YES Problem: a(a(x1)) -> a(b(a(b(a(x1))))) c(a(x1)) -> a(b(a(a(c(x1))))) b(b(b(x1))) -> a(b(x1)) c(b(x1)) -> a(a(c(x1))) c(b(x1)) -> b(a(d(x1))) d(d(x1)) -> d(b(d(b(d(x1))))) c(c(x1)) -> c(d(c(x1))) a(a(a(x1))) -> a(b(b(x1))) Proof: Arctic Interpretation Processor: dimension: 1 interpretation: [d](x0) = x0, [c](x0) = 2x0, [b](x0) = x0, [a](x0) = x0 orientation: a(a(x1)) = x1 >= x1 = a(b(a(b(a(x1))))) c(a(x1)) = 2x1 >= 2x1 = a(b(a(a(c(x1))))) b(b(b(x1))) = x1 >= x1 = a(b(x1)) c(b(x1)) = 2x1 >= 2x1 = a(a(c(x1))) c(b(x1)) = 2x1 >= x1 = b(a(d(x1))) d(d(x1)) = x1 >= x1 = d(b(d(b(d(x1))))) c(c(x1)) = 4x1 >= 4x1 = c(d(c(x1))) a(a(a(x1))) = x1 >= x1 = a(b(b(x1))) problem: a(a(x1)) -> a(b(a(b(a(x1))))) c(a(x1)) -> a(b(a(a(c(x1))))) b(b(b(x1))) -> a(b(x1)) c(b(x1)) -> a(a(c(x1))) d(d(x1)) -> d(b(d(b(d(x1))))) c(c(x1)) -> c(d(c(x1))) a(a(a(x1))) -> a(b(b(x1))) String Reversal Processor: a(a(x1)) -> a(b(a(b(a(x1))))) a(c(x1)) -> c(a(a(b(a(x1))))) b(b(b(x1))) -> b(a(x1)) b(c(x1)) -> c(a(a(x1))) d(d(x1)) -> d(b(d(b(d(x1))))) c(c(x1)) -> c(d(c(x1))) a(a(a(x1))) -> b(b(a(x1))) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [0] [d](x0) = [0 1 0]x0 + [0] [1 0 0] [1], [1 0 0] [c](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [b](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [a](x0) = [0 0 0]x0 [0 0 0] orientation: [1 0 0] [1 0 0] a(a(x1)) = [0 0 0]x1 >= [0 0 0]x1 = a(b(a(b(a(x1))))) [0 0 0] [0 0 0] [1 0 0] [1 0 0] a(c(x1)) = [0 0 0]x1 >= [0 0 0]x1 = c(a(a(b(a(x1))))) [0 0 0] [0 0 0] [1 0 0] [1 0 0] b(b(b(x1))) = [0 0 0]x1 >= [0 0 0]x1 = b(a(x1)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] b(c(x1)) = [0 0 0]x1 >= [0 0 0]x1 = c(a(a(x1))) [0 0 0] [0 0 0] [2 0 1] [1] [1 0 1] [0] d(d(x1)) = [0 1 0]x1 + [0] >= [0 0 0]x1 + [0] = d(b(d(b(d(x1))))) [1 0 1] [1] [1 0 1] [1] [1 0 0] [1 0 0] c(c(x1)) = [0 0 0]x1 >= [0 0 0]x1 = c(d(c(x1))) [0 0 0] [0 0 0] [1 0 0] [1 0 0] a(a(a(x1))) = [0 0 0]x1 >= [0 0 0]x1 = b(b(a(x1))) [0 0 0] [0 0 0] problem: a(a(x1)) -> a(b(a(b(a(x1))))) a(c(x1)) -> c(a(a(b(a(x1))))) b(b(b(x1))) -> b(a(x1)) b(c(x1)) -> c(a(a(x1))) c(c(x1)) -> c(d(c(x1))) a(a(a(x1))) -> b(b(a(x1))) String Reversal Processor: a(a(x1)) -> a(b(a(b(a(x1))))) c(a(x1)) -> a(b(a(a(c(x1))))) b(b(b(x1))) -> a(b(x1)) c(b(x1)) -> a(a(c(x1))) c(c(x1)) -> c(d(c(x1))) a(a(a(x1))) -> a(b(b(x1))) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [d](x0) = [0 0 0]x0 [0 0 0] , [1 1 0] [1] [c](x0) = [0 1 0]x0 + [1] [0 0 1] [0], [1 0 0] [b](x0) = [0 1 0]x0 [1 0 1] , [1 0 0] [a](x0) = [0 1 0]x0 [0 0 0] orientation: [1 0 0] [1 0 0] a(a(x1)) = [0 1 0]x1 >= [0 1 0]x1 = a(b(a(b(a(x1))))) [0 0 0] [0 0 0] [1 1 0] [1] [1 1 0] [1] c(a(x1)) = [0 1 0]x1 + [1] >= [0 1 0]x1 + [1] = a(b(a(a(c(x1))))) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1 0 0] b(b(b(x1))) = [0 1 0]x1 >= [0 1 0]x1 = a(b(x1)) [3 0 1] [0 0 0] [1 1 0] [1] [1 1 