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#175 baaabaabaa=aabba

Original presentation

\( \left\langle a, b \;\middle|\; b a^{3} b a^{2} b a^{2} = a^{2} b^{2} a \right\rangle \)

Modified presentation

\( \left\langle a, b, c, d \;\middle|\; b a d a = a^{2} c a,\; c = b^{2},\; d = a^{2} b a^{2} b a \right\rangle \)

Complete rewriting system

Using recursive path order (RPOCmp) with \(a < c < d < b\):

\( \begin{aligned} b^{2} &\rightarrow c \\ b a d a &\rightarrow a^{2} c a \\ a^{2} b a^{2} b a &\rightarrow d \\ b c &\rightarrow c b \\ b a^{2} c a &\rightarrow c a d a \\ a^{2} b d &\rightarrow d a b a \\ a^{2} b a^{4} c a &\rightarrow d^{2} a \\ b a d^{2} &\rightarrow a^{2} c d \\ b a^{2} c d &\rightarrow c a d^{2} \\ a^{2} b a^{4} c d &\rightarrow d^{3} \\ a^{2} c a d a d a &\rightarrow d a c a \\ b d a c a &\rightarrow c a d a d a d a \\ a^{2} c a d a d^{2} &\rightarrow d a c d \\ a^{2} b a^{2} d a c a &\rightarrow d^{2} a d a d a \\ b d a c d &\rightarrow c a d a d a d^{2} \\ a^{2} b a^{2} d a c d &\rightarrow d^{2} a d a d^{2} \end{aligned} \)