Bob is accused of a crime (let's say murder). Bob is a self-hating man who wants to be punished. He admits to the crime, and hires a lawyer to push forth his guilty plea. The prosecution, however, doesn't want to convict him of the crime, because they believe that the true culprit is still out there, and that Bob is just being ridiculous.
In this case, is it possible to have a trial where the prosecution is trying to prove Bob's innocence, while the defense is trying to prove Bob's guilt?
Edit: if it's so hard to believe that "the state" would prosecute for a case like this, then assume that the case is between two private parties. E.g. Bob admits to murdering Penny's pet cat. Penny trusts Bob, and wants to prove that he didn't murder her cat.