User blog comment:Bubby3/Proof of termination of pair sequence system/@comment-30754445-20181120182103

Ah, I see that you've added an actual rule-set for the notation. Excellent.

Is it correct, though?

It seems that according to your rules, the bad root of (0,0)(1,1)(2,2) is (0,0) (and it needs to be (1,1))

(we scan from the end. 1<2 so we select (1,1). 0<1 so we select (0,0))

Unless you meant the proccess to stop at the first candidate we find? But in that case, the bad root of (0,0)(1,1)(2,1) would be (1,1) (and it needs to be (0,0)).

On a related note, it seems that you aren't actually using the rule-set in your proof. A proper proof will go like this:

(1) Write down a rule-set for the notation.

(2) Prove regorously that there's a 1-to-1 correspondence between the specific rule-set you've written and Buchholz hydras.

(3) Profit.

(and as a bonus, if you do this, you will have a verification that the rule-set you've written indeed works)