User blog comment:LittlePeng9/FORMAL comparison of Hydra battles and Hardy hierarchy/@comment-10429372-20141017061807/@comment-5529393-20141017200028

Nice. The only difficulty is in showing that H_{\alpha[n]+1}(n+1) \le H_{\alpha[n+1](n+1). It's certainly true that $$\alpha[n]+1 \le \alpha[n+1]$$, but that doesn't necessarily imply the previous inequality.