#461 baaaababaa=abaaa

Original presentation

\( \left\langle a, b \;\middle|\; b a^{4} b a b a^{2} = a b a^{3} \right\rangle \)

Modified presentation

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

Complete rewriting system

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

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