YES Problem: a(a(a(a(x1)))) -> b(b(b(b(b(b(x1)))))) b(b(b(b(x1)))) -> c(c(c(c(c(c(x1)))))) c(c(c(c(x1)))) -> d(d(d(d(d(d(x1)))))) b(b(x1)) -> d(d(d(d(x1)))) c(c(d(d(d(d(x1)))))) -> a(a(x1)) Proof: DP Processor: DPs: a#(a(a(a(x1)))) -> b#(x1) a#(a(a(a(x1)))) -> b#(b(x1)) a#(a(a(a(x1)))) -> b#(b(b(x1))) a#(a(a(a(x1)))) -> b#(b(b(b(x1)))) a#(a(a(a(x1)))) -> b#(b(b(b(b(x1))))) a#(a(a(a(x1)))) -> b#(b(b(b(b(b(x1)))))) b#(b(b(b(x1)))) -> c#(x1) b#(b(b(b(x1)))) -> c#(c(x1)) b#(b(b(b(x1)))) -> c#(c(c(x1))) b#(b(b(b(x1)))) -> c#(c(c(c(x1)))) b#(b(b(b(x1)))) -> c#(c(c(c(c(x1))))) b#(b(b(b(x1)))) -> c#(c(c(c(c(c(x1)))))) c#(c(d(d(d(d(x1)))))) -> a#(x1) c#(c(d(d(d(d(x1)))))) -> a#(a(x1)) TRS: a(a(a(a(x1)))) -> b(b(b(b(b(b(x1)))))) b(b(b(b(x1)))) -> c(c(c(c(c(c(x1)))))) c(c(c(c(x1)))) -> d(d(d(d(d(d(x1)))))) b(b(x1)) -> d(d(d(d(x1)))) c(c(d(d(d(d(x1)))))) -> a(a(x1)) TDG Processor: DPs: a#(a(a(a(x1)))) -> b#(x1) a#(a(a(a(x1)))) -> b#(b(x1)) a#(a(a(a(x1)))) -> b#(b(b(x1))) a#(a(a(a(x1)))) -> b#(b(b(b(x1)))) a#(a(a(a(x1)))) -> b#(b(b(b(b(x1))))) a#(a(a(a(x1)))) -> b#(b(b(b(b(b(x1)))))) b#(b(b(b(x1)))) -> c#(x1) b#(b(b(b(x1)))) -> c#(c(x1)) b#(b(b(b(x1)))) -> c#(c(c(x1))) b#(b(b(b(x1)))) -> c#(c(c(c(x1)))) b#(b(b(b(x1)))) -> c#(c(c(c(c(x1))))) b#(b(b(b(x1)))) -> c#(c(c(c(c(c(x1)))))) c#(c(d(d(d(d(x1)))))) -> a#(x1) c#(c(d(d(d(d(x1)))))) -> a#(a(x1)) TRS: a(a(a(a(x1)))) -> b(b(b(b(b(b(x1)))))) b(b(b(b(x1)))) -> c(c(c(c(c(c(x1)))))) c(c(c(c(x1)))) -> d(d(d(d(d(d(x1)))))) b(b(x1)) -> d(d(d(d(x1)))) c(c(d(d(d(d(x1)))))) -> a(a(x1)) graph: c#(c(d(d(d(d(x1)))))) -> a#(a(x1)) -> a#(a(a(a(x1)))) -> b#(b(b(b(b(b(x1)))))) c#(c(d(d(d(d(x1)))))) -> a#(a(x1)) -> a#(a(a(a(x1)))) -> b#(b(b(b(b(x1))))) c#(c(d(d(d(d(x1)))))) -> a#(a(x1)) -> a#(a(a(a(x1)))) -> b#(b(b(b(x1)))) c#(c(d(d(d(d(x1)))))) -> a#(a(x1)) -> a#(a(a(a(x1)))) -> b#(b(b(x1))) c#(c(d(d(d(d(x1)))))) -> a#(a(x1)) -> a#(a(a(a(x1)))) -> b#(b(x1)) c#(c(d(d(d(d(x1)))))) -> a#(a(x1)) -> a#(a(a(a(x1)))) -> b#(x1) c#(c(d(d(d(d(