User blog:Syst3ms/Syst3ms's Array Notation

Syst3ms's Array Notation
Okay, so a while ago I tried to make my own array notation, and it didn't go too well. Having learned quite a bunch of things, I present you my own, better array notation.

Basic notation (S[]) :

Rule #1 : Rule #2 : Rule #3 : Rule #4 : Rule #5 :

represents the (possibly empty) rest of the array.

$$ S[a,b,c] = a\rightarrow b\rightarrow c $$ $$ S[3,2,1,2] = S[3,2,S[3,1,1,2],1] = S[3,2,3] = S[3,S[3,1,3],2] = S[3,3,2] = S[3,S[3,2,2],1] = S[3,S[3,S[3,1,2],1]] = S[3,S[3,3]] = 3^3^3 $$

Limit function :

Extended notation (xS[]) : The comma is a shorthand for.

Rule #6 :

represents the (possibly empty) rest of a chain

$$ S[3\{2\}2\{2\}3] = \\ S[3\{2\}2\{1\}2\{1\}2] = \\ S[3\{2\}2,2,2] = \\ S[3\{1\}3,2,2,2] = \\ S[3,3,2,2,2] $$ $$ S[3\{3\}3] = \\ S[3\{2\}3\{2\}3] =\\ S[3\{2\}3,3,3] =\\ S[3,3,3,3,3,3] $$

The limit function of basic notation can be written as

Limit function : Ex :

Array-extended notation ([x]S[]) :

Rule #6 is "redefined" (not quite, I'm just making it unambiguous with this extension) as such :

Rule #7 :

$$ S[3\{2,2\}2] = \\ S[3\{2,1\}3] = \\ S[3\{3\}3] =\\ S[3,3,3,3,3,3] $$

The limit function of extended notation can be written as

Limit function :

If you have any questions or remarks, make sure to comment below !