#359 ababaaabaa=baaba

Original presentation

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

Modified presentation

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

Complete rewriting system

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

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