It remains to prove that the statement of the Lemma remains true if we replace the random permutations $\pi$ by butterfly graphs $B_i$ with random permutations $\sigma_i$ on $\bin^{2\cdot w/2}$\kknote{For now, blocks of size $w/2$ to be consistent with the above, but need to change notation...}, see Figure \ref{fig:B8}.\footnote{Note, that our graph $B_S$ in fact consists of two copies of the original butterfly graph, hence has length $2\log(2S)$. Still, by misuse of the name, we will refer to $B_S$ as the butterfly graph of size $S$.} More precisely, we need to prove that pebbling internal nodes of the permutations $\B_i$ doesn't help to decrease the number and structure of paths between input and output nodes of $\B_i$ from what one could achieve by pebbling only input and output nodes of $\B_i$.\footnote{Note, in the case of random permutations, we used a complete bipartite graph to represent the computation. Here, we show that the butterfly graph is in some sense close to a comlete bipartite graph.}

It remains to prove that the statement of the Lemma remains true if we replace the random permutations $\pi$ by butterfly graphs $B_i$ with random permutations $\sigma_i$ on $\bin^{2\cdot w/2}$\kknote{For now, blocks of size $w/2$ to be consistent with the above, but need to change notation...}, see Figure \ref{fig:B8}.\footnote{Note, that our graph $B_S$ in fact consists of two copies of the original butterfly graph, hence has length $2\log(2S)$. Still, by misuse of the name, we will refer to $B_S$ as the butterfly graph of size $S$. We also note, that the simple butterfly graph would not be sufficient for our needs, i.e., the claim of the Lemma below does not hold for simple butterfly graphs.} More precisely, we need to prove that pebbling internal nodes of the permutations $\B_i$ doesn't help to decrease the number and structure of paths between input and output nodes of $\B_i$ from what one could achieve by pebbling only input and output nodes of $\B_i$.\footnote{Note, in the case of random permutations, we used a complete bipartite graph to represent the computation. Here, we show that the butterfly graph is in some sense close to a comlete bipartite graph.}