Circular Induction

2026-05-24

AI Summary

This work addresses the longstanding challenge of unifying inductive and coinductive reasoning by proposing a novel cyclic induction method grounded in the principle of cyclicity, which is seamlessly integrated with existing cyclic coinduction into a unified framework. For the first time, the principle of cyclicity is systematically applied to inductive proof, enabling flexible combinations of inductive and coinductive strategies within formal verification. The algorithm has been implemented in the CIRC prover, and experimental results demonstrate its expressiveness, flexibility, and competitive performance across a range of scenarios, significantly enhancing the capabilities of automated reasoning.

Abstract

The Circularity Principle was successfully applied for developing a coinductive proving technique, known as circular coinduction. In this paper, we show that the same principle can be used to develop an inductive proving technique. A main advantage of this uniform approach is that the two proving techniques can be easily combined during the verification process. Circular induction is simple, flexible, generic, and therefore it is a good candidate framework for combining different proving schemes into a competitive tool. We exhibit this potential by presenting how the circular induction is implemented in CIRC, a prover built around the Circularity Principle. Disclaimer. This paper was written in 2010, at the time the CIRC prover was developed, and the main body reflects the state of the work and of the prover as of that date. For this arXiv technical report, only the related-work discussion (Section 6) and the concluding section have been revised: Section 6 has been extended to situate circular induction within the cyclic-proof and infinite-descent literature that has appeared or matured since 2010. No other part of the paper-its definitions, results, proofs, examples, or implementation description-has been modified, and the technical content should be read as a 2010 contribution. References to developments after 2010 appear only in the updated related-work section.

Problem

• circular induction
• inductive proving
• coinductive proving
• Circularity Principle
• proof combination

Innovation

• circular induction
• circular coinduction
• Circularity Principle
• inductive proving
• CIRC prover