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