#477 abbabbbaba=baabba

Original presentation

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

Modified presentation

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

Complete rewriting system

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

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