YES Problem: a(s(x1)) -> s(a(x1)) b(a(b(s(x1)))) -> a(b(s(a(x1)))) b(a(b(b(x1)))) -> c(s(x1)) c(s(x1)) -> a(b(a(b(x1)))) a(b(a(a(x1)))) -> b(a(b(a(x1)))) Proof: DP Processor: DPs: a#(s(x1)) -> a#(x1) b#(a(b(s(x1)))) -> a#(x1) b#(a(b(s(x1)))) -> b#(s(a(x1))) b#(a(b(s(x1)))) -> a#(b(s(a(x1)))) b#(a(b(b(x1)))) -> c#(s(x1)) c#(s(x1)) -> b#(x1) c#(s(x1)) -> a#(b(x1)) c#(s(x1)) -> b#(a(b(x1))) c#(s(x1)) -> a#(b(a(b(x1)))) a#(b(a(a(x1)))) -> b#(a(x1)) a#(b(a(a(x1)))) -> a#(b(a(x1))) a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) TRS: a(s(x1)) -> s(a(x1)) b(a(b(s(x1)))) -> a(b(s(a(x1)))) b(a(b(b(x1)))) -> c(s(x1)) c(s(x1)) -> a(b(a(b(x1)))) a(b(a(a(x1)))) -> b(a(b(a(x1)))) TDG Processor: DPs: a#(s(x1)) -> a#(x1) b#(a(b(s(x1)))) -> a#(x1) b#(a(b(s(x1)))) -> b#(s(a(x1))) b#(a(b(s(x1)))) -> a#(b(s(a(x1)))) b#(a(b(b(x1)))) -> c#(s(x1)) c#(s(x1)) -> b#(x1) c#(s(x1)) -> a#(b(x1)) c#(s(x1)) -> b#(a(b(x1))) c#(s(x1)) -> a#(b(a(b(x1)))) a#(b(a(a(x1)))) -> b#(a(x1)) a#(b(a(a(x1)))) -> a#(b(a(x1))) a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) TRS: a(s(x1)) -> s(a(x1)) b(a(b(s(x1)))) -> a(b(s(a(x1)))) b(a(b(b(x1)))) -> c(s(x1)) c(s(x1)) -> a(b(a(b(x1)))) a(b(a(a(x1)))) -> b(a(b(a(x1)))) graph: c#(s(x1)) -> b#(a(b(x1))) -> b#(a(b(b(x1)))) -> c#(s(x1)) c#(s(x1)) -> b#(a(b(x1))) -> b#(a(b(s(x1)))) -> a#(b(s(a(x1)))) c#(s(x1)) -> b#(a(b(x1))) -> b#(a(b(s(x1)))) -> b#(s(a(x1))) c#(s(x1)) -> b#(a(b(x1))) -> b#(a(b(s(x1)))) -> a#(x1) c#(s(x1)) -> b#(x1) -> b#(a(b(b(x1)))) -> c#(s(x1)) c#(s(x1)) -> b#(x1) -> b#(a(b(s(x1)))) -> a#(b(s(a(x1)))) c#(s(x1)) -> b#(x1) -> b#(a(b(s(x1)))) -> b#(s(a(x1))) c#(s(x1)) -> b#(x1) -> b#(a(b(s(x1)))) -> a#(x1) c#(s(x1)) -> a#(b(a(b(x1)))) -> a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) c#(s(x1)) -> a#(b(a(b(x1)))) -> a#(b(a(a(x1)))) -> a#(b(a(x1))) c#(s(x1)) -> a#(b(a(b(x1)))) -> a#(b(a(a(x1)))) -> b#(a(x1)) c#(s(x1)) -> a#(b(a(b(x1)))) -> a#(s(x1)) -> a#(x1) c#(s(x1)) -> a#(b(x1)) -> a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) c#(s(x1)) -> a#(b(x1)) -> a#(b(a(a(x1)))) -> a#(b(a(x1))) c#(s(x1)) -> a#(b(x1)) -> a#(b(a(a(x1)))) -> b#(a(x1)) c#(s(x1)) -> a#(b(x1)) -> a#(s(x1)) -> a#(x1) b#(a(b(b(x1)))) -> c#(s(x1)) -> c#(s(x1)) -> a#(b(a(b(x1)))) b#(a(b(b(x1)))) -> c#(s(x1)) -> c#(s(x1)) -> b#(a(b(x1))) b#(a(b(b(x1)))) -> c#(s(x1)) -> c#(s(x1)) -> a#(b(x1)) b#(a(b(b(x1)))) -> c#(s(x1)) -> c#(s(x1)) -> b#(x1) b#(a(b(s(x1)))) -> b#(s(a(x1))) -> b#(a(b(b(x1)))