YES Problem: b(a(a(x1))) -> a(b(c(x1))) c(a(x1)) -> a(c(x1)) b(c(a(x1))) -> a(b(c(x1))) c(b(x1)) -> d(x1) d(x1) -> b(a(x1)) a(d(x1)) -> d(a(x1)) a(a(x1)) -> a(b(a(x1))) Proof: DP Processor: DPs: b#(a(a(x1))) -> c#(x1) b#(a(a(x1))) -> b#(c(x1)) b#(a(a(x1))) -> a#(b(c(x1))) c#(a(x1)) -> c#(x1) c#(a(x1)) -> a#(c(x1)) b#(c(a(x1))) -> c#(x1) b#(c(a(x1))) -> b#(c(x1)) b#(c(a(x1))) -> a#(b(c(x1))) c#(b(x1)) -> d#(x1) d#(x1) -> a#(x1) d#(x1) -> b#(a(x1)) a#(d(x1)) -> a#(x1) a#(d(x1)) -> d#(a(x1)) a#(a(x1)) -> b#(a(x1)) a#(a(x1)) -> a#(b(a(x1))) TRS: b(a(a(x1))) -> a(b(c(x1))) c(a(x1)) -> a(c(x1)) b(c(a(x1))) -> a(b(c(x1))) c(b(x1)) -> d(x1) d(x1) -> b(a(x1)) a(d(x1)) -> d(a(x1)) a(a(x1)) -> a(b(a(x1))) TDG Processor: DPs: b#(a(a(x1))) -> c#(x1) b#(a(a(x1))) -> b#(c(x1)) b#(a(a(x1))) -> a#(b(c(x1))) c#(a(x1)) -> c#(x1) c#(a(x1)) -> a#(c(x1)) b#(c(a(x1))) -> c#(x1) b#(c(a(x1))) -> b#(c(x1)) b#(c(a(x1))) -> a#(b(c(x1))) c#(b(x1)) -> d#(x1) d#(x1) -> a#(x1) d#(x1) -> b#(a(x1)) a#(d(x1)) -> a#(x1) a#(d(x1)) -> d#(a(x1)) a#(a(x1)) -> b#(a(x1)) a#(a(x1)) -> a#(b(a(x1))) TRS: b(a(a(x1))) -> a(b(c(x1))) c(a(x1)) -> a(c(x1)) b(c(a(x1))) -> a(b(c(x1))) c(b(x1)) -> d(x1) d(x1) -> b(a(x1)) a(d(x1)) -> d(a(x1)) a(a(x1)) -> a(b(a(x1))) graph: d#(x1) -> a#(x1) -> a#(a(x1)) -> a#(b(a(x1))) d#(x1) -> a#(x1) -> a#(a(x1)) -> b#(a(x1)) d#(x1) -> a#(x1) -> a#(d(x1)) -> d#(a(x1)) d#(x1) -> a#(x1) -> a#(d(x1)) -> a#(x1) d#(x1) -> b#(a(x1)) -> b#(c(a(x1))) -> a#(b(c(x1))) d#(x1) -> b#(a(x1)) -> b#(c(a(x1))) -> b#(c(x1)) d#(x1) -> b#(a(x1)) -> b#(c(a(x1))) -> c#(x1) d#(x1) -> b#(a(x1)) -> b#(a(a(x1))) -> a#(b(c(x1))) d#(x1) -> b#(a(x1)) -> b#(a(a(x1))) -> b#(c(x1)) d#(x1) -> b#(a(x1)) -> b#(a(a(x1))) -> c#(x1) a#(d(x1)) -> d#(a(x1)) -> d#(x1) -> b#(a(x1)) a#(d(x1)) -> d#(a(x1)) -> d#(x1) -> a#(x1) a#(d(x1)) -> a#(x1) -> a#(a(x1)) -> a#(b(a(x1))) a#(d(x1)) -> a#(x1) -> a#(a(x1)) -> b#(a(x1)) a#(d(x1)) -> a#(x1) -> a#(d(x1)) -> d#(a(x1)) a#(d(