YES Problem: a(c(x1)) -> c(b(c(c(a(x1))))) b(b(b(x1))) -> c(b(x1)) d(d(x1)) -> d(b(d(b(d(x1))))) a(a(x1)) -> a(d(a(x1))) a(b(x1)) -> c(c(a(x1))) c(c(x1)) -> c(b(c(b(c(x1))))) c(c(c(x1))) -> c(b(b(x1))) Proof: String Reversal Processor: c(a(x1)) -> a(c(c(b(c(x1))))) b(b(b(x1))) -> b(c(x1)) d(d(x1)) -> d(b(d(b(d(x1))))) a(a(x1)) -> a(d(a(x1))) b(a(x1)) -> a(c(c(x1))) c(c(x1)) -> c(b(c(b(c(x1))))) c(c(c(x1))) -> b(b(c(x1))) Matrix Interpretation Processor: dim=2 interpretation: [1 1] [0] [d](x0) = [0 1]x0 + [1], [1 0] [b](x0) = [0 0]x0, [1 0] [a](x0) = [0 0]x0, [1 0] [c](x0) = [0 0]x0 orientation: [1 0] [1 0] c(a(x1)) = [0 0]x1 >= [0 0]x1 = a(c(c(b(c(x1))))) [1 0] [1 0] b(b(b(x1))) = [0 0]x1 >= [0 0]x1 = b(c(x1)) [1 2] [1] [1 1] [0] d(d(x1)) = [0 1]x1 + [2] >= [0 0]x1 + [1] = d(b(d(b(d(x1))))) [1 0] [1 0] a(a(x1)) = [0 0]x1 >= [0 0]x1 = a(d(a(x1))) [1 0] [1 0] b(a(x1)) = [0 0]x1 >= [0 0]x1 = a(c(c(x1))) [1 0] [1 0] c(c(x1)) = [0 0]x1 >= [0 0]x1 = c(b(c(b(c(x1))))) [1 0] [1 0] c(c(c(x1))) = [0 0]x1 >= [0 0]x1 = b(b(c(x1))) problem: c(a(x1)) -> a(c(c(b(c(x1))))) b(b(b(x1))) -> b(c(x1)) a(a(x1)) -> a(d(a(x1))) b(a(x1)) -> a(c(c(x1))) c(c(x1)) -> c(b(c(b(c(x1))))) c(c(c(x1))) -> b(b(c(x1))) String Reversal Processor: a(c(x1)) -> c(b(c(c(a(x1))))) b(b(b(x1))) -> c(b(x1)) a(a(x1)) -> a(d(a(x1))) a(b(x1)) -> c(c(a(x1))) c(c(x1)) -> c(b(c(b(c(x1))))) c(c(c(x1))) -> c(b(b(x1))) Matrix Interpretation Processor: dim=2 interpretation: [1 0] [1] [d](x0) = [0 0]x0 + [0], [b](x0) = x0, [2 2] [0] [a](x0) = [0 0]x0 + [2], [c](x0) = x0 orientation: [2 2] [0] [2 2] [0] a(c(x1)) = [0 0]x1 + [2] >= [0 0]x1 + [2] = c(b(c(c(a(x1))))) b(b(b(x1))) = x1 >= x1 = c(b(x1)) [4 4] [4] [4 4] [2] a(a(x1)) = [0 0]x1 + [2] >= [0 0]x1 + [2] = a(d(a(x1))) [2 2] [0] [2 2] [0] a(b(x1)) = [0 0]x1 + [2] >= [0 0]x1 + [2] = c(c(a(x1))) c(c(x1)) = x1 >= x1 = c(b(c(b(c(x1))))) c(c(c(x1))) = x1 >= x1 = c(b(b(x1))) problem: a(c(x1)) -> c(b(c(c(a(x1))))) b(b(b(x1))) -> c(b(x1)) a(b(x1)) -> c(c(a(x1))) c(c(x1)) -> c(b(c(b(c(x1))))) c(c(c(x1))) -> c(b(b(x1))) String Reversal Processor: c(a(x1)) -> a(c(c(b(c(x1))))) b(b(b(x1))) -> b(c(x1)) b(a(x1)) -> a(c(c(x1))) c(c(x1)) -> c(b(c(b(c(x1))))) c(c(c(x1))) -> b(b(c(x1))) DP Processor: DPs: