I have the language $\Sigma = \{0,1,+,= \}$ and $$\mathrm{ADD} = \{x = y + z \mid \text{$x$, $y$, $z$ are binary integers and $x$ is the sum of $y$ and $z$}\}$$
And with the pumping lemma I find what a think is a counter example to the $xy^iz \in \mathrm{ADD}$ with $i = 2$, $p = 4$ and $s = 0+1^p$:
$x = 0, y=+1, z= 111 \rightarrow 0+1+1111$
Which becomes the addition of 3 binary integers so I would say that it does not belong in ADD and so the language is not regular. Or is it still regular cause I can just simplify things down to the addition of two binary integers, e.g 1+1111 or 0+10000
Update on comment from Yuval:
So if I can only assuming a P satisfy the lemma then with
s = $0+1^p$ can I still assume that say $x=0$,$y=+1^p$,$z=\{\}$?
Then again in general for any $i \ge 0$ we can have something in the form if $i=2$
$0+1^p+1^p$
which again is brings me back to my first question if that is still considered in ADD or not as it is addition of 3 binary integers.