x1)) -> a#(x1) -> a#(d(x1)) -> a#(x1) a#(a(x1)) -> a#(b(a(x1))) -> a#(a(x1)) -> a#(b(a(x1))) a#(a(x1)) -> a#(b(a(x1))) -> a#(a(x1)) -> b#(a(x1)) a#(a(x1)) -> a#(b(a(x1))) -> a#(d(x1)) -> d#(a(x1)) a#(a(x1)) -> a#(b(a(x1))) -> a#(d(x1)) -> a#(x1) a#(a(x1)) -> b#(a(x1)) -> b#(c(a(x1))) -> a#(b(c(x1))) a#(a(x1)) -> b#(a(x1)) -> b#(c(a(x1))) -> b#(c(x1)) a#(a(x1)) -> b#(a(x1)) -> b#(c(a(x1))) -> c#(x1) a#(a(x1)) -> b#(a(x1)) -> b#(a(a(x1))) -> a#(b(c(x1))) a#(a(x1)) -> b#(a(x1)) -> b#(a(a(x1))) -> b#(c(x1)) a#(a(x1)) -> b#(a(x1)) -> b#(a(a(x1))) -> c#(x1) c#(b(x1)) -> d#(x1) -> d#(x1) -> b#(a(x1)) c#(b(x1)) -> d#(x1) -> d#(x1) -> a#(x1) c#(a(x1)) -> a#(c(x1)) -> a#(a(x1)) -> a#(b(a(x1))) c#(a(x1)) -> a#(c(x1)) -> a#(a(x1)) -> b#(a(x1)) c#(a(x1)) -> a#(c(x1)) -> a#(d(x1)) -> d#(a(x1)) c#(a(x1)) -> a#(c(x1)) -> a#(d(x1)) -> a#(x1) c#(a(x1)) -> c#(x1) -> c#(b(x1)) -> d#(x1) c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> a#(c(x1)) c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> c#(x1) b#(c(a(x1))) -> a#(b(c(x1))) -> a#(a(x1)) -> a#(b(a(x1))) b#(c(a(x1))) -> a#(b(c(x1))) -> a#(a(x1)) -> b#(a(x1)) b#(c(a(x1))) -> a#(b(c(x1))) -> a#(d(x1)) -> d#(a(x1)) b#(c(a(x1))) -> a#(b(c(x1))) -> a#(d(x1)) -> a#(x1) b#(c(a(x1))) -> c#(x1) -> c#(b(x1)) -> d#(x1) b#(c(a(x1))) -> c#(x1) -> c#(a(x1)) -> a#(c(x1)) b#(c(a(x1))) -> c#(x1) -> c#(a(x1)) -> c#(x1) b#(c(a(x1))) -> b#(c(x1)) -> b#(c(a(x1))) -> a#(b(c(x1))) b#(c(a(x1))) -> b#(c(x1)) -> b#(c(a(x1))) -> b#(c(x1)) b#(c(a(x1))) -> b#(c(x1)) -> b#(c(a(x1))) -> c#(x1) b#(c(a(x1))) -> b#(c(x1)) -> b#(a(a(x1))) -> a#(b(c(x1))) b#(c(a(x1))) -> b#(c(x1)) -> b#(a(a(x1))) -> b#(c(x1)) b#(c(a(x1))) -> b#(c(x1)) -> b#(a(a(x1))) -> c#(x1) b#(a(a(x1))) -> a#(b(c(x1))) -> a#(a(x1)) -> a#(b(a(x1))) b#(a(a(x1))) -> a#(b(c(x1))) -> a#(a(x1)) -> b#(a(x1)) b#(a(a(x1))) -> a#(b(c(x1))) -> a#(d(x1)) -> d#(a(x1)) b#(a(a(x1))) -> a#(b(c(x1))) -> a#(d(x1)) -> a#(x1) b#(a(a(x1))) -> c#(x