"Progress" in this case would just be defined as moving forward and actively trying to complete the levels, in strict contrast to standing still in order to run up the timer. The levels are being played throughout the video and Mario reaches the end of each one, so it counts as progressing each time.
I'm gonna respond to this post and a few similar ones (absolutely not picking on this poster at all; just using it as an example since this argument has been made a few times in this thread)
The point of a speedrun isn't to learn how to play the game. The point is to watch the game being beaten fast. If you've never played the game, then there are normal runs on Youtube that you can watch. Ocarina of Time is so massively sequence broken and glitched that even if this same run were in English, you still would not "understand" it or see how it was "supposed to be played". That's why I don't think that this type of argument holds up for voting "no" on a run like this.
A run should be judged solely on its own merits, not how many bits and pieces of the original game you're able to pull out of it.
That is intentional, as the moonwalk is indeed a real technique in Melee.
The current video that Wak just uploaded is good, except the sound of 5 Mr. Game & Watches getting shined at once makes my ears cry.