Not to prove it always-winning; you only have to go down every possible branch for the other player for that (i.e. √ of the number of states you'd otherwise need). To find it, though? Well, unless you hit lucky, you're going to be considering pretty much every possible game or state, except where you can find shortcuts.
It is a bit more than that as it is unlikely that the always win sequence would be a fixed list of moves. You would need to show that for you each move the other player makes you have a reaction that also wins.
