User:Username5243/Linear Array-E Notation

This is my plan to extend xE^ to linear arrays. I've yet to figure out how it's supposed to work beyond that, but here you go.

Array decomposition rules
All rules from xE^ still apply, so products etc. work the same. Here are the rules for decomposing linear array separators. I use letters (a,b,c,...) for individual array entries, % for the rest of the array, $ for a string containing 0 or more entries of "1", and @ is a string of entries each identical to the first entry. BEAF terminology still applies.


 * Rule 1 (Base case): {a,b} = a^b
 * Rule 2 (Trailing Rule): {%,1} = {%}
 * Rule 3 (Prime Rule): {a,1,%} = a
 * Rule 4 (Stack Rule): {a,b,%}>1 = {a,b,%}, {a,b,%}>(x+1) = {{a,b,%}>x,b,%}
 * Rule 5 (Limit Stack Rule): {a,b,%}>x[n] = {a,b,%}>(x[n]) (for limit x)
 * Rule 6 (Limit Prime Rule): {a,b,%}[n] = {a,b[n],%} (if none of the following cases applies)
 * Rule 7 (Creation of Stacks): {a,b*#,%}[1] = {a,b,%}, {a,b*#,%}[n+1] = {a,b,%}>{a,b*#,%}[n]
 * Rule 8 (Second entry is Sum): {a,b+c,%} = {a,b,%}>{a,c,%}
 * Rule 9 (Recursion Rule - In all cases from this point second entry is #, here third entry is a successor): {a,#,b+1,%}[1] = {a,#,b}, {a,#,b+1,%}[n+1] = {a,{a,#,b+1,%}[n],b,%}
 * Rule 10 (Expansion Rule - non-1 successor in fourth or higher entry, all entries between prime and this entry are 1): {a,#,$,1,b+1,%}[1] = a, {a,#,$,1,b+1,%}[n+1] = {a,a,@,{a,#,$,1,b+1,%}[n],b,%}
 * Rule 11 (Third or higher entry is a limit with all entries between prime and this entry being 1): {a,#,$,%,b}[n] = {a,#,$,%,b[n]}

(If there are any cases where the definition doesn't work, please let me know!)