Take your software.

Rigorously define security goals ("no user can see another user's file; no input can cause this handler to access variables other than this specific list").

Convert all or, more likely, part of your program into a model; for instance, predicate logic.

Run analysis tools to verify that the model achieves the security goals. Once expressed in a mathematical model, "proving" it becomes a search problem, achievable with code. Intuition and cleverness can, if your model is accurate and complete, be eliminated.

It's sort of the software security equivalent of teaching a computer to play chess.