0] [1] c(b(x1)) = [0 1 0]x1 + [1] >= [0 1 0]x1 + [1] = a(a(c(x1))) [1 0 1] [0] [0 0 0] [0] [1 2 0] [3] [1 1 0] [2] c(c(x1)) = [0 1 0]x1 + [2] >= [0 0 0]x1 + [1] = c(d(c(x1))) [0 0 1] [0] [0 0 0] [0] [1 0 0] [1 0 0] a(a(a(x1))) = [0 1 0]x1 >= [0 1 0]x1 = a(b(b(x1))) [0 0 0] [0 0 0] problem: a(a(x1)) -> a(b(a(b(a(x1))))) c(a(x1)) -> a(b(a(a(c(x1))))) b(b(b(x1))) -> a(b(x1)) c(b(x1)) -> a(a(c(x1))) a(a(a(x1))) -> a(b(b(x1))) DP Processor: DPs: a#(a(x1)) -> b#(a(x1)) a#(a(x1)) -> a#(b(a(x1))) a#(a(x1)) -> b#(a(b(a(x1)))) a#(a(x1)) -> a#(b(a(b(a(x1))))) c#(a(x1)) -> c#(x1) c#(a(x1)) -> a#(c(x1)) c#(a(x1)) -> a#(a(c(x1))) c#(a(x1)) -> b#(a(a(c(x1)))) c#(a(x1)) -> a#(b(a(a(c(x1))))) b#(b(b(x1))) -> a#(b(x1)) c#(b(x1)) -> c#(x1) c#(b(x1)) -> a#(c(x1)) c#(b(x1)) -> a#(a(c(x1))) a#(a(a(x1))) -> b#(x1) a#(a(a(x1))) -> b#(b(x1)) a#(a(a(x1))) -> a#(b(b(x1))) TRS: a(a(x1)) -> a(b(a(b(a(x1))))) c(a(x1)) -> a(b(a(a(c(x1))))) b(b(b(x1))) -> a(b(x1)) c(b(x1)) -> a(a(c(x1))) a(a(a(x1))) -> a(b(b(x1))) TDG Processor: DPs: a#(a(x1)) -> b#(a(x1)) a#(a(x1)) -> a#(b(a(x1))) a#(a(x1)) -> b#(a(b(a(x1)))) a#(a(x1)) -> a#(b(a(b(a(x1))))) c#(a(x1)) -> c#(x1) c#(a(x1)) -> a#(c(x1)) c#(a(x1)) -> a#(a(c(x1))) c#(a(x1)) -> b#(a(a(c(x1)))) c#(a(x1)) -> a#(b(a(a(c(x1))))) b#(b(b(x1))) -> a#(b(x1)) c#(b(x1)) -> c#(x1) c#(b(x1)) -> a#(c(x1)) c#(b(x1)) -> a#(a(c(x1))) a#(a(a(x1))) -> b#(x1) a#(a(a(x1))) -> b#(b(x1)) a#(a(a(x1))) -> a#(b(b(x1))) TRS: a(a(x1)) -> a(b(a(b(a(x1))))) c(a(x1)) -> a(b(a(a(c(x1))))) b(b(b(x1))) -> a(b(x1)) c(b(x1)) -> a(a(c(x1))) a(a(a(x1))) -> a(b(b(x1))) graph: c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> a#(a(c(x1))) c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> a#(c(x1)) c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> c#(x1) c#(b(x1)) -> c#(x1) -> c#(a(x1)) -> a#(b(a(a(c(x1))))) c#(b(x1)) -> c#(x1) -> c#(a(x1)) -> b#(a(a(c(x1)))) c#(b(x1)) -> c#(x1) -> c#(a(x1)) -> a#(a(c(x1))) c#(b(x1)) -> c#(x1) -> c#(a(x1)) -> a#(c(x1)) c#(b(x1)) -> c#(x1) -> c#(a(x1)) -> c#(x1) c#(b(x1)) -> a#(c(x1)) -> a#(a(a(x1))) -> a#(b(b(x1))) c#(b(x1)) -> a#(c(x1)) -> a#(a(a(x1))) -> b#(b(x1)) c#(b(x1)) -> a#(c(x1)) -> a#(a(a(x1))) -> b#(x1) c#(b(x1)) -> a#(c(x1)) -> a#(a(x1)) -> a#(b(a(b(a(x1))))) c#(b(x1)) -> a#(c(x1)) -> a#(a(x1)) -> b#(a(b(a(x1)))) c#(b(x1)) -> a#(c(x1)) -> a#(a(x1)) -> a#(b(a(x1))) c#(b(x1)) -> a#(c(x1)) -> a#(a(x1)) -> b#(a(x1)) c#(b(x1)) -> a#(a(c(x1))) -> a#(a(a(x1))) -> a#(b(b(x1))) c#(b(x1)) -> a#(a(c(x1))) -> a#(a(a(x1))) -> b#(b(x1)) c#(b(x1)) -> a#(a(c(x1))) -> a#(a(a(x1))) -> b#(x1) c#(b(x1)) -> a#(a(c(x1))) -> a#(a(x1)) -> a#(b(a(b(a(x1))))) c#(b(x1)) -> a#(a(c(x1))) -> a#(a(x1)) -> b#(a(b(a(x1)))) c#(b(x1)) -> a#(a(c(x1))) -> a#(a(x1)) -> a#(b(a(x1))) c#(b(x1)) -> a#(a(c(x1))) -> a#(a(x1)) -> b#(a(x1)) c#(a(x1)) -> c#(x1) -> c#(b(x1)) -> a#(a(c(x1))) c#(a(x1)) -> c#(x1) -> c#(b(x1)) -> a#(c(x1