x1)))))) -> a#(x1) -> a#(a(a(a(x1)))) -> b#(b(b(b(b(b(x1)))))) c#(c(d(d(d(d(x1)))))) -> a#(x1) -> a#(a(a(a(x1)))) -> b#(b(b(b(b(x1))))) c#(c(d(d(d(d(x1)))))) -> a#(x1) -> a#(a(a(a(x1)))) -> b#(b(b(b(x1)))) c#(c(d(d(d(d(x1)))))) -> a#(x1) -> a#(a(a(a(x1)))) -> b#(b(b(x1))) c#(c(d(d(d(d(x1)))))) -> a#(x1) -> a#(a(a(a(x1)))) -> b#(b(x1)) c#(c(d(d(d(d(x1)))))) -> a#(x1) -> a#(a(a(a(x1)))) -> b#(x1) b#(b(b(b(x1)))) -> c#(c(c(c(c(c(x1)))))) -> c#(c(d(d(d(d(x1)))))) -> a#(a(x1)) b#(b(b(b(x1)))) -> c#(c(c(c(c(c(x1)))))) -> c#(c(d(d(d(d(x1)))))) -> a#(x1) b#(b(b(b(x1)))) -> c#(c(c(c(c(x1))))) -> c#(c(d(d(d(d(x1)))))) -> a#(a(x1)) b#(b(b(b(x1)))) -> c#(c(c(c(c(x1))))) -> c#(c(d(d(d(d(x1)))))) -> a#(x1) b#(b(b(b(x1)))) -> c#(c(c(c(x1)))) -> c#(c(d(d(d(d(x1)))))) -> a#(a(x1)) b#(b(b(b(x1)))) -> c#(c(c(c(x1)))) -> c#(c(d(d(d(d(x1)))))) -> a#(x1) b#(b(b(b(x1)))) -> c#(c(c(x1))) -> c#(c(d(d(d(d(x1)))))) -> a#(a(x1)) b#(b(b(b(x1)))) -> c#(c(c(x1))) -> c#(c(d(d(d(d(x1)))))) -> a#(x1) b#(b(b(b(x1)))) -> c#(c(x1)) -> c#(c(d(d(d(d(x1)))))) -> a#(a(x1)) b#(b(b(b(x1)))) -> c#(c(x1)) -> c#(c(d(d(d(d(x1)))))) -> a#(x1) b#(b(b(b(x1)))) -> c#(x1) -> c#(c(d(d(d(d(x1)))))) -> a#(a(x1)) b#(b(b(b(x1)))) -> c#(x1) -> c#(c(d(d(d(d(x1)))))) -> a#(x1) a#(a(a(a(x1)))) -> b#(b(b(b(b(b(x1)))))) -> b#(b(b(b(x1)))) -> c#(c(c(c(c(c(x1)))))) a#(a(a(a(x1)))) -> b#(b(b(b(b(b(x1)))))) -> b#(b(b(b(x1)))) -> c#(c(c(c(c(x1))))) a#(a(a(a(x1)))) -> b#(b(b(b(b(b(x1)))))) -> b#(b(b(b(x1)))) -> c#(c(c(c(x1)))) a#(a(a(a(x1)))) -> b#(b(b(b(b(b(x1)))))) -> b#(b(b(b(x1)))) -> c#(c(c(x1))) a#(a(a(a(x1)))) -> b#(b(b(b(b(b(x1)))))) -> b#(b(b(b(x1)))) -> c#(c(x1)) a#(a(a(a(x1)))) -> b#(b(b(b(b(b(x1)))))) -> b#(b(b(b(x1)))) -> c#(x1) a#(a(a(a(x1)))) -> b#(b(b(b(b(x1))))) -> b#(b(b(b(x1)))) -> c#(c(c(c(c(c(x1)))))) a#(a(a(a(x1)))) -> b#(b(b(b(b(x1))))) -> b#(b(b(b(x1)))) -> c#(c(c(c(c(x1))))) a#(a(a(a(x1)))) -> b#(b(b(b(b(x1))))) -> b#(b(b(b(x1)))) -> c#(c(c(c(x1)))) a#(a(a(a(x1)))) -> b#(b(b(b(b(x1))))) -> b#(b(b