1) -> c#(b(x1)) -> d#(x1) b#(a(a(x1))) -> c#(x1) -> c#(a(x1)) -> a#(c(x1)) b#(a(a(x1))) -> c#(x1) -> c#(a(x1)) -> c#(x1) b#(a(a(x1))) -> b#(c(x1)) -> b#(c(a(x1))) -> a#(b(c(x1))) b#(a(a(x1))) -> b#(c(x1)) -> b#(c(a(x1))) -> b#(c(x1)) b#(a(a(x1))) -> b#(c(x1)) -> b#(c(a(x1))) -> c#(x1) b#(a(a(x1))) -> b#(c(x1)) -> b#(a(a(x1))) -> a#(b(c(x1))) b#(a(a(x1))) -> b#(c(x1)) -> b#(a(a(x1))) -> b#(c(x1)) b#(a(a(x1))) -> b#(c(x1)) -> b#(a(a(x1))) -> c#(x1) EDG Processor: DPs: b#(a(a(x1))) -> c#(x1) b#(a(a(x1))) -> b#(c(x1)) b#(a(a(x1))) -> a#(b(c(x1))) c#(a(x1)) -> c#(x1) c#(a(x1)) -> a#(c(x1)) b#(c(a(x1))) -> c#(x1) b#(c(a(x1))) -> b#(c(x1)) b#(c(a(x1))) -> a#(b(c(x1))) c#(b(x1)) -> d#(x1) d#(x1) -> a#(x1) d#(x1) -> b#(a(x1)) a#(d(x1)) -> a#(x1) a#(d(x1)) -> d#(a(x1)) a#(a(x1)) -> b#(a(x1)) a#(a(x1)) -> a#(b(a(x1))) TRS: b(a(a(x1))) -> a(b(c(x1))) c(a(x1)) -> a(c(x1)) b(c(a(x1))) -> a(b(c(x1))) c(b(x1)) -> d(x1) d(x1) -> b(a(x1)) a(d(x1)) -> d(a(x1)) a(a(x1)) -> a(b(a(x1))) graph: d#(x1) -> a#(x1) -> a#(d(x1)) -> a#(x1) d#(x1) -> a#(x1) -> a#(d(x1)) -> d#(a(x1)) d#(x1) -> a#(x1) -> a#(a(x1)) -> b#(a(x1)) d#(x1) -> a#(x1) -> a#(a(x1)) -> a#(b(a(x1))) d#(x1) -> b#(a(x1)) -> b#(a(a(x1))) -> c#(x1) d#(x1) -> b#(a(x1)) -> b#(a(a(x1))) -> b#(c(x1)) d#(x1) -> b#(a(x1)) -> b#(a(a(x1))) -> a#(b(c(x1))) a#(d(x1)) -> d#(a(x1)) -> d#(x1) -> a#(x1) a#(d(x1)) -> d#(a(x1)) -> d#(x1) -> b#(a(x1)) a#(d(x1)) -> a#(x1) -> a#(d(x1)) -> a#(x1) a#(d(x1)) -> a#(x1) -> a#(d(x1)) -> d#(a(x1)) a#(d(x1)) -> a#(x1) -> a#(a(x1)) -> b#(a(x1)) a#(d(x1)) -> a#(x1) -> a#(a(x1)) -> a#(b(a(x1))) a#(a(x1)) -> a#(b(a(x1))) -> a#(d(x1)) -> a#(x1) a#(a(x1)) -> a#(b(a(x1))) -> a#(d(x1)) -> d#(a(x1)) a#(a(x1)) -> a#(b(a(x1))) -> a#(a(x1)) -> b#(a(x1)) a#(a(x1)) -> a#(b(a(x1))) -> a#(a(x1)) -> a#(b(a(x1))) a#(a(x1)) -> b#(a(x1)) -> b#(a(a(x1))) -> c#(x1) a#(a(x1)) -> b#(a(x1)) -> b#(a(a(x1))) -> b#(c(x1)) a#(a(x1)) -> b#(a(x1)) -> b#(a(a(x1))) -> a#(b(c(x1))