Is a brute force method considered a proof? Say we have some finite set, and some theory about a set, say "All elements of the finite set $X$ satisfy condition $Y$".
If we let a computer check every single member of $X$ and conclude that the condition $Y$ holds for all of them, can we call this a proof? Or is it possibly something else?

Using a computer to brute-force can be the first step to a proof. The next step is to prove that the program is correct!
A few ways you might do this are:


*

*Have the program output a proof for each member of the set. We can then check these proofs without having to trust the program at all. We could even send them all through an automated proof checker, which of course would also need to be proven correct! This may be worth doing, since proof checkers are generally simpler (easier to prove correct) and more general than proof finders; you might output proofs in a format understood by an existing proof checker.

*Prove that the program is correct for each member of the set. This might defeat the point of using a program in the first place!

*Prove that the program is correct for all possible inputs. This can be a good strategy, since the program only needs to simplify the problem, it doesn't need to solve it. For example, we might prove that our program returns "TRUE" if our property holds and "FALSE" if it doesn't; we specifically don't have to prove the more difficult part about which one it will actually return. To do that, we just run it.

