#139 abababbaaa=baaba

Original presentation

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

Modified presentation

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

Complete rewriting system

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

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