) c#(b(x1)) -> d#(x1) -> d#(x1) -> a#(x1) c#(b(x1)) -> d#(x1) -> d#(x1) -> b#(a(x1)) c#(a(x1)) -> a#(c(x1)) -> a#(d(x1)) -> a#(x1) c#(a(x1)) -> a#(c(x1)) -> a#(d(x1)) -> d#(a(x1)) c#(a(x1)) -> a#(c(x1)) -> a#(a(x1)) -> b#(a(x1)) c#(a(x1)) -> a#(c(x1)) -> a#(a(x1)) -> a#(b(a(x1))) c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> c#(x1) c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> a#(c(x1)) c#(a(x1)) -> c#(x1) -> c#(b(x1)) -> d#(x1) b#(c(a(x1))) -> a#(b(c(x1))) -> a#(d(x1)) -> a#(x1) b#(c(a(x1))) -> a#(b(c(x1))) -> a#(d(x1)) -> d#(a(x1)) b#(c(a(x1))) -> a#(b(c(x1))) -> a#(a(x1)) -> b#(a(x1)) b#(c(a(x1))) -> a#(b(c(x1))) -> a#(a(x1)) -> a#(b(a(x1))) b#(c(a(x1))) -> c#(x1) -> c#(a(x1)) -> c#(x1) b#(c(a(x1))) -> c#(x1) -> c#(a(x1)) -> a#(c(x1)) b#(c(a(x1))) -> c#(x1) -> c#(b(x1)) -> d#(x1) b#(c(a(x1))) -> b#(c(x1)) -> b#(a(a(x1))) -> c#(x1) b#(c(a(x1))) -> b#(c(x1)) -> b#(a(a(x1))) -> b#(c(x1)) b#(c(a(x1))) -> b#(c(x1)) -> b#(a(a(x1))) -> a#(b(c(x1))) b#(c(a(x1))) -> b#(c(x1)) -> b#(c(a(x1))) -> c#(x1) b#(c(a(x1))) -> b#(c(x1)) -> b#(c(a(x1))) -> b#(c(x1)) b#(c(a(x1))) -> b#(c(x1)) -> b#(c(a(x1))) -> a#(b(c(x1))) b#(a(a(x1))) -> a#(b(c(x1))) -> a#(d(x1)) -> a#(x1) b#(a(a(x1))) -> a#(b(c(x1))) -> a#(d(x1)) -> d#(a(x1)) b#(a(a(x1))) -> a#(b(c(x1))) -> a#(a(x1)) -> b#(a(x1)) b#(a(a(x1))) -> a#(b(c(x1))) -> a#(a(x1)) -> a#(b(a(x1))) b#(a(a(x1))) -> c#(x1) -> c#(a(x1)) -> c#(x1) b#(a(a(x1))) -> c#(x1) -> c#(a(x1)) -> a#(c(x1)) b#(a(a(x1))) -> c#(x1) -> c#(b(x1)) -> d#(x1) b#(a(a(x1))) -> b#(c(x1)) -> b#(a(a(x1))) -> c#(x1) b#(a(a(x1))) -> b#(c(x1)) -> b#(a(a(x1))) -> b#(c(x1)) b#(a(a(x1))) -> b#(c(x1)) -> b#(a(a(x1))) -> a#(b(c(x1))) b#(a(a(x1))) -> b#(c(x1)) -> b#(c(a(x1))) -> c#(x1) b#(a(a(x1))) -> b#(c(x1)) -> b#(c(a(x1))) -> b#(c(x1)) b#(a(a(x1))) -> b#(c(x1)) -> b#(c(a(x1))) -> a#(b(c(x1))) Arctic Interpretation Processor: dimension: 1 usable rules: b(a(a(x1))) -> a(b(c(x1))) c(a(x1)) -> a(c(x1)) b(c(a(x1))) -> a(b(c(x1))) c(b(x1)) -> d(x1) d(x1) -> b(a(x1)) a(d(x1)) -> d(a(x1)) a(a(x1)) -> a(b(a(x1))) interpretation: [b#](x0) = x0, [b](x0) = x0, [d](x0) = 2x0, [c#](x0) = 4x0, [a](x0) = 2x0, [d#](x0) = 2x0, [a#](x0) = 2x0, [c](x0) = 2x0 orientation: b#(a(a(x1))) = 4x1 >= 4x1 = c#(x1) b#(a(a(x1))) = 4x1 >= 2x1 = b#(c(x1)) b#(a(a(x1))) = 4x1 >= 4x1 = a#(b(c(x1))) c#(a(x1)) = 6x1 >= 4x1 = c#(x1) c#(a(x1)) = 6x1 >= 4x1 = a#(c(x1)) b#(c(a(x1))) = 4x1 >= 4x1 = c#(x1) b#(c(a(x1))) = 4x1 >= 2x1 = b#(c(x1)) b#(c(a(x1))) = 4x1 >= 4x1 = a#(b(c(x1))) c#(b(x1)) = 4x1 >= 2x1 = d#(x1) d#(x1) = 2x1 >= 2x1 = a#(x1) d#(x1) = 2x1 >= 2x1 = b#(a(x1)) a#(d(x1)) = 4x1 >= 2x1 = a#(x1) a#(d(x1)) = 4x1 >= 4x1 = d#(a(x1)) a#(a(x1)) = 4x1 >= 2x1 = b#(a(x1)) a#(a(x1)) = 4x1 >= 4x1 = a#(b(a(x1))) b(a(a(x1))) = 4x1 >= 4x1 = a(b(c(x1))) c(a(x1)) = 4x1 >= 4x1 = a(c(x1)) b(c(a(x1))) = 4x1 >= 4x1 = a(b(c(x1))) c(b(x1)) = 2x1 >= 2x1 = d(x1) d(x1) = 2x1 >= 2x1 = b(a(x1)) a(d(x1)) = 4x1 >= 4x1 = d(a(x1)) a(a(x1)) = 4x1 >= 4x1 = a(b(a(x1))) problem: DPs: b#(a(a(x1))) -> c#(x1) b#(a(a(x1))) -> a#(b(c(x1))) b#(c(a(x1))) -> c#(x1) b#(c(a(x1))) -> a#(b(c(x1))) d#(x1) -> a#(x1) d#(x1) -> b#(a(x1)) a#(d(x1)) -> d#(a(x1)) a#(a(x1)) -> a#(b(a(x1))) TRS: b(a(a(x1))) -> a(b(c(x1))) c(a(x1)) -> a(c(x1)) b(c(a(x1))) -> a(b(c(x1))) c(b(x1)) -> d(x1) d(x1) -> b(a(x1)) a(d(x1)) -> d(a(x1)) a(a(x1)) -> a(b(a(x1))) Restore Modifier: DPs: b#(a(a(x1))) -> c#(x1) b#(a(a(x1))) -> a#(b(c(x1))) b#(c(a(x1))) -> c#(x1) b#(c(a(x1))) -> a#(b(c(x1))) d#(x1) -> a#(x1) d#(x1) -> b#(a(x1)) a#(d(x1)) -> d#(a(x1)) a#(a(x1)) -> a#(b(a(x1))) TRS: b(a(a(x1))) -> a(b(c(x1))) c(a(x1)) -> a(c(x1)) b(c(a(x1))) -> a(b(c(x1))) c(b(x1)) -> d(x1) d(x1) -> b(a(x1)) a(d(x1)) -> d(a(x1)) a(a(x1)) -> a(b(a(x1))) EDG Processor: DPs: b#(a(a(x1))) -> c#(x1) b#(a(a(x1))) -> a#(b(c(x1))) b#(c(a(x1))) -> c#(x1) b#(c(a(x1))) -> a#(b(c(x1))) d#(x1) -> a#(x1) d#(x1) -> b#(a(x1)) a#(d(x1)) -> d#(a(x1)) a#(a(x1)) -> a#(b(a(x1))) TRS: b(a(a(x1))) -> a(b(c(x1))) c(a(x1)) -> a(c(x1)) b(c(a(x1))) -> a(b(c(x1))) c(b(x1)) -> d(x1) d(x1) -> b(a(x1)) a(d(x1)) -> d(a(x1)) a(a(x1)) -> a(b(a(x1))) graph: d#(x1) -> a#(x1) -> a#(a(x1)) -> a#(b(a(x1))) d#(x1) -> a#(x1) -> a#(d(x1)) -> d#(a(x1)) d#(x1) -> b#(a(x1)) -> b#(a(a(x1))) -> a#(b(c(x1))) d#(x1) -> b#(a(x1)) -> b#(a(a(x1))) -> c#(x1) a#(d(x1)) -> d#(a(x1)) -> d#(x1) -> b#(a(x1)) a#(d(x1)) -> d#(a(x1)) -> d#(x1) -> a#(x1) a#(a(x1)) -> a#(b(a(x1))) -> a#(a(x1)) -> a#(b(a(x1))) a#(a(x1)) -> a#(b(a(x1))) -> a#(d(x1)) -> d#(a(x1)) b#(c(a(x1))) -> a#(b(c(x1))) -> a#(a(x1)) -> a#(b(a(x1))) b#(c(a(x1))) -> a#(b(c(x1))) -> a#(d(x1)) -> d#(a(x1)) b#(a(a(x1))) -> a#(b(c(x1))) -> a#(a(x1)) -> a#(b(a(x1))) b#(a(a(x1))) -> a#(b(c(x1))) -> a#(d(x1)) -> d#(a(x1)) SCC Processor: #sccs: 1 #rules: 5 #arcs: 12/64 DPs: d#(x1) -> a#(x1) a#(d(x1)) -> d#(a(x1)) d#(x1) -> b#(a(x1)) b#(a(a(x1))) -> a#(b(c(x1))) a#(a(x1)) -> a#(b(a(x1))) TRS: b(a(a(x1))) -> a(b(c(x1))) c(a(x1)) -> a(c(x1)) b(c(a(x1))) -> a(b(c(x1))) c(b(x1)) -> d(x1) d(x1) -> b(a(x1)) a(d(x1)) -> d(a(x1)) a(a(x1)) -> a(b(a(x1))) Arctic Interpretation Processor: dimension: 1 usable rules: b(a(a(x1))) -> a(b(c(x1))) b(c(a(x1))) -> a(b(c(x1))) d(x1) -> b(a(x1)) a(d(x1)) -> d(a(x1)) a(a(x1)) -> a(b(a(x1))) interpretation: [b#](x0) = 2, [b](x0) = 1, [d](x0) = x0 + 2, [a](x0) = x0 + 1, [d#](x0) = x0 + 2, [a#](x0) = x0 + 0, [c](x0) = x0 + 4 orientation: d#(x1) = x1 + 2 >= x1 + 0 = a#(x1) a#(d(x1)) = x1 + 2 >= x1 + 2 = d#(a(x1)) d#(x1) = x1 + 2 >= 2 = b#(a(x1)) b#(a(a(x1))) = 2 >= 1 = a#(b(c(x1))) a#(a(x1)) = x1 + 1 >= 1 = a#(b(a(x1))) b(a(a(x1))) = 1 >= 1 = a(b(c(x1))) c(a(x1)) = x1 + 4 >= x1 + 4 = a(c(x1)) b(c(a(x1))) = 1 >= 1 = a(b(c(x1))) c(b(x1)) = 4 >= x1 + 2 = d(x1) d(x1) = x1 + 2 >= 1 = b(a(x1)) a(d(x1)) = x1 + 2 >= x1 + 2 = d(a(x1)) a(a(x1)) = x1 + 1 >= 1 = a(b(a(x1))) problem: DPs: d#(x1) -> a#(x1) a#(d(x1)) -> d#(a(x1)) d#(x1) -> b#(a(x1)) a#(a(x1)) -> a#(b(a(x1))) TRS: b(a(a(x1))) -> a(b(c(x1))) c(a(x1)) -> a(c(x1)) b(c(a(x1))) -> a(b(c(x1))) c(b(x1)) -> d(x1) d(x1) -> b(a(x1)) a(d(x1)) -> d(a(x1)) a(a(x1)) -> a(b(a(x1))) Restore Modifier: DPs: d#(x1) -> a#(x1) a#(d(x1)) -> d#(a(x1)) d#(x1) -> b#(a(x1)) a#(a(x1)) -> a#(b(a(x1))) TRS: b(a(a(x1))) -> a(b(c(x1))) c(a(x1)) -> a(c(x1)) b(c(a(x1))) -> a(b(c(x1))) c(b(x1)) -> d(x1) d(x1) -> b(a(x1)) a(d(x1)) -> d(a(x1)) a(a(x1)) -> a(b(a(x1))) EDG Processor: DPs: d#(x1) -> a#(x1) a#(d(x1)) -> d#(a(x1)) d#(x1) -> b#(a(x1)) a#(a(x1)) -> a#(b(a(x1))) TRS: b(a(a(x1))) -> a(b(c(x1))) c(a(x1)) -> a(c(x1)) b(c(a(x1))) -> a(b(c(x1))) c(b(x1)) -> d(x1) d(x1) -> b(a(x1)) a(d(x1)) -> d(a(x1)) a(a(x1)) -> a(b(a(x1))) graph: d#(x1) -> a#(x1) -> a#(d(x1)) -> d#(a(x1)) d#(x1) -> a#(x1) -> a#(a(x1)) -> a#(b(a(x1))) a#(d(x1)) -> d#(a(x1)) -> d#(x1) -> a#(x1) a#(d(x1)) -> d#(a(x1)) -> d#(x1) -> b#(a(x1)) a#(a(x1)) -> a#(b(a(x1))) -> a#(d(x1)) -> d#(a(x1)) a#(a(x1)) -> a#(b(a(x1))) -> a#(a(x1)) -> a#(b(a(x1))) SCC Processor: #sccs: 1 #rules: 3 #arcs: 6/16 DPs: d#(x1) -> a#(x1) a#(a(x1)) -> a#(b(a(x1))) a#(d(x1)) -> d#(a(x1)) TRS: b(a(a(x1))) -> a(b(c(x1))) c(a(x1)) -> a(c(x1)) b(c(a(x1))) -> a(b(c(x1))) c(b(x1)) -> d(x1) d(x1) -> b(a(x1)) a(d(x1)) -> d(a(x1)) a(a(x1)) -> a(b(a(x1))) Arctic Interpretation Processor: dimension: 1 usable rules: b(a(a(x1))) -> a(b(c(x1))) b(c(a(x1))) -> a(b(c(x1))) d(x1) -> b(a(x1)) a(d(x1)) -> d(a(x1)) a(a(x1)) -> a(b(a(x1))) interpretation: [b](x0) = 0, [d](x0) = 1x0 + 1, [a](x0) = x0 + 0, [d#](x0) = x0 + 0, [a#](x0) = x0 + 0, [c](x0) = 14x0 + 2 orientation: d#(x1) = x1 + 0 >= x1 + 0 = a#(x1) a#(a(x1)) = x1 + 0 >= 0 = a#(b(a(x1))) a#(d(x1)) = 1x1 + 1 >= x1 + 0 = d#(a(x1)) b(a(a(x1))) = 0 >= 0 = a(b(c(x1))) c(a(x1)) = 14x1 + 14 >= 14x1 + 2 = a(c(x1)) b(c(a(x1))) = 0 >= 0 = a(b(c(x1))) c(b(x1)) = 14 >= 1x1 + 1 = d(x1) d(x1) = 1x1 + 1 >= 0 = b(a(x1)) a(d(x1)) = 