Which word could I use for the Pumping Lemma proof?
If we want to use the usual pumping lemma for regular languages as introduced in most textbooks to prove a given language is not regular, the way to avoid or reduce case distinction in showing a word cannot be pumped 
is to 


select that word so that its front part is in a very restricted form and at least at long as the pumping length. 
combine all cases into very few or only one case conceptually.


Let me take the language in the question as an example.

To show $A_{1} \triangleq\left\{a w c^{l} d^{m}\mid l \in \mathbb{N} \wedge m \in \mathbb{N}^{+} \wedge w \in\{a, b\}^{*} \wedge |\left.w\right|_{a}=l+m\right\}$ is not regular, let us suppose our adversary Adriana claims $A$ is regular with pumping length $p$. In order to beat Adriana, we select word $w=aa^pd^p\in A_1$. Adriana claims that $w=xyz$ for some $x,y,z$ such that $|xy|\le p$, $|y|\ge1$ and for all $n\ge0$, $xy^nz\in A$.

It looks like we have to make a case distinction, because Adriana could decompose the word in many ways. For example, Adriana could have any one of the following decompositions.


$ x=\epsilon, y=a^{j}$ and $  z=a^{p+1-j} d^{p}.$
$ x=a, y=a^{j}$ and $  z=d^{p-j} d^{p}.$
$x=aa^i, y=a^{j}$ and $  z=d^{p-i-j} d^{p}.$
$\cdots$.


We have to prove Adriana is wrong in every possible case. There is no way around that, as Adrianna is free to choose any one of them.

However, a little dose of abstraction and intelligence will enable us to treat all cases simultaneously.

Since $|xy|\le p$ and all $p$ letters at the front of $w$ are $a$'s, $y$ must contain $a$ only. Suppose $y=a^i$ for some $i&gt;0$. Then $xz=xy^0z$ contains less number of $a$'s than $w$ but the same number of $c$ and $d$ as $w$. That means $xz\not=A_1$. We beat Adriana as if she has only one case.

Hopefully this strategy answers your question. Indeed, it should be applicable to almost all similar problems in the course of your study where you want to prove a language is not regular.



Here are a few exercises about the languages you have encountered.

Exercise 1. Prove the language $C \triangleq\left\{a^{m} b^{n} c d^{n} d^{m} \mid n, m \in \mathbb{N} \wedge m \leq 3\right\}$ is not regular "without case distinction".


Exercise 2. Prove the language $B \triangleq\left\{w \in \Sigma^{*} \mid |\left.w\right|_{b} \neq|w|_{c} \right\}$ is not regular "without case distinction".


Exercise 3. Prove the language $A \triangleq\left\{(a b)^{n} c^{n} | n \in \mathbb{N}\right\}$ is not regular using two combined cases.