User blog:Bubby3/BM2 pair sequence system proof of termination.

You can imgaine pair sequence expressions as labeled hydras, where (a,b) corresponds to a node that is a+1 braches away from the root node, and has label b, and is a child of the last node whose value of a is less than it's value of a.

In this proof, I will be using Bucholoz psi notation, and showing that it is actually equivlant to it by the rules. The a value not being fixed means that the expressions can be on any level of the hydra. Now, time for comparisons. [m](n) corresponds to an expression where it's value of a is a+m and it's level in n (n is usually a variable)

Empty expression corresponds to 0.

(a,0)...(a,0) with n "(a,0)"'s for finite n has level n

(a,0)[1](n) has level \(\psi_0(n)\)

(a,1)[1](n) has level \(\psi_1(n)\) (a must be at least 1)

and in general (a,b)[1](n) has level \(\psi_b(n)\) (a must be at least b)

The reason this works is that in BM2 searches for the most recent ancestor of the rightmost node with a label less than it's label, which means that (a,b) with b >= 0 corresponds to  \(\Omega_b\) or \(\psi_b(0)\), which is a diagnolizer of \(\psi_b-1(n)\), or nodes with label b-1. The nodes with label 0 act like a Kirby-Paris hydra, so (a,b)[1](n)(b+1,0) corresponds to \(\psi_b(n+1)\).

The problem with BM1, is that sometimes the bad part would not start with an ancestor of therightmost node, therefore causing an infinite loop.