(I Don't speak English)
In my last post, I demonstrated a very powerful factorial, but weaker than HAN and said that I would try to create a more powerful factorial than HAN and I believe I did!
Let's start by demonstrating how it works (Your basic idea):
[m; n] ^! = n ^ ... ^ n
with n ^ ... ^ n arrows
with n ^ ... ^ n arrows
.
. (repeated by m times)
.
with n ^ ... ^ n arrows
with n ^ ... ^ n arrows
.
. (repeats by (m-1) times)
.
...
with n ^ ... ^ n arrows (repeats 1 time)
with (n-1) ^ ... ^ n arrows
with (n-1) ^ ... ^ n arrows
.
. (repeats by m times)
.
with (n-1) ^ ... ^ n arrows
with (n-1) ^ ... ^ n arrows
.
. (is repeated for (m-1) times)
.
...
with 1 ^ n arrows (repeats 1 time)
First we have "m", after that it has been repeated "m" times, the "m" is reduced by 1, and this is repeated until it becomes 1. When it becomes 1, the "n" of the left is reduced by 1, and the process returns.
We can add another entry like:
[a; b; c]^! = [[a; b]; c]^! = [[a; b]^!; c]^!
we can add "n" inputs
However, this is still very "weak", we can make it stronger:
(a) (a-1) (a-1)
[m; n]^ ! = [m; n] ^ ! ^ ... ^ [m; n] ^ !
(a-1) (a-1)
with [m; n]^ ! ^ ... ^ [m; n]^ ! arrows
. (a-1)
. (repeats ([m; n] ^ !) times)
.
(a-1) (a-1)
with [m; n]^ ! ^ ... ^ [m; n]^ ! arrows
...
and this continues in the same way that has already been explained
It is important to say that the rule [m; n]^(with "a" arrows) ! only applies when a> 1, since a = 1 has already been shown how it is done.