c#(a(x1)) -> c#(x1) c#(a(x1)) -> b#(c(x1)) c#(a(x1)) -> c#(b(c(x1))) c#(a(x1)) -> c#(c(b(c(x1)))) b#(b(b(x1))) -> c#(x1) b#(b(b(x1))) -> b#(c(x1)) b#(a(x1)) -> c#(x1) b#(a(x1)) -> c#(c(x1)) c#(c(x1)) -> b#(c(x1)) c#(c(x1)) -> c#(b(c(x1))) c#(c(x1)) -> b#(c(b(c(x1)))) c#(c(x1)) -> c#(b(c(b(c(x1))))) c#(c(c(x1))) -> b#(c(x1)) c#(c(c(x1))) -> b#(b(c(x1))) TRS: c(a(x1)) -> a(c(c(b(c(x1))))) b(b(b(x1))) -> b(c(x1)) b(a(x1)) -> a(c(c(x1))) c(c(x1)) -> c(b(c(b(c(x1))))) c(c(c(x1))) -> b(b(c(x1))) TDG Processor: DPs: c#(a(x1)) -> c#(x1) c#(a(x1)) -> b#(c(x1)) c#(a(x1)) -> c#(b(c(x1))) c#(a(x1)) -> c#(c(b(c(x1)))) b#(b(b(x1))) -> c#(x1) b#(b(b(x1))) -> b#(c(x1)) b#(a(x1)) -> c#(x1) b#(a(x1)) -> c#(c(x1)) c#(c(x1)) -> b#(c(x1)) c#(c(x1)) -> c#(b(c(x1))) c#(c(x1)) -> b#(c(b(c(x1)))) c#(c(x1)) -> c#(b(c(b(c(x1))))) c#(c(c(x1))) -> b#(c(x1)) c#(c(c(x1))) -> b#(b(c(x1))) TRS: c(a(x1)) -> a(c(c(b(c(x1))))) b(b(b(x1))) -> b(c(x1)) b(a(x1)) -> a(c(c(x1))) c(c(x1)) -> c(b(c(b(c(x1))))) c(c(c(x1))) -> b(b(c(x1))) graph: b#(b(b(x1))) -> b#(c(x1)) -> b#(a(x1)) -> c#(c(x1)) b#(b(b(x1))) -> b#(c(x1)) -> b#(a(x1)) -> c#(x1) b#(b(b(x1))) -> b#(c(x1)) -> b#(b(b(x1))) -> b#(c(x1)) b#(b(b(x1))) -> b#(c(x1)) -> b#(b(b(x1))) -> c#(x1) b#(b(b(x1))) -> c#(x1) -> c#(c(c(x1))) -> b#(b(c(x1))) b#(b(b(x1))) -> c#(x1) -> c#(c(c(x1))) -> b#(c(x1)) b#(b(b(x1))) -> c#(x1) -> c#(c(x1)) -> c#(b(c(b(c(x1))))) b#(b(b(x1))) -> c#(x1) -> c#(c(x1)) -> b#(c(b(c(x1)))) b#(b(b(x1))) -> c#(x1) -> c#(c(x1)) -> c#(b(c(x1))) b#(b(b(x1))) -> c#(x1) -> c#(c(x1)) -> b#(c(x1)) b#(b(b(x1))) -> c#(x1) -> c#(a(x1)) -> c#(c(b(c(x1)))) b#(b(b(x1))) -> c#(x1) -> c#(a(x1)) -> c#(b(c(x1))) b#(b(b(x1))) -> c#(x1) -> c#(a(x1)) -> b#(c(x1)) b#(b(b(x1))) -> c#(x1) -> c#(a(x1)) -> c#(x1) b#(a(x1)) -> c#(c(x1)) -> c#(c(c(x1))) -> b#(b(c(x1))) b#(a(x1)) -> c#(c(x1)) -> c#(c(c(x1))) -> b#(c(x1)) b#(a(x1)) -> c#(c(x1)) -> c#(c(x1)) -> c#(b(c(b(c(x1))))) b#(a(x1)) -> c#(c(x1)) -> c#(c(x1)) -> b#(c(b(c(x1)))) b#(a(x1)) -> c#(c(x1)) -> c#(c(x1)) -> c#(b(c(x1))) b#(a(x1)) -> c#(c(x1)) -> c#(c(x1)) -> b#(c(x1)) b#(a(x1)) -> c#(c(x1)) -> c#(a(x1)) -> c#(c(b(c(x1)))) b#(a(x1)) -> c#(c(x1)) -> c#(a(x1)) -> c#(b(c(x1))) b#(a(x1)) -> c#(c(x1)) -> c#(a(x1)) -> b#(c(x1)) b#(a(x1)) -> c#(c(x1)) -> c#(a(x1)) -> c#(x1) b#(a(x1)) -> c#(x1) -> c#(c(c(x1))) -> b#(b(c(x1))) b#(a(x1)) -> c#(x1) -> c#(c(c(x1))) -> b#(c(x1)) b#(a(x1)) -> c#(x1) -> c#(c(x1)) -> c#(b(c(b(c(x1))))) b#(a(x1)) -> c#(x1) -> c#(c(x1)) -> b#(c(b(c(x1)))) b#(a(x1)) -> c#(x1) -> c#(c(x1)) -> c#(b(c(x1))) b#(a(x1)) -> c#(x1) -> c#(c(x1)) -> b#(c(x1)) b#(a(x1)) -> c#(x1) -> c#(a(x1)) -> c#(c(b(c(x1)))) b#(a(x1)) -> c#(x1) -> c#(a(x1)) -> c#(b(c(x1))) b#(a(x1)) -> c#(x1) -> c#(a(x1)) -> b#(c(x1)) b#(a(x1)) -> c#(x1) -> c#(a(x1)) -> c#(x1) c#(a(x1)) -> b#(c(x1)) -> b#(a(x1)) -> c#(c(x1)) c#(a(x1)) -> b#(c(x1)) -> b#(a(x1)) -> c#(x1) c#(a(x1)) -> b#(c(x1)) -> b#(b(b(x1))) -> b#(c(x1)) c#(a(x1)) -> b#(c(x1)) -> b#(b(b(x1))) -> c#(x1) c#(a(x1)) -> c#(b(c(x1))) -> c#(c(c(x1))) -> b#(b(c(x1))) c#(a(x1)) -> c#(b(c(x1))) -> c#(c(c(x1))) -> b#(c(x1)) c#(a(x1)) -> c#(b(c(x1))) -> c#(c(x1)) -> c#(b(c(b(c(x1))))) c#(a(x1)) -> c#(b(c(x1))) -> c#(c(x1)) -> b#(c(b(c(x1)))) c#(a(x1)) -> c#(b(c(x1))) -> c#(c(x1)) -> c#(b(c(x1))) c#(a(x1)) -> c#(b(c(x1))) -> c#(c(x1)) -> b#(c(x1)) c#(a(x1)) -> c#(b(c(x1))) -> c#(a(x1)) -> c#(c(b(c(x1)))) c#(a(x1)) -> c#(b(c(x1))) -> c#(a(x1)) -> c#(b(c(x1))) c#(a(x1)) -> c#(b(c(x1))) -> c#(a(x1)) -> b#(c(x1)) c#(a(x1)) -> c#(b(c(x1))) -> c#(a(x1)) -> c#(x1) c#(a(x1)) -> c#(c(b(c(x1)))) -> c#(c(c(x1))) -> b#(b(c(x1))) c#(a(x1)) -> c#(c(b(c(x1)))) -> c#(c(c(x1))) -> b#(c(x1)) c#(a(x1)) -> c#(c(b(c(x1)))) -> c#(c(x1)) -> c#(b(c(b(c(x1))))) c#(a(x1)) -> c#(c(b(c(x1)))) -> c#(c(x1)) -> b#(c(b(c(x1)))) c#(a(x1)) -> c#(c(b(c(x1)))) -> c#(c(x1)) -> c#(b(c(x1))) c#(a(x1)) -> c#(c(b(c(x1)))) -> c#(c(x1)) -> b#(c(x1)) c#(a(x1)) -> c#(c(b(c(x1)))) -> c#(a(x1)) -> c#(c(b(c(x1)))) c#(a(x1)) -> c#(c(b(c(x1)))) -> c#(a(x1)) -> c#(b(c(x1))) c#(a(x1)) -> c#(c(b(c(x1)))) -> c#(a(x1)) -> b#(c(x1)) c#(a(x1)) -> c#(c(b(c(x1)))) -> c#(a(x1)) -> c#(x1) c#(a(x1)) -> c#(x1) -> c#(c(c(x1))) -> b#(b(c(x1))) c#(a(x1)) -> c#(x1) -> c#(c(c(x1))) -> b#(c(x1)) c#(a(x1)) -> c#(x1) -> c#(c(x1)) -> c#(b(c(b(c(x1))))) c#(a(x1)) -> c#(x1) -> c#(c(x1)) -> b#(c(b(c(x1)))) c#(a(x1)) -> c#(x1) -> c#(c(x1)) -> c#(b(c(x1))) c#(a(x1)) -> c#(x1) -> c#(c(x1)) -> b#(c(x1)) c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> c#(c(b(c(x1)))) c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> c#(b(c(x1))) c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> b#(c(x1)) c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> c#(x1) c#(c(c(x1))) -> b#(b(c(x1))) -> b#(a(x1)) -> c#(c(x1)) c#(c(c(x1))) -> b#(b(c(x1))) -> b#(a(x1)) -> c#(x1) c#(c(c(x1))) -> b#(b(c(x1))) -> b#(b(b(x1))) -> b#(c(x1)) c#(c(c(x1))) -> b#(b(c(x1))) -