Prove is a strong word. There are few cases in real-world programming where you can prove anything.

I prefer to make this probabilistic: use testing to reduce the probability that your code isn't correct, for the situations in which it is expected to be deployed. In this sense, coding and testing is much like doing experimental physics: we never really prove a theory or disprove it, we just invalidate clearly wrong ones.

Testing must cover all cases else a 10 LOC LLM created PR is inherently more dangerous than a human 100 LOC PR because the LLM will likely also have written the test cases and it will try to make it all balance out with all passing; instead of making sure the test cases actually cover everything with the type of logic a human would apply.