User blog comment:Wythagoras/Dollar function formal definition/@comment-5529393-20130619222303

Does rule 6 now serve as the general rule for linear arrays in brackets of type > 1? If so, you need to add more dots and such;  right now you have

a$[[  △     b]_(c+1) = a$[[   △     [...[   △     [   △     [   △     b-1]_(c+1)]_(c)...]_(c)]_(c)

which means there can only be one nonzero entry in the array (the rightmost one), and there can be nothing outside the (c+1) bracket. So you can't evaluate [1,1]_2 or [1][0,1]_2, for instance.

Similarly, your rule 8 says

a$[  △     ,0,b   ∙     ] = a$[   △     ,[   <span style="position: absolute; clip: rect(1.746em, 1000em, 2.831em, -0.495em); top: -2.646em; left: 0em;"><span class="mi" id="MathJax-Span-147" style="font-family: MathJax_Main;">△     ,0,b-1   <span style="position: absolute; clip: rect(2.018em, 1000em, 2.776em, -0.499em); top: -2.646em; left: 0em;"><span class="mo" id="MathJax-Span-150" style="font-family: MathJax_Main;">∙     ]_[   <span style="position: absolute; clip: rect(1.746em, 1000em, 2.831em, -0.495em); top: -2.646em; left: 0em;"><span class="mi" id="MathJax-Span-153" style="font-family: MathJax_Main;">△     ,0,b-1   <span style="position: absolute; clip: rect(2.018em, 1000em, 2.776em, -0.499em); top: -2.646em; left: 0em;"><span class="mo" id="MathJax-Span-156" style="font-family: MathJax_Main;">∙     ]...[   <span style="position: absolute; clip: rect(1.746em, 1000em, 2.831em, -0.495em); top: -2.646em; left: 0em;"><span class="mi" id="MathJax-Span-159" style="font-family: MathJax_Main;">△     ,0,b-1   <span style="position: absolute; clip: rect(2.018em, 1000em, 2.776em, -0.499em); top: -2.646em; left: 0em;"><span class="mo" id="MathJax-Span-162" style="font-family: MathJax_Main;">∙     ]_[   <span style="position: absolute; clip: rect(1.746em, 1000em, 2.831em, -0.495em); top: -2.646em; left: 0em;"><span class="mi" id="MathJax-Span-165" style="font-family: MathJax_Main;">△     ,0,b-1   <span style="position: absolute; clip: rect(2.018em, 1000em, 2.776em, -0.499em); top: -2.646em; left: 0em;"><span class="mo" id="MathJax-Span-168" style="font-family: MathJax_Main;">∙     ],b-1   <span style="position: absolute; clip: rect(2.018em, 1000em, 2.776em, -0.499em); top: -2.646em; left: 0em;"><span class="mo" id="MathJax-Span-171" style="font-family: MathJax_Main;">∙     ]

so there can be nothing outside the bracket again.

Also, what is the purpose of rule 7? I thought perhaps it meant that we can drop 0's on the right of arrays, but from rule 9 this doesn't appear to be the case.

How does a$[0,0...,0] reduce?