Difference between revisions of "User:Moosey/Googology"

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===Notating enormous arrays===
 
===Notating enormous arrays===
 
ah_n{$_0//(b+1),$_1} = ah_n((($_0)(@^_n)ah_n{$_0//b})#{//b,$_1})
 
ah_n{$_0//(b+1),$_1} = ah_n((($_0)(@^_n)ah_n{$_0//b})#{//b,$_1})
 
+
<br>
 
ah_n{$_0//0} = ah_n{$_0}
 
ah_n{$_0//0} = ah_n{$_0}
 
+
<br>
 
ah_n{$_0//b,$_1} = ah_n{$_0//(b[n]),$_1} for lim ord b
 
ah_n{$_0//b,$_1} = ah_n{$_0//(b[n]),$_1} for lim ord b
 
+
<br>
 
Else: apply ah's 7 rules, starting after legion bar. (e.g. ah_{$_0//,0,0,w+1} =  
 
Else: apply ah's 7 rules, starting after legion bar. (e.g. ah_{$_0//,0,0,w+1} =  
 
+
<br>
 
ah_{$_0//,w+1,w+1,w})
 
ah_{$_0//,w+1,w+1,w})
  
 
($_1)(@^_n)a =
 
($_1)(@^_n)a =
 
+
<br>
 
($_1)@(ah_n(($_1)(@^_n)(a-1))), a > 1,
 
($_1)@(ah_n(($_1)(@^_n)(a-1))), a > 1,
 
+
<br>
 
ah_n{$_1}, a=1
 
ah_n{$_1}, a=1
  
 
ah_n{$_0@@0} = ah_n{$_0}
 
ah_n{$_0@@0} = ah_n{$_0}
 
+
<br>
 
ah_n{$_0@@a,$_1} = ah_n{$_0@@a[n],$_1},a a lim ord
 
ah_n{$_0@@a,$_1} = ah_n{$_0@@a[n],$_1},a a lim ord
 
+
<br>
 
ah_n{$_0@@(a+1),$_1} = ah_n({$_0//}#{$_0@@(a),$_1}
 
ah_n{$_0@@(a+1),$_1} = ah_n({$_0//}#{$_0@@(a),$_1}
 
+
<br>
 
Apply basic ah rules otherwise to everything after the @@
 
Apply basic ah rules otherwise to everything after the @@
  
 
ah_n{$_0(@@b,$_2)0} = ah_n{$_0}
 
ah_n{$_0(@@b,$_2)0} = ah_n{$_0}
 
+
<br>
 
ah_n{$_0((@@b,$_2)a,$_1} = ah_n{$_0(@@b)a[n],$_1},a a lim ord
 
ah_n{$_0((@@b,$_2)a,$_1} = ah_n{$_0(@@b)a[n],$_1},a a lim ord
 
+
<br>
 
ah_n{$_0(@@0)(a+1),$_1} = ah_n({$_0//}#{$_0(@@0)(a),$_1}
 
ah_n{$_0(@@0)(a+1),$_1} = ah_n({$_0//}#{$_0(@@0)(a),$_1}
 
+
<br>
 
ah_n{$_0(@@b+1,$_2)(a+1),$_1} = ah_n({$_0(@@b,$_2)}#{$_0@@(a),$_1}
 
ah_n{$_0(@@b+1,$_2)(a+1),$_1} = ah_n({$_0(@@b,$_2)}#{$_0@@(a),$_1}
 
+
<br>
 
ah_n{$_0(@@b,$_2)$_3} = ah_n{$_0(@@b[n],$_2)$_3}, if b is a lim ord
 
ah_n{$_0(@@b,$_2)$_3} = ah_n{$_0(@@b[n],$_2)$_3}, if b is a lim ord
 
+
<br>
 
Apply basic ah rules otherwise to everything after the (@@$) or everything inside the (@@$)  
 
Apply basic ah rules otherwise to everything after the (@@$) or everything inside the (@@$)  
 
+
<br>
 
depending on what is necessary
 
depending on what is necessary
 
+
<br>
 
These array rules can be applied without the ah_n of the array, but any n refers to the n in the ah  
 
These array rules can be applied without the ah_n of the array, but any n refers to the n in the ah  
 
+
<br>
 
subscript.
 
subscript.
  
