There are two meanings of prove. The one that type theorists are using is roughly "to derive your statement from axioms with pure logic." This sense of the term can never apply to things in the real world, like programs (of course, a program also exists both as an abstraction about which things can be proved, but when you're talking about programs which are actually doing things in the real world, you can't treat them as being pure logic).

The second sense is the scientific "gather enough supporting evidence that your are reasonably sure". In this sense, you can prove a lot of things.