#370 baaababaaa=aababaa

Original presentation

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

Modified presentation

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

Complete rewriting system

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

\( \begin{aligned} b a^{2} d &\rightarrow a c a^{2} \\ a b a b &\rightarrow c \\ c a^{3} &\rightarrow d \\ a b c &\rightarrow c a b \\ d b a b &\rightarrow c a^{2} c \\ a^{2} c a^{2} &\rightarrow c a^{2} d a \\ b c a^{2} d a c &\rightarrow a c a c \\ d b c &\rightarrow c a^{2} c a b \\ c a b a^{3} &\rightarrow a b d \\ b c a^{2} d a d &\rightarrow a c a d \\ a^{2} c^{2} a^{2} d a &\rightarrow c a^{2} d a c a^{2} \\ a^{2} c a c &\rightarrow c a^{2} d c \\ a b d a c a^{2} &\rightarrow c^{2} a^{2} d^{2} a \\ c a b a c a^{2} d a &\rightarrow a b d c a^{2} \\ c a^{2} d a^{2} &\rightarrow a^{2} d \\ c^{2} a^{2} d a d c &\rightarrow d a c a c \\ c^{2} a^{2} d a d a &\rightarrow d a c a^{2} \\ c a c a^{2} d a &\rightarrow d c a^{2} \\ a b d a^{3} d &\rightarrow c^{2} a^{2} d^{2} a d a^{2} \\ a b d a d &\rightarrow c^{2} a^{2} d^{2} a^{2} \\ a b d a c^{2} a^{2} d a &\rightarrow c^{2} a^{2} d^{2} a c a^{2} \\ a b d a c^{2} a^{2} d c &\rightarrow c^{2} a^{2} d^{2} a c a c \\ c a b a c a^{2} d c &\rightarrow a b d c a c \\ c a^{2} d c a^{2} d c &\rightarrow a^{2} d c a c \\ a^{2} c^{2} a^{2} d c &\rightarrow c a^{2} d a c a c \\ c^{2} a^{2} d a d^{2} &\rightarrow d a c a d \\ a^{2} c a d &\rightarrow c a^{2} d^{2} \\ a b d a c a c &\rightarrow c^{2} a^{2} d^{2} c \\ c a^{2} d a d a^{2} &\rightarrow a^{4} d \\ c a^{2} d a c a^{2} d a &\rightarrow a^{2} d a c a^{2} \\ c a^{2} d a c a^{2} d c &\rightarrow a^{2} d a c a c \\ c a^{2} d c a^{2} d a &\rightarrow a^{2} d c a^{2} \\ d b d a c a c &\rightarrow c^{2} a^{2} d a c a d c \\ c a c a^{2} d c &\rightarrow d c a c \\ d b d a c a^{2} &\rightarrow c^{2} a^{2} d a c a d a \\ a b d a c^{2} a^{2} d^{2} &\rightarrow c^{2} a^{2} d^{2} a c a d \\ c a b a c a^{2} d^{2} &\rightarrow a b d c a d \\ c a^{2} d c a^{2} d^{2} &\rightarrow a^{2} d c a d \\ d b d a^{3} d &\rightarrow c^{2} a^{2} d a c a d a d a^{2} \\ d b d a d &\rightarrow c^{2} a^{2} d a c a d a^{2} \\ d b d a c^{2} a^{2} d a &\rightarrow c^{2} a^{2} d a c a d a c a^{2} \\ d b d a c^{2} a^{2} d c &\rightarrow c^{2} a^{2} d a c a d a c a c \\ c a^{2} d a d c a^{2} d c &\rightarrow a^{4} d c a c \\ b a^{4} d &\rightarrow a c a d a^{2} \\ a b d a^{5} d &\rightarrow c^{2} a^{2} d^{2} a d a d a^{2} \\ a b d a c a d &\rightarrow c^{2} a^{2} d^{3} \\ a^{6} d &\rightarrow c a^{2} d a d a d a^{2} \\ a^{2} c^{2} a^{2} d^{2} &\rightarrow c a^{2} d a c a d \\ c a^{2} d a d a c a^{2} d a &\rightarrow a^{4} d a c a^{2} \\ c a^{2} d a d a c a^{2} d^{2} &\rightarrow a^{4} d a c a d \\ c a^{2} d a d a c a^{2} d c &\rightarrow a^{4} d a c a c \\ c a^{2} d a d c a^{2} d a &\rightarrow a^{4} d c a^{2} \\ c a^{2} d a d c a^{2} d^{2} &\rightarrow a^{4} d c a d \\ c a^{2} d a c a^{2} d^{2} &\rightarrow a^{2} d a c a d \\ d b d a c a d &\rightarrow c^{2} a^{2} d a c a d^{2} \\ c a c a^{2} d^{2} &\rightarrow d c a d \\ d b d a^{5} d &\rightarrow c^{2} a^{2} d a c a d a d a d a^{2} \\ d b d a c^{2} a^{2} d^{2} &\rightarrow c^{2} a^{2} d a c a d a c a d \end{aligned} \)