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\@writefile{toc}{\contentsline {section}{\numberline {1}Lower bound for bounded-depth Frege proofs of $PHP_n^{n+1}$}{1}}
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\@writefile{toc}{\contentsline {subsection}{\numberline {1.2}Analogy}{2}}
\@writefile{toc}{\contentsline {subsection}{\numberline {1.3}$PHP_n^{n+1}$ consists of 0-trees}{2}}
\@writefile{toc}{\contentsline {subsection}{\numberline {1.4}All formulas in a bounded-depth Frege proofs get assigned 1-trees}{3}}