Minor Revision with suggestion of an alternative proof by reviewer

Question

I submitted a paper to a mathematical journal, and recently received a Minor Revision decision from the Editor. One of the reviewers suggested an alternative and much simpler proof to my main theorem. The suggested proof greatly shortens my paper. Now, I am in a dilemma.

1) Should I retain my original proof, which I must admit in hindsight, was overly complicated? And then simply acknowledge the reviewer...

2) Should I write down the shorter and simpler proof suggested by the reviewer, and explicitly mention in the acknowledgemnts that I have used one of the reviewers' proofs?

The reason I ask is that I am worried whether going with Option 2 reduces the value/contributions of my paper (the revised one is not my proof after all!), although the Theorem and its implications stand nonetheless. Do I have a better chance of acceptance with Option 1? Is not seeing a simpler proof a ground for rejection of the paper? Any suggestions are welcome (since its a minor revision, it's due in a couple of weeks)!!

Answer 1

I would not worry about the paper getting rejected one way or the other: The reviewers gave you a shorter proof, but did not suggest that the theorem is obvious or trivial. This isn't going to change once you put their proof into the paper.

So the question is how to acknowledge the reviewer. It is not uncommon in mathematics papers to show the shortened proof and then, in the acknowledgements say "We appreciate the contributions of reviewer 2 who provided the shortened proof of theorem 4." On the other hand, if you think that coming up with this proof really required some hard work even though the reviewer has seen your proof, then maybe it is appropriate to ask the reviewer (through the editor) whether they want to become a co-author.

The final option may be to show both proofs. If you think that your proof is interesting despite being complicated because it shows connections to other areas that are of independent interest, then it's worth keeping it in. In such a case, it may be useful to show the alternative and shorter proof given by the referee in an appendix, and explain why you think it's worthwhile to show both. Remember that a paper isn't just about showing a result, but also teaching others how that result came about and what it means and implies. As such, there is a place for papers with two independent proofs of the same statement.

Answer 2

Ideally option 2 is better and it need not lessen your impact, unless that was the main theorem in the paper. However, in order to use the reviewers proof you need to be able to credit it properly and it may be hard to do that with blind review. Perhaps the editor can put you in contact with the reviewer for this purpose.

One way to present it (one I've used myself) is to give the reviewer's proof, with proper citation-attribution and then follow it with a statement that your own proof is omitted here for reasons of length.

But, I hope you are also aware that in many ways the proof technique is as important as the theorem itself as it may suggest approaches to other problems. So, it may be that your own proof is interesting in itself, even if complicated.

But, for the paper, say that you have an alternate, longer, proof, than the one presented. I see no real downside as long as you can properly cite.

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If you can't use the other proof for reasons of citation, then present your own, if the editor agrees, but say that a shorter proof was suggested by an anonymous reviewer.

Answer 3

You said that this theorem is the main theorem of the paper. I'll assume that means it's a very important part of the paper. (That's not automatically the case. A paper could present a single main theorem and dozens of consequences, and the latter might be the main point of the paper.) If my assumption is correct, it seems appropriate to offer the reviewer (through the editor) several options: (1) become a co-author of your paper, with the new proof replacing your older one, (2) allow you to use the new proof in your paper with only you as an author but with appropriate credit ("I thank the referee for greatly simplifying my original proof of Theorem 4 and for allowing me to include the simplified proof here."), or (3) insist that you use only your own proof (e.g., in case the referee wants to publish the new proof, perhaps as part of a larger study that led to it).