User blog comment:Naruyoko/My Utterly Obscure and Slowly Evolve-Growing Function-Notation-Whatever/@comment-35470197-20190228224550/@comment-35470197-20190301222429

The updated rule still contains a contradiction. I guess that you need to add the restriction \(s_2 < s_1\) with respect to the order corresponding to \(\omega^{\omega}\) so that the rule is not applicable to the case \(s_1 = s_2\).