YES Problem: b(a(b(b(x1)))) -> b(b(b(a(b(x1))))) b(a(a(b(b(x1))))) -> b(a(b(b(a(a(b(x1))))))) b(a(a(a(b(b(x1)))))) -> b(a(a(b(b(a(a(a(b(x1))))))))) Proof: String Reversal Processor: b(b(a(b(x1)))) -> b(a(b(b(b(x1))))) b(b(a(a(b(x1))))) -> b(a(a(b(b(a(b(x1))))))) b(b(a(a(a(b(x1)))))) -> b(a(a(a(b(b(a(a(b(x1))))))))) DP Processor: DPs: b#(b(a(b(x1)))) -> b#(b(x1)) b#(b(a(b(x1)))) -> b#(b(b(x1))) b#(b(a(b(x1)))) -> b#(a(b(b(b(x1))))) b#(b(a(a(b(x1))))) -> b#(a(b(x1))) b#(b(a(a(b(x1))))) -> b#(b(a(b(x1)))) b#(b(a(a(b(x1))))) -> b#(a(a(b(b(a(b(x1))))))) b#(b(a(a(a(b(x1)))))) -> b#(a(a(b(x1)))) b#(b(a(a(a(b(x1)))))) -> b#(b(a(a(b(x1))))) b#(b(a(a(a(b(x1)))))) -> b#(a(a(a(b(b(a(a(b(x1))))))))) TRS: b(b(a(b(x1)))) -> b(a(b(b(b(x1))))) b(b(a(a(b(x1))))) -> b(a(a(b(b(a(b(x1))))))) b(b(a(a(a(b(x1)))))) -> b(a(a(a(b(b(a(a(b(x1))))))))) EDG Processor: DPs: b#(b(a(b(x1)))) -> b#(b(x1)) b#(b(a(b(x1)))) -> b#(b(b(x1))) b#(b(a(b(x1)))) -> b#(a(b(b(b(x1))))) b#(b(a(a(b(x1))))) -> b#(a(b(x1))) b#(b(a(a(b(x1))))) -> b#(b(a(b(x1)))) b#(b(a(a(b(x1))))) -> b#(a(a(b(b(a(b(x1))))))) b#(b(a(a(a(b(x1)))))) -> b#(a(a(b(x1)))) b#(b(a(a(a(b(x1)))))) -> b#(b(a(a(b(x1))))) b#(b(a(a(a(b(x1)))))) -> b#(a(a(a(b(b(a(a(b(x1))))))))) TRS: b(b(a(b(x1)))) -> b(a(b(b(b(x1))))) b(b(a(a(b(x1))))) -> b(a(a(b(b(a(b(x1))))))) b(b(a(a(a(b(x1)))))) -> b(a(a(a(b(b(a(a(b(x1))))))))) graph: b#(b(a(a(a(b(x1)))))) -> b#(b(a(a(b(x1))))) -> b#(b(a(a(b(x1))))) -> b#(a(b(x1))) b#(b(a(a(a(b(x1)))))) -> b#(b(a(a(b(x1))))) -> b#(b(a(a(b(x1))))) -> b#(b(a(b(x1)))) b#(b(a(a(a(b(x1)))))) -> b#(b(a(a(b(x1))))) -> b#(b(a(a(b(x1))))) -> b#(a(a(b(b(a(b(x1))))))) b#(b(a(a(b(x1))))) -> b#(b(a(b(x1)))) -> b#(b(a(b(x1)))) -> b#(b(x1)) b#(b(a(a(b(x1))))) -> b#(b(a(b(x1)))) -> b#(b(a(b(x1)))) -> b#(b(b(x1))) b#(b(a(a(b(x1))))) -> b#(b(a(b(x1)))) -> b#(b(a(b(x1)))) -> b#(a(b(b(b(x1))))) b#(b(a(b(x1)))) -> b#(b(b(x1))) -> b#(b(a(b(x1)))) -> b#(b(x1)) b#(b(a(b(x1)))) -> b#(b(b(x1))) -> b#(b(a(b(x1)))) -> b#(b(b(x1))) b#(b(a(b(x1)))) -> b#(b(b(x1))) -> b#(b(a(b(x1)))) -> b#(a(b(b(b(x1))))) b#(b(a(b(x1)))) -> b#