which looks for a non-counting violation of $\M$’s commands (which is a simple regular check). Once reaching a violation it goes to $heaven$. All of its transitions are with weight $0$. If it does not find a violation, it cannot continue the run.

-   The state $q_3$ is a positive-cheat checker for $\M$’s counter $x$. It has a self loop over with weight $+1$ and over with weight $-1$. Over it can nondeterministically choose between a self loop with weight $0$ and a transition to $q_0$ with weight $-1$. Over the rest of the alphabet lettres, except for and $\#$, it has a self loop with weight $0$. (Over and $\#$ it cannot continue the run.)

-   The state $q_4$ is a negative-cheat checker for $\M$’s counter $x$. It has a self loop over with weight $-1$ and over with weight $+1$. Over it can nondeterministically choose between a self loop with weight $0$ and a transition to $q_0$ with weight $0$. Over the rest of the alphabet lettres, except for and $\#$, it has a self loop with weight $0$.

-   The states $q_5$ and $q_6$ provide positive-cheat checker and negative-cheat checker for $\M$’s counter $y$, respectively, analogously to 