in the solution of exercise 4 question 1b you say:
Halt* <=m ALLTM ( f(<M>)=<M0> where M0 on input y simulates M on input y moving qr when M moves to qa.
shouldn't qr and qa be replaced?
because in this form:
M halts on all inputs »> M0 does not accept anything
M does not halt on all inputs »> M0 don't halts on all inputs and if it halts it rejects.
and in the replaced form:
M halts on all inputs »> M0 accepts anything ( its language is sigma star)
M does not halt on all inputs »> M0 don't halts on all inputs and if it halts it accepts.(its language is not sigma star)