> b#(b(b(x1))) -> c#(x1) c#(c(c(x1))) -> b#(c(x1)) -> b#(a(x1)) -> c#(c(x1)) c#(c(c(x1))) -> b#(c(x1)) -> b#(a(x1)) -> c#(x1) c#(c(c(x1))) -> b#(c(x1)) -> b#(b(b(x1))) -> b#(c(x1)) c#(c(c(x1))) -> b#(c(x1)) -> b#(b(b(x1))) -> c#(x1) c#(c(x1)) -> b#(c(b(c(x1)))) -> b#(a(x1)) -> c#(c(x1)) c#(c(x1)) -> b#(c(b(c(x1)))) -> b#(a(x1)) -> c#(x1) c#(c(x1)) -> b#(c(b(c(x1)))) -> b#(b(b(x1))) -> b#(c(x1)) c#(c(x1)) -> b#(c(b(c(x1)))) -> b#(b(b(x1))) -> c#(x1) c#(c(x1)) -> b#(c(x1)) -> b#(a(x1)) -> c#(c(x1)) c#(c(x1)) -> b#(c(x1)) -> b#(a(x1)) -> c#(x1) c#(c(x1)) -> b#(c(x1)) -> b#(b(b(x1))) -> b#(c(x1)) c#(c(x1)) -> b#(c(x1)) -> b#(b(b(x1))) -> c#(x1) c#(c(x1)) -> c#(b(c(b(c(x1))))) -> c#(c(c(x1))) -> b#(b(c(x1))) c#(c(x1)) -> c#(b(c(b(c(x1))))) -> c#(c(c(x1))) -> b#(c(x1)) c#(c(x1)) -> c#(b(c(b(c(x1))))) -> c#(c(x1)) -> c#(b(c(b(c(x1))))) c#(c(x1)) -> c#(b(c(b(c(x1))))) -> c#(c(x1)) -> b#(c(b(c(x1)))) c#(c(x1)) -> c#(b(c(b(c(x1))))) -> c#(c(x1)) -> c#(b(c(x1))) c#(c(x1)) -> c#(b(c(b(c(x1))))) -> c#(c(x1)) -> b#(c(x1)) c#(c(x1)) -> c#(b(c(b(c(x1))))) -> c#(a(x1)) -> c#(c(b(c(x1)))) c#(c(x1)) -> c#(b(c(b(c(x1))))) -> c#(a(x1)) -> c#(b(c(x1))) c#(c(x1)) -> c#(b(c(b(c(x1))))) -> c#(a(x1)) -> b#(c(x1)) c#(c(x1)) -> c#(b(c(b(c(x1))))) -> c#(a(x1)) -> c#(x1) c#(c(x1)) -> c#(b(c(x1))) -> c#(c(c(x1))) -> b#(b(c(x1))) c#(c(x1)) -> c#(b(c(x1))) -> c#(c(c(x1))) -> b#(c(x1)) c#(c(x1)) -> c#(b(c(x1))) -> c#(c(x1)) -> c#(b(c(b(c(x1))))) c#(c(x1)) -> c#(b(c(x1))) -> c#(c(x1)) -> b#(c(b(c(x1)))) c#(c(x1)) -> c#(b(c(x1))) -> c#(c(x1)) -> c#(b(c(x1))) c#(c(x1)) -> c#(b(c(x1))) -> c#(c(x1)) -> b#(c(x1)) c#(c(x1)) -> c#(b(c(x1))) -> c#(a(x1)) -> c#(c(b(c(x1)))) c#(c(x1)) -> c#(b(c(x1))) -> c#(a(x1)) -> c#(b(c(x1))) c#(c(x1)) -> c#(b(c(x1))) -> c#(a(x1)) -> b#(c(x1)) c#(c(x1)) -> c#(b(c(x1))) -> c#(a(x1)) -> c#(x1) Arctic Interpretation Processor: dimension: 1 interpretation: [b#](x0) = x0, [c#](x0) = x0, [b](x0) = x0, [a](x0) = 15x0, [c](x0) = x0 orientation: c#(a(x1)) = 15x1 >= x1 = c#(x1) c#(a(x1)) = 15x1 >= x1 = b#(c(x1)) c#(a(x1)) = 15x1 >= x1 = c#(b(c(x1))) c#(a(x1)) = 15x1 >= x1 = c#(c(b(c(x1)))) b#(b(b(x1))) = x1 >= x1 = c#(x1) b#(b(b(x1))) = x1 >= x1 = b#(c(x1)) b#(a(x1)) = 15x1 >= x1 = c#(x1) b#(a(x1)) = 15x1 >= x1 = c#(c(x1)) c#(c(x1)) = x1 >= x1 = b#(c(x1)) c#(c(x1)) = x1 >= x1 = c#(b(c(x1))) c#(c(x1)) = x1 >= x1 = b#(c(b(c(x1)))) c#(c(x1)) = x1 >= x1 = c#(b(c(b(c(x1))))) c#(c(c(x1))) = x1 >= x1 = b#(c(x1)) c#(c(c(x1))) = x1 >= x1 = b#(b(c(x1))) c(a(x1)) = 15x1 >= 15x1 = a(c(c(b(c(x1))))) b(b(b(x1))) = x1 >= x1 = b(c(x1)) b(a(x1)) = 15x1 >= 15x1 = a(c(c(x1))) c(c(x1)) = x1 >= x1 = c(b(c(b(c(x1))))) c(c(c(x1))) = x1 >= x1 = b(b(c(x1))) problem: DPs: b#(b(b(x1))) -> c#(x1) b#(b(b(x1))) -> b#(c(x1)) c#(c(x1)) -> b#(c(x1)) c#(c(x1)) -> c#(b(c(x1))) c#(c(x1)) -> b#(c(b(c(x1)))) c#(c(x1)) -> c#(b(c(b(c(x1))))) c#(c(c(x1))) -> b#(c(x1)) c#(c(c(x1))) -> b#(b(c(x1))) TRS: c(a(x1)) -> a(c(c(b(c(x1))))) b(b(b(x1))) -> b(c(x1)) b(a(x1)) -> a(c(c(x1))) c(c(x1)) -> c(b(c(b(c(x1))))) c(c(c(x1))) -> b(b(c(x1))) EDG Processor: DPs: b#(b(b(x1))) -> c#(x1) b#(b(b(x1))) -> b#(c(x1)) c#(c(x1)) -> b#(c(x1)) c#(c(x1)) -> c#(b(c(x1))) c#(c(x1)) -> b#(c(b(c(x1)))) c#(c(x1)) -> c#(b(c(b(c(x1))))) c#(c(c(x1))) -> b#(c(x1)) c#(c(c(x1))) -> b#(b(c(x1))) TRS: c(a(x1)) -> a(c(c(b(c(x1))))) b(b(b(x1))) -> b(c(x1)) b(a(x1)) -> a(c(c(x1))) c(c(x1)) -> c(b(c(b(c(x1))))) c(c(c(x1))) -> b(b(c(x1))) graph: b#(b(b(x1))) -> b#(c(x1)) -> b#(b(b(x1))) -> c#(x1) b#(b(b(x1))) -> b#(c(x1)) -> b#(b(b(x1))) -> b#(c(x1)) b#(b(b(x1))) -> c#(x1) -> c#(c(x1)) -> b#(c(x1)) b#(b(b(x1))) -> c#(x1) -> c#(c(x1)) -> c#(b(c(x1))) b#(b(b(x1))) -> c#(x1) -> c#(c(x1)) -> b#(c(b(c(x1)))) b#(b(b(x1))) -> c#(x1) -> c#(c(x1)) -> c#(b(c(b(c(x1))))) b#(b(b(x1))) -> c#(x1) -> c#(c(c(x1))) -> b#(c(x1)) b#(b(b(x1))) -> c#(x1) -> c#(c(c(x1))) -> b#(b(c(x1))) c#(c(c(x1))) -> b#(b(c(x1))) -> b#(b(b(x1))) -> c#(x1) c#(c(c(x1))) -> b#(b(c(x1))) -> b#(b(b(x1))) -> b#(c(x1)) c#(c(c(x1))) -> b#(c(x1)) -> b#(b(b(x1))) -> c#(x1) c#(c(c(x1))) -> b#(c(x1)) -> b#(b(b(x1))) -> b#(c(x1)) c#(c(x1)) -> b#(c(b(c(x1)))) -> b#(b(b(x1))) -> c#(x1) c#(c(x1)) -> b#(c(b(c(x1)))) -> b#(b(b(x1))) -> b#(c(x1)) c#(c(x1)) -> b#(c(x1)) -> b#(b(b(x1))) -> c#(x1) c#(c(x1)) -> b#(c(x1)) -> b#(b(b(x1))) -> b#(c(x1)) c#(c(x1)) -> c#(b(c(b(c(x1))))) -> c#(c(x1)) -> b#(c(x1)) c#(c(x1)) -> c#(b(c(b(c(x1))))) -> c#(c(x1)) -> c#(b(c(x1))) c#(c(x1)) -> c#(b(c(b(c(x1))))) -> c#(c(x1)) -> b#(c(b(c(x1)))) c#(c(x1)) -> c#(b(c(b(c(x1))))) -> c#(c(x1)) -> c#(b(c(b(c(x1))))) c#(c(x1)) -> c#(b(c(b(c(x1))))) -> c#(c(c(x1))) -> b#(c(x1)) c#(c(x1)) -> c#(b(c(b(c(x1))))) -> c#(c(c(x1))) -> b#(b(c(x1))) c#(c(x1)) -> c#(b(c(x1))) -> c#(c(x1)) -> b#(c(x1)) c#(c(x1)) -> c#(b(c(x1))) -> c#(c(x1)) -> c#(b(c(x1))) c#(c(x1)) -> c#(b(c(x1))) -> c#(c(x1)) -> b#(c(b(c(x1)))) c#(c(x1)) -> c#(b(c(x1))) -> c#(c(x1)) -> c#(b(c(b(c(x1))))) c#(c(x1)) -> c#(b(c(x1))) -> c#(c(c(x1))) -> b#(c(x1)) c#(c(x1)) -> c#(b(c(x1))) -> c#(c(c(x1))) -> b#(b(c(x1))) Arctic