#475 baaabbaaba=abbaa

Original presentation

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

Modified presentation

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

Complete rewriting system

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

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