User blog:ArtismScrub/Applying Cascading-E logic to up-arrows

"Inspired", I guess, by Matthew's Function. ''note: i'm not going to use carets like Saibian does in higher ExE. combining carets and up-arrows looks very awkward''

a ↑( ↑ b) c = a ↑bc (this notation already exists for arbitrary amounts of up-arrows, just pointing out how this fits)

a ↑( ↑) ↑b or a ↑↑b  = a ↑ b a

a ↑ ↑ × ↑ b =  a ↑↑ a ↑↑ a ↑↑... ↑↑ a ↑↑ a ↑↑ a with b "a"s

a ↑ ↑ × ↑ ↑ b =  a ↑ ↑  × ↑ a  ↑ ↑ × ↑ a  ↑ ↑  × ↑ ... ↑ ↑ × ↑ a  ↑ ↑ × ↑ a  ↑ ↑  × ↑ a with b "a"s

a ↑ ↑ × ↑ b c =  a ↑ ↑  ×  ↑ b-1 a  ↑ ↑ × ↑ b-1 a  ↑ ↑  ×  ↑ <sup style="font-weight:400;">b-1 ... ↑ <sup style="font-weight:400;">↑ ×  ↑ <sup style="font-weight:400;">b-1 a  ↑ <sup style="font-weight:400;">↑ × ↑ <sup style="font-weight:400;">b-1 a  ↑ <sup style="font-weight:400;">↑  ×  ↑ <sup style="font-weight:400;">b-1 a with c "a"s

a ↑ <sup style="font-weight:400;">↑ × ↑ <sup style="font-weight:400;">↑ b =  a ↑ <sup style="font-weight:400;">↑ × ↑ <sup style="font-weight:400;">b a

etc.

a ↑ <sup style="font-weight:400;">↑↑ b =  a ↑ <sup style="font-weight:400;">↑ × ↑ <sup style="font-weight:400;">↑  ×  ↑ <sup style="font-weight:400;">↑  ×... × ↑ <sup style="font-weight:400;">↑  ×  ↑ <sup style="font-weight:400;">↑  ×  ↑ <sup style="font-weight:400;">↑  a with b " ↑ <sup style="font-weight:400;">↑ "s

alright i'm done with ridiculous superscripts

a ↑( ↑) ↑b =  a ↑↑b

a ↑( ↑) ↑ ↑ b =  a ↑ <sup style="font-weight:400;">↑↑ b

etc.

a ↑( ↑) ↑ ( ↑b)c =  a ↑( ↑) ↑ ( ↑b-1) a ↑( ↑) ↑ ( ↑b-1) a ↑( ↑) ↑ ( ↑b-1)... ↑( ↑) ↑ ( ↑b-1) a ↑( ↑) ↑ ( ↑b-1) a ↑( ↑) ↑ ( ↑b-1) a with c "a"s

a ↑( ↑) ↑ ( ↑) ↑b =  a ↑( ↑) ↑ ( ↑b) a

etc.

a ↑( ↑ ↑) ↑b =  a ↑( ↑) ↑ ( ↑) ↑ ( ↑)... ( ↑) ↑ ( ↑) ↑ ( ↑) ↑a with b-1 " ( ↑) "s

a( ↑( ↑ ↑) ↑) ( ↑) ↑ b =  a ↑( ↑ ↑) ↑ × ↑( ↑ ↑) ↑ × ↑( ↑ ↑) ↑ ×... × ↑( ↑ ↑) ↑ × ↑( ↑ ↑) ↑ × ↑( ↑ ↑) ↑ a with b " ↑( ↑ ↑) ↑"s

a( ↑( ↑ ↑) ↑) ( ↑ ↑ ) ↑ b = a ( ↑( ↑ ↑) ↑) ( ↑) ( ↑( ↑ ↑) ↑) ( ↑) ( ↑( ↑ ↑) ↑) ( ↑)... ( ↑) ( ↑( ↑ ↑) ↑) ( ↑) ( ↑( ↑ ↑) ↑) ( ↑) ( ↑( ↑ ↑) ↑)a with b " ( ↑( ↑ ↑) ↑)"s

a( ↑( ↑ ↑) ↑) → ↑b = a(((...(( ( ↑( ↑ ↑) ↑) ( ↑ ↑) ↑) ( ↑ ↑) ↑)...) ( ↑ ↑) ↑) ( ↑ ↑) ↑) ( ↑ ↑) ↑a  (no, these aren't chained arrows)

etc.

a ↑( ↑ ↑) ↑ ↑b = a ( ↑( ↑ ↑) ↑) → ( ↑( ↑ ↑) ↑) →... → ( ↑( ↑ ↑) ↑) → ( ↑( ↑ ↑) ↑)a with b " ( ↑( ↑ ↑) ↑)"s

<p style="font-weight:400;"> a ↑( ↑ ↑) ↑ ↑ ↑ b = a ( ↑( ↑ ↑) ↑ ↑ ) → ( ↑( ↑ ↑) ↑ ↑ ) →... → ( ↑( ↑ ↑) ↑ ↑ ) → ( ↑( ↑ ↑) ↑ ↑ )a with b " ( ↑( ↑ ↑) ↑ ↑ )"s

<p style="font-weight:400;"> etc.

