#390 abaaabbaba=babaaaba

Original presentation

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

Modified presentation

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

Complete rewriting system

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

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