#174 ababaaaba=baba

Original presentation

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

Modified presentation

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

Complete rewriting system

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

\( \begin{aligned} c &\rightarrow b a \\ a^{3} b &\rightarrow d \\ a d a b d a &\rightarrow d a b a \\ a d a b d^{2} &\rightarrow d a b d \\ b a b d &\rightarrow d a b d^{4} \\ b a b a &\rightarrow d a b d^{3} a \\ b d a b d^{3} &\rightarrow d a b d^{3} a b d \\ b d a b d^{2} a &\rightarrow d a b d^{3} a b a \\ a d^{2} a b d^{2} a b d &\rightarrow d^{2} a b d^{5} \\ a d^{2} a b d^{2} a b a &\rightarrow d^{2} a b d^{4} a \\ a d^{2} a b d^{3} a b d^{3} &\rightarrow d^{2} a b d^{4} a b d \\ a d^{2} a b d^{3} a b d^{2} a &\rightarrow d^{2} a b d^{4} a b a \end{aligned} \)