What went into Tool?

Well, I am trying to get away from my obsession with Tool - by trying to understand in which way they are "prog". The point of all this stupid exercise is that I want to find out why the hipster press - at least here in Germany - behaves so strangely on the prog matter. They utterly ignore prog bands such as Dream Theater or Spock's Beard, yet review bands that are labeled "prog rock" or "prog metal" and more or less all sound like Tool, who - it seems to me; I may be wrong about that - were the pioneers of this trend.

I have come up with various ideas why Tool are called "prog", most of which failed. It doesn't seem to have anything to do with the way "prog" is used in classifying electronic dance music, so much I have found out. Apparently, Tool are considered "prog" because they go beyond the format of the standard rock song, sometimes use odd time signatures, and their playing is also influenced by King Crimson and Rush.

Yet, I feel that there are some things essential to prog - the changeful dramaturgy, the progressive mind set - that are absent from Tool and their ilk. But that is surely a subjective notion. However, I feel that if Tool are "prog", many other artists that are usually not considered "prog", such as Velvet Underground, could also put claims on being "prog".

My latest idea on music classification is that within "artistic rock music" there is a dark "yin" to the bright "yang" of prog. This "yin" is the tradition that started with VU, and moved on through industrial and noise to shoegazing, post-rock and whatever came after that. And I feel that Tool are closer to this "yin" than to the "yang" of prog. This may be utter hogwash, though.

^Definitely.

The OP seems obsessed with this band.

I just don't know where to put this stuff. What are in your opinion the main musical influences and tradition that went into the music of Tool? Velvet Underground? Industrial? Noise rock? The Melvins? King Crimson? What else?