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A Divergence Critic for Inductive Proof
 Journal of Artificial Intelligence Research
, 1996
"... Inductive theorem provers often diverge. This paper describes a simple critic, a computer program which monitors the construction of inductive proofs attempting to identify diverging proof attempts. Divergence is recognized by means of a "difference matching" procedure. The critic then pro ..."
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Inductive theorem provers often diverge. This paper describes a simple critic, a computer program which monitors the construction of inductive proofs attempting to identify diverging proof attempts. Divergence is recognized by means of a "difference matching" procedure. The critic then proposes lemmas and generalizations which "ripple" these differences away so that the proof can go through without divergence. The critic enables the theorem prover Spike to prove many theorems completely automatically from the definitions alone. 1. Introduction Two key problems in inductive theorem proving are proposing lemmas and generalizations. A prover's divergence often suggests to the user an appropriate lemma or generalization that will enable the proof to go through without divergence. As a simple example, consider the theorem, 8n : dbl(n) = n + n: This is part of a simple program verification problem (Dershowitz & Pinchover, 1990). Addition and doubling are defined recursively by means of th...
A Divergence Critic
 Procedings of the Twelfth International Conference on Automated Deduction
, 1994
"... Abstract. Inductive theorem provers often diverge. This paper describes a critic which monitors the construction of inductive proofs attempting to identify diverging proof attempts. The critic proposes lemmas and generalizations which hopefully allow the proof to go through without divergence. The c ..."
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Abstract. Inductive theorem provers often diverge. This paper describes a critic which monitors the construction of inductive proofs attempting to identify diverging proof attempts. The critic proposes lemmas and generalizations which hopefully allow the proof to go through without divergence. The critic enables the system SPIKE to prove many theorems completely automatically from the denitions alone. 1
Proof Plans for the Correction of False Conjectures
 5TH INTERNATIONAL CONFERENCE ON LOGIC PROGRAMMING AND AUTOMATED REASONING, LPAR'94, LECTURE NOTES IN ARTIFICIAL INTELLIGENCE, V. 822
, 1994
"... Theorem proving is the systematic derivation of a mathematical proof from a set of axioms by the use of rules of inference. We are interested in a related but far less explored problem: the analysis and correction of false conjectures, especially where that correction involves finding a collection o ..."
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Cited by 12 (7 self)
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Theorem proving is the systematic derivation of a mathematical proof from a set of axioms by the use of rules of inference. We are interested in a related but far less explored problem: the analysis and correction of false conjectures, especially where that correction involves finding a collection of antecedents that, together with a set of axioms, transform nontheorems into theorems. Most failed search trees are huge, and special care is to be taken in order to tackle the combinatorial explosion phenomenon. Fortunately, the planning search space generated by proof plans, see [1], are moderately small. We have explored the possibility of using this technique in the implementation of an abduction mechanism to correct nontheorems.
Termination Orderings for Rippling
 In Proceedings of the 12th International Conference on Automated Deduction (CADE12), LNAI 814
, 1994
"... Abstract. Rippling is a special type of rewriting developed for inductive theorem proving. Bundy et. al. have shown that rippling terminates by providing a wellfounded order for the annotated rewrite rules used by rippling. Here, we simplify and generalize this order, thereby enlarging the class of ..."
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Abstract. Rippling is a special type of rewriting developed for inductive theorem proving. Bundy et. al. have shown that rippling terminates by providing a wellfounded order for the annotated rewrite rules used by rippling. Here, we simplify and generalize this order, thereby enlarging the class of rewrite rules that can be used. In addition, we extend the power of rippling by proposing new domain dependent orders. These extensions elegantly combine rippling with more conventional term rewriting. Such combinations offer the flexibility and uniformity of conventional rewriting with the highly goal directed nature of rippling. Finally, we show how our orders simplify implementation of provers based on rippling.
Generalization Discovery for Proofs by Induction in Conditional Theories
, 1999
"... Several induction provers have been developed to automate inductive proofs (see for instance: Nqthm, RRL INKA, LP, SPIKE, CLAMOyster, ...). However, inductive theorem provers very often fail to terminate. A proof to go through requires either additional lemmas, a generalization, a suitable ind ..."
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Several induction provers have been developed to automate inductive proofs (see for instance: Nqthm, RRL INKA, LP, SPIKE, CLAMOyster, ...). However, inductive theorem provers very often fail to terminate. A proof to go through requires either additional lemmas, a generalization, a suitable induction variable to induce upon , or a case split. The aim of this paper is to present a simple and powerful heuristic that allows to overcome, in many cases, the divergence of induction provers when working with conditional theories. We first provide a new definition of induction variables and then formalize a new transition rule for induction (named CGTrule). The essential idea behind it is to propose a generalized form of the conclusion just before another induction is attempted and failure begins. This generalized form is based on the induction hypothesis and the current goal. CGTrule enables to prove many theorems completely automatically from the functions definitions alo...
The Problem
"... Several induction provers have been developed to automate inductive proofs (see for instance: Nqthm, RRL INKA, LP, SPIKE, CLAMOyster,...). However, inductive theorem provers very often fail to terminate. A proof to go through requires either additional lemmas, a generalization, a suitable induction ..."
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Several induction provers have been developed to automate inductive proofs (see for instance: Nqthm, RRL INKA, LP, SPIKE, CLAMOyster,...). However, inductive theorem provers very often fail to terminate. A proof to go through requires either additional lemmas, a generalization, a suitable induction variable to induce upon, or a case split. The aim of this paper is to present a simple and powerful heuristic that allows to overcome, in many cases, the divergence of induction provers when working with conditional theories. We first provide a new definition of induction variables and then formalize a new transition rule for induction (named CGTrule). The essential idea behind it is to propose a generalized form of the conclusion just before another induction i s attempted and failure begins. This generalized form i s based on the induction hypothesis and the current goal. CGTrule enables to prove many theorems completely automatically from the functions definitions alone. We illustrate computer applications to the correctness proof of the insertion sorting algorithm and other programs computing on lists and numbers. All of them have never proved before without userprovided generalizations and/or lemmas. (Content Areas: Automated Reasoning, Theorem Proving)