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#386 baabaaabaa=aba

Original presentation

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

Modified presentation

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

Complete rewriting system

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

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