YES Problem: d(a(x1)) -> b(d(x1)) b(x1) -> a(a(a(x1))) c(d(c(x1))) -> a(d(x1)) b(d(d(x1))) -> c(c(d(d(c(x1))))) Proof: DP Processor: DPs: d#(a(x1)) -> d#(x1) d#(a(x1)) -> b#(d(x1)) c#(d(c(x1))) -> d#(x1) b#(d(d(x1))) -> c#(x1) b#(d(d(x1))) -> d#(c(x1)) b#(d(d(x1))) -> d#(d(c(x1))) b#(d(d(x1))) -> c#(d(d(c(x1)))) b#(d(d(x1))) -> c#(c(d(d(c(x1))))) TRS: d(a(x1)) -> b(d(x1)) b(x1) -> a(a(a(x1))) c(d(c(x1))) -> a(d(x1)) b(d(d(x1))) -> c(c(d(d(c(x1))))) TDG Processor: DPs: d#(a(x1)) -> d#(x1) d#(a(x1)) -> b#(d(x1)) c#(d(c(x1))) -> d#(x1) b#(d(d(x1))) -> c#(x1) b#(d(d(x1))) -> d#(c(x1)) b#(d(d(x1))) -> d#(d(c(x1))) b#(d(d(x1))) -> c#(d(d(c(x1)))) b#(d(d(x1))) -> c#(c(d(d(c(x1))))) TRS: d(a(x1)) -> b(d(x1)) b(x1) -> a(a(a(x1))) c(d(c(x1))) -> a(d(x1)) b(d(d(x1))) -> c(c(d(d(c(x1))))) graph: c#(d(c(x1))) -> d#(x1) -> d#(a(x1)) -> b#(d(x1)) c#(d(c(x1))) -> d#(x1) -> d#(a(x1)) -> d#(x1) b#(d(d(x1))) -> c#(c(d(d(c(x1))))) -> c#(d(c(x1))) -> d#(x1) b#(d(d(x1))) -> c#(d(d(c(x1)))) -> c#(d(c(x1))) -> d#(x1) b#(d(d(x1))) -> c#(x1) -> c#(d(c(x1))) -> d#(x1) b#(d(d(x1))) -> d#(c(x1)) -> d#(a(x1)) -> b#(d(x1)) b#(d(d(x1))) -> d#(c(x1)) -> d#(a(x1)) -> d#(x1) b#(d(d(x1))) -> d#(d(c(x1))) -> d#(a(x1)) -> b#(d(x1)) b#(d(d(x1))) -> d#(d(c(x1))) -> d#(a(x1)) -> d#(x1) d#(a(x1)) -> b#(d(x1)) -> b#(d(d(x1))) -> c#(c(d(d(c(x1))))) d#(a(x1)) -> b#(d(x1)) -> b#(d(d(x1))) -> c#(d(d(c(x1)))) d#(a(x1)) -> b#(d(x1)) -> b#(d(d(x1))) -> d#(d(c(x1))) d#(a(x1)) -> b#(d(x1)) -> b#(d(d(x1))) -> d#(c(x1)) d#(a(x1)) -> b#(d(x1)) -> b#(d(d(x1))) -> c#(x1) d#(a(x1)) -> d#(x1) -> d#(a(x1)) -> b#(d(x1)) d#(a(x1)) -> d#(x1) -> d#(a(x1)) -> d#(x1) EDG Processor: DPs: d#(a(x1)) -> d#(x1) d#(a(x1)) -> b#(d(x1)) c#(d(c(x1))) -> d#(x1) b#(d(d(x1))) -> c#(x1) b#(d(d(x1))) -> d#(c(x1)) b#(d(d(x1))) -> d#(d(c(x1))) b#(d(d(x1))) -> c#(d(d(c(x1)))) b#(d(d(x1))) -> c#(c(d(d(c(x1))))) TRS: d(a(x1)) -> b(d(x1)) b(x1) -> a(a(a(x1))) c(d(c(x1))) -> a(d(x1)) b(d(d(x1))) -> c(c(d(d(c(x1))))) graph: c#(d(c(x1))) -> d#(x1) -> d#(a(x1)) -> d#(x1) c#(d(c(x1))) -> d#(x1) -> d#(a(x1)) -> b#(d(x1)) b#(d(d(x1))) -> c#(d(d(c(x1)))) -> c#(d(c(x1))) -> d#(x1) b#(d(d(x1))) -> c#(x1) -> c#(d(c(x1))) -> d#(x1) b#(d(d(x1))) -> d#(c(x1)) -> d#(a(x1)) -> d#(x1) b#(d(d(x1))) -> d#(c(x1)) -> d#(a(x1)) -> b#(d(x1)) b#(d(d(x1))) -> d#(d(c(x1))) -> d#(a(x1)) -> d#(x1) b#(d(d(x1))) -> d#(d(c(x1))) -> d#(a(x1)) -> b#(d(x1)) d#(a(x1)) -> b#(d(x1)) -> b#(d(d(x1))) -> c#(x1) d#(a(x1)) -> b#(d(x1)) -> b#(d(d(x1))) -> d#(c(x1)) d#(a(x1)) -> b#(d(x1)) -> b#(d(d(x1))) -> d#(d(c(x1))) d#(a(x1)) -> b#(d(x1)) -> b#(d(d(x1))) -> c#(d(d(c(x1)))) d#(a(x1)) -> b#(d(x1)) -> b#(d(d(x1))) -> c#(c(d(d(c(x1))))) d#(a(x1)) -> d#(x1) -> d#(a(x1)) -> d#(x1) d#(a(x1)) -> d#(x1) -> d#(a(x1)) -> b#(d(x1)) SCC Processor: #sccs: 1 #rules: 7 #arcs: 15/64 DPs: