#152 babaaaabba=aaabbaba

Original presentation

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

Modified presentation

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

Complete rewriting system

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

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