(b(b(x1))) -> b#(b(a(a(b(x1))))) -> b#(a(b(x1))) b#(b(a(b(x1)))) -> b#(b(b(x1))) -> b#(b(a(a(b(x1))))) -> b#(b(a(b(x1)))) b#(b(a(b(x1)))) -> b#(b(b(x1))) -> b#(b(a(a(b(x1))))) -> b#(a(a(b(b(a(b(x1))))))) b#(b(a(b(x1)))) -> b#(b(b(x1))) -> b#(b(a(a(a(b(x1)))))) -> b#(a(a(b(x1)))) b#(b(a(b(x1)))) -> b#(b(b(x1))) -> b#(b(a(a(a(b(x1)))))) -> b#(b(a(a(b(x1))))) b#(b(a(b(x1)))) -> b#(b(b(x1))) -> b#(b(a(a(a(b(x1)))))) -> b#(a(a(a(b(b(a(a(b(x1))))))))) b#(b(a(b(x1)))) -> b#(b(x1)) -> b#(b(a(b(x1)))) -> b#(b(x1)) b#(b(a(b(x1)))) -> b#(b(x1)) -> b#(b(a(b(x1)))) -> b#(b(b(x1))) b#(b(a(b(x1)))) -> b#(b(x1)) -> b#(b(a(b(x1)))) -> b#(a(b(b(b(x1))))) b#(b(a(b(x1)))) -> b#(b(x1)) -> b#(b(a(a(b(x1))))) -> b#(a(b(x1))) b#(b(a(b(x1)))) -> b#(b(x1)) -> b#(b(a(a(b(x1))))) -> b#(b(a(b(x1)))) b#(b(a(b(x1)))) -> b#(b(x1)) -> b#(b(a(a(b(x1))))) -> b#(a(a(b(b(a(b(x1))))))) b#(b(a(b(x1)))) -> b#(b(x1)) -> b#(b(a(a(a(b(x1)))))) -> b#(a(a(b(x1)))) b#(b(a(b(x1)))) -> b#(b(x1)) -> b#(b(a(a(a(b(x1)))))) -> b#(b(a(a(b(x1))))) b#(b(a(b(x1)))) -> b#(b(x1)) -> b#(b(a(a(a(b(x1)))))) -> b#(a(a(a(b(b(a(a(b(x1))))))))) SCC Processor: #sccs: 1 #rules: 4 #arcs: 24/81 DPs: b#(b(a(a(a(b(x1)))))) -> b#(b(a(a(b(x1))))) b#(b(a(a(b(x1))))) -> b#(b(a(b(x1)))) b#(b(a(b(x1)))) -> b#(b(b(x1))) b#(b(a(b(x1)))) -> b#(b(x1)) TRS: b(b(a(b(x1)))) -> b(a(b(b(b(x1))))) b(b(a(a(b(x1))))) -> b(a(a(b(b(a(b(x1))))))) b(b(a(a(a(b(x1)))))) -> b(a(a(a(b(b(a(a(b(x1))))))))) Matrix Interpretation Processor: dim=4 interpretation: [0 0 1 1] [0] [0 0 0 0] [1] [a](x0) = [1 1 0 0]x0 + [0] [0 0 0 0] [0], [0 0 0 0] [1 0 0 1] [b](x0) = [0 0 0 0]x0 [1 0 0 1] , [b#](x0) = [0 1 0 0]x0 orientation: b#(b(a(a(a(b(x1)))))) = [1 0 0 1]x1 + [1] >= [1 0 0 1]x1 = b#(b(a(a(b(x1))))) b#(b(a(a(b(x1))))) = [1 0 0 1]x1 >= [1 0 0 1]x1 = b#(b(a(b(x1)))) b#(b(a(b(x1)))) = [1 0 0 1]x1 >= [1 0 0 1]x1 = b#(b(b(x1))) b#(b(a(b(x1)))) = [1 0 0 1]x1 >= [1 0 0 1]x1 = b#(b(x1)) [0 0 0 0] [0 0 0 0] [1 0 0 1] [1 0 0 1] b(b(a(b(x1)))) = [0 0 0 0]x1 >= [0 0 0 0]x1 = b(a(b(b(b(x1))))) [1 0 0 1] [1 0 0 1] [0 0 0 0] [0 0 0 0] [1 0 0 1] [1 0 0 