Lets say I am doing a reduction from $\mathrm{HALT}_{\mathrm{TM}}$ to another language $S$, in order to prove that $S$ is not decidable. For this I need to build a new Turing machine, $M'$. Can I create $M'$ in such a way that the next step is depended upon whether or not a simulation stops? Take the following two examples:
$M' = $ "on input $M, w$:
- Simulate $M$ on input $w$. If $M$ stops, go to step 2. Else: go to step 3.
- ....
- ....."
And
$M'' = $ "on input $M, w$:
- ...
- Run $M$ on input $w$.
- accept."
The first one requires the decider for $S$ to solve $\mathrm{HALT}_{\mathrm{TM}}$ more explicitly then the second one. Is the first one wrong?