)) c#(a(x1)) -> c#(x1) -> c#(b(x1)) -> c#(x1) c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> a#(b(a(a(c(x1))))) c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> b#(a(a(c(x1)))) c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> a#(a(c(x1))) c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> a#(c(x1)) c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> c#(x1) c#(a(x1)) -> b#(a(a(c(x1)))) -> b#(b(b(x1))) -> a#(b(x1)) c#(a(x1)) -> a#(c(x1)) -> a#(a(a(x1))) -> a#(b(b(x1))) c#(a(x1)) -> a#(c(x1)) -> a#(a(a(x1))) -> b#(b(x1)) c#(a(x1)) -> a#(c(x1)) -> a#(a(a(x1))) -> b#(x1) c#(a(x1)) -> a#(c(x1)) -> a#(a(x1)) -> a#(b(a(b(a(x1))))) c#(a(x1)) -> a#(c(x1)) -> a#(a(x1)) -> b#(a(b(a(x1)))) c#(a(x1)) -> a#(c(x1)) -> a#(a(x1)) -> a#(b(a(x1))) c#(a(x1)) -> a#(c(x1)) -> a#(a(x1)) -> b#(a(x1)) c#(a(x1)) -> a#(b(a(a(c(x1))))) -> a#(a(a(x1))) -> a#(b(b(x1))) c#(a(x1)) -> a#(b(a(a(c(x1))))) -> a#(a(a(x1))) -> b#(b(x1)) c#(a(x1)) -> a#(b(a(a(c(x1))))) -> a#(a(a(x1))) -> b#(x1) c#(a(x1)) -> a#(b(a(a(c(x1))))) -> a#(a(x1)) -> a#(b(a(b(a(x1))))) c#(a(x1)) -> a#(b(a(a(c(x1))))) -> a#(a(x1)) -> b#(a(b(a(x1)))) c#(a(x1)) -> a#(b(a(a(c(x1))))) -> a#(a(x1)) -> a#(b(a(x1))) c#(a(x1)) -> a#(b(a(a(c(x1))))) -> a#(a(x1)) -> b#(a(x1)) c#(a(x1)) -> a#(a(c(x1))) -> a#(a(a(x1))) -> a#(b(b(x1))) c#(a(x1)) -> a#(a(c(x1))) -> a#(a(a(x1))) -> b#(b(x1)) c#(a(x1)) -> a#(a(c(x1))) -> a#(a(a(x1))) -> b#(x1) c#(a(x1)) -> a#(a(c(x1))) -> a#(a(x1)) -> a#(b(a(b(a(x1))))) c#(a(x1)) -> a#(a(c(x1))) -> a#(a(x1)) -> b#(a(b(a(x1)))) c#(a(x1)) -> a#(a(c(x1))) -> a#(a(x1)) -> a#(b(a(x1))) c#(a(x1)) -> a#(a(c(x1))) -> a#(a(x1)) -> b#(a(x1)) b#(b(b(x1))) -> a#(b(x1)) -> a#(a(a(x1))) -> a#(b(b(x1))) b#(b(b(x1))) -> a#(b(x1)) -> a#(a(a(x1))) -> b#(b(x1)) b#(b(b(x1))) -> a#(b(x1)) -> a#(a(a(x1))) -> b#(x1) b#(b(b(x1))) -> a#(b(x1)) -> a#(a(x1)) -> a#(b(a(b(a(x1))))) b#(b(b(x1))) -> a#(b(x1)) -> a#(a(x1)) -> b#(a(b(a(x1)))) b#(b(b(x1))) -> a#(b(x1)) -> a#(a(x1)) -> a#(b(a(x1))) b#(b(b(x1))) -> a#(b(x1)) -> a#(a(x1)) -> b#(a(x1)) a#(a(a(x1))) -> b#(b(x1)) -> b#(b(b(x1))) -> a#(b(x1)) a#(a(a(x1))) -> b#(x1) -> b#(b(b(x1))) -> a#(b(x1)) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> a#(b(b(x1))) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> b#(b(x1)) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> b#(x1) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(x1)) -> a#(b(a(b(a(x1))))) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(x1)) -> b#(a(b(a(x1)))) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(x1)) -> a#(b(a(x1))) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(x1)) -> b#(a(x1)) a#(a(x1)) -> b#(a(b(a(x1)))) -> b#(b(b(x1))) -> a#(b(x1)) a#(a(x1)) -> b#(a(x1)) -> b#(b(b(x1))) -> a#(b(x1)) a#(a(x1)) -> a#(b(a(b(a(x1))))) -> a#(a(a(x1))) -> a#(b(b(x1))) a#(a(x1)) -> a#(b(a(b(a(x1))))) -> a#(a(a(x1))) -> b#(b(