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As a reviewer suppose you receive a paper (Paper 1) where one of main results is claimed to be concurrent work with another recently published paper (Paper 2). There is sufficient evidence to establish that this claim is true: (1) The proofs of the main result in Paper 1 and Paper 2 are completely different, (2) Paper 2 became available on arXiv only a few days before Paper 1 was submitted to the conference, and (3) Paper 2's authors are all fine with the claimed concurrency.

As a reviewer, should you judge Paper 1 based on its entire contributions, or should you judge it based on the novelty compared to Paper 2? If the former, it would be a clear accept. If the latter, it would be borderline reject.

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Based on the information you provide I would recommend acceptance. The result may not be new (but only by a short time lag), but the other papers authors are OK with the coincidence. The result is interesting (else you would not clearly accept on its own) so it's good to have two different proofs.

Your response to the editor can include your argument for acceptance in spite of the recent appearance of the same result independently.

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Presumably this is mathematics or something similar. In that case, having completely different proofs is, in fact, sufficiently novel. The same might be true for other fields as well.

Evaluate the paper independently, based on its merits, but mention the other work to the editor and to the authors.

In many fields, the techniques of proof are actually more important than the results.

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