FANDOM


Dollar function part 1

Rules

1. a$b \(\bullet\) = (a+b)$ \(\bullet\)

2. a\(\circ\)[0 \(\bullet\)]\(\circ\) = a$\(\circ\)a[ \(\bullet\)]\(\circ\)

3. a$\(\circ\)[b+1\(\bullet\)]\(\circ\) = a$\(\circ\)[b\(\bullet\)][b\(\bullet\)]...[b\(\bullet\)][b\(\bullet\)]\(\circ\) a [b\(\bullet\)]'s Where 0 and b are the less nested numbers

\(\bullet\) is the remainder of the array and \(\circ\) contains only brackets

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$[1[0]] = a$[[0]][[0]]...[[0]][[0]] > 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

NEW!!! Extended Definition

Normal brackets have level 1

1. a$b \(\bullet\) = (a+b)$ \(\bullet\)

2. a\(\circ\)[0 \(\bullet\)]\(\circ\) = a$\(\circ\)a[ \(\bullet\)]\(\circ\)

3. a$\(\circ\)[b+1\(\bullet\)]\(\circ\) = a$\(\circ\)[b\(\bullet\)][b\(\bullet\)]...[b\(\bullet\)][b\(\bullet\)]\(\circ\) a [b\(\bullet\)]'s Where 0 and b are the less nested numbers

4. a$[\(\diamond\)]_(b+1)\(\bullet\)  = a$[[...[[\(\diamond\)]_b ]...]_b ]_b\(\bullet\)  where there are a b-brackets and the b-bracket is the bracket with the lowest level.

here contains \(\diamond\) only brackets with level bigger than b and zeroes

5. a$\(\circ\)[\(\diamond\)]\(\circ\) = a$\(\circ\)\(\diamond\)\(\diamond\)...\(\diamond\)\(\diamond\)\(\circ\) where there are a \(\diamond\)'s  ( here is 'b' 1 )

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


\(\bullet\) is the remainder of the array and \(\circ\) contains only brackets

Example

2$[[1]_2] = 2$[1]_2[1]_2 = 2$[[[0]_2]][1]_2  = 2$[[0]_2[0]_2][1]_2 = 2$[0[0]_2][1]_2 =

2$[[2][0]_2][1]_2  = 2$[[1][1][0]_2][1]_2 = 2$[[0][0][1][0]_2][1]_2 = 2$[2[0][1][0]_2][1]_2 =

2$[2[0][1][0]_2][1]_2 = 2$[1[0][1][0]_2][1[0][1][0]_2][1]_2 =

2$[0[0][1][0]_2][0[0][1][0]_2][1[0][1][0]_2][1]_2 = 2$2[[0][1][0]_2][0[0][1][0]_2][1[0][1][0]_2][1]_2 =

4$[[0][1][0]_2][0[0][1][0]_2][1[0][1][0]_2][1]_2 = 8$[[0][1][0]_2][[0][1][0]_2][1[0][1][0]_2][1]_2 = huge

the changing part is bold

FGH

a$[0]_2 ~ f_e_0(a)

a$[1]_2 ~ f_e_1(a)

a$[2]_2 ~ f_e_2(a)

a$[[1]_2]_2 ~ f_e_e_0(a)

a$[0]_3 ~ f_z_0(a)

a$[0]_4 ~ f_phi(3,0)(a)

a$[0]_5 ~ f_phi(4,0)(a)

a$[0]_[0]_2 ~ f_phi(e_0,0)(a)

a$[0]_[0]_[0] ~ f_phi(phi(w,0),0)(a)

a$[0]_[0]_[0]_[0] ~ f_phi(phi(phi(w,0),0),0)(a)

a$[0]_[0]...[0]_[0] ~ f_gamma_0(a) there are a [0]'s

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