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I made this grammar:

$S \rightarrow ASa$

$S \rightarrow c$

$A \rightarrow a|b$

And I want to check that it accepts words like $aacaa$, $abcaa$, $babcaaa$, I formed the grammar by thinking about the typical $ a ^ nb ^ n $,but I added what is necessary for my interest.

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It looks like the grammar indeed accepts all words of the form $[b+a]^nca^n$ (which means, all words that start with any sequence of $n$ $b$'s and $a$'s, and then a single $c$ and afterward exactly $n$ times the letter $a$).

To show why to try to show the two following things:

  1. every word accepted by the grammar must be in such form

  2. every word with such form has a derivation sequence in the grammar.

The first statement can be easily proved by induction (over sequence derivation length) if you notice that each derivation of $S$ adds only one element to both sides.

The second statement can be much more easily proved, try to think of what derivations are necessary to create such words.

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