YES Problem: a(a(d(d(x1)))) -> d(d(b(b(x1)))) a(a(x1)) -> b(b(b(b(b(b(x1)))))) b(b(d(d(b(b(x1)))))) -> a(a(c(c(x1)))) c(c(x1)) -> d(d(x1)) Proof: DP Processor: DPs: a#(a(d(d(x1)))) -> b#(x1) a#(a(d(d(x1)))) -> b#(b(x1)) a#(a(x1)) -> b#(x1) a#(a(x1)) -> b#(b(x1)) a#(a(x1)) -> b#(b(b(x1))) a#(a(x1)) -> b#(b(b(b(x1)))) a#(a(x1)) -> b#(b(b(b(b(x1))))) a#(a(x1)) -> b#(b(b(b(b(b(x1)))))) b#(b(d(d(b(b(x1)))))) -> c#(x1) b#(b(d(d(b(b(x1)))))) -> c#(c(x1)) b#(b(d(d(b(b(x1)))))) -> a#(c(c(x1))) b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) TRS: a(a(d(d(x1)))) -> d(d(b(b(x1)))) a(a(x1)) -> b(b(b(b(b(b(x1)))))) b(b(d(d(b(b(x1)))))) -> a(a(c(c(x1)))) c(c(x1)) -> d(d(x1)) TDG Processor: DPs: a#(a(d(d(x1)))) -> b#(x1) a#(a(d(d(x1)))) -> b#(b(x1)) a#(a(x1)) -> b#(x1) a#(a(x1)) -> b#(b(x1)) a#(a(x1)) -> b#(b(b(x1))) a#(a(x1)) -> b#(b(b(b(x1)))) a#(a(x1)) -> b#(b(b(b(b(x1))))) a#(a(x1)) -> b#(b(b(b(b(b(x1)))))) b#(b(d(d(b(b(x1)))))) -> c#(x1) b#(b(d(d(b(b(x1)))))) -> c#(c(x1)) b#(b(d(d(b(b(x1)))))) -> a#(c(c(x1))) b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) TRS: a(a(d(d(x1)))) -> d(d(b(b(x1)))) a(a(x1)) -> b(b(b(b(b(b(x1)))))) b(b(d(d(b(b(x1)))))) -> a(a(c(c(x1)))) c(c(x1)) -> d(d(x1)) graph: b#(b(d(d(b(b(x1)))))) -> a#(c(c(x1))) -> a#(a(x1)) -> b#(b(b(b(b(b(x1)))))) b#(b(d(d(b(b(x1)))))) -> a#(c(c(x1))) -> a#(a(x1)) -> b#(b(b(b(b(x1))))) b#(b(d(d(b(b(x1)))))) -> a#(c(c(x1))) -> a#(a(x1)) -> b#(b(b(b(x1)))) b#(b(d(d(b(b(x1)))))) -> a#(c(c(x1))) -> a#(a(x1)) -> b#(b(b(x1))) b#(b(d(d(b(b(x1)))))) -> a#(c(c(x1))) -> a#(a(x1)) -> b#(b(x1)) b#(b(d(d(b(b(x1)))))) -> a#(c(c(x1))) -> a#(a(x1)) -> b#(x1) b#(b(d(d(b(b(x1)))))) -> a#(c(c(x1))) -> a#(a(d(d(x1)))) -> b#(b(x1)) b#(b(d(d(b(b(x1)))))) -> a#(c(c(x1))) -> a#(a(d(d(x1)))) -> b#(x1) b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) -> a#(a(x1)) -> b#(b(b(b(b(b(x1)))))) b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) -> a#(a(x1)) -> b#(b(b(b(b(x1))))) b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) -> a#(a(x1)) -> b#(b(b(b(x1)))) b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) -> a#(a(x1)) -> b#(b(b(x1))) b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) -> a#(a(x1)) -> b#(b(x1)) b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) -> a#(a(x1)) -> b#(x1) b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) -> a#(a(d(d(x1)))) -> b#(b(x1)) b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