1] b(b(a(a(b(x1))))) = [0 0 0 0]x1 >= [0 0 0 0]x1 = b(a(a(b(b(a(b(x1))))))) [1 0 0 1] [1 0 0 1] [0 0 0 0] [0] [0 0 0 0] [0] [1 0 0 1] [1] [1 0 0 1] [1] b(b(a(a(a(b(x1)))))) = [0 0 0 0]x1 + [0] >= [0 0 0 0]x1 + [0] = b(a(a(a(b(b(a(a(b(x1))))))))) [1 0 0 1] [1] [1 0 0 1] [1] problem: DPs: b#(b(a(a(b(x1))))) -> b#(b(a(b(x1)))) b#(b(a(b(x1)))) -> b#(b(b(x1))) b#(b(a(b(x1)))) -> b#(b(x1)) TRS: b(b(a(b(x1)))) -> b(a(b(b(b(x1))))) b(b(a(a(b(x1))))) -> b(a(a(b(b(a(b(x1))))))) b(b(a(a(a(b(x1)))))) -> b(a(a(a(b(b(a(a(b(x1))))))))) Restore Modifier: DPs: b#(b(a(a(b(x1))))) -> b#(b(a(b(x1)))) b#(b(a(b(x1)))) -> b#(b(b(x1))) b#(b(a(b(x1)))) -> b#(b(x1)) TRS: b(b(a(b(x1)))) -> b(a(b(b(b(x1))))) b(b(a(a(b(x1))))) -> b(a(a(b(b(a(b(x1))))))) b(b(a(a(a(b(x1)))))) -> b(a(a(a(b(b(a(a(b(x1))))))))) EDG Processor: DPs: b#(b(a(a(b(x1))))) -> b#(b(a(b(x1)))) b#(b(a(b(x1)))) -> b#(b(b(x1))) b#(b(a(b(x1)))) -> b#(b(x1)) TRS: b(b(a(b(x1)))) -> b(a(b(b(b(x1))))) b(b(a(a(b(x1))))) -> b(a(a(b(b(a(b(x1))))))) b(b(a(a(a(b(x1)))))) -> b(a(a(a(b(b(a(a(b(x1))))))))) graph: b#(b(a(a(b(x1))))) -> b#(b(a(b(x1)))) -> b#(b(a(b(x1)))) -> b#(b(b(x1))) b#(b(a(a(b(x1))))) -> b#(b(a(b(x1)))) -> b#(b(a(b(x1)))) -> b#(b(x1)) b#(b(a(b(x1)))) -> b#(b(b(x1))) -> b#(b(a(a(b(x1))))) -> b#(b(a(b(x1)))) b#(b(a(b(x1)))) -> b#(b(b(x1))) -> b#(b(a(b(x1)))) -> b#(b(b(x1))) b#(b(a(b(x1)))) -> b#(b(b(x1))) -> b#(b(a(b(x1)))) -> b#(b(x1)) b#(b(a(b(x1)))) -> b#(b(x1)) -> b#(b(a(a(b(x1))))) -> b#(b(a(b(x1)))) b#(b(a(b(x1)))) -> b#(b(x1)) -> b#(b(a(b(x1)))) -> b#(b(b(x1))) b#(b(a(b(x1)))) -> b#(b(x1)) -> b#(b(a(b(x1)))) -> b#(b(x1)) Matrix Interpretation Processor: dim=4 interpretation: [0 0 0 0] [1] [0 0 0 0] [0] [a](x0) = [1 1 0 0]x0 + [0] [0 0 1 0] [0], [0 0 0 1] [0 1 1 0] [b](x0) = [0 0 0 0]x0 [0 0 0 1] , [b#](x0) = [1 1 0 0]x0 orientation: b#(b(a(a(b(x1))))) = [0 1 1 1]x1 + [1] >= [0 1 1 1]x1 = b#(b(a(b(x1)))) b#(b(a(b(x1)))) = [0 1 1 1]x1 >= [0 1 1 1]x1 = b#(b(b(x1))) b#(b(a(b(x1)))) = [0 1 1 1]x1 >= [0 1 1 1]x1 = b#(b(x1)) [0 0 0 0] [0 0 0 0] [0 1 1 1] [0 1 1 1] b(b(a(b(x1)))) = [0 0 0 0]x1 >= [0 0 0 0]x1 = b(a(b(b(b(x1))))) [0 0 0 0] [0 0 0 0] [0 1 1 1] [0] [0 1 1 1] [0] [0 0 0 0] [1] [0 0 0 0] [1] b(b(a(a(b(x1))))) = [0 0 0 0]x1 + [0] >= [0 0 0 0]x1 + [0] = b(a(a(b(b(a(b(x1))))))) [0 1 1 1] [0] [0 1 1 1] [0] [1] [1] [1] [1] b(b(a(a(a(b(x1)))))) = [0] >= [0] = b(a(a(a(b(b(a(a(b(x1))))))))) [1] [1] problem: DPs: b#(b(a(b(x1)))) -> b#(b(b(x1))) b#(b(a(b(x1)))) -> b#(b(x1)) TRS: b(b(a(b(x1)))) -> b(a(b(b(b(x1))))) b(b(a(a(b(x1))))) -> b(a(a(b(b(a(b(x1))))))) b(b(a(a(a(b(x1)))))) -> b(a(a(a(b(b(a(a(b(x1))))))))) Restore Modifier: DPs: b#(b(a(b(x1)))) -> b#(b(b(x1))) b#(b(a(b(x1)))) -> b#(b(x1)) TRS: b(b(a(b(x1)))) -> b(a(b(b(b(x1))))) b(b(a(a(b(x1))))) -> b(a(a(b(b(a(b(x1))))))) b(b(a(a(a(b(x1)))))) -> b(a(a(a(b(b(a(a(b(x1))))))))) EDG Processor: DPs: b#(b(a(b(x1)))) -> b#(b(b(x1))) b#(b(a(b(x1)))) -> b#(b(x1)) TRS: b(b(a(b(x1)))) -> b(a(b(b(b(x1))))) b(b(a(a(b(x1))))) -> b(a(a(b(b(a(b(x1))))))) b(b(a(a(a(b(x1)))))) -> b(a(a(a(b(b(a(a(b(x1))))))))) graph: b#(b(a(b(x1)))) -> b#(b(b(x1))) -> b#(b(a(b(x1)))) -> b#(b(x1)) b#(b(a(b(x1)))) -> b#(b(b(x1))) -> b#(b(a(b(x1)))) -> b#(b(b(x1))) b#(b(a(b(x1)))) -> b#(b(x1)) -> b#(b(a(b(x1)))) -> b#(b(x1)) b#(b(a(b(x1)))) -> b#(b(x1)) -> b#(b(a(b(x1)))) -> b#(b(b(x1))) Matrix Interpretation Processor: dim=4 interpretation: [0 1 0 0] [1] [0 0 0 0] [0] [a](x0) = [0 0 0 1]x0 + [0] [0 0 0 0] [0], [0 0 0 0] [0 1 1 0] [b](x0) = [0 0 0 0]x0 [1 0 0 1] , [b#](x0) = [0 1 0 1]x0 orientation: b#(b(a(b(x1)))) = [1 1 1 1]x1 + [1] >= [1 1 1 1]x1 = b#(b(b(x1))) b#(b(a(b(x1)))) = [1 1 1 1]x1 + [1] >= [1 1 1 1]x1 = b#(b(x1)) [0 0 0 0] [0] [0 0 0 0] [0] [1 0 0 1] [0] [1 0 0 1] [0] b(b(a(b(x1)))) = [0 0 0 0]x1 + [0] >= [0 0 0 0]x1 + [0] = b(a(b(b(b(x1))))) [0 1 1 0] [1] [0 1 1 0] [1] [0] [0] [0] [0] b(b(a(a(b(x1))))) = [0] >= [0] = b(a(a(b(b(a(b(x1))))))) [1] [1] [0] [0] [0] [0] b(b(a(a(a(b(x1)))))) = [0] >= [0] = b(a(a(a(b(b(a(a(b(x1))))))))) [1] [1] problem: DPs: TRS: b(b(a(b(x1)))) -> b(a(b(b(b(x1))))) b(b(a(a(b(x1))))) -> b(a(a(b(b(a(b(x1))))))) b(b(a(a(a(b(x1)))))) -> b(a(a(a(b(b(a(a(b(x1))))))))) Qed