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Sometimes when one receives referee's reports, the referee supplies alternative proofs for the results of the paper. Sometimes the supplied proof is much shorter and more elegant than the original proof.

Is it acceptable to include these proofs in the paper (without formally asking the permission from the reviewer) if the author emphasizes that this proof comes from the anonymous reviewer?

Many times I have done this, and the reviewer has never objected. But I am wondering if one should be more cautious and first formally asks a permission and if the referee permits then includes the referee's proof.

If that matters the subject is Mathematics.

Edit: Side information (following the comment). Assuming that the supplied proofs are for the auxiliary results of the paper, not the main results.

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I would think that if the referee shared an alternative proof in their comments to the author, their very reason for doing so was so that you could include it in the paper. What else would be the point?

I don't think you need to ask for permission. It would be sort of like asking for permission to take a mint from that little bowl by the door of a restaurant - that's what it's there for. You can ask if you really feel it's important, but it creates a slight amount of extra work for the referee and the editor who has to relay your communications.

If the alternative proof is such a major contribution that you think the referee ought to become a co-author, then of course that is a separate discussion.

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You can definitely include the superior proof, with attribution to the anonymous referee. Referee comments are made for the purpose of improving the paper, and revisions based on referee comments are expected.

Giving you a better proof of a theorem is above and beyond the call of duty, so I can understand why you might want to give a greater acknowledgement in that case. If you decide you would like to give a named acknowledgement for the referee (or even make them a co-author) then write back to the editor and propose that you would like to include a named acknowledgement for this part, and ask if the referee is willing to identify himself/herself so that you can give an acknowledgement by name.

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Since the result in question is a side result, not a main one, I don't think you're required to ask permission, but I would suggest that you do so anyway, as a matter of courtesy. And the paper should say explicitly what the referee contributed --- a simplified (or improved or ...) proof of such-and-such result.

Looking at it from the referee's point of view, if an author asked me for permission, I'd give it. If an author used the proof from my report without asking for permission, I wouldn't worry about it. If the paper didn't say that this proof came from a referee, I'd be unhappy, not because I lose credit (an acknowledgement for an anonymous referee doesn't gain me any credit anyway) but because the author falsely makes it look as if (s)he invented that proof.

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