) -> c#(s(x1)) b#(a(b(s(x1)))) -> b#(s(a(x1))) -> b#(a(b(s(x1)))) -> a#(b(s(a(x1)))) b#(a(b(s(x1)))) -> b#(s(a(x1))) -> b#(a(b(s(x1)))) -> b#(s(a(x1))) b#(a(b(s(x1)))) -> b#(s(a(x1))) -> b#(a(b(s(x1)))) -> a#(x1) b#(a(b(s(x1)))) -> a#(b(s(a(x1)))) -> a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) b#(a(b(s(x1)))) -> a#(b(s(a(x1)))) -> a#(b(a(a(x1)))) -> a#(b(a(x1))) b#(a(b(s(x1)))) -> a#(b(s(a(x1)))) -> a#(b(a(a(x1)))) -> b#(a(x1)) b#(a(b(s(x1)))) -> a#(b(s(a(x1)))) -> a#(s(x1)) -> a#(x1) b#(a(b(s(x1)))) -> a#(x1) -> a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) b#(a(b(s(x1)))) -> a#(x1) -> a#(b(a(a(x1)))) -> a#(b(a(x1))) b#(a(b(s(x1)))) -> a#(x1) -> a#(b(a(a(x1)))) -> b#(a(x1)) b#(a(b(s(x1)))) -> a#(x1) -> a#(s(x1)) -> a#(x1) a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) -> b#(a(b(b(x1)))) -> c#(s(x1)) a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) -> b#(a(b(s(x1)))) -> a#(b(s(a(x1)))) a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) -> b#(a(b(s(x1)))) -> b#(s(a(x1))) a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) -> b#(a(b(s(x1)))) -> a#(x1) a#(b(a(a(x1)))) -> b#(a(x1)) -> b#(a(b(b(x1)))) -> c#(s(x1)) a#(b(a(a(x1)))) -> b#(a(x1)) -> b#(a(b(s(x1)))) -> a#(b(s(a(x1)))) a#(b(a(a(x1)))) -> b#(a(x1)) -> b#(a(b(s(x1)))) -> b#(s(a(x1))) a#(b(a(a(x1)))) -> b#(a(x1)) -> b#(a(b(s(x1)))) -> a#(x1) a#(b(a(a(x1)))) -> a#(b(a(x1))) -> a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) a#(b(a(a(x1)))) -> a#(b(a(x1))) -> a#(b(a(a(x1)))) -> a#(b(a(x1))) a#(b(a(a(x1)))) -> a#(b(a(x1))) -> a#(b(a(a(x1)))) -> b#(a(x1)) a#(b(a(a(x1)))) -> a#(b(a(x1))) -> a#(s(x1)) -> a#(x1) a#(s(x1)) -> a#(x1) -> a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) a#(s(x1)) -> a#(x1) -> a#(b(a(a(x1)))) -> a#(b(a(x1))) a#(s(x1)) -> a#(x1) -> a#(b(a(a(x1)))) -> b#(a(x1)) a#(s(x1)) -> a#(x1) -> a#(s(x1)) -> a#(x1) EDG Processor: DPs: a#(s(x1)) -> a#(x1) b#(a(b(s(x1)))) -> a#(x1) b#(a(b(s(x1)))) -> b#(s(a(x1))) b#(a(b(s(x1)))) -> a#(b(s(a(x1)))) b#(a(b(b(x1)))) -> c#(s(x1)) c#(s(x1)) -> b#(x1) c#(s(x1)) -> a#(b(x1)) c#(s(x1)) -> b#(a(b(x1))) c#(s(x1)) -> a#(b(a(b(x1)))) a#(b(a(a(x1)))) -> b#(a(x1)) a#(b(a(a(x1)))) -> a#(b(a(x1))) a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) TRS: a(s(x1)) -> s(a(x1)) b(a(b(s(x1)))) -> a(b(s(a(x1)))) b(a(b(b(x1)))) -> c(s(x1)) c(s(x1)) -> a(b(a(b(x1)))) a(b(a(a(x1)))) -> b(a(b(a(x1)))) graph: c#(s(x1)) -> b#(a(b(x1))) -> b#(a(b(s(x1)))) -> a#(x1) c#(s(x1)) -> b#(a(b(x1))) -> b#(a(b(s(x1)))) -> b#(s(a(x1))) c#(s(x1)) -> b#(a(b(x1))) -> b#(a(b(s(x1)))) -> a#(b(s(a(x1)))) c#(s(x1)) -> b#(a(b(x1))) -> b#(a(b(b(x1)))) -> c#(s(x1)) c#(s(x1)) -> b#(x1) -> b#(a(b(s(x1)))) -> a#(x1) c#(s(x1)) -> b#(x1) -> b#(a(b(s(x1)))) -> b#(s(a(x1))) c#(s(x1)) -> b#(x1) -> b#(a(b(s(x1)))) -> a#(b(s(a(x1)))) c#(s(x1)) -> b#(x1) -> b#(a(b(b(x1)))) -> c#(s(x1)) c#(s(x1)) -> a#(b(a(b(x1)))) -> a#(s(x1)) -> a#(x1) c#(s(x1)) -> a#(b(a(b(x1)))) -> a#(b(a(a(x1)))) -> b#(a(x1)) c#(s(x1)) -> a#(b(a(b(x1)))) -> a#(b(a(a(x1)))) -> a#(b(a(x1))) c#(s(x1)) -> a#(b(a(b(x1)))) -> a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) c#(s(x1)) -> a#(b(x1)) -> a#(s(x1)) -> a#(x1) c#(s(x1)) -> a#(b(x1)) -> a#(b(a(a(x1)))) -> b#(a(x1)) c#(s(x1)) -> a#(b(x1)) -> a#(b(a(a(x1)))) -> a#(b(a(x1))) c#(s(x1)) -> a#(b(x1)) -> a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) b#(a(b(b(x1)))) -> c#(s(x1)) -> c#(s(x1)) -> b#(x1) b#(a(b(b(x1)))) -> c#(s(x1)) -> c#(s(x1)) -> a#(b(x1)) b#(a(b(b(x1)))) -> c#(s(x1)) -> c#(s(x1)) -> b#(a(b(x1))) b#(a(b(b(x1)))) -> c#(s(x1)) -> c#(s(x1)) -> a#(b(a(b(x1)))) b#(a(b(s(x1)))) -> a#(x1) -> a#(s(x1)) -> a#(x1) b#(a(b(s(x1)))) -> a#(x1) -> a#(b(a(a(x1)))) -> b#(a(x1)) b#(a(b(s(x1)))) -> a#(x1) -> a#(b(a(a(x1)))) -> a#(b(a(x1))) b#(a(b(s(x1)))) -> a#(x1) -> a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) -> b#(a(b(s(x1)))) -> a#(x1) a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) -> b#(a(b(s(x1)))) -> b#(s(a(x1))) a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) -> b#(a(b(s(x1)))) -> a#(b(s(a(x1)))) a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) -> b#(a(b(b(x1)))) -> c#(s(x1)) a#(b(a(a(x1)))) -> b#(a(x1)) -> b#(a(b(s(x1)))) -> a#(x1) a#(b(a(a(x1)))) -> b#(a(x1)) -> b#(a(b(s(x1)))) -> b#(s(a(x1))) a#(b(a(a(x1)))) -> b#(a(x1)) -> b#(a(b(s(x1)))) -> a#(b(s(a(x1)))) a#(b(a(a(x1)))) -> b#(a(x1)) -> b#(a(b(b(x1)))) -> c#(s(x1)) a#(b(a(a(x1)))) -> a#(b(a(x1))) -> a#(s(x1)) -> a#(x1) a#(b(a(a(x1)))) -> a#(b(a(x1))) -> a#(b(a(a(x1)))) -> b#(a(x1)) a#(b(a(a(x1)))) -> a#(b(a(x1))) -> a#(b(a(a(x1)))) -> a#(b(a(x1))) a#(b(a(a(x1)))) -> a#(b(a(x1))) -> a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) a#(s(x1)) -> a#(x1) -> a#(s(x1)) -> a#(x1) a#(s(x1)) -> a#(x1) -> a#(b(a(a(x1)))) -> b#(a(x1)) a#(s(x1)) -> a#(x1) -> a#(b(a(a(x1)))) -> a#(b(a(x1))) a#(s(x1)) -> a#(x1) -> a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) SCC