advice - Oneupsmanship and Publishing Etiquette

I largely agree with Igor Pak's response, and I think he has the right perspective on the matter.

Two comments: [X is the author of the first paper, Y is the OP!]

1) I'm not sure it's critical to show that your result completely subsumes the result of X if that verification is going to, say, double the length of your paper and require you to introduce a lot of terminology and notation that is far away from your approach to the problem. It is not clear what value you are adding to the community by doing this: is your proof going to be any more pleasant to read or insightful than X's?

On the other hand, I find it slightly odd that it should be as hard or harder to demonstrate that your results imply X's than it is to prove your (in fact more general) result. I'm having trouble thinking of an example of this phenomenon from my own experience. Is it possible that there's more insight to gained here, and perhaps a(n obvious!) common generalization of your two approaches?

2) A good rule about describing your own work -- borrowed from the creative writing community -- is: show, don't tell. Specifically, you should seek to minimize the number of times in which you tell the reader how she should feel about your work. Rather, you want to present the mathematical information which brings the reader to this conclusion. In this case, I would recommend putting a lot of effort into the writing of the example(s) of the novelty of your approach. The more clear and detailed your writing is, the smaller your sales pitch needs to be. Some direct comparison may be necessary, but if you stick to comparing one mathematical statement to another, then "stronger than" has an objective meaning. It seems that there is little in such a practice that could reasonably cause offense.

Unfortunately 1) and 2) are somewhat at cross-purposes to each other. Regarding 1), in lieu of writing out all the gory details it is tempting to write something like "It can be shown that in every case where X's theorem applies, so does mine". If this takes pages and pages to show, then the author might well not see / believe it at first, and it could be annoying to have to work for hours to verify someone's claim that they have trumped you.

Maybe the best solution -- although not the easiest -- is for you to carefully write up the implication that your result implies X's result in a separate document, which you submit along with the paper itself and post a copy on your webpage. This also places some of the judgment of whether this gory extra part should be included in the paper in the hands of the referee and the editor, which is perhaps as it should be.