) -> a#(a(d(d(x1)))) -> b#(x1) a#(a(d(d(x1)))) -> b#(b(x1)) -> b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) a#(a(d(d(x1)))) -> b#(b(x1)) -> b#(b(d(d(b(b(x1)))))) -> a#(c(c(x1))) a#(a(d(d(x1)))) -> b#(b(x1)) -> b#(b(d(d(b(b(x1)))))) -> c#(c(x1)) a#(a(d(d(x1)))) -> b#(b(x1)) -> b#(b(d(d(b(b(x1)))))) -> c#(x1) a#(a(d(d(x1)))) -> b#(x1) -> b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) a#(a(d(d(x1)))) -> b#(x1) -> b#(b(d(d(b(b(x1)))))) -> a#(c(c(x1))) a#(a(d(d(x1)))) -> b#(x1) -> b#(b(d(d(b(b(x1)))))) -> c#(c(x1)) a#(a(d(d(x1)))) -> b#(x1) -> b#(b(d(d(b(b(x1)))))) -> c#(x1) a#(a(x1)) -> b#(b(b(b(b(b(x1)))))) -> b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) a#(a(x1)) -> b#(b(b(b(b(b(x1)))))) -> b#(b(d(d(b(b(x1)))))) -> a#(c(c(x1))) a#(a(x1)) -> b#(b(b(b(b(b(x1)))))) -> b#(b(d(d(b(b(x1)))))) -> c#(c(x1)) a#(a(x1)) -> b#(b(b(b(b(b(x1)))))) -> b#(b(d(d(b(b(x1)))))) -> c#(x1) a#(a(x1)) -> b#(b(b(b(b(x1))))) -> b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) a#(a(x1)) -> b#(b(b(b(b(x1))))) -> b#(b(d(d(b(b(x1)))))) -> a#(c(c(x1))) a#(a(x1)) -> b#(b(b(b(b(x1))))) -> b#(b(d(d(b(b(x1)))))) -> c#(c(x1)) a#(a(x1)) -> b#(b(b(b(b(x1))))) -> b#(b(d(d(b(b(x1)))))) -> c#(x1) a#(a(x1)) -> b#(b(b(b(x1)))) -> b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) a#(a(x1)) -> b#(b(b(b(x1)))) -> b#(b(d(d(b(b(x1)))))) -> a#(c(c(x1))) a#(a(x1)) -> b#(b(b(b(x1)))) -> b#(b(d(d(b(b(x1)))))) -> c#(c(x1)) a#(a(x1)) -> b#(b(b(b(x1)))) -> b#(b(d(d(b(b(x1)))))) -> c#(x1) a#(a(x1)) -> b#(b(b(x1))) -> b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) a#(a(x1)) -> b#(b(b(x1))) -> b#(b(d(d(b(b(x1)))))) -> a#(c(c(x1))) a#(a(x1)) -> b#(b(b(x1))) -> b#(b(d(d(b(b(x1)))))) -> c#(c(x1)) a#(a(x1)) -> b#(b(b(x1))) -> b#(b(d(d(b(b(x1)))))) -> c#(x1) a#(a(x1)) -> b#(b(x1)) -> b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) a#(a(x1)) -> b#(b(x1)) -> b#(b(d(d(b(b(x1)))))) -> a#(c(c(x1))) a#(a(x1)) -> b#(b(x1)) -> b#(b(d(d(b(b(x1)))))) -> c#(c(x1)) a#(a(x1)) -> b#(b(x1)) -> b#(b(d(d(b(b(x1)))))) -> c#(x1) a#(a(x1)) -> b#(x1) -> b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) a#(a(x1)) -> b#(x1) -> b#(b(d(d(b(b(x1)))))) -> a#(c(c(x1))) a#(a(x1)) -> b#(x1) -> b#(b(d(d(b(b(x1)))))) -> c#(c(x1)) a#(a(x1)) -> b#(x1) -> b#(b(d(d(b(b(x1)))))) -> c#(x1) SCC Processor: #sccs: 1 #rules: 10 #arcs: 48/144 DPs: b#(b(d(d(b(b(x1)))))) -> a#(c(c(x1))) a#(a(d(d(x1)))) -> b#(x1) b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) a#(a(d(d(x1)))) -> b#(b(x1)) a#(a(x1)) -> b#(x1) a#(a(x1)) -> b#(b(x1)) a#(a(x1)) -> b#(b(b(x1))) a#(a(x1)) -> b#(b(b(b(x1)))) a#(a(x1)) -> b#(b(b(b(b(x1))))) a#(a(x1)) -> b#(b(b(b(b(b(x1)))))) TRS: a(a(d(d(x1)))) -> d(d(b(b(x1)))) a(a(x1)) -> b(b(b(b(b(b(x1)))))) b(b(d(d(b(b(x1)))))) -> a(a(c(c(x1)))) c(c(x1)) -> d(d(x1)) Arctic Interpretation Processor: dimension: 1 interpretation: [b#](x0) = x0, [a#](x0) = x0, [c](x0) = 8x0, [b](x0) = x0, [a](x0) = x0, [d](x0) = 8x0 orientation: b#(b(d(d(b(b(x1)))))) = 16x1 >= 16x1 = a#(c(c(x1))) a#(a(d(d(x1)))) = 16x1 >= x1 = b#(x1) b#(b(d(d(b(b(x1)))))) = 16x1 >= 16x1 = a#(a(c(c(x1)))) a#(a(d(d(x1)))) = 16x1 >= x1 = b#(b(x1)) a#(a(x1)) = x1 >= x1 = b#(x1) a#(a(x1)) = x1 >= x1 = b#(b(x1)) a#(a(x1)) = x1 >= x1 = b#(b(b(x1))) a#(a(x1)) = x1 >= x1 = b#(b(b(b(x1)))) a#(a(x1)) = x1 >= x1 = b#(b(b(b(b(x1))))) a#(a(x1)) = x1 >= x1 = b#(b(b(b(b(b(x1)))))) a(a(d(d(x1)))) = 16x1 >= 16x1 = d(d(b(b(x1)))) a(a(x1)) = x1 >= x1 = b(b(b(b(b(b(x1)))))) b(b(d(d(b(b(x1)))))) = 16x1 >= 16x1 = a(a(c(c(x1)))) c(c(x1)) = 16x1 >= 16x1 = d(d(x1)) problem: DPs: b#(b(d(d(b(b(x1)))))) -> a#(c(c(x1))) b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) a#(a(x1)) -> b#(x1) a#(a(x1)) -> b#(b(x1)) a#(a(x1)) -> b#(b(b(x1))) a#(a(x1)) -> b#(b(b(b(x1)))) a#(a(x1)) -> b#(b(b(b(b(x1))))) a#(a(x1)) -> b#(b(b(b(b(b(x1)))))) TRS: a(a(d(d(x1)))) -> d(d(b(b(x1)))) a(a(x1)) -> b(b(b(b(b(b(x1)))))) b(b(d(d(b(b(x1)))))) -> a(a(c(c(x1)))) c(c(x1)) -> d(d(x1)) EDG Processor: DPs: b#(b(d(d(b(b(x1)))))) -> a#(c(c(x1))) b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) a#(a(x1)) -> b#(x1) a#(a(x1)) -> b#(b(x1)) a#(a(x1)) -> b#(b(b(x1))) a#(a(x1)) -> b#(b(b(b(x1)))) a#(a(x1)) -> b#(b(b(b(b(x1))))) a#(a(x1)) -> b#(b(b(b(b(b(x1)))))) TRS: a(a(d(d(x1)))) -> d(d(b(b(x1)))) a(a(x1)) -> b(b(b(b(b(b(x1)))))) b(b(d(d(b(b(x1)))))) -> a(a(c(c(x1)))) c(c(x1)) -> d(d(x1)) graph: b#(b(d(d(b(b(x1)))))) -> a#(c(c(x1))) -> a#(a(x1)) -> b#(x1) b#(b(d(d(b(b(x1)))))) -> a#(c(c(x1))) -> a#(a(x1)) -> b#(b(x1)) b#(b(d(d(b(b(x1)))))) -> a#(c(c(x1))) -> a#(a(x1)) -> b#(b(b(x1))) b#(b(d(d(b(b(x1)))))) -> a#(c(c(x1))) -> a#(a(x1)) -> b#(b(b(b(x1)))) b#(b(d(d(b(b(x1)))))) -> a#(c(c(x1))) -> a#(a(x1)) -> b#(b(b(b(b(x1))))) b#(b(d(d(b(b(x1)))))) -> a#(c(c(x1))) -> a#(a(x1)) -> b#(b(b(b(b(b(x1)))))) b