(b(x1)))) -> c#(c(c(x1))) a#(a(a(a(x1)))) -> b#(b(b(b(b(x1))))) -> b#(b(b(b(x1)))) -> c#(c(x1)) a#(a(a(a(x1)))) -> b#(b(b(b(b(x1))))) -> b#(b(b(b(x1)))) -> c#(x1) a#(a(a(a(x1)))) -> b#(b(b(b(x1)))) -> b#(b(b(b(x1)))) -> c#(c(c(c(c(c(x1)))))) a#(a(a(a(x1)))) -> b#(b(b(b(x1)))) -> b#(b(b(b(x1)))) -> c#(c(c(c(c(x1))))) a#(a(a(a(x1)))) -> b#(b(b(b(x1)))) -> b#(b(b(b(x1)))) -> c#(c(c(c(x1)))) a#(a(a(a(x1)))) -> b#(b(b(b(x1)))) -> b#(b(b(b(x1)))) -> c#(c(c(x1))) a#(a(a(a(x1)))) -> b#(b(b(b(x1)))) -> b#(b(b(b(x1)))) -> c#(c(x1)) a#(a(a(a(x1)))) -> b#(b(b(b(x1)))) -> b#(b(b(b(x1)))) -> c#(x1) a#(a(a(a(x1)))) -> b#(b(b(x1))) -> b#(b(b(b(x1)))) -> c#(c(c(c(c(c(x1)))))) a#(a(a(a(x1)))) -> b#(b(b(x1))) -> b#(b(b(b(x1)))) -> c#(c(c(c(c(x1))))) a#(a(a(a(x1)))) -> b#(b(b(x1))) -> b#(b(b(b(x1)))) -> c#(c(c(c(x1)))) a#(a(a(a(x1)))) -> b#(b(b(x1))) -> b#(b(b(b(x1)))) -> c#(c(c(x1))) a#(a(a(a(x1)))) -> b#(b(b(x1))) -> b#(b(b(b(x1)))) -> c#(c(x1)) a#(a(a(a(x1)))) -> b#(b(b(x1))) -> b#(b(b(b(x1)))) -> c#(x1) a#(a(a(a(x1)))) -> b#(b(x1)) -> b#(b(b(b(x1)))) -> c#(c(c(c(c(c(x1)))))) a#(a(a(a(x1)))) -> b#(b(x1)) -> b#(b(b(b(x1)))) -> c#(c(c(c(c(x1))))) a#(a(a(a(x1)))) -> b#(b(x1)) -> b#(b(b(b(x1)))) -> c#(c(c(c(x1)))) a#(a(a(a(x1)))) -> b#(b(x1)) -> b#(b(b(b(x1)))) -> c#(c(c(x1))) a#(a(a(a(x1)))) -> b#(b(x1)) -> b#(b(b(b(x1)))) -> c#(c(x1)) a#(a(a(a(x1)))) -> b#(b(x1)) -> b#(b(b(b(x1)))) -> c#(x1) a#(a(a(a(x1)))) -> b#(x1) -> b#(b(b(b(x1)))) -> c#(c(c(c(c(c(x1)))))) a#(a(a(a(x1)))) -> b#(x1) -> b#(b(b(b(x1)))) -> c#(c(c(c(c(x1))))) a#(a(a(a(x1)))) -> b#(x1) -> b#(b(b(b(x1)))) -> c#(c(c(c(x1)))) a#(a(a(a(x1)))) -> b#(x1) -> b#(b(b(b(x1)))) -> c#(c(c(x1))) a#(a(a(a(x1)))) -> b#(x1) -> b#(b(b(b(x1)))) -> c#(c(x1)) a#(a(a(a(x1)))) -> b#(x1) -> b#(b(b(b(x1)))) -> c#(x1) Matrix Interpretation Processor: dim=1 interpretation: [a#](x0) = x0 + 7, [b](x0) = x0 + 9, [d](x0) = x0 + 4, [b#](x0) = x0 + 4, [a](x0) = x0 + 14, [c#](x0) = x0 + 1, [c](x0) = x0 + 6 orientation: a#(a(a(a(x1)))) = x1 + 49 >= x1 + 4 = b#(x1) a#(a(a(a(x1)))) = x1 + 49 >= x1 + 13 = b#(b(x1)) a#(a(a(a(x1)))) = x1 + 49 >= x1 + 22 = b#(b(b(x1))) a#(a(a(a(x1)))) = x1 + 49 >= x1 + 31 = b#(b(b(b(x1)))) a#(a(a(a(x1)))) = x1 + 49 >= x1 + 40 = b#(b(b(b(b(x1))))) a#(a(a(a(x1)))) = x1 + 49 >= x1 + 49 = b#(b(b(b(b(b(x1)))))) b#(b(b(b(x1)))) = x1 + 31 >= x1 + 1 = c#(x1) b#(b(b(b(x1)))) = x1 + 31 >= x1 + 7 = c#(c(x1)) b#(b(b(b(x1)))) = x1 + 31 >= x1 + 13 = c#(c(c(x1))) b#(b(b(b(x1)))) = x1 + 31 >= x1 + 19 = c#(c(c(c(x1)))) b#(b(b(b(x1)))) = x1 + 31 >= x1 + 25 = c#(c(c(c(c(x1))))) b#(b(b(b(x1)))) = x1 + 31 >= x1 + 31 = c#(c(c(c(c(c(x1)))))) c#(c(d(d(d(d(x1)))))) = x1 + 23 >= x1 + 7 = a#(x1) c#(c(d(d(d(d(x1)))))) = x1 + 23 >= x1 + 21 = a#(a(x1)) a(a(a(a(x1)))) = x1 + 56 >= x1 + 54 = b(b(b(b(b(b(x1)))))) b(b(b(b(x1)))) = x1 + 36 >= x1 + 36 = c(c(c(c(c(c(x1)))))) c(c(c(c(x1)))) = x1 + 24 >= x1 + 24 = d(d(d(d(d(d(x1)))))) b(b(x1)) = x1 + 18 >= x1 + 16 = d(d(d(d(x1)))) c(c(d(d(d(d(x1)))))) = x1 + 28 >= x1 + 28 = a(a(x1)) problem: DPs: a#(a(a(a(x1)))) -> b#(b(b(b(b(b(x1)))))) b#(b(b(b(x1)))) -> c#(c(c(c(c(c(x1)))))) TRS: a(a(a(a(x1)))) -> b(b(b(b(b(b(x1)))))) b(b(b(b(x1)))) -> c(c(c(c(c(c(x1)))))) c(c(c(c(x1)))) -> d(d(d(d(d(d(x1)))))) b(b(x1)) -> d(d(d(d(x1)))) c(c(d(d(d(d(x1)))))) -> a(a(x1)) Restore Modifier: DPs: a#(a(a(a(x1)))) -> b#(b(b(b(b(b(x1)))))) b#(b(b(b(x1)))) -> c#(c(c(c(c(c(x1)))))) TRS: a(a(a(a(x1)))) -> b(b(b(b(b(b(x1)))))) b(b(b(b(x1)))) -> c(c(c(c(c(c(x1)))))) c(c(c(c(x1)))) -> d(d(d(d(d(d(x1)))))) b(b(x1)) -> d(d(d(d(x1)))) c(c(d(d(d(d(x1)))))) -> a(a(x1)) EDG Processor: DPs: a#(a(a(a(x1)))) -> b#(b(b(b(b(b(x1)))))) b#(b(b(b(x1)))) -> c#(c(c(c(c(c(x1)))))) TRS: a(a(a(a(x1)))) -> b(b(b(b(b(b(x1)))))) b(b(b(b(x1)))) -> c(c(c(c(c(c(x1)))))) c(c(c(c(x1)))) -> d(d(d(d(d(d(x1)))))) b(b(x1)) -> d(d(d(d(x1)))) c(c(d(d(d(d(x1)))))) -> a(a(x1)) graph: a#(a(a(a(x1)))) -> b#(b(b(b(b(b(x1)))))) -> b#(b(b(b(x1)))) -> c#(c(c(c(c(c(x1)))))) SCC Processor: #sccs: 0 #rules: 0 #arcs: 1/4