1x1 + 1 >= 1x1 + 1 = d(a(x1)) a(a(x1)) = x1 + 0 >= 0 = a(b(a(x1))) problem: DPs: d#(x1) -> a#(x1) a#(a(x1)) -> a#(b(a(x1))) TRS: b(a(a(x1))) -> a(b(c(x1))) c(a(x1)) -> a(c(x1)) b(c(a(x1))) -> a(b(c(x1))) c(b(x1)) -> d(x1) d(x1) -> b(a(x1)) a(d(x1)) -> d(a(x1)) a(a(x1)) -> a(b(a(x1))) Restore Modifier: DPs: d#(x1) -> a#(x1) a#(a(x1)) -> a#(b(a(x1))) TRS: b(a(a(x1))) -> a(b(c(x1))) c(a(x1)) -> a(c(x1)) b(c(a(x1))) -> a(b(c(x1))) c(b(x1)) -> d(x1) d(x1) -> b(a(x1)) a(d(x1)) -> d(a(x1)) a(a(x1)) -> a(b(a(x1))) EDG Processor: DPs: d#(x1) -> a#(x1) a#(a(x1)) -> a#(b(a(x1))) TRS: b(a(a(x1))) -> a(b(c(x1))) c(a(x1)) -> a(c(x1)) b(c(a(x1))) -> a(b(c(x1))) c(b(x1)) -> d(x1) d(x1) -> b(a(x1)) a(d(x1)) -> d(a(x1)) a(a(x1)) -> a(b(a(x1))) graph: d#(x1) -> a#(x1) -> a#(a(x1)) -> a#(b(a(x1))) a#(a(x1)) -> a#(b(a(x1))) -> a#(a(x1)) -> a#(b(a(x1))) SCC Processor: #sccs: 1 #rules: 1 #arcs: 2/4 DPs: a#(a(x1)) -> a#(b(a(x1))) TRS: b(a(a(x1))) -> a(b(c(x1))) c(a(x1)) -> a(c(x1)) b(c(a(x1))) -> a(b(c(x1))) c(b(x1)) -> d(x1) d(x1) -> b(a(x1)) a(d(x1)) -> d(a(x1)) a(a(x1)) -> a(b(a(x1))) Arctic Interpretation Processor: dimension: 1 usable rules: b(a(a(x1))) -> a(b(c(x1))) c(a(x1)) -> a(c(x1)) b(c(a(x1))) -> a(b(c(x1))) c(b(x1)) -> d(x1) d(x1) -> b(a(x1)) a(d(x1)) -> d(a(x1)) a(a(x1)) -> a(b(a(x1))) interpretation: [b](x0) = -1x0 + 0, [d](x0) = 3x0 + 4, [a](x0) = 4x0 + 1, [a#](x0) = x0 + 0, [c](x0) = 4x0 orientation: a#(a(x1)) = 4x1 + 1 >= 3x1 + 0 = a#(b(a(x1))) b(a(a(x1))) = 7x1 + 4 >= 7x1 + 4 = a(b(c(x1))) c(a(x1)) = 8x1 + 5 >= 8x1 + 1 = a(c(x1)) b(c(a(x1))) = 7x1 + 4 >= 7x1 + 4 = a(b(c(x1))) c(b(x1)) = 3x1 + 4 >= 3x1 + 4 = d(x1) d(x1) = 3x1 + 4 >= 3x1 + 0 = b(a(x1)) a(d(x1)) = 7x1 + 8 >= 7x1 + 4 = d(a(x1)) a(a(x1)) = 8x1 + 5 >= 7x1 + 4 = a(b(a(x1))) problem: DPs: TRS: b(a(a(x1))) -> a(b(c(x1))) c(a(x1)) -> a(c(x1)) b(c(a(x1))) -> a(b(c(x1))) c(b(x1)) -> d(x1) d(x1) -> b(a(x1)) a(d(x1)) -> d(a(x1)) a(a(x1)) -> a(b(a(x1))) Qed