I was trying to prove that Parikh Image of every regular language is semi-linear. Even though it is true for CFL, but this question was about regular languages. To prove this, I decided to proceed as follows:
Lemma 1: Every regular expression can be written as sum of products with star, i.e., $R = R_1 + R_2+ ... +R_n $ where each $R_i$ is a regular expression not involving any $+$.
Lemma 2: The Parikh image of a language corresponding to a regular expression that doesn't involve any $+$ is linear.
I proved the lemma 2 by induction on the structure of this specific type of regular expression. (which was easy and correct, by the way).
After proving (allegedly) these, it was easy to state that the Parikh image of the language defined by $R$ is just the union of linear sets, hence it is semi-linear.
I proved the lemma 1, again, using induction on the structure of the regular expression, but I am not sure if it is correct. Moreover, I am not sure if the Lemma 1 is even valid or not!
My proof was as follows:
$R = a $ where $ a \in \Sigma$ or $a=\epsilon$. This is already in the same form we required.
$R = R_1.R_2$ . This is also in required form.
$R = (R_1 + R_2).R_3$. Replace by $R_1.R_3 + R_2.R_3$.
$R = (R_1 + R_2)^* $ Replace by $(R_1^*.R_2^*)^*$
Do you think this proof is wrong, or even worse, the statement if wrong!