User blog:Wythagoras/Dollar function part 1

Rules (1)
1. a$b# = (a+b)$#

2. a$#_[][0#]#_[] = a$#_[]a[#]#_[]

FGH
a$[0] = a$a = 2a = f_1(a)

a$[0][0] = a$a[0] = 2a$[0] = 4a = f_1(f_1(a))

a$[1] = f_2(a)

a$[2] = f_3(a)

a$[b] = f_(b+1)(a)

a$0 = a$[a] > f_w(a)

a$[1[0]] = a$00...00 > f_w+1(a)

a$[2[0]]  > f_w+2(a)

a$0][0 > f_2w(a)

a$1  > f_w^2(a)

a$2  > f_w^3(a)

a$[0]  > f_w^w(a)

a$1[0]  > f_w^(w+1)(a)

a$[0][0]  > f_w^(2w)(a)

a$[1]  > f_w^w^2(a)

a$[2]  > f_w^w^3(a)

a$[[0]]  > f_w^w^w(a)

a$[[[0]]]  > f_w^w^w^w(a)

a$[[[[0]]]]  > f_w^w^w^w^w(a)

limit at e_0