Processor: #sccs: 1 #rules: 10 #arcs: 40/144 DPs: c#(s(x1)) -> b#(a(b(x1))) b#(a(b(b(x1)))) -> c#(s(x1)) c#(s(x1)) -> a#(b(a(b(x1)))) a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) b#(a(b(s(x1)))) -> a#(x1) a#(b(a(a(x1)))) -> a#(b(a(x1))) a#(b(a(a(x1)))) -> b#(a(x1)) a#(s(x1)) -> a#(x1) c#(s(x1)) -> a#(b(x1)) c#(s(x1)) -> b#(x1) TRS: a(s(x1)) -> s(a(x1)) b(a(b(s(x1)))) -> a(b(s(a(x1)))) b(a(b(b(x1)))) -> c(s(x1)) c(s(x1)) -> a(b(a(b(x1)))) a(b(a(a(x1)))) -> b(a(b(a(x1)))) Arctic Interpretation Processor: dimension: 1 interpretation: [c#](x0) = 6x0, [b#](x0) = x0, [a#](x0) = x0, [c](x0) = 8x0, [b](x0) = 2x0, [a](x0) = 2x0, [s](x0) = x0 orientation: c#(s(x1)) = 6x1 >= 4x1 = b#(a(b(x1))) b#(a(b(b(x1)))) = 6x1 >= 6x1 = c#(s(x1)) c#(s(x1)) = 6x1 >= 6x1 = a#(b(a(b(x1)))) a#(b(a(a(x1)))) = 6x1 >= 6x1 = b#(a(b(a(x1)))) b#(a(b(s(x1)))) = 4x1 >= x1 = a#(x1) a#(b(a(a(x1)))) = 6x1 >= 4x1 = a#(b(a(x1))) a#(b(a(a(x1)))) = 6x1 >= 2x1 = b#(a(x1)) a#(s(x1)) = x1 >= x1 = a#(x1) c#(s(x1)) = 6x1 >= 2x1 = a#(b(x1)) c#(s(x1)) = 6x1 >= x1 = b#(x1) a(s(x1)) = 2x1 >= 2x1 = s(a(x1)) b(a(b(s(x1)))) = 6x1 >= 6x1 = a(b(s(a(x1)))) b(a(b(b(x1)))) = 8x1 >= 8x1 = c(s(x1)) c(s(x1)) = 8x1 >= 8x1 = a(b(a(b(x1)))) a(b(a(a(x1)))) = 8x1 >= 8x1 = b(a(b(a(x1)))) problem: DPs: b#(a(b(b(x1)))) -> c#(s(x1)) c#(s(x1)) -> a#(b(a(b(x1)))) a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) a#(s(x1)) -> a#(x1) TRS: a(s(x1)) -> s(a(x1)) b(a(b(s(x1)))) -> a(b(s(a(x1)))) b(a(b(b(x1)))) -> c(s(x1)) c(s(x1)) -> a(b(a(b(x1)))) a(b(a(a(x1)))) -> b(a(b(a(x1)))) EDG Processor: DPs: b#(a(b(b(x1)))) -> c#(s(x1)) c#(s(x1)) -> a#(b(a(b(x1)))) a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) a#(s(x1)) -> a#(x1) TRS: a(s(x1)) -> s(a(x1)) b(a(b(s(x1)))) -> a(b(s(a(x1)))) b(a(b(b(x1)))) -> c(s(x1)) c(s(x1)) -> a(b(a(b(x1)))) a(b(a(a(x1)))) -> b(a(b(a(x1)))) graph: c#(s(x1)) -> a#(b(a(b(x1)))) -> a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) c#(s(x1)) -> a#(b(a(b(x1)))) -> a#(s(x1)) -> a#(x1) b#(a(b(b(x1)))) -> c#(s(x1)) -> c#(s(x1)) -> a#(b(a(b(x1)))) a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) -> b#(a(b(b(x1)))) -> c#(s(x1)) a#(s(x1)) -> a#(x1) -> a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) a#(s(x1)) -> a#(x1) -> a#(s(x1)) -> a#(x1) CDG Processor: DPs: b#(a(b(b(x1)))) -> c#(s(x1)) c#(s(x1)) -> a#(b(a(b(x1)))) a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) a#(s(x1)) -> a#(x1) TRS: