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I'm trying to prove that:
$L=\{w\in\{a,b,c\}^*\Big|\#_a(u)=\#_b(v),\ \ w=ucv,\ \ \ u,v\in\{a,b\}^*\}$ is irregular, so I'm trying to use the Pumping Lemma.

This is what I tried so far:
$w=a^ncb^n$, then $|w|\ge n$ and $w\in L$.

By, condition 1 and 2 (of the Lemma) we will get that: $w=xyz$ - such that: $x=a^r,\ y=a^s, \ z=a^{n-r-s}cb^n$. ($r+s\le n,\ s>0$)
Then, by condition 3: for any $i\ge 0$ : $xy^iz\in L$.
We will take $i=0$ then - we will get: $xz=a^ra^{n-r-s}cb^n\in L$.
That's means that: $a^{n-s}cb^n\in L$ - contradict!!

This what I did, but I think that I miss something...
Can you please guide me and tell me where to look at?

Raphael
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stud1
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    You may be interested in our reference question. – Raphael Dec 13 '15 at 00:42
  • Oh, I saw it! :-) but I'm still thinking that I miss something, I look at my proof and saying to myself "It's too good to be truth". I don't see why this incorrect, but I feel that I miss something at my proof, and this why I wrote this question - because I'd like to get little help :-) – stud1 Dec 13 '15 at 00:52

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