#334 abbabbbaaa=baaabba

Original presentation

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

Modified presentation

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

Complete rewriting system

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

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