peer review - Minor Revision with suggestion of an alternative proof by reviewer

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

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.