Exploding Dollar Notation

Exploding Dollar Notation (EDN):

Rules:
1) n$x# = n+1$x-1#
2) n$#[0]# = n+1$#n#
3) n$#[#[#]0[#]#] = n+1$#[0][#[#][#]#]#
4) n$#[#[#]x[#]#]# = n+1$#[n][#[#]x-1[#]#]#
5) n$#[x+1]# = n+1$#[x][x]...[x][x]# (n [x]'s)
6) Always solve by the left.


3$[2] = 4$[1][1][1] = 5$[0][0][0][1][1][1]
= 6$[0][0][1][1][1] = 7$[0][1][1][1]
= 8$[1][1][1] = 9$[0][0][0][0][0][0][0][0][1][1]
= 17$[1][1] = 18$[0][0][0][0][0][0][0][0][0][0][0][0][0][0][0][0][0][1]
= 35$[1] = 70$[0] = 71$70 = 141

2$[[0]] = 3$[2] = 141

2$[[1]] = 3$[[0][0]] = 4$[3[0]] = 5$[4][2[0]]
= 6$[3][3][3][3][3][2[0]] = ...
= n$[2[0]] = n+2$[n-1][n-1][n-1]...[n-1][n-1][n-1][1[0]]
= x$[1[0]] = x+2$[x-1][x-1][x-1]...[x-1][x-1][x-1][0[0]]
= y$[0[0]] = y+2$[y-1][y-1][y-1]...[y-1][y-1][y-1][[0]]
= ... (it's getting huge!!!)

2$[[[1]]] = 3$[[[0][0]]] = 4[[4[0]]] = 5[[4][3[0]]] = ...

F(n) = n$[[[...[[[n]]]...]]] with n nestings is ε0-recursive!

Rules (next):
7) n$#,0# = n+1$##
8) n$#,x# = n+1$[[[...[[[#,x-1#],#,x-1#],#,x-1#]...],#,x-1#],#,x-1#],#,x-1#
9) n$x,[0]# = n+1$#,n,n,...,n,n# with n n's
10) n$#,#[#[#]0[#]#] = n+1$#,#[0],[#[#][#]#]#
11) n$#,#[#[#]x[#]#]# = n+1$#,#[x],[#[#]x-1[#]#]#
12) n$#,#[x+1]# = n+1$#,#,[x],[x],...,[x],[x]# (n [x]'s)
13) n$#,[#],[#],...: start with the first one
14) n$x,y,z,...,[#],[#],[#],...: start with the numbers (like if there wasn't any [#])

2$1,2 = 3$[[1,1],1],1 = 4$[[[[[[1]]]]],1],1
= n$[x,1],1 = n+1$[[[[...[[[x]]]...]]]],1
= y$z,1 = y+1$[[[[...[[[z]]]...]]]]

2$1,[0] = 2$1,2,2
2$1,[1] = 3$1,[0],[0]
2$1,[[1]] = 3$1,[[0][0]]
= 4$1,[3[0]]
= 5$1,[4],[2[0]]
= 6$1,[3],[3],[3],[3],[3],[2[0]]
= 7$1,[2],[2],[2],[2],[2],[2],[3],[3],[3],[3],[2[0]]
= 8$1,[1],[1],[1],[1],[1],[1],[1],[2],[2],[2],[2],[2],[3],[3],[3],[3],[2[0]]
= 9$8,8,8,8,8,8,8,8,[1],[1],[1],[1],[1],[1],[2],[2],[2],[2],[2],[3],[3],[3],[3],[2[0]]
= ...

[x]_2 behave exactly like [x], but instead of [0] => n we have [0]_2 = A,A,A,...,A,A,A with A = [[[...[[n]]...]]] with n nestings
and also [x+1]_2 => A,A,A,...,A,A,A with A = [x]_2
Then, [0]_(k+1) => A,A,A,...,A,A,A with A = [[[...[[n]_k]_k..]_k]_k]_k with n nestings
And [x+1]_k => A,A,A,...,A,A,A with A = [x]_k

now, [x]_(1,2) = [x]_2. From there, normal comma rules applies, which means that [x]_[y] is possible!

Finally,
<0> => [n]_[n]_..._[n]_[n] (n nestings)
<x+1> => [<x>]_[<x>]_..._[<x>]_<x> (n nestings)

χ(n) = n$<<<...<<<n>>>...>>> with n nestings!
Veraxorris = χ(χ(χ(...(χ(10^^100))...))) with χ(10^^100) χ's