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) -> a#(a(x1)) -> b#(x1) b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) -> a#(a(x1)) -> b#(b(x1)) b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) -> a#(a(x1)) -> b#(b(b(x1))) b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) -> a#(a(x1)) -> b#(b(b(b(x1)))) b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) -> a#(a(x1)) -> b#(b(b(b(b(x1))))) b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) -> a#(a(x1)) -> b#(b(b(b(b(b(x1)))))) a#(a(x1)) -> b#(b(b(b(b(b(x1)))))) -> b#(b(d(d(b(b(x1)))))) -> a#(c(c(x1))) a#(a(x1)) -> b#(b(b(b(b(b(x1)))))) -> b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) a#(a(x1)) -> b#(b(b(b(b(x1))))) -> b#(b(d(d(b(b(x1)))))) -> a#(c(c(x1))) a#(a(x1)) -> b#(b(b(b(b(x1))))) -> b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) a#(a(x1)) -> b#(b(b(b(x1)))) -> b#(b(d(d(b(b(x1)))))) -> a#(c(c(x1))) a#(a(x1)) -> b#(b(b(b(x1)))) -> b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) a#(a(x1)) -> b#(b(b(x1))) -> b#(b(d(d(b(b(x1)))))) -> a#(c(c(x1))) a#(a(x1)) -> b#(b(b(x1))) -> b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) a#(a(x1)) -> b#(b(x1)) -> b#(b(d(d(b(b(x1)))))) -> a#(c(c(x1))) a#(a(x1)) -> b#(b(x1)) -> b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) a#(a(x1)) -> b#(x1) -> b#(b(d(d(b(b(x1)))))) -> a#(c(c(x1))) a#(a(x1)) -> b#(x1) -> b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) Arctic Interpretation Processor: dimension: 1 interpretation: [b#](x0) = 3, [a#](x0) = 1x0 + -16, [c](x0) = 1, [b](x0) = 2, [a](x0) = 2, [d](x0) = 0 orientation: b#(b(d(d(b(b(x1)))))) = 3 >= 2 = a#(c(c(x1))) b#(b(d(d(b(b(x1)))))) = 3 >= 3 = a#(a(c(c(x1)))) a#(a(x1)) = 3 >= 3 = b#(x1) a#(a(x1)) = 3 >= 3 = b#(b(x1)) a#(a(x1)) = 3 >= 3 = b#(b(b(x1))) a#(a(x1)) = 3 >= 3 = b#(b(b(b(x1)))) a#(a(x1)) = 3 >= 3 = b#(b(b(b(b(x1))))) a#(a(x1)) = 3 >= 3 = b#(b(b(b(b(b(x1)))))) a(a(d(d(x1)))) = 2 >= 0 = d(d(b(b(x1)))) a(a(x1)) = 2 >= 2 = b(b(b(b(b(b(x1)))))) b(b(d(d(b(b(x1)))))) = 2 >= 2 = a(a(c(c(x1)))) c(c(x1)) = 1 >= 0 = d(d(x1)) problem: DPs: b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) a#(a(x1)) -> b#(x1) a#(a(x1)) -> b#(b(x1)) a#(a(x1)) -> b#(b(b(x1))) a#(a(x1)) -> b#(b(b(b(x1)))) a#(a(x1)) -> b#(b(b(b(b(x1))))) a#(a(x1)) -> b#(b(b(b(b(b(x1)))))) TRS: a(a(d(d(x1)))) -> d(d(b(b(x1)))) a(a(x1)) -> b(b(b(b(b(b(x1)))))) b(b(d(d(b(b(x1)))))) -> a(a(c(c(x1)))) c(c(x1)) -> d(d(x1)) EDG