# Proving mutual termination

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## Abstract

Two programs are said to be *mutually terminating* if they terminate on exactly the same inputs. We suggest inference rules and a proof system for proving mutual termination of a given pair of procedures \(\langle \) \(f\), \(f'\) \(\rangle \) and the respective subprograms that they call under a free context. Given a (possibly partial) mapping between the procedures of the two programs, the premise of the rule requires proving that given the same arbitrary input *in*, *f*(*in*) and \(f'(in)\) call procedures mapped in the mapping with the same arguments. A variant of this proof rule with a weaker premise allows to prove termination of one of the programs if the other is *known* to terminate. In addition, we suggest various techniques for battling the inherent incompleteness of our solution, including a case in which the interface of the two procedures is not identical, and a case in which partial equivalence (the equivalence of their input/output behavior) has only been proven for some, but not all, the outputs of the two given procedures. We present an algorithm for decomposing the verification problem of whole programs to that of proving mutual termination of individual procedures, based on our suggested inference rules. The reported prototype implementation of this algorithm is the first to deal with the mutual termination problem.

## Keywords

Regression-verification Program termination Mutual termination## Notes

### Acknowledgments

This material is based on research sponsored by the Air Force Research Laboratory, under agreement number FA8655-11-1-3006. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon.

### Compliance with Ethical Standards

### Conflicts of interest

The authors declare that they have no conflict of interest.

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