User blog comment:DrCeasium/Hyperfactorial array notation: Analysis part 2/@comment-5529393-20130527105642/@comment-5529393-20130530150341

I still need explanation with regard to the disappearing and reappearing ...'s, and the order of evaluation. Say we have

[a,....,a] w/ [b,...,b] w/[2,2]

The rule says to reduce to

[a,...,a] w/ [b,b] w/[1,2]

Now, do we consider the [b,b] to still have a ..., and evaluate the [1,2] until the first entry is greater than 1? Or do we consider [b,b] to not have a ..., and reduce to

[a,...a] (with b a's) w/ [1,b] w/ [1,2]

If it's the first case, what is the point of making [b,b] if we are going back to [b...b] and changing the number of b's? In this case, it would be better just to change [b,...,b] to have [2,2] b's, since using your rule you reduce the [2,2], which is bad. But, using that rule, remember, [1 (1) 2] is still just \(\phi(\omega) + 1\). I'm not seeing how you get greater than that.