Another AHAN

This is a variant of AHAN, but different rules by the definition of ABHAN, which Boris' notation doesn't work thanks to Deedlit.

ALPHA STAGE

Linear

Let get started! All Linear Array rules are same as AAN, but let replace a() into A2<>.

Dimensional

All Dimensional Arrays rules are same as AAN.

Nested

All Expanding Arrays rules are same as AAN, but replace ; with \.

Dimensional Nested

Now, \ will be legion mark for {_}. Then #\ will be legion mark for #{_}. #{n} works like {n}, but {0} changed into #.

Hyper Nested

Set #{A#/B} as #\\\\...{A} with B \s. That's mean / is hyper nested mark for \\\... and #/ is hyper nested mark for #\\\...

{A/B} works like \(A,B) so it is stronger.

Diagonalisers

I think I will redo this. Let {0,B:1} will be legion mark for {(_),B-1:1}, but {0:1} is legion mark for {_} and {A{# 0,C:1}B} can be reduced into {A,B # C-1:1}.

Then: {0:2} is legion mark for {_:1}, {0,A:2} is legion mark for {(_:1),A-1:2}, and {A{0 # 0,C:2}B:1} can be reduced into {A # B,C-1:2}.

We can repeat this: {0:X} is legion mark for {_:X-1}, {0,A:X} is legion mark for {(_:X-1),A-1:X}, and {A{0 # 0,C:X}B:X-1} can be reduced into {A # B,C-1:X}.

Before we move on, every {A:X} adds the (_:X-1) layers. And (A:B) works like {A:B} but in the entry.

Then, {0:# 0,X} will be legion mark for {0:# (_),X-1} and {0 # 0:1} will be legion mark for {0 # _}.

Higher Diagonalisers

Now the mayhem began... HAHAHA! Let ●_0 will be comma, ●_1 will be colon, and ●_2 will be... something.

Before we begin, let #: works like : but the comma replaced into #. Then /_: works like / but \ replaced into :.

Then ●_2 works like :, but \ replaced into :, and / replaced into /_:.

To go further:

• #●_A works like ●_A but the comma replaced into #. Then /_●_A works like / but \ replaced into ●_A, :_●_A works like : but \ replaced into ●_A, (●_2)_●_A works like ●_2 but \ replaced into ●_A, etc.
• Then ●_A works like :, but \ replaced into ●_(A-1), / replaced into /_●_(A-1), : replaced into :_●_(A-1), etc.

Now set diagonalisers of diagonalisers as ●, its works like : but {0,N●1} is equal into ●_N.

3rd Order Diagonalisers

●● already exist so we need to define bracket that A on [_] is stronger than normal A. Let call that as [], the square brackets.

Let [●●_0] as :, [●●_1] as ●, [●●_A] as ●, but it uses [●●_(A-1)] instead of :, then [●●] works like ●, but with [●●_x] instead of ●_x.

Then we can define more diagonalisers: Let [#●●_0] as [#], [#●●_1] as [#●], [●●_A] as [#●], but it uses [#●_(A-1)] instead of [#], then [#●] works like ●, but with [#_x] instead of ●_x.

Higher Order Diagonalisers

Let:

• [\_0] = \
• [\_0\_0] = /
• [\_1] = :
• [\_1\_1] = ●_2
• [\_2] = ●
• [\_2\_2] = [●●]
• [\_2\_2...] = [●●...] (Limit of 3rd Order Diagonalisers)
• [#\_N] works like ● but let uses [#\_(N-1)\_(N-1)\_(N-1)...] with x \_(N-1)s instead of ●_x.

Simple as that?

BETA STAGE

Ordered Order Diagonalisers

Now we can define diagonalisers of \_n! As - using in square brackets, so: [0,N-1] = [\_(N+1)].

Diagonalisers-into-Diagonalisers

Coming soon.

Now we can reach the final extension in AAHAN...

Cascaders!

NOTE: These symbols on this extension is not defined on ABHAN. This will replaced when a new extension come out.

Let call X as a cascader, and [#+X] is a diagonaliser of [#+(_)]. But wait how? We can do this:

• : ~ [X] is a diagonaliser of {_}
• ●_2 ~ [X*2] is a diagonaliser of :{_}
• ● ~ [X^2] is a diagonaliser of ●_(_)
• [●●_2] ~ [X^2+X] is a diagonaliser of ●{_}
• [●●] ~ [X^2*2] is a diagonaliser of [●●_(_)]
• [●●●] ~ [X^2*3] is a diagonaliser of [●●●_(_)]
• [\_3] ~ [X^3] is a diagonaliser of [●●●...] with (_) ●s
• [\_N] ~ [X^N] is a diagonaliser of [\_(N-1)\_(N-1)\_(N-1)...] with (_) \_(N-1)s
• - ~ [X^X] is a diagonaliser of [\_(_)]

Then we can define more operators:

• #^^A = #^#^#^... with A ^s.
• #^^(0,1) = #^#^#^... with the second entry ^s.
• #^^(A,B) = (#^^(0,B-1))^^(A+1)
• Else, () in ^^s work the same as normal arrays.
• ^^^ works like ^^ but with ^^s instead of ^s
• ^^^^ works like ^^ but with ^^^s instead of ^s
• etc.

Since Alemagno defined the limit of X^^^^...X is H_X<X,(1,2)>, we do the same as AAHAN.

• A2<X,A,n> = X^^^...^^^A with N ^s
• Rest are defined as normal AAHAN.

Limit of AAHAN is A2<n,n{0[A2<X,X{0[A2<X,X{0[A2<X,X{0[..]1}1>]1}1>]1}1>]1}1>