User blog:Googleaarex/Aarex's Array Notation Rebooted

Here is reboot version of my array notation, simpler than ever.

Simple
The simplest and weakest part in AANR (Aarex's Array Notation Rebooted).
 * \(r(a)\) = \(a+1\)

2-Entry
Let [A]^B = AAA... with Bs, but [A] evaluates, even inside of any [_]s.


 * \(r(a,0)\) = \(a+1\)
 * \(r(a,b)\) = \([r(]^aa[,[b-1])]^a\)

Linear
Let \(\#\) be the rest of the array, and stay the same while outputting.


 * \(r(a_1)\) = \(a_1+1\)
 * \(r(\#,0)\) = \(\#\)
 * \(r(a_1,a_2,\#)\) = \([r(]^{a_1}a_1[,[a_2-1],\#)]^{a_1}\)
 * \(r(a_1,0,0,...0,0,a_2,\#)\) = \(r(a_1,0,0,...0,a_1,a_2-1,\#)\)

Dimensional
Let \(\{0\}\) is first and weakest separator as comma is and \(\{a+1\}\) is next and stronger separator than \(\{a\}\).


 * \(r(a_1)\) = \(a_1+1\)
 * \(r(\#,0)\) = \(\#\)
 * \(r(a_1,a_2,\#)\) = \([r(]^{a_1}a_1[,[a_2-1],\#)]^{a_1}\)
 * \(r(a_1,0...0\{a_2\}0,a_4,\#)\) = \(r(a_1,0...0\{a_2\}a_1,a_4-1,\#)\)
 * \(r(a_1,0...0\{a_2\}0\{a_3\}a_4,\#)\) = \(r(a_1,0...0\{a_2\}[0\{[a_3-1]\}^{a_1}]1\{a_3\}a_4-1,\#)\)

Nested
Let \(\#_2\) be the rest of the array, and stay the same while outputting, different than \(\#\).

Let \(\#_3\) be the rest of the array, and stay the same while outputting, different than \(\#\) and \(\#_2\).


 * \(r(a_1)\) = \(a_1+1\)
 * \(r(\#,0)\) = \(\#\)
 * \(r(a_1,a_2,\#)\) = \([r(]^{a_1}a_1[,[a_2-1],\#)]^{a_1}\)
 * \(r(a_1,0...0\{\#\}0,a_4,\#_2\) = \(r(a_1,0...0\{\#\}a_1,a_4-1,\#_2)\)
 * \(r(a_1,0...0\{\#\}0\{a_3\#_2\}a_4,\#_3)\) = \(r(a_1,0...0\{\#\}[0\{[a_3-1]\#_2\}]^{a_1}1\{a_3\#_2\}a_4-1,\#_3)\)
 * \(r(a_1,0...0\{\#\}0\{0\#_2\}a_4,\#_3)\) = \(r(a_1,0...0\{\#\}0\{b\}1\{a_3\#_2\}a_4-1,\#_3)\) such that \(r(a_1,0,#\_2)\) = \(r(a_1,b)\)