#439 baaaabbaaa=aaaabba

Original presentation

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

Modified presentation

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

Complete rewriting system

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

\( \begin{aligned} b^{2} &\rightarrow d \\ a^{2} &\rightarrow c \\ d c^{3} &\rightarrow e \\ b d &\rightarrow d b \\ a c &\rightarrow c a \\ d b c^{3} &\rightarrow b e \\ c^{2} d c &\rightarrow d c^{2} e \\ d^{2} c^{2} e c a &\rightarrow c^{2} d a \\ b c^{2} e &\rightarrow d c^{2} e c \\ b c^{2} d a &\rightarrow d^{2} c^{2} e a \\ a d c^{2} e &\rightarrow c^{2} a d c \\ c^{2} d^{2} c^{2} e &\rightarrow d c^{2} e c d c \\ b e c d c &\rightarrow d^{2} c^{2} e^{2} \\ d b c d c^{2} e &\rightarrow b e d c \\ d c^{2} e c^{2} &\rightarrow c^{2} e \\ d^{2} c^{2} e c e &\rightarrow e c d c \\ d c d c^{2} e &\rightarrow e d c \\ b e c^{3} e &\rightarrow d^{2} c^{2} e^{2} c e c^{2} \\ b e c e &\rightarrow d^{2} c^{2} e^{2} c^{2} \\ b e c d^{2} c^{2} e &\rightarrow d^{2} c^{2} e^{2} c d c \\ d c^{2} e c e c^{2} &\rightarrow c^{4} e \\ d c^{2} e c d c^{2} a &\rightarrow c^{4} d a \\ d c^{2} e c d c^{2} e &\rightarrow c^{2} e c d c \\ d c^{2} e d c^{2} e &\rightarrow c^{2} e d c \\ b e c^{5} e &\rightarrow d^{2} c^{2} e^{2} c e c e c^{2} \\ b e c^{5} d a &\rightarrow d^{2} c^{2} e^{2} c e c d c^{2} a \\ b e c^{3} d a &\rightarrow d^{2} c^{2} e^{2} c d c^{2} a \\ c^{6} e &\rightarrow d c^{2} e c e c e c^{2} \\ c^{6} d a &\rightarrow d c^{2} e c e c d c^{2} a \\ b c^{4} e &\rightarrow d c^{2} e^{2} c^{2} \\ b c^{4} d a &\rightarrow d c^{2} e d c^{2} a \\ d c^{2} e c e c d c^{2} e &\rightarrow c^{4} e c d c \\ d c^{2} e c e d c^{2} e &\rightarrow c^{4} e d c \\ b e c d a &\rightarrow d^{2} c^{2} e d c^{2} a \end{aligned} \)