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{$_0//(b+1),$_1} = ((($_0)(@^_n)ah_n{$_0//b})#{//b,$_1})
 
{$_0//(b+1),$_1} = ((($_0)(@^_n)ah_n{$_0//b})#{//b,$_1})
 
+
<br>
 
{$_0//0} = {$_0}
 
{$_0//0} = {$_0}
 
+
<br>
 
{$_0//b,$_1} = {$_0//(b[n]),$_1} for lim ord b
 
{$_0//b,$_1} = {$_0//(b[n]),$_1} for lim ord b
 
+
<br>
Else: apply ah’s 7 rules, starting after legion bar. This includes nesting and incrementing the subscript (e.g. ah_n{$_0//,0,0,w+1} = ah_(n+1){$_0//,w+1,w+1,w})
+
Else: apply ah’s 7 rules, starting after legion bar. This includes nesting and incrementing the subscript (e.g. ah_n{$_0//,0,0,w+1} = ah_(n+1)
 +
{$_0//,w+1,w+1,w})
  
 
($_1)(@^_n)a =
 
($_1)(@^_n)a =
 
+
<br>
 
($_1)@(ah_n(($_1)(@^_n)(a-1))), a > 1,
 
($_1)@(ah_n(($_1)(@^_n)(a-1))), a > 1,
 
+
<br>
 
ah_n{$_1}, a=1
 
ah_n{$_1}, a=1
  
 
{$_0@@0} = {$_0}
 
{$_0@@0} = {$_0}
 
+
<br>
 
{$_0@@a,$_1} = {$_0@@a[n],$_1},a a lim ord
 
{$_0@@a,$_1} = {$_0@@a[n],$_1},a a lim ord
 
+
<br>
 
{$_0@@(a+1),$_1} = ({$_0//}#{$_0@@(a),$_1}
 
{$_0@@(a+1),$_1} = ({$_0//}#{$_0@@(a),$_1}
 
+
<br>
 
Apply basic ah rules otherwise to everything after the @@. See legion bar notes for more details
 
Apply basic ah rules otherwise to everything after the @@. See legion bar notes for more details
  
 
{$_0(@@b,$_2)0} = {$_0}
 
{$_0(@@b,$_2)0} = {$_0}
 
+
<br>
 
{$_0((@@b,$_2)a,$_1} = {$_0(@@b)a[n],$_1},a a lim ord
 
{$_0((@@b,$_2)a,$_1} = {$_0(@@b)a[n],$_1},a a lim ord
 
+
<br>
 
{$_0(@@0)(a+1),$_1} = ({$_0//}#{$_0(@@0)(a),$_1}
 
{$_0(@@0)(a+1),$_1} = ({$_0//}#{$_0(@@0)(a),$_1}
 
+
<br>
 
{$_0(@@b+1,$_2)(a+1),$_1} = ({$_0(@@b,$_2)}#{$_0@@(a),$_1}
 
{$_0(@@b+1,$_2)(a+1),$_1} = ({$_0(@@b,$_2)}#{$_0@@(a),$_1}
 
+
<br>
 
{$_0(@@b,$_2)$_3} = {$_0(@@b[n],$_2)$_3}, if b is a lim ord
 
{$_0(@@b,$_2)$_3} = {$_0(@@b[n],$_2)$_3}, if b is a lim ord
 
+
<br>
 
Apply basic ah rules otherwise to everything after the (@@$) or everything inside the (@@$)  
 
Apply basic ah rules otherwise to everything after the (@@$) or everything inside the (@@$)  
 
depending on what is necessary. See legion bar notes for more details.
 
depending on what is necessary. See legion bar notes for more details.
 
