YES Problem: a(a(b(x1))) -> b(a(b(x1))) b(a(x1)) -> a(b(b(x1))) b(c(a(x1))) -> c(c(a(a(b(x1))))) Proof: DP Processor: DPs: a#(a(b(x1))) -> b#(a(b(x1))) b#(a(x1)) -> b#(x1) b#(a(x1)) -> b#(b(x1)) b#(a(x1)) -> a#(b(b(x1))) b#(c(a(x1))) -> b#(x1) b#(c(a(x1))) -> a#(b(x1)) b#(c(a(x1))) -> a#(a(b(x1))) TRS: a(a(b(x1))) -> b(a(b(x1))) b(a(x1)) -> a(b(b(x1))) b(c(a(x1))) -> c(c(a(a(b(x1))))) TDG Processor: DPs: a#(a(b(x1))) -> b#(a(b(x1))) b#(a(x1)) -> b#(x1) b#(a(x1)) -> b#(b(x1)) b#(a(x1)) -> a#(b(b(x1))) b#(c(a(x1))) -> b#(x1) b#(c(a(x1))) -> a#(b(x1)) b#(c(a(x1))) -> a#(a(b(x1))) TRS: a(a(b(x1))) -> b(a(b(x1))) b(a(x1)) -> a(b(b(x1))) b(c(a(x1))) -> c(c(a(a(b(x1))))) graph: b#(c(a(x1))) -> b#(x1) -> b#(c(a(x1))) -> a#(a(b(x1))) b#(c(a(x1))) -> b#(x1) -> b#(c(a(x1))) -> a#(b(x1)) b#(c(a(x1))) -> b#(x1) -> b#(c(a(x1))) -> b#(x1) b#(c(a(x1))) -> b#(x1) -> b#(a(x1)) -> a#(b(b(x1))) b#(c(a(x1))) -> b#(x1) -> b#(a(x1)) -> b#(b(x1)) b#(c(a(x1))) -> b#(x1) -> b#(a(x1)) -> b#(x1) b#(c(a(x1))) -> a#(a(b(x1))) -> a#(a(b(x1))) -> b#(a(b(x1))) b#(c(a(x1))) -> a#(b(x1)) -> a#(a(b(x1))) -> b#(a(b(x1))) b#(a(x1)) -> b#(b(x1)) -> b#(c(a(x1))) -> a#(a(b(x1))) b#(a(x1)) -> b#(b(x1)) -> b#(c(a(x1))) -> a#(b(x1)) b#(a(x1)) -> b#(b(x1)) -> b#(c(a(x1))) -> b#(x1) b#(a(x1)) -> b#(b(x1)) -> b#(a(x1)) -> a#(b(b(x1))) b#(a(x1)) -> b#(b(x1)) -> b#(a(x1)) -> b#(b(x1)) b#(a(x1)) -> b#(b(x1)) -> b#(a(x1)) -> b#(x1) b#(a(x1)) -> b#(x1) -> b#(c(a(x1))) -> a#(a(b(x1))) b#(a(x1)) -> b#(x1) -> b#(c(a(x1))) -> a#(b(x1)) b#(a(x1)) -> b#(x1) -> b#(c(a(x1))) -> b#(x1) b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> a#(b(b(x1))) b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> b#(b(x1)) b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> b#(x1) b#(a(x1)) -> a#(b(b(x1))) -> a#(a(b(x1))) -> b#(a(b(x1))) a#(a(b(x1))) -> b#(a(b(x1))) -> b#(c(a(x1))) -> a#(a(b(x1))) a#(a(b(x1))) -> b#(a(b(x1))) -> b#(c(a(x1))) -> a#(b(x1)) a#(a(b(x1))) -> b#(a(b(x1))) -> b#(c(a(x1))) -> b#(x1) a#(a(b(x1))) -> b#(a(b(x1))) -> b#(a(x1)) -> a#(b(b(x1))) a#(a(b(x1))) -> b#(a(b(x1))) -> b#(a(x1)) -> b#(b(x1)) a#(a(b(x1))) -> b#(a(b(x1))) -> b#(a(x1)) -> b#(x1) Matrix