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    \$\begingroup\$ Is the last rule really needed ? It is needed for minimizing the length of the word, but we only need to minimize the number of \$a\$s. I can't find a example where it is needed. \$\endgroup\$ Commented Dec 22, 2021 at 7:44
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    \$\begingroup\$ Save 1 byte by replacing tr with y. \$\endgroup\$ Commented Dec 22, 2021 at 14:34
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    \$\begingroup\$ @alephalpha Not sure. It’s required by the current correctness proof, at least. \$\endgroup\$ Commented Dec 22, 2021 at 19:05