c#(d(c(x1))) -> d#(x1) d#(a(x1)) -> b#(d(x1)) b#(d(d(x1))) -> c#(d(d(c(x1)))) b#(d(d(x1))) -> d#(d(c(x1))) d#(a(x1)) -> d#(x1) b#(d(d(x1))) -> d#(c(x1)) b#(d(d(x1))) -> c#(x1) TRS: d(a(x1)) -> b(d(x1)) b(x1) -> a(a(a(x1))) c(d(c(x1))) -> a(d(x1)) b(d(d(x1))) -> c(c(d(d(c(x1))))) Arctic Interpretation Processor: dimension: 1 interpretation: [c#](x0) = 4x0, [b#](x0) = 4x0, [d#](x0) = 5x0, [c](x0) = x0, [b](x0) = x0, [d](x0) = 1x0, [a](x0) = x0 orientation: c#(d(c(x1))) = 5x1 >= 5x1 = d#(x1) d#(a(x1)) = 5x1 >= 5x1 = b#(d(x1)) b#(d(d(x1))) = 6x1 >= 6x1 = c#(d(d(c(x1)))) b#(d(d(x1))) = 6x1 >= 6x1 = d#(d(c(x1))) d#(a(x1)) = 5x1 >= 5x1 = d#(x1) b#(d(d(x1))) = 6x1 >= 5x1 = d#(c(x1)) b#(d(d(x1))) = 6x1 >= 4x1 = c#(x1) d(a(x1)) = 1x1 >= 1x1 = b(d(x1)) b(x1) = x1 >= x1 = a(a(a(x1))) c(d(c(x1))) = 1x1 >= 1x1 = a(d(x1)) b(d(d(x1))) = 2x1 >= 2x1 = c(c(d(d(c(x1))))) problem: DPs: c#(d(c(x1))) -> d#(x1) d#(a(x1)) -> b#(d(x1)) b#(d(d(x1))) -> c#(d(d(c(x1)))) b#(d(d(x1))) -> d#(d(c(x1))) d#(a(x1)) -> d#(x1) TRS: d(a(x1)) -> b(d(x1)) b(x1) -> a(a(a(x1))) c(d(c(x1))) -> a(d(x1)) b(d(d(x1))) -> c(c(d(d(c(x1))))) EDG Processor: DPs: c#(d(c(x1))) -> d#(x1) d#(a(x1)) -> b#(d(x1)) b#(d(d(x1))) -> c#(d(d(c(x1)))) b#(d(d(x1))) -> d#(d(c(x1))) d#(a(x1)) -> d#(x1) TRS: d(a(x1)) -> b(d(x1)) b(x1) -> a(a(a(x1))) c(d(c(x1))) -> a(d(x1)) b(d(d(x1))) -> c(c(d(d(c(x1))))) graph: c#(d(c(x1))) -> d#(x1) -> d#(a(x1)) -> b#(d(x1)) c#(d(c(x1))) -> d#(x1) -> d#(a(x1)) -> d#(x1) b#(d(d(x1))) -> c#(d(d(c(x1)))) -> c#(d(c(x1))) -> d#(x1) b#(d(d(x1))) -> d#(d(c(x1))) -> d#(a(x1)) -> b#(d(x1)) b#(d(d(x1))) -> d#(d(c(x1))) -> d#(a(x1)) -> d#(x1) d#(a(x1)) -> b#(d(x1)) -> b#(d(d(x1))) -> c#(d(d(c(x1)))) d#(a(x1)) -> b#(d(x1)) -> b#(d(d(x1))) -> d#(d(c(x1))) d#(a(x1)) -> d#(x1) -> d#(a(x1)) -> b#(d(x1)) d#(a(x1)) -> d#(x1) -> d#(a(x1)) -> d#(x1) Matrix Interpretation Processor: dim=1 interpretation: [c#](x0) = x0, [b#](x0) = x0 + 16, [d#](x0) = 4x0, [c](x0) = x0 + 1, [b](x0) = x0 + 20, [d](x0) = 4x0, [a](x0) = x0 + 5 orientation: c#(d(c(x1))) = 4x1 + 4 >= 4x1 = d#(x1) d#(a(x1)) = 4x1 + 20 >= 4x1 + 16 = b#(d(x1)) b#(d(d(x1))) = 16x1 + 16 >= 16x1 + 16 = c#(d(d(c(x1)))) b#(d(d(x1))) = 16x1 + 16 >= 16x1 + 16 = d#(d(c(x1))) d#(a(x1)) = 4x1 + 20 >= 4x1 = d#(x1) d(a(x1)) = 4x1 + 20 >= 4x1 + 20 = b(d(x1)) b(x1) = x1 + 20 >= x1 + 15 = a(a(a(x1))) c(d(c(x1))) = 4x1 + 5 >= 4x1 + 5 = a(d(x1)) b(d(d(x1))) = 16x1 + 20 >= 16x1 + 18 = c(c(d(d(c(x1))))) problem: DPs: b#(d(d(x1))) -> c#(d(d(c(x1)))) b#(d(d(x1))) -> d#(d(c(x1))) TRS: d(a(x1)) -> b(d(x1)) b(x1) -> a(a(a(x1))) c(d(c(x1))) -> a(d(x1)) b(d(d(x1))) -> c(c(d(d(c(x1))))) EDG Processor: DPs: b#(d(d(x1))) -> c#(d(d(c(x1)))) b#(d(d(x1))) -> d#(d(c(x1))) TRS: d(a(x1)) -> b(d(x1)) b(x1) -> a(a(a(x1))) c(d(c(x1))) -> a(d(x1)) b(d(d(x1))) -> c(c(d(d(c(x1))))) graph: Qed