a(s(x1)) -> s(a(x1)) b(a(b(s(x1)))) -> a(b(s(a(x1)))) b(a(b(b(x1)))) -> c(s(x1)) c(s(x1)) -> a(b(a(b(x1)))) a(b(a(a(x1)))) -> b(a(b(a(x1)))) graph: c#(s(x1)) -> a#(b(a(b(x1)))) -> a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) b#(a(b(b(x1)))) -> c#(s(x1)) -> c#(s(x1)) -> a#(b(a(b(x1)))) a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) -> b#(a(b(b(x1)))) -> c#(s(x1)) a#(s(x1)) -> a#(x1) -> a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) SCC Processor: #sccs: 1 #rules: 3 #arcs: 4/16 DPs: c#(s(x1)) -> a#(b(a(b(x1)))) a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) b#(a(b(b(x1)))) -> c#(s(x1)) TRS: a(s(x1)) -> s(a(x1)) b(a(b(s(x1)))) -> a(b(s(a(x1)))) b(a(b(b(x1)))) -> c(s(x1)) c(s(x1)) -> a(b(a(b(x1)))) a(b(a(a(x1)))) -> b(a(b(a(x1)))) Matrix Interpretation Processor: dim=3 interpretation: [c#](x0) = [0 1 1]x0 + [1], [b#](x0) = [1 0 0]x0 + [1], [a#](x0) = [0 0 1]x0, [0 1 1] [1] [c](x0) = [0 1 0]x0 + [1] [0 1 1] [1], [0 0 1] [0] [b](x0) = [0 1 0]x0 + [0] [0 0 1] [1], [1 0 0] [1] [a](x0) = [1 0 0]x0 + [0] [1 0 0] [1], [1 0 0] [1] [s](x0) = [0 0 1]x0 + [0] [0 0 0] [1] orientation: c#(s(x1)) = [0 0 1]x1 + [2] >= [0 0 1]x1 + [2] = a#(b(a(b(x1)))) a#(b(a(a(x1)))) = [1 0 0]x1 + [3] >= [1 0 0]x1 + [3] = b#(a(b(a(x1)))) b#(a(b(b(x1)))) = [0 0 1]x1 + [3] >= [0 0 1]x1 + [2] = c#(s(x1)) [1 0 0] [2] [1 0 0] [2] a(s(x1)) = [1 0 0]x1 + [1] >= [1 0 0]x1 + [1] = s(a(x1)) [1 0 0] [2] [0 0 0] [1] [2] [2] b(a(b(s(x1)))) = [1] >= [1] = a(b(s(a(x1)))) [3] [2] [0 0 1] [2] [0 0 1] [2] b(a(b(b(x1)))) = [0 0 1]x1 + [1] >= [0 0 1]x1 + [1] = c(s(x1)) [0 0 1] [3] [0 0 1] [2] [0 0 1] [2] [0 0 1] [2] c(s(x1)) = [0 0 1]x1 + [1] >= [0 0 1]x1 + [1] = a(b(a(b(x1)))) [0 0 1] [2] [0 0 1] [2] [1 0 0] [3] [1 0 0] [2] a(b(a(a(x1)))) = [1 0 0]x1 + [2] >= [1 0 0]x1 + [1] = b(a(b(a(x1)))) [1 0 0] [3] [1 0 0] [3] problem: DPs: c#(s(x1)) -> a#(b(a(b(x1)))) a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) TRS: a(s(x1)) -> s(a(x1)) b(a(b(s(x1)))) -> a(b(s(a(x1)))) b(a(b(b(x1)))) -> c(s(x1)) c(s(x1)) -> a(b(a(b(x1)))) a(b(a(a(x1)))) -> b(a(b(a(x1)))) EDG Processor: DPs: c#(s(x1)) -> a#(b(a(b(x1)))) a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) TRS: a(s(x1)) -> s(a(x1)) b(a(b(s(x1)))) -> a(b(s(a(x1)))) b(a(b(b(x1)))) -> c(s(x1)) c(s(x1)) -> a(b(a(b(x1)))) a(b(a(a(x1)))) -> b(a(b(a(x1)))) graph: c#(s(x1)) -> a#(b(a(b(x1)))) -> a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) SCC Processor: #sccs: 0 #rules: 0 #arcs: 1/4