#143 baaaabbbba=abaa

Original presentation

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

Modified presentation

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

Complete rewriting system

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

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