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I am struggling understanding how to apply to pumping lemma to a CF string.

I've got this string:

$$ a^{n}b^{n}c^{m} $$

I would like to understand the steps to apply the pumpuing lemma for this particular string and hence for any other.

At the moment I have tried to decompose the string to $u,v,w,x,y$ with a trial and error approach but I can't figure out what is the procedure to retrive the result systematically.

I know that pumping lemma is condition neccessary but not sufficient to assert that a string belgons to CF, but I would like to understand how to apply the pumpuing lemma to this particular string.

G M
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    It is trial and error; there is no algorithm to come up with suitable partitionings. Doing lots of Pumping-lemma proofs will give you the experience you need to "see" them faster. See also many questions tagged [tag:pumping-lemma]. Starting with the regular version may be advisable. – Raphael Sep 07 '16 at 09:42
  • @Raphael I have already seen that question, my question is different is about finding the solution. So if you say that it is a trial and error approch that would be an answer that is what I am actually doing but I would like also to know if other users have other answers, I really don't see why my question is a duplicate. – G M Sep 07 '16 at 10:46
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    I'm sure we answered that dozens of times already; I won't spend dozens of minutes to track one down. There are many explanations of the Pumping lemma around, on this site and elsewhere, that make clear that its application is not an algorithm. – Raphael Sep 07 '16 at 11:45
  • thanks @Raphael! However the fact that you have to spend dozens of minutes to track one down, suggests that there isn't a specific question with that answer. If you are not willing to help me let at least the other users help me. – G M Sep 07 '16 at 12:03
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    Oh no, we have a list of reference questions for those types of questions that get asked over and over again. Plus, form completion in the duplicate dialogue has a good selection of the standards in store for me. (Note against that there are plenty of questions tagged [tag:pumping-lemma] that are very similar do to yours.) – Raphael Sep 07 '16 at 12:56
  • Thanks @Raphael for the list, but noone of the specific questions that I have found face this issue; I asked a specific question for a specific string. it's easy to answer questions saying: " go and look on your own you will find something interesting somewhere else in the site". I think that this is just another case of elitism. I'll take your first comment as an answer, although the stackexchange guidelines forbid to answer questions using comments. Best regards. – G M Sep 07 '16 at 13:16
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    @GM It's not elitism, it's a desire to avoid the site filling up with questions that differ only on the specific example they ask about. The fact that every string is its own special snowflake is exactly why we don't want hundreds of questions about different specific languages: each of those questions would only be interesting to the single person who asked it; everybody else wants to know about some other language. The general principles, as explained in our reference question, are generally applicable. Specific examples aren't. – David Richerby Sep 07 '16 at 13:25
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    Also, I don't understand your question. Your reference to strings $u$, $v$, $w$, $x$, $y$ suggests that you're asking about the pumping lemma for context-free languages. However, the language ${a^nb^nc^m\mid m,n\geq 0}$ is context-free so you can't use the pumping lemma (or anything else) to prove that it's not. Are you supposed to be using the pumping lemma for regular languages to prove that your language isn't regular? – David Richerby Sep 07 '16 at 13:28
  • Plus, the language at hand can be dealt with in exactly the same way as The Example of Pumping lemma applications $a^n b^n$. You'll find full writeups in any number of places. So no, I don't think it's "elitist" at all to say that we don't need that again. (Also, I don't view elitism as bad in the context of this site: very few humans on earth, relative speaking, pursue so I don't think we should be expected to account for the needs of all the others.) – Raphael Sep 07 '16 at 14:43
  • @DavidRicherby as already written in the question: I know that pumping lemma is condition neccessary but not sufficient to assert that a string belgons to CF, but I would like to understand how to apply the pumpuing lemma to this particular string. – G M Sep 07 '16 at 15:39
  • @Raphael I clearly yes it is elitarism, because there is clearly a gap of knowledge between me and you and you are not doing anything to fill it. stackexchange was supposed to exchange knowledge not to keep it between some pro-user. There is nothing wrong in specific example, you are not supposed to answer it if you don't want but if I user want to why he can't because you decided that he hasn't to do... My question followed the guidelines of the site and it wasn't a duplicate. – G M Sep 07 '16 at 15:52

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