction/guidance while mechanically attempting proofs by induction. An approach for generating intermediate lemmas is developed, and its effectiveness is demonstrated while proving properties of recursively defined functions. The approach is guided by the paradigm of attempting to generate a proof of the conclusion subgoal in an induction step by the application of an induction hypothesis (es). Generation of intermediate conjectures is motivated by attempts to find appropriate instantiations for non-induction variables in the main conjecture. In case, the main conjecture does not have any non-induction variables, such variables are introduced by attempting its generalization. A constraint based paradigm is proposed for guessing the missing side of an intermediate conjecture by identifying constraints on the term schemes introduced for the missing side. Definitions and properties of functions are judiciously used for generating instantiations and intermediate conjectures. Heuristics are identified for performing such analysis. The approach fails if appropriate instantiations of non-induction variables cannot be generated. Otherwise, proofs of intermediate conjectures are attempted and the proposed method is recursively applied. The method has proven to be surprisingly effective in speculating intermediate lemmas for tail-recursive programs.

UR - http://www.scopus.com/inward/record.url?scp=84957615401&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84957615401&partnerID=8YFLogxK

U2 - 10.1007/3-540-61511-3_112

DO - 10.1007/3-540-61511-3_112

M3 - Conference contribution

AN - SCOPUS:84957615401

SN - 3540615113

SN - 9783540615118

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 538

EP - 552

BT - Automated Deduction – Cade-13 - 13th International Conference on Automated Deduction, Proceedings

A2 - Slaney, John K.

A2 - McRobbie, Michael A.

PB - Springer Verlag

ER -