<p style="font-weight:400;"> a ↑( ↑ ↑) ↑(  ↑) ↑ b =   a ↑( ↑ ↑) ↑(  ↑b)a

<p style="font-weight:400;"> a ↑( ↑ ↑) ↑(  ↑ ↑ ) ↑ b =   a ↑( ↑ ↑) ↑(  ↑)   ↑(  ↑)... ( ↑)   ↑    (  ↑)   ↑   a with b-1 " ( ↑)"s

<p style="font-weight:400;"> etc.

<p style="font-weight:400;"> a ↑( ↑ ↑ ↑ ) ↑b =  a ↑( ↑ ↑) ↑(  ↑ ↑ )   ↑(  ↑  ↑  )... ( ↑ ↑ )   ↑    (  ↑ ↑ )   ↑   a with b-1 " ( ↑ ↑ )"s

<p style="font-weight:400;"> a ↑( ↑ ↑ ↑ ↑ ) ↑b =  a ↑( ↑ ↑  ↑  ) ↑(  ↑ ↑ ↑ )   ↑(  ↑  ↑ ↑  )... ( ↑ ↑ ↑ )   ↑    (  ↑ ↑ ↑ )   ↑   a with b-1 " ( ↑ ↑ ↑ )"s

<p style="font-weight:400;"> etc.

<p style="font-weight:400;"> a ↑{ ↑} ↑b = a ↑( ↑( ↑) ↑) ↑b =  a ↑( ↑( ↑b) ) ↑a

<p style="font-weight:400;"> a ↑{ ↑+1} ↑b =  a ↑{ ↑} ↑ { ↑} ↑ { ↑}... { ↑} ↑ { ↑} ↑ { ↑} ↑ b with b-1 " { ↑}"s

<p style="font-weight:400;"> a ↑{ ↑+n} ↑b =  a ↑{ ↑+n-1} ↑ { ↑ +n-1 } ↑ { ↑ +n-1 }... { ↑ +n-1 } ↑ { ↑ +n-1 } ↑ { ↑ +n-1 } ↑ b with b-1 " { ↑ +n-1 }"s

<p style="font-weight:400;"> a ↑{ ↑+ ↑ } ↑b =  a ↑{ ↑+ b } ↑a

<p style="font-weight:400;"> a ↑{ ↑ × ↑} ↑ b =  a ↑{ ↑ + ↑ + ↑ +... + ↑ + ↑ + ↑ } ↑ a with b-1 "+"s

a ↑{ ↑ ( ↑) ↑ } ↑ b =  a ↑{ ↑ × ↑ × ↑ × ... × ↑ × ↑  × ↑ } ↑ a with b-1 " × "s

a ↑{ ↑ ( ↑ ↑ ) ↑ } ↑ b =  a ↑{ ↑ ( ↑) ↑ ( ↑) ↑ ( ↑)... ( ↑) ↑ ( ↑) ↑ ( ↑) ↑ } ↑ a with b-1 " ( ↑)"s

etc.

a ↑{ ↑{ ↑} ↑ } ↑ b =  a ↑{ ↑( ↑( ↑) ↑) ↑ } ↑ b =  a ↑{ ↑( ↑( ↑b) ) ↑ } ↑ a

continue with  a ↑{ ↑{ ↑+1} ↑ } ↑ b,  a ↑{ ↑{ ↑+ ↑ } ↑ } ↑ b,  a ↑{ ↑{ ↑ × ↑ } ↑ } ↑ b,  a ↑{ ↑{ ↑ ( ↑) ↑ } ↑ } ↑ b, etc.

...  a ↑{ ↑{ ↑{ ↑} ↑ } ↑ } ↑ b,  a ↑{ ↑{ ↑{ ↑{ ↑} ↑ } ↑ } ↑ } ↑ b,  a ↑{ ↑{ ↑{ ↑{ ↑} ↑ } ↑ } ↑ } ↑ b,  a ↑{ ↑{ ↑{ ↑{ ↑{ ↑} ↑ } ↑ } ↑ } ↑ } ↑ b, etc.

... a ↑{ {1} } ↑b.

Beyond that, ExE is a bunch of ad-hoc filler, and { {2} } and beyond are not defined anyway. (can't input double brackets as normal because trashy text edit system)

Why did I do this? I dunno. It's actually weaker than normal ExE, since  a ↑( ↑) ↑a is comparable to fω(a), whereas Ea#^#a is comparable to  fωω(a). I wonder where "cascading up-arrow notation" would catch up to ExE, though?