we saw at class that HTM-all is not in RE nor CORE . not in CORE by red from HTM. and not in RE by red from HTM^c.
at the HTM^c reduction we do the following:
given <M,w> we construct M' that on input x:
1. run M on w for |x| steps
2. if M stops whitin this time M' enters loop
3. if M dont stops whitin this time then M' break and accepts.
regarding the correctness of this reduction: the side (no=>no) "if M stops on w in less then |x| steps then M' enters loop " is clear.
the side (yes=>yes) "if M dont stops on w in less then |x| steps then M' break and accepts " is not clear. what if M stops on W at X^2 step.
then <M,w> does not belong to HTM^c and not as we assumed!
can someone please explain ?