User blog comment:Mh314159/A hopefully powerful new system/@comment-39585023-20190630030857/@comment-35470197-20190630142723

I analysed sn and ss in order to compute o, but just omitted the correspondence. Now I have more strict upper bounds. Let $M$ denote the sum of variables of the left hand sides. For example, we have \begin{eqnarray*} [a,1][n] & \leq & [f_{\omega^2}(M)][\underbrace{n,\ldots,n}_{[f_{\omega^2}(M),0][n]}] \sim f_{\omega^2}^2(M) \\ [a,2][n] & \leq & [f_{\omega^2}(M)][\underbrace{n,\ldots,n}_{f_{\omega^2}^2(M)}] \sim f_{\omega^2}^3(M) \\ [a,b][n] & \leq & f_{\omega^2}^{b+1}(M) \\ [a,b,1][n] & \leq & [f_{\omega^2}(M)][\underbrace{n,\ldots,n}_{f_{\omega^2 + 1}(f_{\omega^2}(M))}] \sim f_{\omega^2 + 1}(f_{\omega^2}(M)) \\ [a,b,2][n] & \leq & [f_{\omega^2}(M)][\underbrace{n,\ldots,n}_{f_{\omega^2 + 1}(f_{\omega^2}^2(M))}] \sim f_{\omega^2 + 1}(f_{\omega^2}^2(M)) \\ [a,b,c][n] & \leq & f_{\omega^2 + 1}(f_{\omega^2}^c(M)) \\ [n,\ldots,n][n] & \leq & f_{\omega^2 + n}(n) = f_{\omega^2 + \omega}(n) \\ [x][1,1] & \leq & [\underbrace{x,\ldots,x}_{f_{\omega^2 + \omega}(M)}][f_{\omega^2 + \omega}(M)] \sim f_{\omega^2 + \omega}^2(M) \\ [x][2,1] & \leq & [\underbrace{x,\ldots,x}_{f_{\omega^2 + \omega}^2(M)}][f_{\omega^2 + \omega}^2(M)] \sim f_{\omega^2 + \omega}^3(M) \\ [x][a,1] & \leq & f_{\omega^2 + \omega}^{a+1}(M) \\ [x][a,2] & \leq & [\underbrace{x,\ldots,x}_{f_{\omega^2 + \omega + 1}(M)}][f_{\omega^2 + \omega + 1}(M),1] \sim f_{\omega^2 + \omega + 1}^2(M)) \\ [x][a,b] & \leq & f_{\omega^2 + \omega + 1}^b(M) \\ [x][a,b,1] & \leq & [\underbrace{x,\ldots,x}_{f_{\omega^2 + \omega + 2}(M)}][f_{\omega^2 + \omega + 2}(M),f_{\omega^2 + \omega + 2}(M)] \sim f_{\omega^2 + \omega + 2}^2(M) \\ [x][a,b,c] & \leq & f_{\omega^2 + \omega + 2}^{c+1}(M) \\ [n,\ldots,n][n,\ldots,n] & \leq & f_{\omega^2 + \omega + n}(n) = f_{\omega^2 + \omega \times 2}(n) \end{eqnarray*} if I am correct. My previous analysis gave the upper bound \(\omega^{\omega}\) in FGH because I just calculated the ordinal type of the well-ordering associated to the recursion, but since the rule set is actually weaker than FGH, the precise value is much smaller than that rough upper bound. I might be wrong.

Then I guess that \([n] \backslash [n]\) is bounded by \(f_{\omega^3}(n)\).