In the same way that writing a unit test for something proves your logic is correct? This is not intended as a snark or something. Just stating the obvious that your unit tests are no silver bullet to a working correct piece of software.

My 2 cents: Combine static(ish) typing with tests and a number of (semi-manual) test scenario's and you get a few steps closer to a correctly working piece of software.

Manual testing is what really confirms that your code is working properly. Automated testing verifies the conditions necessary for your code to pass manual testing. The real value of automated tests is for when you need (or someone else needs) to come back and change something.

I don't think static typing is necessary in that case, but I understand it has benefits in some situations.