+
<br>
 
(Now things such as ah_n{a(@@(a(@@a)a))a} are well defined.)
 
(Now things such as ah_n{a(@@(a(@@a)a))a} are well defined.)
  
 
{$_0@@@(a+1),$_1} = {$_0(@@($_0@@@a,$_1))$_0,$_1}
 
{$_0@@@(a+1),$_1} = {$_0(@@($_0@@@a,$_1))$_0,$_1}
 
+
<br>
 
{$_0@@@0} = {$_0}
 
{$_0@@@0} = {$_0}
 
+
<br>
 
{$_0@@@a,$_1} = {$_0@@@a[n],$_1}, a a lim ord.
 
{$_0@@@a,$_1} = {$_0@@@a[n],$_1}, a a lim ord.
 
+
<br>
 
Apply ah rules otherwise
 
Apply ah rules otherwise
  
Line 146: Line 147:
  
 
<nowiki>{$_0[[1]]0} = {$_0}
 
<nowiki>{$_0[[1]]0} = {$_0}
 
+
</nowiki><br><nowiki>
 
{$_0[[1]]a+1,$_1} = {$_0@@@$_0[[1]]a,$_1}
 
{$_0[[1]]a+1,$_1} = {$_0@@@$_0[[1]]a,$_1}
 
+
</nowiki><br><nowiki>
 
{$_0[[1]]a,$_1} = {$_0,a[[1]]a[n],$_1}, a a lim ord
 
{$_0[[1]]a,$_1} = {$_0,a[[1]]a[n],$_1}, a a lim ord
 
+
</nowiki><br><nowiki>
 
{$_0[[1]]0@m,a+1,$_1} = {$_0[[1]](ah_n{$_0[[1]](a+1)@m,a,$_1})@m,a,$_1}
 
{$_0[[1]]0@m,a+1,$_1} = {$_0[[1]](ah_n{$_0[[1]](a+1)@m,a,$_1})@m,a,$_1}
 
+
</nowiki><br><nowiki>
 
{$_0[[1]]0@m,a,$_1} = {$_0[[1]]a@m,a[n],$_1}, a a lim ord</nowiki>
 
{$_0[[1]]0@m,a,$_1} = {$_0[[1]]a@m,a[n],$_1}, a a lim ord</nowiki>

Revision as of 23:16, 20 November 2019

ah

Part one: Starter notations

define $_n as any entries (including no entries) in an array. It’s my symbol for we-don’t-care entries. In any one use of any one rule, if n is the same, $_n is the same

a#b = concatenation of a and b

n@m =
n, m = 1
{n}#(n@(m-1)), m > 1, m not a lim ord
n@(m[n]) if m is a lim ord

g(a,n,B) =
ah_g(a-1,n,B) {B}, a > 1;
n, a = 0

Basic ah rules

Rule 1.
ah^a_n {$_0} = g(a,n,$_0)
Rule 2.
ah_n ({$_1}#{z}) = ah_n {$_1}, z = 0
Rule 3.
ah_n{} = n+1
Rule 4.
ah_n{a+1,$_2} = ah^n_n{a,$_2}
Rule 5.
ah_n{a,$_3} = ah_n{a[n],$_3}, a a lim ord
Rule 6.
ah_n ((0@b)#{a+1,$_4}) = ah_(n+1) (((a+1)@b)#{a,$_4}), b > 0
Rule 7.
ah_n ((0@b)#{a,$_5}) = ah_n ((a@b)#{a[n],$_5}), a a lim ord & b > 0

Notating enormous arrays

ah_n{$_0//(b+1),$_1} = ah_n((($_0)(@^_n)ah_n{$_0//b})#{//b,$_1})
ah_n{$_0//0} = ah_n{$_0}
ah_n{$_0//b,$_1} = ah_n{$_0//(b[n]),$_1} for lim ord b
Else: apply ah's 7 rules, starting after legion bar. (e.g. ah_{$_0//,0,0,w+1} =
ah_{$_0//,w+1,w+1,w})

