User blog comment:P進大好きbot/Please Help me on study of Pair Sequence System (2-rowed Bashicu Matrix System)/@comment-35392788-20180813124110/@comment-35392788-20180813151407

You said that \(\psi_0(\psi_1(\psi_2(0))) = \psi_0(\psi_2(0))\). By lemma 1.2 (d), it follows that \(\psi_1(\psi_2(0)) < \psi_1(\psi_3(0))\), and thus, \(\psi_0(\psi_1(\psi_2(0))) < \psi_0(\psi_1(\psi_3(0)))\).

I think there is something else wrong. Have you checked if the lemmas apply provided specific criteria ?