How to prove it has been subject to some debate for the past century, the answer is that it's context dependent to what degree you will or even can prove the program and exposed identifiers correct. Programming is a communication problem as well as a math problem, often an engineering problem too. Only the math portion can be proved, the a small by critical amount engineering portion tested.
Communication is the most important for velocity it's the difference between hand rolling machine code and sshing into a computer halfway across the world having every tool you expect. If you don't trust that webdevs know what they are doing then you can be the most amazing dev in the world you but your actual ability to contribute will be hampered. The same is true of vibe coding, if people aren't on the same page as to what is and isn't acceptable velocity starts to slow down.
Languages have not caught up to AI tools, since AI operates well above the function level, what level would be appropriate to be named and signed off on? pull request and link to the chat as a commit? (what is wrong with that that could be fixed at the name level)
Honest communication is the most important. Amazon telling investors that they use TLA+ is just signaling that they "for realz take uptime very seriously guize", "we know distributed systems" and engineering culture. The honest reality is that they could prove all their code and not IMprove their uptime one lick, because most of what they run isn't their code. It's a communication breakdown if effort gets spent on that outside a research department.