I'd imagine you want something defined recursively like multiplication
- ( 0x = 0 )
- ( xy = x(y-1)+ x ) ( y > 0 ).
So it needs to be
- ( x^0 = c ) (c is some constant)
- ( x^y = xx^{y-1} ) (( y > 0 ) (to see why, replace multiplication with exponentiation and addition with multiplication). So what could ( c ) be? Well, the recursive exponentiation definition we want refers to ( x^0 ) in ( x^1 ). ( x^1 ) must be ( x ) by the thing we wish to capture in the formalism (multiplication repeated a single time). So the proposed formalism has ( x = x^1 = xx^0 = xc ). So ( cx = x ) hence ( c = 1 ), the multiplicative identity. Anything else would leave exponentiation to a zeroth power undefined, require a special case for a zeroth power and make the base definition that of ( x^1 ), or violate the intuition that exponentiation is repeated multiplication.
On an unrelated note, it'd be nice if Lemmy had Mathjax. I just wrote all this on mobile with that assumption, and I'm not rewriting now that I know better.
Checks I Should Have Done Before Posting
Sorry for the self-posting. I just wanted to share my post-hoc file checks since it was due-diligence I didn't think of until after I shared.
TLDR: I redirected into a file and inspected it at least enough to say I received an mkv container with an h264 video and opus audio.
Caveats
Details
I ran the command from my post in a world-readable directory with
>mystery_video_file
substituted for| mpv -- -
and inspected the download withsudo --user=nobody -- file -- mystery_video_file
which output
I ran
rename --last -- '' '.mkv' mystery_video_file # the '' is the empty string delimited with apostrophes
and thensudo --user=nobody -- ffprobe -hide_banner -- mystery_video_file.mkv
which output
If you trust me and not the presenter for some inexplicable reason, the SHA-512 checksum for the video is “24345bd3ca8015c14a7d5d63d6b2a40f9d0f8c0307a65996226a496f121fa5ae934718cf58090f43ee67bc250b06804f23c73688cc871c15c1ba18d79b1a82a8”.