#052 abaabbabba=babbaaba

Original presentation

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

Modified presentation

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

Complete rewriting system

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

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