Interpretation Processor: dim=4 interpretation: [b#](x0) = [0 1 0 0]x0 + [1], [a#](x0) = [1 1 0 0]x0, [0 0 0 0] [1 1 0 0] [c](x0) = [0 0 0 0]x0 [0 0 1 0] , [0 0 1 0] [0] [0 1 0 1] [0] [a](x0) = [0 0 0 1]x0 + [1] [0 0 0 0] [0], [0 0 1 0] [0] [0 0 0 1] [0] [b](x0) = [0 0 0 0]x0 + [1] [0 0 0 0] [0] orientation: a#(a(b(x1))) = [0 0 0 1]x1 + [1] >= [0 0 0 1]x1 + [1] = b#(a(b(x1))) b#(a(x1)) = [0 1 0 1]x1 + [1] >= [0 1 0 0]x1 + [1] = b#(x1) b#(a(x1)) = [0 1 0 1]x1 + [1] >= [0 0 0 1]x1 + [1] = b#(b(x1)) b#(a(x1)) = [0 1 0 1]x1 + [1] >= [1] = a#(b(b(x1))) b#(c(a(x1))) = [0 1 1 1]x1 + [1] >= [0 1 0 0]x1 + [1] = b#(x1) b#(c(a(x1))) = [0 1 1 1]x1 + [1] >= [0 0 1 1]x1 = a#(b(x1)) b#(c(a(x1))) = [0 1 1 1]x1 + [1] >= [0 0 0 1]x1 + [1] = a#(a(b(x1))) [0 0 0 0] [1] [1] [0 0 0 1] [0] [0] a(a(b(x1))) = [0 0 0 0]x1 + [1] >= [1] = b(a(b(x1))) [0 0 0 0] [0] [0] [0 0 0 1] [1] [1] [0 0 0 0] [0] [0] b(a(x1)) = [0 0 0 0]x1 + [1] >= [1] = a(b(b(x1))) [0 0 0 0] [0] [0] [0 0 0 0] [0] [0 0 0 0] [0] [0 0 0 1] [1] [0 0 0 1] [1] b(c(a(x1))) = [0 0 0 0]x1 + [1] >= [0 0 0 0]x1 + [0] = c(c(a(a(b(x1))))) [0 0 0 0] [0] [0 0 0 0] [0] problem: DPs: a#(a(b(x1))) -> b#(a(b(x1))) b#(a(x1)) -> b#(x1) b#(a(x1)) -> b#(b(x1)) b#(a(x1)) -> a#(b(b(x1))) b#(c(a(x1))) -> b#(x1) b#(c(a(x1))) -> a#(a(b(x1))) TRS: a(a(b(x1))) -> b(a(b(x1))) b(a(x1)) -> a(b(b(x1))) b(c(a(x1))) -> c(c(a(a(b(x1))))) EDG Processor: DPs: a#(a(b(x1))) -> b#(a(b(x1))) b#(a(x1)) -> b#(x1) b#(a(x1)) -> b#(b(x1)) b#(a(x1)) -> a#(b(b(x1))) b#(c(a(x1))) -> b#(x1) b#(c(a(x1))) -> a#(a(b(x1))) TRS: a(a(b(x1))) -> b(a(b(x1))) b(a(x1)) -> a(b(b(x1))) b(c(a(x1))) -> c(c(a(a(b(x1))))) graph: b#(c(a(x1))) -> b#(x1) -> b#(a(x1)) -> b#(x1) b#(c(a(x1))) -> b#(x1) -> b#(a(x1)) -> b#(b(x1)) b#(c(a(x1))) -> b#(x1) -> b#(a(x1)) -> a#(b(b(x1))) b#(c(a(x1))) -> b#(x1) -> b#(c(a(x1))) -> b#(x1) b#(c(a(x1))) -> b#(x1) -> b#(c(a(x1))) -> a#(a(b(x1))) b#(c(a(x1))) -> a#(a(b(x1))) -> a#(a(b(x1))) -> b#(a(b(x1))) b#(a(x1)) -> b#(b(x1)) -> b#(a(x1)) -> b#(x1) b#(a(x1)) -> b#(b(x1)) -> b#(a(x1)) -> b#(b(x1)) b#(a(x1)) -> b#(b(x1)) -> b#(a(x1)) -> a#(b(b(x1))) b#(a(x1)) -> b#(b(x1)) -> b#(c(a(x1))) -> b#(x1) b#(a(x1)) -> b#(b(x1)) -> b#(c(a(x1))) -> a#(a(b(x1))) b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> b#(x1) b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> b#(b(x1)) b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> a#(b(b(x1))) b#(a(x1)) -> b#(x1) -> b#(c(a(x1))) -> b#(x1) b#(a(x1)) -> b#(x1) -> b#(c(a(x1))) -> a#(a(b(x1))) b#(a(x1)) -> a#(b(b(x1))) -> a#(a(b(x1))) -> b#(a(b(x1))) a#(a(b(x1))) -> b#(a(b(x1))) -> b#(a(x1)) -> b#(x1) a#(a(b(x1))) -> b#(a(b(x1))) -> b#(a(x1)) -> b#(b(x1)) a#(a(b(x1))) -> b#(a(b(x1))) -> b#(a(x1)) -> a#(b(b(x1))) a#(a(b(x1))) -> b#(a(b(x1))) -> b#(c(a(x1))) -> b#(x1) a#(a(b(x1))) -> b#(a(b(x1))) -> b#(c(a(x1))) -> a#(a(b(x1))) Matrix Interpretation Processor: dim=4 interpretation: [b#](x0) = [1 0 1 0]x0, [a#](x0) = [0 0 0 1]x0, [0 0 0 0] [1] [0 0 0 0] [1] [c](x0) = [0 0 1 0]x0 + [0] [1 0 0 1] [0], [0 0 0 0] [0 0 0 0] [a](x0) = [1 1 1 0]x0 [1 0 0 1] , [0 1 0 0] [1 0 0 0] [b](x0) = [0 0 1 0]x0 [1 0 1 0] orientation: a#(a(b(x1))) = [1 1 1 0]x1 >= [1 1 1 0]x1 = b#(a(b(x1))) b#(a(x1)) = [1 1 1 0]x1 >= [1 0 1 0]x1 = b#(x1) b#(a(x1)) = [1 1 1 0]x1 >= [0 1 1 0]x1 = b#(b(x1)) b#(a(x1)) = [1 1 1 0]x1 >= [0 1 1 0]x1 = a#(b(b(x1))) b#(c(a(x1))) = [1 1 1 0]x1 + [1] >= [1 0 1 0]x1 = b#(x1) b#(c(a(x1))) = [1 1 1 0]x1 + [1] >= [1 1 1 0]x1 = a#(a(b(x1))) [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] a(a(b(x1))) = [1 1 1 0]x1 >= [1 1 1 0]x1 = b(a(b(x1))) [1 1 1 0] [1 1 1 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] b(a(x1)) = [1 1 1 0]x1 >= [1 1 1 0]x1 = a(b(b(x1))) [1 1 1 0] [1 1 1 0] [0 0 0 0] [1] [0 0 0 0] [1] [0 0 0 0] [1] [0 0 0 0] [1] b(c(a(x1))) = [1 1 1 0]x1 + [0] >= [1 1 1 0]x1 + [0] = c(c(a(a(b(x1))))) [1 1 1 0] [1] [1 1 1 0] [1] problem: DPs: a#(a(b(x1))) -> b#(a(b(x1))) b#(a(x1)) -> b#(x1) b#(a(x1)) -> b#(b(x1)) b#(a(x1)) -> a#(b(b(x1))) TRS: a(a(b(x1))) -> b(a(b(x1))) b(a(x1)) -> a(b(b(x1))) b(c(a(x1))) -> c(c(a(a(b(x1))))) EDG Processor: DPs: a#(a(b(x1))) -> b#(a(b(x1))) b#(a(x1)) -> b#(x1) b#(a(x1)) -> b#(b(x1)) b#(a(x1)) -> a#(b(b(x1))) TRS: a(a(b(x1))) -> b(a(b(x1))) b(a(x1)) -> a(b(b(x1))) b(c(a(x1))) -> c(c(a(a(b(x1))))) graph: b#(a(x1)) -> b#(b(x1)) -> b#(a(x1)) -> a#(b(b(x1))) b#(a(x1)) -> b#(b(x1)) -> b#(a(x1)) -> b#(b(x1)) b#(a(x1)) -> b#(b(x1)) -> b#(a(x1)) -> b#(x1) b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> a#(b(b(x1))) b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> b#(b(x1)) b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> b#(x1) b#(a(x1)) -> a#(b(b(x1))) -> a#(a(b(x1))) -> b#(a(b(x1))) a#(a(b(x1))) -> b#(a(b(x1))) -> b#(a(x1)) -> a#(b(b(x1))) a#(a(b(x1))) -> b#(a(b(x1))) -> b#(a(x1)) -> b#(b(x1)) a#(a(b(x1))) -> b#(a(b(x1))) -> b#(a(x1)) -> b#(x1) Arctic Interpretation Processor: dimension: 1 interpretation: [b#](x0) = x0 + -16, [a#](x0) = x0 + 0, [c](x0) = 0, [a](x0) = 1x0 + 0, [b](x0) = x0 orientation: a#(a(b(x1))) = 1x1 + 0 >= 1x1 + 0 = b#(a(b(x1))) b#(a(x1)) = 1x1 + 0 >= x1 + -16 = b#(x1) b#(a(x1)) = 1x1 + 0 >= x1 + -16 = b#(b(x1)) b#(a(x1)) = 1x1 + 0 >= x1 + 0 = a#(b(b(x1))) a(a(b(x1))) = 2x1 + 1 >= 1x1 + 0 = b(a(b(x1))) b(a(x1)) = 1x1 + 0 >= 1x1 + 0 = a(b(b(x1))) b(c(a(x1))) = 0 >= 0 = c(c(a(a(b(x1))))) problem: DPs: a#(a(b(x1))) -> b#(a(b(x1))) b#(a(x1)) -> a#(b(b(x1))) TRS: a(a(b(x1))) -> b(a(b(x1))) b(a(x1)) -> a(b(b(x1))) b(c(a(x1))) -> c(c(a(a(b(x1))))) EDG Processor: DPs: a#(a(b(x1))) -> b#(a(b(x1))) b#(a(x1)) -> a#(b(b(x1))) TRS: a(a(b(x1))) -> b(a(b(x1))) b(a(x1)) -> a(b(b(x1))) b(c(a(x1))) -> c(c(a(a(b(x1))))) graph: b#(a(x1)) -> a#(b(b(x1))) -> a#(a(b(x1))) -> b#(a(b(x1))) a#(a(b(x1))) -> b#(a(b(x1))) -> b#(a(x1)) -> a#(b(b(x1))) Arctic Interpretation Processor: dimension: 1 interpretation: [b#](x0) = x0, [a#](x0) = x0, [c](x0) = 14, [a](x0) = 6x0, [b](x0) = x0 orientation: a#(a(b(x1))) = 6x1 >= 6x1 = b#(a(b(x1))) b#(a(x1)) = 6x1 >= x1 = a#(b(b(x1))) a(a(b(x1))) = 12x1 >= 6x1 = b(a(b(x1))) b(a(x1)) = 6x1 >= 6x1 = a(b(b(x1))) b(c(a(x1))) = 14 >= 14 = c(c(a(a(b(x1))))) problem: DPs: a#(a(b(x1))) -> b#(a(b(x1))) TRS: a(a(b(x1))) -> b(a(b(x1))) b(a(x1)) -> a(b(b(x1))) b(c(a(x1))) -> c(c(a(a(b(x1))))) EDG Processor: DPs: a#(a(b(x1))) -> b#(a(b(x1))) TRS: a(a(b(x1))) -> b(a(b(x1))) b(a(x1)) -> a(b(b(x1))) b(c(a(x1))) -> c(c(a(a(b(x1))))) graph: Qed