x1)) a#(a(x1)) -> a#(b(a(b(a(x1))))) -> a#(a(a(x1))) -> b#(x1) a#(a(x1)) -> a#(b(a(b(a(x1))))) -> a#(a(x1)) -> a#(b(a(b(a(x1))))) a#(a(x1)) -> a#(b(a(b(a(x1))))) -> a#(a(x1)) -> b#(a(b(a(x1)))) a#(a(x1)) -> a#(b(a(b(a(x1))))) -> a#(a(x1)) -> a#(b(a(x1))) a#(a(x1)) -> a#(b(a(b(a(x1))))) -> a#(a(x1)) -> b#(a(x1)) a#(a(x1)) -> a#(b(a(x1))) -> a#(a(a(x1))) -> a#(b(b(x1))) a#(a(x1)) -> a#(b(a(x1))) -> a#(a(a(x1))) -> b#(b(x1)) a#(a(x1)) -> a#(b(a(x1))) -> a#(a(a(x1))) -> b#(x1) a#(a(x1)) -> a#(b(a(x1))) -> a#(a(x1)) -> a#(b(a(b(a(x1))))) a#(a(x1)) -> a#(b(a(x1))) -> a#(a(x1)) -> b#(a(b(a(x1)))) a#(a(x1)) -> a#(b(a(x1))) -> a#(a(x1)) -> a#(b(a(x1))) a#(a(x1)) -> a#(b(a(x1))) -> a#(a(x1)) -> b#(a(x1)) EDG Processor: DPs: a#(a(x1)) -> b#(a(x1)) a#(a(x1)) -> a#(b(a(x1))) a#(a(x1)) -> b#(a(b(a(x1)))) a#(a(x1)) -> a#(b(a(b(a(x1))))) c#(a(x1)) -> c#(x1) c#(a(x1)) -> a#(c(x1)) c#(a(x1)) -> a#(a(c(x1))) c#(a(x1)) -> b#(a(a(c(x1)))) c#(a(x1)) -> a#(b(a(a(c(x1))))) b#(b(b(x1))) -> a#(b(x1)) c#(b(x1)) -> c#(x1) c#(b(x1)) -> a#(c(x1)) c#(b(x1)) -> a#(a(c(x1))) a#(a(a(x1))) -> b#(x1) a#(a(a(x1))) -> b#(b(x1)) a#(a(a(x1))) -> a#(b(b(x1))) TRS: a(a(x1)) -> a(b(a(b(a(x1))))) c(a(x1)) -> a(b(a(a(c(x1))))) b(b(b(x1))) -> a(b(x1)) c(b(x1)) -> a(a(c(x1))) a(a(a(x1))) -> a(b(b(x1))) graph: c#(b(x1)) -> c#(x1) -> c#(a(x1)) -> c#(x1) c#(b(x1)) -> c#(x1) -> c#(a(x1)) -> a#(c(x1)) c#(b(x1)) -> c#(x1) -> c#(a(x1)) -> a#(a(c(x1))) c#(b(x1)) -> c#(x1) -> c#(a(x1)) -> b#(a(a(c(x1)))) c#(b(x1)) -> c#(x1) -> c#(a(x1)) -> a#(b(a(a(c(x1))))) c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> c#(x1) c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> a#(c(x1)) c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> a#(a(c(x1))) c#(b(x1)) -> a#(c(x1)) -> a#(a(x1)) -> b#(a(x1)) c#(b(x1)) -> a#(c(x1)) -> a#(a(x1)) -> a#(b(a(x1))) c#(b(x1)) -> a#(c(x1)) -> a#(a(x1)) -> b#(a(b(a(x1)))) c#(b(x1)) -> a#(c(x1)) -> a#(a(x1)) -> a#(b(a(b(a(x1))))) c#(b(x1)) -> a#(c(x1)) -> a#(a(a(x1))) -> b#(x1) c#(b(x1)) -> a#(c(x1)) -> a#(a(a(x1))) -> b#(b(x1)) c#(b(x1)) -> a#(c(x1)) -> a#(a(a(x1))) -> a#(b(b(x1))) c#(b(x1)) -> a#(a(c(x1))) -> a#(a(x1)) -> b#(a(x1)) c#(b(x1)) -> a#(a(c(x1))) -> a#(a(x1)) -> a#(b(a(x1))) c#(b(x1)) -> a#(a(c(x1))) -> a#(a(x1)) -> b#(a(b(a(x1)))) c#(b(x1)) -> a#(a(c(x1))) -> a#(a(x1)) -> a#(b(a(b(a(x1))))) c#(b(x1)) -> a#(a(c(x1))) -> a#(a(a(x1))) -> b#(x1) c#(b(x1)) -> a#(a(c(x1))) -> a#(a(a(x1))) -> b#(b(x1)) c#(b(x1)) -> a#(a(c(x1))) -> a#(a(a(x1))) -> a#(b(b(x1))) c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> c#(x1) c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> a#(c(x1)) c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> a#(a(c(x1))) c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> b#(a(a(c(x1)))) c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> a#(b(a(a(c(x1))))) c#(a(x1)) -> c#(x1) -> c#(b(x1)) -> c#(x1) c#(a(x1)) -> c#(x1) -> c#(b(x1)) -> a#(c(x1)) c#(a(x1)) -> c#(x1) -> c#(b(x1)) -> a#(a(c(x1))) c#(a(x1)) -> a#(c(x1)) -> a#(a(x1)) -> b#(a(x1)) c#(a(x1)) -> a#(c(x1)) -> a#(a(x1)) -> a#(b(a(x1))) c#(a(x1)) -> a#(c(x1)) -> a#(a(x1)) -> b#(a(b(a(x1)))) c#(a(x1)) -> a#(c(x1)) -> a#(a(x1)) -> a#(b(a(b(a(x1))))) c#(a(x1)) -> a#(c(x1)) -> a#(a(a(x1))) -> b#(x1) c#(a(x1)) -> a#(c(x1)) -> a#(a(a(x1))) -> b#(b(x1)) c#(a(x1)) -> a#(c(x1)) -> a#(a(a(x1))) -> a#(b(b(x1))) c#(a(x1)) -> a#(b(a(a(c(x1))))) -> a#(a(x1)) -> b#(a(x1)) c#(a(x1)) -> a#(b(a(a(c(x1))))) -> a#(a(x1)) -> a#(b(a(x1))) c#(a(x1)) -> a#(b(a(a(c(x1))))) -> a#(a(x1)) -> b#(a(b(a(x1)))) c#(a(x1)) -> a#(b(a(a(c(x1))))) -> a#(a(x1)) -> a#(b(a(b(a(x1))))) c#(a(x1)) -> a#(b(a(a(c(x1))))) -> a#(a(a(x1))) -> b#(x1) c#(a(x1)) -> a#(b(a(a(c(x1))))) -> a#(a(a(x1))) -> b#(b(x1)) c#(a(x1)) -> a#(b(a(a(c(x1))))) -> a#(a(a(x1))) -> a#(b(b(x1))) c#(a(x1)) -> a#(a(c(x1))) -> a#(a(x1)) -> b#(a(x1)) c#(a(x1)) -> a#(a(c(x1))) -> a#(a(x1)) -> a#(b(a(x1))) c#(a(x1)) -> a#(a(c(x1))) -> a#(a(x1)) -> b#(a(b(a(x1)))) c#(a(x1)) -> a#(a(c(x1))) -> a#(a(x1)) -> a#(b(a(b(a(x1))))) c#(a(x1)) -> a#(a(c(x1))) -> a#(a(a(x1))) -> b#(x1) c#(a(x1)) -> a#(a(c(x1))) -> a#(a(a(x1))) -> b#(b(x1)) c#(a(x1)) -> a#(a(c(x1))) -> a#(a(a(x1))) -> a#(b(b(x1))) b#(b(b(x1))) -> a#(b(x1)) -> a#(a(x1)) -> b#(a(x1)) b#(b(b(x1))) -> a#(b(x1)) -> a#(a(x1)) -> a#(b(a(x1))) b#(b(b(x1))) -> a#(b(x1)) -> a#(a(x1)) -> b#(a(b(a(x1)))) b#(b(b(x1))) -> a#(b(x1)) -> a#(a(x1)) -> a#(b(a(b(a(x1))))) b#(b(b(x1))) -> a#(b(x1)) -> a#(a(a(x1))) -> b#(x1) b#(b(b(x1))) -> a#(b(x1)) -> a#(a(a(x1))) -> b#(b(x1)) b#(b(b(x1))) -> a#(b(x1)) -> a#(a(a(x1))) -> a#(b(b(x1))) a#(a(a(x1))) -> b#(b(x1)) -> b#(b(b(x1))) -> a#(b(x1)) a#(a(a(x1))) -> b#(x1) -> b#(b(b(x1))) -> a#(b(x1)) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(x1)) -> b#(a(x1)) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(x1)) -> a#(b(a(x1))) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(x1)) -> b#(a(b(a(x1)))) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(x1)) -> a#(b(a(b(a(x1))))) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> b#(x1) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> b#(b(x1)) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> a#(b(b(x1))) a#(a(x1)) -> a#(b(a(b(a(x1))))) -> a#(a(x1)) -> b#(a(x1)) a#(a(x1)) -> a#(b(a(b(a(x1))))) -> a#(a(x1)) -> a#(b(a(x1))) a#(a(x1)) -> a#(b(a(b(a(x1))))) -> a#(a(x1)) -> b#(a(b(a(x1)))) a#(a(x1)) -> a#(b(a(b(a(x1))))) -> a#(a(x1)) -> a#(b(a(b(a(x1))))) a#(a(x1)) -> a#(b(a(b(a(x1))))) -> a#(a(a(x1))) -> b#(x1) a#(a(x1)) -> a#(b(a(b(a(x1))))) -> a#(a(a(x1))) -> b#(b(x1)) a#(a(x1)) -> a#(b(a(b(a(x1))))) -> a#(a(a(x1))) -> a#(b(b(x1))) a#(a(x1)) -> a#(b(a(x1))) -> a#(a(x1)) -> b#(a(x1)) a#(a(x1)) -> a#(b(a(x1))) -> a#(a(x1)) -> a#(b(a(x1))) a#(a(x1)) -> a#(b(a(x1))) -> a#(a(x1)) -> b#(a(b(a(x1)))) a#(a(x1)) -> a#(b(a(x1))) -> a#(a(x1)) -> a#(b(a(b(a(x1))))) a#(a(x1)) -> a#(b(a(x1))) -> a#(a(a(x1))) -> b#(x1) a#(a(x1)) -> a#(b(a(x1))) -> a#(a(a(x1))) -> b#(b(x1)) a#(a(x1)) -> a#(b(a(x1))) -> a#(a(a(x1))) -> a#(b(b(x1))) CDG