Extended First-order Array Notation (ExFoAN) is the sixth part of almighty array notation. A valid expression in ExFoAN is of the form a*(a, b X c X d X e X ... X n) where a, b, c, d, e, ..., n ≥ 1, and X are the separators.
The Extended First-order Array Notation has the following form:
a*(a, b X c X d X e X ... X n) where a, b, c, d, e, ..., n ≥ 1, and X are the separators.
The separator can come in many forms, such as {3}, {1, 4, 5}, {1, 1, 3 {2} 2}.
The first entry of the array is the base, the number after it is the iterator.
The {1} separator stands for comma.
In this extension, the separators are the comma, the exclamation mark (!) [not to be confused with the factorial operation], and {x A n} where x is an expression, A is a separator, and n is an integer > 1.
Empty array rule: a*() = 1, since a*(1) = a*() using the tailing rule.
Base rule: a*(a) = a and a*(a, b) = a^b
Tailing rule 1: a*(# 1 X #) = a*(#) where # indicates the rest of array (X can be any separators)
Tailing rule 2: a*(# X 1) = a*(#)
Tailing rule 3: a*(# {% 1} #) = a*(# {%} #) where % indicates the rest of array in the separator
Prime rule: a*(a, 1 #) = a
Recursion rule: a*(a, b, 1, 1, ..., 1, 1, c #) = a*(a, a, a, a, ..., a, a*(a, b - 1, 1, 1, ..., 1, 1, c #), c - 1 #) (w/ b number of a's) for a, b, c > 1 and the length of the array ≥ 4
If rules 1 - 7 do not apply: a*(a, b, c #) = a*(a, a*(a, b - 1, c #), c - 1 #)
Comma rule: a*(a, b {1} c #) = a*(a, b, c #)
If more than one rule(s) apply to an array, start from step 1, and find the nearest rule that applies to that array.
So a*(2, 1, 1, 1, 1, 1, 2, 2) = a*(2) = 2 (by rule 6) instead of a*(2, 2, 2, 2, ..., 2, a*(2, 0, 1, 1, ..., 1, 1, 2, 2), 1, 2) (by rule 7), which is ill-formed due to the occurrence of 0.
Followed by some previous rules by following:
Reuse the rules for linear arrays to that row (after separators other than commas)
a*(a, b {2} c) a*(a, a, 1, 1, 1, ..., 1, 1, 2 {2} c - 1) with b string of 1's using comma as a separator.
a*(a, b {2} 1 {2} ... {2} 1 {2} c) = a*(a, a, {2} 1 {2} ... {2} 1, 1, 1, ..., 1, 1, 2 {2} c - 1) with b string of 1's using comma as a separator.
a*(a, b {d} 1 {d} ... {d} 1 {d} c) = a*(a, a, {d - 1} 1 {d - 1} ... {d - 1} 1, 1, 1, ..., 1, 1, 2 {d - 1} c - 1) with b string of 1's using {d - 1} as a separator.
a*(a, b {1, 1, 1, ..., 1, 1, c, d%} 2) = a*(a, a {1, 1, 1, ..., 1, b, c - 1, d%} 2), which is different from a*(a, b, 1, 1, ..., 1, 1, c #) = a*(a, a, a, a, ..., a, a*(a, b - 1, 1, 1, ..., 1, 1, c #), c - 1 #).
a*(a, b {1 {2} c} 2) = a*(a, a {1 {1, 1, 1, ..., 1, 1, 2 {2} c - 1} 2), and so on. Nested arrays reuse the rules 2 through 5.
If n < m, a*(# {n} 1 {m} #) = a*(# {m} #).
At this part, we define the new rules by following:
Reuse the previous rules above and ignore the exclamation mark separators.
Define the simplest expression for that part as a*(a, b {1 ! 2} 2) = a*(a, a {1 { ... 1 {1, 2} 2 ... } 2} 2)
If the first entry before an exclamation mark is just 1, change the {1 ! n + 1 #} to Sb (b is the iterator), where S1 is {1 ! n #}, Sn + 1 = {1 {Sn} 2 ! n #}, then change the iterator to the base.
Now we expand the First-order Array Notation. The new general rule is:
If there is a ! before it, then change {# 1 ! m + 1 %} in Sn, while S1 = {# 1 ! m %} and S(n + 1) = {# 1 {Sn} 2 ! m %}.