Interpretation Processor: dimension: 2 interpretation: [b#](x0) = [3 3]x0 + [0], [c#](x0) = [0 3]x0 + [0], [0 0 ] [3] [b](x0) = [-& -&]x0 + [1], [2] [a](x0) = [0], [0 0] [1] [c](x0) = [0 0]x0 + [3] orientation: b#(b(b(x1))) = [3 3]x1 + [6] >= [0 3]x1 + [0] = c#(x1) b#(b(b(x1))) = [3 3]x1 + [6] >= [3 3]x1 + [6] = b#(c(x1)) c#(c(x1)) = [3 3]x1 + [6] >= [3 3]x1 + [6] = b#(c(x1)) c#(c(x1)) = [3 3]x1 + [6] >= [0 0]x1 + [4] = c#(b(c(x1))) c#(c(x1)) = [3 3]x1 + [6] >= [3 3]x1 + [6] = b#(c(b(c(x1)))) c#(c(x1)) = [3 3]x1 + [6] >= [0 0]x1 + [4] = c#(b(c(b(c(x1))))) c#(c(c(x1))) = [3 3]x1 + [6] >= [3 3]x1 + [6] = b#(c(x1)) c#(c(c(x1))) = [3 3]x1 + [6] >= [3 3]x1 + [6] = b#(b(c(x1))) [2] [2] c(a(x1)) = [3] >= [0] = a(c(c(b(c(x1))))) [0 0 ] [3] [0 0 ] [3] b(b(b(x1))) = [-& -&]x1 + [1] >= [-& -&]x1 + [1] = b(c(x1)) [3] [2] b(a(x1)) = [1] >= [0] = a(c(c(x1))) [0 0] [3] [0 0] [3] c(c(x1)) = [0 0]x1 + [3] >= [0 0]x1 + [3] = c(b(c(b(c(x1))))) [0 0] [3] [0 0 ] [3] c(c(c(x1))) = [0 0]x1 + [3] >= [-& -&]x1 + [1] = b(b(c(x1))) problem: DPs: b#(b(b(x1))) -> c#(x1) b#(b(b(x1))) -> b#(c(x1)) c#(c(x1)) -> b#(c(x1)) c#(c(x1)) -> b#(c(b(c(x1)))) c#(c(c(x1))) -> b#(c(x1)) c#(c(c(x1))) -> b#(b(c(x1))) TRS: c(a(x1)) -> a(c(c(b(c(x1))))) b(b(b(x1))) -> b(c(x1)) b(a(x1)) -> a(c(c(x1))) c(c(x1)) -> c(b(c(b(c(x1))))) c(c(c(x1))) -> b(b(c(x1))) EDG Processor: DPs: b#(b(b(x1))) -> c#(x1) b#(b(b(x1))) -> b#(c(x1)) c#(c(x1)) -> b#(c(x1)) c#(c(x1)) -> b#(c(b(c(x1)))) c#(c(c(x1))) -> b#(c(x1)) c#(c(c(x1))) -> b#(b(c(x1))) TRS: c(a(x1)) -> a(c(c(b(c(x1))))) b(b(b(x1))) -> b(c(x1)) b(a(x1)) -> a(c(c(x1))) c(c(x1)) -> c(b(c(b(c(x1))))) c(c(c(x1))) -> b(b(c(x1))) graph: b#(b(b(x1))) -> b#(c(x1)) -> b#(b(b(x1))) -> b#(c(x1)) b#(b(b(x1))) -> b#(c(x1)) -> b#(b(b(x1))) -> c#(x1) b#(b(b(x1))) -> c#(x1) -> c#(c(c(x1))) -> b#(b(c(x1))) b#(b(b(x1))) -> c#(x1) -> c#(c(c(x1))) -> b#(c(x1)) b#(b(b(x1))) -> c#(x1) -> c#(c(x1)) -> b#(c(b(c(x1)))) b#(b(b(x1))) -> c#(x1) -> c#(c(x1)) -> b#(c(x1)) c#(c(c(x1))) -> b#(b(c(x1))) -> b#(b(b(x1))) -> b#(c(x1)) c#(c(c(x1))) -> b#(b(c(x1))) -> b#(b(b(x1))) -> c#(x1) c#(c(c(x1))) -> b#(c(x1)) -> b#(b(b(x1))) -> b#(c(x1)) c#(c(c(x1))) -> b#(c(x1)) -> b#(b(b(x1))) -> c#(x1) c#(c(x1)) -> b#(c(b(c(x1)))) -> b#(b(b(x1))) -> b#(c(x1)) c#(c(x1)) -> b#(c(b(c(x1)))) -> b#(b(b(x1))) -> c#(x1) c#(c(x1)) -> b#(c(x1)) -> b#(b(b(x1))) -> b#(c(x1)) c#(c(x1)) -> b#(c(x1)) -> b#(b(b(x1))) -> c#(x1) Arctic