($_1)(@^_n)a =
($_1)@(ah_n(($_1)(@^_n)(a-1))), a > 1,
ah_n{$_1}, a=1

ah_n{$_0@@0} = ah_n{$_0}
ah_n{$_0@@a,$_1} = ah_n{$_0@@a[n],$_1},a a lim ord
ah_n{$_0@@(a+1),$_1} = ah_n({$_0//}#{$_0@@(a),$_1}
Apply basic ah rules otherwise to everything after the @@

ah_n{$_0(@@b,$_2)0} = ah_n{$_0}
ah_n{$_0((@@b,$_2)a,$_1} = ah_n{$_0(@@b)a[n],$_1},a a lim ord
ah_n{$_0(@@0)(a+1),$_1} = ah_n({$_0//}#{$_0(@@0)(a),$_1}
ah_n{$_0(@@b+1,$_2)(a+1),$_1} = ah_n({$_0(@@b,$_2)}#{$_0@@(a),$_1}
ah_n{$_0(@@b,$_2)$_3} = ah_n{$_0(@@b[n],$_2)$_3}, if b is a lim ord
Apply basic ah rules otherwise to everything after the (@@$) or everything inside the (@@$)
depending on what is necessary
These array rules can be applied without the ah_n of the array, but any n refers to the n in the ah
subscript.

Enormous array update

if unspecified, n is the subscript in the first (closest) ah that is in front of the array.

{$_0//(b+1),$_1} = ((($_0)(@^_n)ah_n{$_0//b})#{//b,$_1})
{$_0//0} = {$_0}
{$_0//b,$_1} = {$_0//(b[n]),$_1} for lim ord b
Else: apply ah’s 7 rules, starting after legion bar. This includes nesting and incrementing the subscript (e.g. ah_n{$_0//,0,0,w+1} = ah_(n+1) {$_0//,w+1,w+1,w})

($_1)(@^_n)a =
($_1)@(ah_n(($_1)(@^_n)(a-1))), a > 1,
ah_n{$_1}, a=1

{$_0@@0} = {$_0}
{$_0@@a,$_1} = {$_0@@a[n],$_1},a a lim ord
{$_0@@(a+1),$_1} = ({$_0//}#{$_0@@(a),$_1}
Apply basic ah rules otherwise to everything after the @@. See legion bar notes for more details

{$_0(@@b,$_2)0} = {$_0}
{$_0((@@b,$_2)a,$_1} = {$_0(@@b)a[n],$_1},a a lim ord
{$_0(@@0)(a+1),$_1} = ({$_0//}#{$_0(@@0)(a),$_1}
{$_0(@@b+1,$_2)(a+1),$_1} = ({$_0(@@b,$_2)}#{$_0@@(a),$_1}
{$_0(@@b,$_2)$_3} = {$_0(@@b[n],$_2)$_3}, if b is a lim ord
Apply basic ah rules otherwise to everything after the (@@$) or everything inside the (@@$) depending on what is necessary. See legion bar notes for more details.
(Now things such as ah_n{a(@@(a(@@a)a))a} are well defined.)

{$_0@@@(a+1),$_1} = {$_0(@@($_0@@@a,$_1))$_0,$_1}
{$_0@@@0} = {$_0}
{$_0@@@a,$_1} = {$_0@@@a[n],$_1}, a a lim ord.
Apply ah rules otherwise

Dimensional arrays: 2D

{$_0[[1]]0} = {$_0}
{$_0[[1]]a+1,$_1} = {$_0@@@$_0[[1]]a,$_1}
{$_0[[1]]a,$_1} = {$_0,a[[1]]a[n],$_1}, a a lim ord
{$_0[[1]]0@m,a+1,$_1} = {$_0[[1]](ah_n{$_0[[1]](a+1)@m,a,$_1})@m,a,$_1}
{$_0[[1]]0@m,a,$_1} = {$_0[[1]]a@m,a[n],$_1}, a a lim ord