Example: a*(3, 3 {1 ! 1 ! 2} 2) = a*(3, 3 {1 ! 1 {1 ! 1 {1 ! 2} 2} 2} 2); a*(3, 3 {1 ! 1 ! 2, 2} 2) = a*(3, 3 {1 ! 1 {1 ! 1 {1 ! 1 {1 ! 1 ! 1, 2} 2 ! 1, 2} 2 ! 1, 2} 2 ! 1, 2} 2).
a*(a, b {1 ! 1 ! 2} 2) = a*(a, a {1 ! 1 {1 ! 1 ... 1 {1 ! 2} 2 ... 2} 2} 2) with b 1{1's inside the outermost {}. FGH level ζ0.
a*(a, b, 2 {1 ! 1 ! 2} 2) ~ FGH level ζ0 + 1.
a*(a, b {1 ! 1 ! 2} 3) ~ FGH level (ζ0)2.
a*(a, b {1 ! 1 ! 2} 1, 2} ~ FGH level (ζ0)ω.
a*(a, b {1 ! 1 ! 2} 1 {1 ! 1 ! 2} 2) ~ FGH level ζ0^2.
a*(a, b {2 ! 1 ! 2} 2) ~ FGH level ζ0^ω.
a*(a, b {1 {1 ! 1 ! 2} 2 ! 1 ! 2} 2) ~ FGH level ζ0^ζ0.
a*(a, b {1 ! 2 ! 2} 2) = a*(a, a {1 {1 ... 1 {1 ! 1 ! 2} 2 ... 2 ! 1 ! 2} 2 ! 1 ! 2} 2) with b 1{1's inside the outermost {}. FGH level ε(ζ0 + 1).
a*(a, b {1 ! 3 ! 2} 2) = a*(a, a {1 {1 ... 1 {1 ! 2 ! 2} 2 ... 2 ! 2 ! 2} 2 ! 2 ! 2} 2) with b 1{1's inside the outermost {}. FGH level ε(ζ0 + 2).
a*(a, b {1 ! 1, 2 ! 2} 2) = a*(a, a {1 ! b ! 2} 2). FGH level ε(ζ0 + ω).
a*(a, b {1 ! 1 {1 ! 2} 2 ! 2} 2) ~ FGH level ε(ζ0 + ε0).
a*(a, b {1 ! 1 {1 ! 1 ! 2} 2 ! 2} 2) ~ FGH level ε((ζ0)2).
a*(a, b {1 ! 1 ! 3} 2) = a*(a, a {1 ! 1 {1 ! 1 ... 1 {1 ! 1 ! 2} 2 ... 2 ! 2} 2 ! 2} 2) with b 1{1's inside the outermost {}. FGH level ζ1.
a*(a, b {1 ! 2 ! 3} 2) ~ FGH level ε(ζ1 + 1).
a*(a, b {1 ! 1 ! 4} 2) = a*(a, a {1 ! 1 {1 ! 1 ... 1 {1 ! 1 ! 3} 2 ... 2 ! 3} 2 ! 3} with b 1{1's inside the outermost {}. FGH level ζ2.
a*(a, b {1 ! 1 ! 1, 2} 2) = a*(a, a {1 ! 1 ! b} 2). FGH level ζω.
a*(a, b {1 ! 1 ! 1 {1 ! 1 ! 2} 2} 2) ~ FGH level ζ(ε0).
a*(a, b {1 ! 1 ! 1 ! 2} 2) = a*(a, a {1 ! 1 ! 1 {1 ! 1 ! 1 ... 1 {1 ! 1 ! 2} 2 ... 2} 2} 2) with b 1{1's inside the outermost {}. FGH level η0 = φ(3, 0).
a*(a, b {2 ! 1 ! 1 ! 2} 2) ~ FGH level η0^ω.
a*(a, b {1 ! 2 ! 1 ! 2} 2) ~ FGH level ε(η0 + 1).
a*(a, b {1 ! 1 ! 2 ! 2} 2) ~ FGH level ζ(η0 + 1).
a*(a, b {1 ! 1 ! 1 ! 3} 2) = a*(a, a {1 ! 1 ! 1 {1 ! 1 ! 1 ... 1 {1 ! 1 ! 1 ! 2} 2 ... 2 ! 2} 2 ! 2} 2) with b 1{1's inside the outermost {}. FGH level η1 = φ(3, 1).
a*(a, b {1 ! 1 ! 1 ! 1, 2} 2) = a*(a, a {1 ! 1 ! 1 ! b} 2). FGH level ηω = φ(3, ω).
a*(a, b {1 ! 1 ! 1 ! 1 {1 ! 1 ! 1 ! 2} 2} 2) ~ FGH level η(η0) = φ(3, φ(3, 0)).
a*(a, b {1 ! 1 ! 1 ! 1 ! 2} 2) = a*(a, a {1 ! 1 ! 1 ! 1 ... {1 ! 1 ! 1 ! 2} ... 2} 2) with b 1{1's inside the outermost {}. FGH level φ(4, 0).
a*(a, b {1 ! 1 ! 1 ! 1 ! 1 ! 2} 2) = a*(a, a {1 ! 1 ! 1 ! 1 ! 1 ... {1 ! 1 ! 1 ! 1 ! 2} ... 2} 2) with b 1{1's inside the outermost {}. FGH level φ(5, 0).
The limit of this level is φ(ω, 0) = ψ(Ω^ω).