Interpretation Processor: dimension: 2 interpretation: [b#](x0) = [-& 0 ]x0 + [1], [c#](x0) = [0 0]x0, [-& -&] [0] [b](x0) = [0 0 ]x0 + [2], [0 ] [a](x0) = [-&], [0 0 ] [2] [c](x0) = [0 -&]x0 + [0] orientation: b#(b(b(x1))) = [0 0]x1 + [2] >= [0 0]x1 = c#(x1) b#(b(b(x1))) = [0 0]x1 + [2] >= [0 -&]x1 + [1] = b#(c(x1)) c#(c(x1)) = [0 0]x1 + [2] >= [0 -&]x1 + [1] = b#(c(x1)) c#(c(x1)) = [0 0]x1 + [2] >= [1] = b#(c(b(c(x1)))) c#(c(c(x1))) = [0 0]x1 + [2] >= [0 -&]x1 + [1] = b#(c(x1)) c#(c(c(x1))) = [0 0]x1 + [2] >= [0 0]x1 + [2] = b#(b(c(x1))) [2] [0 ] c(a(x1)) = [0] >= [-&] = a(c(c(b(c(x1))))) [-& -&] [0] [-& -&] [0] b(b(b(x1))) = [0 0 ]x1 + [2] >= [0 0 ]x1 + [2] = b(c(x1)) [0] [0 ] b(a(x1)) = [2] >= [-&] = a(c(c(x1))) [0 0] [2] [0 0 ] [2] c(c(x1)) = [0 0]x1 + [2] >= [-& -&]x1 + [0] = c(b(c(b(c(x1))))) [0 0] [2] [-& -&] [0] c(c(c(x1))) = [0 0]x1 + [2] >= [0 0 ]x1 + [2] = b(b(c(x1))) problem: DPs: b#(b(b(x1))) -> c#(x1) b#(b(b(x1))) -> b#(c(x1)) c#(c(x1)) -> b#(c(x1)) c#(c(c(x1))) -> b#(c(x1)) c#(c(c(x1))) -> b#(b(c(x1))) TRS: c(a(x1)) -> a(c(c(b(c(x1))))) b(b(b(x1))) -> b(c(x1)) b(a(x1)) -> a(c(c(x1))) c(c(x1)) -> c(b(c(b(c(x1))))) c(c(c(x1))) -> b(b(c(x1))) EDG Processor: DPs: b#(b(b(x1))) -> c#(x1) b#(b(b(x1))) -> b#(c(x1)) c#(c(x1)) -> b#(c(x1)) c#(c(c(x1))) -> b#(c(x1)) c#(c(c(x1))) -> b#(b(c(x1))) TRS: c(a(x1)) -> a(c(c(b(c(x1))))) b(b(b(x1))) -> b(c(x1)) b(a(x1)) -> a(c(c(x1))) c(c(x1)) -> c(b(c(b(c(x1))))) c(c(c(x1))) -> b(b(c(x1))) graph: b#(b(b(x1))) -> b#(c(x1)) -> b#(b(b(x1))) -> c#(x1) b#(b(b(x1))) -> b#(c(x1)) -> b#(b(b(x1))) -> b#(c(x1)) b#(b(b(x1))) -> c#(x1) -> c#(c(x1)) -> b#(c(x1)) b#(b(b(x1))) -> c#(x1) -> c#(c(c(x1))) -> b#(c(x1)) b#(b(b(x1))) -> c#(x1) -> c#(c(c(x1))) -> b#(b(c(x1))) c#(c(c(x1))) -> b#(b(c(x1))) -> b#(b(b(x1))) -> c#(x1) c#(c(c(x1))) -> b#(b(c(x1))) -> b#(b(b(x1))) -> b#(c(x1)) c#(c(c(x1))) -> b#(c(x1)) -> b#(b(b(x1))) -> c#(x1) c#(c(c(x1))) -> b#(c(x1)) -> b#(b(b(x1))) -> b#(c(x1)) c#(c(x1)) -> b#(c(x1)) -> b#(b(b(x1))) -> c#(x1) c#(c(x1)) -> b#(c(x1)) -> b#(b(b(x1))) -> b#(c(x1)) Arctic