User:Hyp cos/Mixed Arrow Notation

Mixed arrow notation is a superset of up-arrow notation and down-arrow notation. You can write mixed arrows in it. e.g. \(3\uparrow\uparrow\downarrow\uparrow\downarrow4\). It's defined as follows:

Rule 1 - Only an arrow: \(a\uparrow b=a^b\), and \(a\downarrow b=a^b\)

Rule 2 - Second argument is 1: \(a\Diamond 1=a\)

Rule 3 - Last arrow is an up-arrow: \(a\Diamond\uparrow(b+1)=a\Diamond(a\Diamond\uparrow b)\)

Rule 4 - Last arrow is a down-arrow: \(a\Diamond\downarrow(b+1)=(a\Diamond\uparrow b)\Diamond a\)

Where \(\Diamond\) stands up or down arrows (not empty).