#186 baabaaaba=abaa

Original presentation

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

Modified presentation

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

Complete rewriting system

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

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