my escalating array notation

DEAN1

definition of DEAN1

the # only contain numbers and commas unless specified that it is the remaining part of the array

the variables are non-negative integers

basically linear arrays with a different bracket

rule A1 [a,1,#]=a where # is the remaining part of the array

rule A2 [#]=[#,1]

rule A3 [a,b]=a^b

rule A4 [a,b,c,#]=[a,[a,b-1,c,#],c-1,#] where # is the remaining part of the array

rule A5 [a,b,1,1,...,1,1,1,c,#₂]=[a,a,a,a,...,a,a,[a,b-1,1,1,...,1,1,1,c,#₂],c-1,#₂] where #₂ is the remaining part of the array

merging (my analogue of dimensional arrays)

rule B1 [a,b(1)]=[a,a,...,a,a] with b a's

rule B2a [#(f)]=[#,1(f)]

rule B2b [#(f)d₁(g₁)d₂(g₂)...d_n(g_n)]=[#,1(f)d₁(g₁)d₂(g₂)...d_n(g_n)]

rule B3a [a,b,(f+1)]=[a,a(f)a(f)...a(f)a(f)] with b a(f)'s

rule B3b [a,b,(f+1)d₁(g₁)d₂(g₂)...d_n(g_n)]=[a,a(f)a(f)...a(f)a(f)d₁(g₁)d₂(g₂)...d_n(g_n)] with b a(f)'s

rule B4a [a,b(1)c(f)]=[a,b,b,...,b,b(f)] with c b's

rule B4b [a,b(1)c(f)d₁(g₁)d₂(g₂)...d_n(g_n)]=[a,b,b,...,b,b(f)d₁(g₁)d₂(g₂)...d_n(g_n)] with c b's

a,#(f) would become a,<#>,f,X

a,#₃(d)b(f) would become a,<b,<#₃>,d,X>,f,X where #₃ is the remaining part of the array

"structure of" operation

for any structure on the right side of the "structure of" operator below N^^N or $\varepsilon_0$, N and $\omega$ are interchangeable

let % represent the remaining part of the structure

let a\b=[a,b(1)]

a\N^b=[a,a(b)]

a\N^c₁+N^c₂+...+N^c_n-1+N^c_n=[a,a(c_n)a(c_n-1)...a(c₂)a(c₁)]

a\$\omega^{\omega}$=[a,2(0,2)]

a\$\omega ^{\omega }+b$=[a,b(1)a(0,2)] for any b>0

a\N^(N+b)=[a,a(b,2)]

a\N^(N*b+c+1)=[a,a(c+1,b+1)]

a\N^(Nb)=[a,2(0,b+1)]

a\%+N^(%+(N^b)*a)=a\%+N^(%+N^(b+1))

a\%+N^(%+c)*a=a\%+N^(%+c+1)

a\%+N^(%+N^(...(%+(N^b)*a))...))=a\%+N^(%+N^(...(%+(N^b+1)))...))

if a\S=[a,a#] for all a, then a\S+1=[a,a,a#] where # is the remaining part of the array

if a\S=[a,2(#)#] for all a, then a\S+1=[a,a(#)#] where # are the remaining parts of the array

if a\%₁=[a,b#₁] and a\%₂=[a,a#₂] for all a, then a\%₂+%₁=[a,b#₁a#₂]

nesting (my analogue of bower's tetrational arrays?)

a\N^N^b=[a,2(0,1,1,...,1,1,2)] with b-1 1's

a\N^(c*N^b)=[a,2(0,1,1,...,1,1,c+1)] with b-1 1's

a\N^(d_nN^n+1+d_n-1N^(n-1)+...+d₁N+d₀)=[a,a(d₀+1,d₁+1,...,d_n-1,d_n)]

a\N^(d_nN^n+...+d₂N^(2)+d₁N)=[a,2(0,d₁+1,...,d_n-1,d_n)]

if a\N^%=[a,2(#)] for all a, then a\N^N^N^%=[a,2(a(#))] where # is the remaining part of the array

if a\N^%=[a,a(#)] for all a, then a\N^N^N^%=[a,a(a(#))] where # is the remaining part of the array

[a,b+1(0,n+2)]=a\N^(N*(n)+[a,b(0,n+2)])

if a\N^%=[a,2(a(#₁)#₂)] for all a, then [a,b+1(a(#₁)#₂)]=a\N^(%-N+[a,b(a(#₁)#₂)])

if a\N^%=[a,b(#)], then a\N^N^N^%=[a,b(a(#))]

[a,b(a,c+2,#(#)#)]=[a,b-1(a,c+2,#(#)#)a(a,c+1,#(#)#)]

[a,b(c,0(#)#)]=[a,b(c(#)#)]

if a\N^%=[a,a(a(#)#)], then a\N^(%+d_nN^n+d_n-1N^(N-1)+...+d₁N+d₀)=[a,a(a,d₀+1,d₁+1,...,d_n-1,d_n(#)#)]

if a\N^%=[a,a(a(#)#)], then a\N^(%+d_nN^n+d_n-1N^(N-1)+...+d₁N)=[a,2(a,0,d₁+1,...,d_n-1,d_n(#)#)]

creating more space (abandoned)

a\$\varepsilon_0$=a\N^^a

a\$\varepsilon _{\alpha +1}$=a\$\varepsilon_\alpha$*(N^$\varepsilon_\alpha$)^^(a-1)

a\$\varepsilon_{\omega}$=a\$\varepsilon _{a-1}$, other than that, if a\%₁=a\%₂ then a\$\varepsilon _{\%_{1}}$=a\$\varepsilon _{\%_{2}}$

if a\%=[a,b(#)] and % is not a finite variable, then a\$\varepsilon _{\%}$=[a,b(0;#)]

a\$\varepsilon _{b}$=[a,a(0;b+1)]

a\N^^b=[a,b(0;1)]

if a\N^(%)=[a,b(#)] and %>N, then a\$\varepsilon _{b}$+N^(%)=[a,b(#;b+1)] where # is the remaining part of the array

if a\%=[a,b(#₁)] and %>N, then a\$\varepsilon _{\%}$=[a,b(0;#₁)] where #₁ is the remaining part of the array

if a\%=[a,c(#₁)], %>N and a\N^(%₁)=[a,b(#)] and %₁>N, then a\$\varepsilon _{\%}$+N^(%₁)=[a,b(#;#₁)]