Interpretation Processor: dimension: 2 interpretation: [b#](x0) = [0 0]x0 + [0], [c#](x0) = [1 0]x0 + [1], [1 1 ] [0 ] [b](x0) = [-& -&]x0 + [-&], [1 ] [a](x0) = [-&], [0 2] [0] [c](x0) = [0 2]x0 + [0] orientation: b#(b(b(x1))) = [2 2]x1 + [1] >= [1 0]x1 + [1] = c#(x1) b#(b(b(x1))) = [2 2]x1 + [1] >= [0 2]x1 + [0] = b#(c(x1)) c#(c(x1)) = [1 3]x1 + [1] >= [0 2]x1 + [0] = b#(c(x1)) c#(c(c(x1))) = [3 5]x1 + [3] >= [0 2]x1 + [0] = b#(c(x1)) c#(c(c(x1))) = [3 5]x1 + [3] >= [1 3]x1 + [1] = b#(b(c(x1))) [1] [1 ] c(a(x1)) = [1] >= [-&] = a(c(c(b(c(x1))))) [3 3 ] [2 ] [1 3 ] [1 ] b(b(b(x1))) = [-& -&]x1 + [-&] >= [-& -&]x1 + [-&] = b(c(x1)) [2 ] [1 ] b(a(x1)) = [-&] >= [-&] = a(c(c(x1))) [2 4] [2] [2 4] [2] c(c(x1)) = [2 4]x1 + [2] >= [2 4]x1 + [2] = c(b(c(b(c(x1))))) [4 6] [4] [2 4 ] [2 ] c(c(c(x1))) = [4 6]x1 + [4] >= [-& -&]x1 + [-&] = b(b(c(x1))) problem: DPs: b#(b(b(x1))) -> c#(x1) b#(b(b(x1))) -> b#(c(x1)) TRS: c(a(x1)) -> a(c(c(b(c(x1))))) b(b(b(x1))) -> b(c(x1)) b(a(x1)) -> a(c(c(x1))) c(c(x1)) -> c(b(c(b(c(x1))))) c(c(c(x1))) -> b(b(c(x1))) EDG Processor: DPs: b#(b(b(x1))) -> c#(x1) b#(b(b(x1))) -> b#(c(x1)) TRS: c(a(x1)) -> a(c(c(b(c(x1))))) b(b(b(x1))) -> b(c(x1)) b(a(x1)) -> a(c(c(x1))) c(c(x1)) -> c(b(c(b(c(x1))))) c(c(c(x1))) -> b(b(c(x1))) graph: b#(b(b(x1))) -> b#(c(x1)) -> b#(b(b(x1))) -> b#(c(x1)) b#(b(b(x1))) -> b#(c(x1)) -> b#(b(b(x1))) -> c#(x1) SCC Processor: #sccs: 1 #rules: 1 #arcs: 2/4 DPs: b#(b(b(x1))) -> b#(c(x1)) TRS: c(a(x1)) -> a(c(c(b(c(x1))))) b(b(b(x1))) -> b(c(x1)) b(a(x1)) -> a(c(c(x1))) c(c(x1)) -> c(b(c(b(c(x1))))) c(c(c(x1))) -> b(b(c(x1))) Arctic Interpretation Processor: dimension: 2 interpretation: [b#](x0) = [0 2]x0 + [3], [-& 0 ] [0] [b](x0) = [1 0 ]x0 + [2], [0] [a](x0) = [0], [1 -&] [1] [c](x0) = [0 -&]x0 + [0] orientation: b#(b(b(x1))) = [3 3]x1 + [4] >= [2 -&]x1 + [3] = b#(c(x1)) [1] [0] c(a(x1)) = [0] >= [0] = a(c(c(b(c(x1))))) [1 1] [2] [0 -&] [0] b(b(b(x1))) = [2 1]x1 + [3] >= [2 -&]x1 + [2] = b(c(x1)) [0] [0] b(a(x1)) = [2] >= [0] = a(c(c(x1))) [2 -&] [2] [1 -&] [1] c(c(x1)) = [1 -&]x1 + [1] >= [0 -&]x1 + [0] = c(b(c(b(c(x1))))) [3 -&] [3] [2 -&] [2] c(c(c(x1))) = [2 -&]x1 + [2] >= [2 -&]x1 + [2] = b(b(c(x1))) problem: DPs: TRS: c(a(x1)) -> a(c(c(b(c(x1))))) b(b(b(x1))) -> b(c(x1)) b(a(x1)) -> a(c(c(x1))) c(c(x1)) -> c(b(c(b(c(x1))))) c(c(c(x1))) -> b(b(c(x1))) Qed