Processor: DPs: b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) a#(a(x1)) -> b#(x1) a#(a(x1)) -> b#(b(x1)) a#(a(x1)) -> b#(b(b(x1))) a#(a(x1)) -> b#(b(b(b(x1)))) a#(a(x1)) -> b#(b(b(b(b(x1))))) a#(a(x1)) -> b#(b(b(b(b(b(x1)))))) TRS: a(a(d(d(x1)))) -> d(d(b(b(x1)))) a(a(x1)) -> b(b(b(b(b(b(x1)))))) b(b(d(d(b(b(x1)))))) -> a(a(c(c(x1)))) c(c(x1)) -> d(d(x1)) graph: b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) -> a#(a(x1)) -> b#(b(b(b(b(b(x1)))))) b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) -> a#(a(x1)) -> b#(b(b(b(b(x1))))) b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) -> a#(a(x1)) -> b#(b(b(b(x1)))) b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) -> a#(a(x1)) -> b#(b(b(x1))) b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) -> a#(a(x1)) -> b#(b(x1)) b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) -> a#(a(x1)) -> b#(x1) a#(a(x1)) -> b#(b(b(b(b(b(x1)))))) -> b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) a#(a(x1)) -> b#(b(b(b(b(x1))))) -> b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) a#(a(x1)) -> b#(b(b(b(x1)))) -> b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) a#(a(x1)) -> b#(b(b(x1))) -> b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) a#(a(x1)) -> b#(b(x1)) -> b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) a#(a(x1)) -> b#(x1) -> b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) Arctic Interpretation Processor: dimension: 1 interpretation: [b#](x0) = x0 + 0, [a#](x0) = 2x0 + 7, [c](x0) = 0, [b](x0) = x0 + 7, [a](x0) = 3x0 + 4, [d](x0) = 0 orientation: b#(b(d(d(b(b(x1)))))) = 7 >= 7 = a#(a(c(c(x1)))) a#(a(x1)) = 5x1 + 7 >= x1 + 0 = b#(x1) a#(a(x1)) = 5x1 + 7 >= x1 + 7 = b#(b(x1)) a#(a(x1)) = 5x1 + 7 >= x1 + 7 = b#(b(b(x1))) a#(a(x1)) = 5x1 + 7 >= x1 + 7 = b#(b(b(b(x1)))) a#(a(x1)) = 5x1 + 7 >= x1 + 7 = b#(b(b(b(b(x1))))) a#(a(x1)) = 5x1 + 7 >= x1 + 7 = b#(b(b(b(b(b(x1)))))) a(a(d(d(x1)))) = 7 >= 0 = d(d(b(b(x1)))) a(a(x1)) = 6x1 + 7 >= x1 + 7 = b(b(b(b(b(b(x1)))))) b(b(d(d(b(b(x1)))))) = 7 >= 7 = a(a(c(c(x1)))) c(c(x1)) = 0 >= 0 = d(d(x1)) problem: DPs: b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) a#(a(x1)) -> b#(b(x1)) a#(a(x1)) -> b#(b(b(x1))) a#(a(x1)) -> b#(b(b(b(x1)))) a#(a(x1)) -> b#(b(b(b(b(x1))))) a#(a(x1)) -> b#(b(b(b(b(b(x1)))))) TRS: a(a(d(d(x1)))) -> d(d(b(b(x1)))) a(a(x1)) -> b(b(b(b(b(b(x1)))))) b(b(d(d(b(b(x1)))))) -> a(a(c(c(x1)))) c(c(x1)) -> d(d(x1)) EDG Processor: DPs: b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) a#(a(x1)) -> b#(b(x1)) a#(a(x1)) -> b#(b(b(x1))) a#(a(x1)) -> b#(b(b(b(x1)))) a#(a(x1)) -> b#(b(b(b(b(x1))))) a#(a(x1)) -> b#(b(b(b(b(b(x1)))))) TRS: a(a(d(d(x1)))) -> d(d(b(b(x1)))) a(a(x1)) -> b(b(b(b(b(b(x1)))))) b(b(d(d(b(b(x1)))))) -> a(a(c(c(x1)))) c(c(x1)) -> d(d(x1)) graph: b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) -> a#(a(x1)) -> b#(b(x1)) b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) -> a#(a(x1)) -> b#(b(b(x1))) b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) -> a#(a(x1)) -> b#(b(b(b(x1)))) b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) -> a#(a(x1)) -> b#(b(b(b(b(x1))))) b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) -> a#(a(x1)) -> b#(b(b(b(b(b(x1)))))) a#(a(x1)) -> b#(b(b(b(b(b(x1)))))) -> b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) a#(a(x1)) -> b#(b(b(b(b(x1))))) -> b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) a#(a(x1)) -> b#(b(b(b(x1)))) -> b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) a#(a(x1)) -> b#(b(b(x1))) -> b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) a#(a(x1)) -> b#(b(x1)) -> b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) Arctic Interpretation Processor: dimension: 2 interpretation: [b#](x0) = [1 0]x0 + [0], [a#](x0) = [1 2]x0 + [0], [0 -&] [-&] [c](x0) = [1 3 ]x0 + [3 ], [-& 0 ] [0] [b](x0) = [0 -&]x0 + [0], [-& 0 ] [0] [a](x0) = [0 -&]x0 + [0], [0 -&] [-&] [d](x0) = [1 3 ]x0 + [3 ] orientation: b#(b(d(d(b(b(x1)))))) = [5 7]x1 + [7] >= [5 7]x1 + [7] = a#(a(c(c(x1)))) a#(a(x1)) = [2 1]x1 + [2] >= [0 1]x1 + [1] = b#(b(x1)) a#(a(x1)) = [2 1]x1 + [2] >= [1 0]x1 + [1] = b#(b(b(x1))) a#(a(x1)) = [2 1]x1 + [2] >= [0 1]x1 + [1] = b#(b(b(b(x1)))) a#(a(x1)) = [2 1]x1 + [2] >= [1 0]x1 + [1] = b#(b(b(b(b(x1))))) a#(a(x1)) = [2 1]x1 + [2] >= [0 1]x1 + [1] = b#(b(b(b(b(b(x1)))))) [0 -&] [0] [0 -&] [0] a(a(d(d(x1)))) = [4 6 ]x1 + [6] >= [4 6 ]x1 + [6] = d(d(b(b(x1)))) [0] [0] a(a(x1)) = x1 + [0] >= x1 + [0] = b(b(b(b(b(b(x1)))))) [0 -&] [0] [0 -&] [0] b(b(d(d(b(b(x1)))))) = [4 6 ]x1 + [6] >= [4 6 ]x1 + [6] = a(a(c(c(x1)))) [0 -&] [-&] [0 -&] [-&] c(c(x1)) = [4 6 ]x1 + [6 ] >= [4 6 ]x1 + [6 ] = d(d(x1)) problem: DPs: b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) a#(a(x1)) -> b#(b(x1)) a#(a(x1)) -> b#(b(b(b(x1)))) a#(a(x1)) -> b#(b(b(b(b(b(x1)))))) TRS: a(a(d(d(x1)))) -> d(d(b(b(x1)))) a(a(x1)) -> b(b(b(b(b(b(x1)))))) b(b(d(d(b(b(x1)))))) -> a(a(c(c(x1)))) c(c(x1)) -> d(d(x1)) EDG Processor: DPs: b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) a#(a(x1)) -> b#(b(x1)) a#(a(x1)) -> b#(b(b(b(x1)))) a#(a(x1)) -> b#(b(b(b(b(b(x1)))))) TRS: a(a(d(d(x1)))) -> d(d(b(b(x1)))) a(a(x1)) -> b(b(b(b(b(b(x1)))))) b(b(d(d(b(b(x1)))))) -> a(a(c(c(x1)))) c(c(x1)) -> d(d(x1)) graph: b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) -> a#(a(x1)) -> b#(b(b(b(b(b(x1)))))) b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) -> a#(a(x1)) -> b#(b(b(b(x1)))) b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) -> a#(a(x1)) -> b#(b(x1)) a#(a(x1)) -> b#(b(b(b(b(b(x1)))))) -> b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) a#(a(x1)) -> b#(b(b(b(x1)))) -> b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) a#(a(x1)) -> b#(b(x1)) -> b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) Matrix Interpretation Processor: dim=3 interpretation: [b#](x0) = [0 1 0]x0, [a#](x0) = [1 0 1]x0 + [1], [1 1 1] [c](x0) = [1 0 1]x0 [0 0 0] , [0 1 0] [0] [b](x0) = [1 0 1]x0 + [0] [0 0 0] [1], [1 0 1] [1] [a](x0) = [0 1 1]x0 + [1] [0 0 0] [1], [1 1 1] [d](x0) = [1 0 1]x0 [0 0 0] orientation: b#(b(d(d(b(b(x1)))))) = [2 1 2]x1 + [3] >= [2 1 2]x1 + [3] = a#(a(c(c(x1)))) a#(a(x1)) = [1 0 1]x1 + [3] >= [1 0 1]x1 = b#(b(x1)) a#(a(x1)) = [1 0 1]x1 + [3] >= [1 0 1]x1 + [1] = b#(b(b(b(x1)))) a#(a(x1)) = [1 0 1]x1 + [3] >= [1 0 1]x1 + [2] = b#(b(b(b(b(b(x1)))))) [2 1 2] [3] [2 1 2] [3] a(a(d(d(x1)))) = [1 1 1]x1 + [3] >= [1 1 1]x1 + [2] = d(d(b(b(x1)))) [0 0 0] [1] [0 0 0] [0] [1 0 1] [3] [1 0 1] [2] a(a(x1)) = [0 1 1]x1 + [3] >= [0 1 0]x1 + [3] = b(b(b(b(b(b(x1)))))) [0 0 0] [1] [0 0 0] [1] [2 1 2] [3] [2 1 2] [3] b(b(d(d(b(b(x1)))))) = [1 1 1]x1 + [3] >= [1 1 1]x1 + [3] = a(a(c(c(x1)))) [0 0 0] [1] [0 0 0] [1] [2 1 2] [2 1 2] c(c(x1)) = [1 1 1]x1 >= [1 1 1]x1 = d(d(x1)) [0 0 0] [0 0 0] problem: DPs: b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) TRS: a(a(d(d(x1)))) -> d(d(b(b(x1)))) a(a(x1)) -> b(b(b(b(b(b(x1)))))) b(b(d(d(b(b(x1)))))) -> a(a(c(c(x1)))) c(c(x1)) -> d(d(x1)) EDG Processor: DPs: b#(b(d(d(b(b(x1)))))) -> a#(a(c(c(x1)))) TRS: a(a(d(d(x1)))) -> d(d(b(b(x1)))) a(a(x1)) -> b(b(b(b(b(b(x1)))))) b(b(d(d(b(b(x1)))))) -> a(a(c(c(x1)))) c(c(x1)) -> d(d(x1)) graph: Qed