Processor: DPs: a#(a(x1)) -> b#(a(x1)) a#(a(x1)) -> a#(b(a(x1))) a#(a(x1)) -> b#(a(b(a(x1)))) a#(a(x1)) -> a#(b(a(b(a(x1))))) c#(a(x1)) -> c#(x1) c#(a(x1)) -> a#(c(x1)) c#(a(x1)) -> a#(a(c(x1))) c#(a(x1)) -> b#(a(a(c(x1)))) c#(a(x1)) -> a#(b(a(a(c(x1))))) b#(b(b(x1))) -> a#(b(x1)) c#(b(x1)) -> c#(x1) c#(b(x1)) -> a#(c(x1)) c#(b(x1)) -> a#(a(c(x1))) a#(a(a(x1))) -> b#(x1) a#(a(a(x1))) -> b#(b(x1)) a#(a(a(x1))) -> a#(b(b(x1))) TRS: a(a(x1)) -> a(b(a(b(a(x1))))) c(a(x1)) -> a(b(a(a(c(x1))))) b(b(b(x1))) -> a(b(x1)) c(b(x1)) -> a(a(c(x1))) a(a(a(x1))) -> a(b(b(x1))) graph: c#(b(x1)) -> a#(c(x1)) -> a#(a(a(x1))) -> a#(b(b(x1))) c#(b(x1)) -> a#(c(x1)) -> a#(a(a(x1))) -> b#(b(x1)) c#(b(x1)) -> a#(c(x1)) -> a#(a(a(x1))) -> b#(x1) c#(b(x1)) -> a#(c(x1)) -> a#(a(x1)) -> a#(b(a(b(a(x1))))) c#(b(x1)) -> a#(c(x1)) -> a#(a(x1)) -> b#(a(b(a(x1)))) c#(b(x1)) -> a#(c(x1)) -> a#(a(x1)) -> a#(b(a(x1))) c#(b(x1)) -> a#(c(x1)) -> a#(a(x1)) -> b#(a(x1)) c#(b(x1)) -> a#(a(c(x1))) -> a#(a(a(x1))) -> a#(b(b(x1))) c#(b(x1)) -> a#(a(c(x1))) -> a#(a(a(x1))) -> b#(b(x1)) c#(b(x1)) -> a#(a(c(x1))) -> a#(a(a(x1))) -> b#(x1) c#(b(x1)) -> a#(a(c(x1))) -> a#(a(x1)) -> a#(b(a(b(a(x1))))) c#(b(x1)) -> a#(a(c(x1))) -> a#(a(x1)) -> b#(a(b(a(x1)))) c#(b(x1)) -> a#(a(c(x1))) -> a#(a(x1)) -> a#(b(a(x1))) c#(b(x1)) -> a#(a(c(x1))) -> a#(a(x1)) -> b#(a(x1)) c#(a(x1)) -> a#(c(x1)) -> a#(a(a(x1))) -> a#(b(b(x1))) c#(a(x1)) -> a#(c(x1)) -> a#(a(a(x1))) -> b#(b(x1)) c#(a(x1)) -> a#(c(x1)) -> a#(a(a(x1))) -> b#(x1) c#(a(x1)) -> a#(c(x1)) -> a#(a(x1)) -> a#(b(a(b(a(x1))))) c#(a(x1)) -> a#(c(x1)) -> a#(a(x1)) -> b#(a(b(a(x1)))) c#(a(x1)) -> a#(c(x1)) -> a#(a(x1)) -> a#(b(a(x1))) c#(a(x1)) -> a#(c(x1)) -> a#(a(x1)) -> b#(a(x1)) c#(a(x1)) -> a#(a(c(x1))) -> a#(a(a(x1))) -> a#(b(b(x1))) c#(a(x1)) -> a#(a(c(x1))) -> a#(a(a(x1))) -> b#(b(x1)) c#(a(x1)) -> a#(a(c(x1))) -> a#(a(a(x1))) -> b#(x1) c#(a(x1)) -> a#(a(c(x1))) -> a#(a(x1)) -> a#(b(a(b(a(x1))))) c#(a(x1)) -> a#(a(c(x1))) -> a#(a(x1)) -> b#(a(b(a(x1)))) c#(a(x1)) -> a#(a(c(x1))) -> a#(a(x1)) -> a#(b(a(x1))) c#(a(x1)) -> a#(a(c(x1))) -> a#(a(x1)) -> b#(a(x1)) b#(b(b(x1))) -> a#(b(x1)) -> a#(a(a(x1))) -> a#(b(b(x1))) b#(b(b(x1))) -> a#(b(x1)) -> a#(a(a(x1))) -> b#(b(x1)) b#(b(b(x1))) -> a#(b(x1)) -> a#(a(a(x1))) -> b#(x1) b#(b(b(x1))) -> a#(b(x1)) -> a#(a(x1)) -> a#(b(a(b(a(x1))))) b#(b(b(x1))) -> a#(b(x1)) -> a#(a(x1)) -> b#(a(b(a(x1)))) b#(b(b(x1))) -> a#(b(x1)) -> a#(a(x1)) -> a#(b(a(x1))) b#(b(b(x1))) -> a#(b(x1)) -> a#(a(x1)) -> b#(a(x1)) a#(a(a(x1))) -> b#(b(x1)) -> b#(b(b(x1))) -> a#(b(x1)) a#(a(a(x1))) -> b#(x1) -> b#(b(b(x1))) -> a#(b(x1)) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> a#(b(b(x1))) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> b#(b(x1)) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> b#(x1) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(x1)) -> a#(b(a(b(a(x1))))) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(x1)) -> b#(a(b(a(x1)))) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(x1)) -> a#(b(a(x1))) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(x1)) -> b#(a(x1)) a#(a(x1)) -> a#(b(a(x1))) -> a#(a(a(x1))) -> a#(b(b(x1))) a#(a(x1)) -> a#(b(a(x1))) -> a#(a(a(x1))) -> b#(b(x1)) a#(a(x1)) -> a#(b(a(x1))) -> a#(a(a(x1))) -> b#(x1) a#(a(x1)) -> a#(b(a(x1))) -> a#(a(x1)) -> a#(b(a(b(a(x1))))) a#(a(x1)) -> a#(b(a(x1))) -> a#(a(x1)) -> b#(a(b(a(x1)))) a#(a(x1)) -> a#(b(a(x1))) -> a#(a(x1)) -> a#(b(a(x1))) a#(a(x1)) -> a#(b(a(x1))) -> a#(a(x1)) -> b#(a(x1)) SCC Processor: #sccs: 1 #rules: 5 #arcs: 51/256 DPs: a#(a(x1)) -> a#(b(a(x1))) a#(a(a(x1))) -> b#(x1) b#(b(b(x1))) -> a#(b(x1)) a#(a(a(x1))) -> b#(b(x1)) a#(a(a(x1))) -> a#(b(b(x1))) TRS: a(a(x1)) -> a(b(a(b(a(x1))))) c(a(x1)) -> a(b(a(a(c(x1))))) b(b(b(x1))) -> a(b(x1)) c(b(x1)) -> a(a(c(x1))) a(a(a(x1))) -> a(b(b(x1))) Usable Rule Processor: DPs: a#(a(x1)) -> a#(b(a(x1))) a#(a(a(x1))) -> b#(x1) b#(b(b(x1))) -> a#(b(x1)) a#(a(a(x1))) -> b#(b(x1)) a#(a(a(x1))) -> a#(b(b(x1))) TRS: a(a(x1)) -> a(b(a(b(a(x1))))) a(a(a(x1))) -> a(b(b(x1))) b(b(b(x1))) -> a(b(x1)) Arctic Interpretation Processor: dimension: 2 usable rules: a(a(x1)) -> a(b(a(b(a(x1))))) a(a(a(x1))) -> a(b(b(x1))) b(b(b(x1))) -> a(b(x1)) interpretation: [b#](x0) = [0 2]x0 + [0], [a#](x0) = [0 2]x0 + [0], [1 0 ] [2] [b](x0) = [0 -&]x0 + [2], [-& 0 ] [0] [a](x0) = [0 1 ]x0 + [2] orientation: a#(a(x1)) = [2 3]x1 + [4] >= [0 2]x1 + [4] = a#(b(a(x1))) a#(a(a(x1))) = [3 4]x1 + [5] >= [0 2]x1 + [0] = b#(x1) b#(b(b(x1))) = [3 2]x1 + [4] >= [2 0]x1 + [4] = a#(b(x1)) a#(a(a(x1))) = [3 4]x1 + [5] >= [2 0]x1 + [4] = b#(b(x1)) a#(a(a(x1))) = [3 4]x1 + [5] >= [3 2]x1 + [4] = a#(b(b(x1))) [0 1] [2] [-& 0 ] [2] a(a(x1)) = [1 2]x1 + [3] >= [0 1 ]x1 + [3] = a(b(a(b(a(x1))))) [1 2] [3] [1 0] [2] a(a(a(x1))) = [2 3]x1 + [4] >= [2 1]x1 + [3] = a(b(b(x1))) [3 2] [4] [0 -&] [2] b(b(b(x1))) = [2 1]x1 + [3] >= [1 0 ]x1 + [3] = a(b(x1)) problem: DPs: a#(a(x1)) -> a#(b(a(x1))) b#(b(b(x1))) -> a#(b(x1)) a#(a(a(x1))) -> a#(b(b(x1))) TRS: a(a(x1)) -> a(b(a(b(a(x1))))) a(a(a(x1))) -> a(b(b(x1))) b(b(b(x1))) -> a(b(x1)) Restore Modifier: DPs: a#(a(x1)) -> a#(b(a(x1))) b#(b(b(x1))) -> a#(b(x1)) a#(a(a(x1))) -> a#(b(b(x1))) TRS: a(a(x1)) -> a(b(a(b(a(x1))))) c(a(x1)) -> a(b(a(a(c(x1))))) b(b(b(x1))) -> a(b(x1)) c(b(x1)) -> a(a(c(x1))) a(a(a(x1))) -> a(b(b(x1))) EDG Processor: DPs: a#(a(x1)) -> a#(b(a(x1))) b#(b(b(x1))) -> a#(b(x1)) a#(a(a(x1))) -> a#(b(b(x1))) TRS: