With all due respect I think you dodged the hypothetical. In my hypothetical the homeless bum will amount to nothing, and the other child will cure cancer.

These are certainties. Of course in real life there are no such certainties but my hypothetical merely showed that one life may be worth more than another.

Your example of the "simple housewife who gives birth to the child that cures cancer" only serves to prove my point, that ultimately some lives are worth more than others.

I wasn't using that as the basis of arguing for increased protections, because I don't believe in greater punishment because one life is worth more than another. My argument for greater punishments is more subtle than that.

I don't think you were unsuccessful in making your point, I understood it, I simply questioned whether it's naive to assume that the greater sentences imply that the life taken was more "important". That's not the case at all.

As a lawyer, and as one who is already not inclined to go into public prosecutions I'd be even less inclined to do so if killing a prosecutor didn't result in greater punishment. If you make a living putting criminals in prison you take the risk of reprisal attacks on your life. The increased punishments act as an incentive for people to take on those jobs.

You're correct to mention the added training and protections police officers get. Nonetheless, what about the situation where an officer is trying to negotiate the surrender of a murderer? If that murderer knows that they'll face no greater punishment for shooting the police officer than for the murders they have already committed they're less likely to surrender peacefully. That's the reason for the protections.

Going back to my earlier point, what sort of body armour and training do public prosecutors (or judges) get? As far as I'm aware in most jurisdictions they have no greater rights with regards to weapons or protection than anyone else. The greater protection is not there because these lives are worth more, but simply to make sure that we have people willing to become judges and prosecutors.

Whilst I appreciate your point as an ideal I think it is predicated upon a misunderstanding over why these greater punishments exist. I think your analysis is a tad naive.