#369 ababbabbaa=babbaaba

Original presentation

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

Modified presentation

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

Complete rewriting system

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

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