a(a(x1)) -> a(b(a(b(a(x1))))) c(a(x1)) -> a(b(a(a(c(x1))))) b(b(b(x1))) -> a(b(x1)) c(b(x1)) -> a(a(c(x1))) a(a(a(x1))) -> a(b(b(x1))) graph: b#(b(b(x1))) -> a#(b(x1)) -> a#(a(x1)) -> a#(b(a(x1))) b#(b(b(x1))) -> a#(b(x1)) -> a#(a(a(x1))) -> a#(b(b(x1))) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(x1)) -> a#(b(a(x1))) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> a#(b(b(x1))) a#(a(x1)) -> a#(b(a(x1))) -> a#(a(x1)) -> a#(b(a(x1))) a#(a(x1)) -> a#(b(a(x1))) -> a#(a(a(x1))) -> a#(b(b(x1))) SCC Processor: #sccs: 1 #rules: 2 #arcs: 6/9 DPs: a#(a(a(x1))) -> a#(b(b(x1))) a#(a(x1)) -> a#(b(a(x1))) TRS: a(a(x1)) -> a(b(a(b(a(x1))))) c(a(x1)) -> a(b(a(a(c(x1))))) b(b(b(x1))) -> a(b(x1)) c(b(x1)) -> a(a(c(x1))) a(a(a(x1))) -> a(b(b(x1))) Usable Rule Processor: DPs: a#(a(a(x1))) -> a#(b(b(x1))) a#(a(x1)) -> a#(b(a(x1))) TRS: b(b(b(x1))) -> a(b(x1)) a(a(x1)) -> a(b(a(b(a(x1))))) a(a(a(x1))) -> a(b(b(x1))) Arctic Interpretation Processor: dimension: 2 usable rules: b(b(b(x1))) -> a(b(x1)) a(a(x1)) -> a(b(a(b(a(x1))))) a(a(a(x1))) -> a(b(b(x1))) interpretation: [a#](x0) = [-& 0 ]x0 + [0], [-& 3 ] [0] [b](x0) = [0 -&]x0 + [0], [-& -&] [0] [a](x0) = [-& 2 ]x0 + [0] orientation: a#(a(a(x1))) = [-& 4 ]x1 + [2] >= [-& 3 ]x1 + [0] = a#(b(b(x1))) a#(a(x1)) = [-& 2 ]x1 + [0] >= [0] = a#(b(a(x1))) [-& 6 ] [3] [-& -&] [0] b(b(b(x1))) = [3 -&]x1 + [3] >= [2 -&]x1 + [2] = a(b(x1)) [-& -&] [0] [0] a(a(x1)) = [-& 4 ]x1 + [2] >= [2] = a(b(a(b(a(x1))))) [-& -&] [0] [-& -&] [0] a(a(a(x1))) = [-& 6 ]x1 + [4] >= [-& 5 ]x1 + [2] = a(b(b(x1))) problem: DPs: a#(a(x1)) -> a#(b(a(x1))) TRS: b(b(b(x1))) -> a(b(x1)) a(a(x1)) -> a(b(a(b(a(x1))))) a(a(a(x1))) -> a(b(b(x1))) Restore Modifier: DPs: a#(a(x1)) -> a#(b(a(x1))) TRS: a(a(x1)) -> a(b(a(b(a(x1))))) c(a(x1)) -> a(b(a(a(c(x1))))) b(b(b(x1))) -> a(b(x1)) c(b(x1)) -> a(a(c(x1))) a(a(a(x1))) -> a(b(b(x1))) EDG Processor: DPs: a#(a(x1)) -> a#(b(a(x1))) TRS: a(a(x1)) -> a(b(a(b(a(x1))))) c(a(x1)) -> a(b(a(a(c(x1))))) b(b(b(x1))) -> a(b(x1)) c(b(x1)) -> a(a(c(x1))) a(a(a(x1))) -> a(b(b(x1))) graph: a#(a(x1)) -> a#(b(a(x1))) -> a#(a(x1)) -> a#(b(a(x1))) Usable Rule Processor: DPs: a#(a(x1)) -> a#(b(a(x1))) TRS: a(a(x1)) -> a(b(a(b(a(x1))))) a(a(a(x1))) -> a(b(b(x1))) b(b(b(x1))) -> a(b(x1)) Arctic Interpretation Processor: dimension: 2 usable rules: a(a(x1)) -> a(b(a(b(a(x1))))) a(a(a(x1))) -> a(b(b(x1))) b(b(b(x1))) -> a(b(x1)) interpretation: [a#](x0) = [-& 0 ]x0 + [0], [2 0 ] [2 ] [b](x0) = [0 -&]x0 + [-&], [-& 0 ] [0] [a](x0) = [0 2 ]x0 + [2] orientation: a#(a(x1)) = [0 2]x1 + [2] >= [-& 0 ]x1 + [0] = a#(b(a(x1))) [0 2] [2] [-& 0 ] [0] a(a(x1)) = [2 4]x1 + [4] >= [0 2 ]x1 + [2] = a(b(a(b(a(x1))))) [2 4] [4] [2 0] [2] a(a(a(x1))) = [4 6]x1 + [6] >= [4 2]x1 + [4] = a(b(b(x1))) [6 4] [6] [0 -&] [0] b(b(b(x1))) = [4 2]x1 + [4] >= [2 0 ]x1 + [2] = a(b(x1)) problem: DPs: TRS: a(a(x1)) -> a(b(a(b(a(x1))))) a(a(a(x1))) -> a(b(b(x1))) b(b(b(x1))) -> a(b(x1)) Qed