Prove that the algorithm works. That's your proof.
Gödel and Turing make strong cases that proving the algorithm works for some inputs that are correct proofs doesn't count as proof it will work for all correct proof inputs. So no, even if you "prove the algorithm works" it is not the same as a rigorous mathematical proof.
Transfering trains in the BART system, aside from timing, is essentially arbitrary. When transfers are timed you sometimes have to wait less time to get on a different train going in the same or opposite direction, but behavior is the same. Only if the train you're getting on is the last one of the night are you required to transfer at one of those stations, and that's only because those are generally the only places they wait. Any other time of day you can transfer anywhere there are two lines together, but it will probably involve waiting. Yes, less waiting than going all the way out of one's way to a transfer station, but still (for many) too much to make the gain worth the effort.
This whole mess would only make any sense at all if timing were just such that one person's train passes another's exit station at just the right time so they could swap cards through the open door, and BART is not that predictable (though it's WAY more so than any local bus system).
Back when I was poor and could barely afford public transportation when I had to use it, I thought about keeping multiple BART tickets I would use to make it look like all my longer-distance trips were actually short-distance ones that just took a long time, but I never implemented my plan. There are plenty of moral issues with stealing from a public service, but truth is those seem less important when you're a teenager with no money, but I think practicality is what ultimately prevented me from doing it.
Even if the system didn't care that a ticket was used to exit on a different day than it was used to enter (this is a big if, since BART is not a 24-hour service, making it easy to prevent such fraud), I would have to label and keep track of so many tickets and carry them with me whenever I wanted to use the train that it was really never worth it. No way was I going to get someone else involved, and the trouble of having to intentionally get off at the wrong stop sometimes was just too much trouble.
A friend of mine would just buy red children's tickets and use an x-acto knife to cut the magnetic stripe off and glue them onto standard adult blue tickets. Still stealing, still wrong, but much easier than anything suggested here.
I do not think it means what you think it means
Specifically, a dichotomy is a separation, usually a splitting of one thing into two separate and distinct parts. It usually requires that there be a choice, A or B.
It does not mean "hey, that's interesting."
Regarding the actions of the police under the covers, are these activities in general efficient uses of police time and taxpayer money? Why are undercover police spending seven years infiltrating environmental activists? Not terrorists, mind you, but activists. Another officer spent 4 years infiltrating an anti-racist group. Not racists, but people against racism. Really?
Seems like in seven years agents could infiltrate various government or corporate entities and expose enough graft that the program could pay for itself.
It's a naive, domestic operating system without any breeding, but I think you'll be amused by its presumption.