YES Problem: c(c(c(a(x1)))) -> d(d(x1)) d(b(x1)) -> c(c(x1)) c(x1) -> a(a(a(a(x1)))) d(x1) -> b(b(b(b(x1)))) b(d(x1)) -> c(c(x1)) a(c(c(c(x1)))) -> d(d(x1)) Proof: DP Processor: DPs: c#(c(c(a(x1)))) -> d#(x1) c#(c(c(a(x1)))) -> d#(d(x1)) d#(b(x1)) -> c#(x1) d#(b(x1)) -> c#(c(x1)) c#(x1) -> a#(x1) c#(x1) -> a#(a(x1)) c#(x1) -> a#(a(a(x1))) c#(x1) -> a#(a(a(a(x1)))) d#(x1) -> b#(x1) d#(x1) -> b#(b(x1)) d#(x1) -> b#(b(b(x1))) d#(x1) -> b#(b(b(b(x1)))) b#(d(x1)) -> c#(x1) b#(d(x1)) -> c#(c(x1)) a#(c(c(c(x1)))) -> d#(x1) a#(c(c(c(x1)))) -> d#(d(x1)) TRS: c(c(c(a(x1)))) -> d(d(x1)) d(b(x1)) -> c(c(x1)) c(x1) -> a(a(a(a(x1)))) d(x1) -> b(b(b(b(x1)))) b(d(x1)) -> c(c(x1)) a(c(c(c(x1)))) -> d(d(x1)) TDG Processor: DPs: c#(c(c(a(x1)))) -> d#(x1) c#(c(c(a(x1)))) -> d#(d(x1)) d#(b(x1)) -> c#(x1) d#(b(x1)) -> c#(c(x1)) c#(x1) -> a#(x1) c#(x1) -> a#(a(x1)) c#(x1) -> a#(a(a(x1))) c#(x1) -> a#(a(a(a(x1)))) d#(x1) -> b#(x1) d#(x1) -> b#(b(x1)) d#(x1) -> b#(b(b(x1))) d#(x1) -> b#(b(b(b(x1)))) b#(d(x1)) -> c#(x1) b#(d(x1)) -> c#(c(x1)) a#(c(c(c(x1)))) -> d#(x1) a#(c(c(c(x1)))) -> d#(d(x1)) TRS: c(c(c(a(x1)))) -> d(d(x1)) d(b(x1)) -> c(c(x1)) c(x1) -> a(a(a(a(x1)))) d(x1) -> b(b(b(b(x1)))) b(d(x1)) -> c(c(x1)) a(c(c(c(x1)))) -> d(d(x1)) graph: b#(d(x1)) -> c#(c(x1)) -> c#(x1) -> a#(a(a(a(x1)))) b#(d(x1)) -> c#(c(x1)) -> c#(x1) -> a#(a(a(x1))) b#(d(x1)) -> c#(c(x1)) -> c#(x1) -> a#(a(x1)) b#(d(x1)) -> c#(c(x1)) -> c#(x1) -> a#(x1) b#(d(x1)) -> c#(c(x1)) -> c#(c(c(a(x1)))) -> d#(d(x1)) b#(d(x1)) -> c#(c(x1)) -> c#(c(c(a(x1)))) -> d#(x1) b#(d(x1)) -> c#(x1) -> c#(x1) -> a#(a(a(a(x1)))) b#(d(x1)) -> c#(x1) -> c#(x1) -> a#(a(a(x1))) b#(d(x1)) -> c#(x1) -> c#(x1) -> a#(a(x1)) b#(d(x1)) -> c#(x1) -> c#(x1) -> a#(x1) b#(d(x1)) -> c#(x1) -> c#(c(c(a(x1)))) -> d#(d(x1)) b#(d(x1)) -> c#(x1) -> c#(c(c(a(x1)))) -> d#(x1) a#(c(c(c(x1)))) -> d#(d(x1)) -> d#(x1) -> b#(b(b(b(x1)))) a#(c(c(c(x1)))) -> d#(d(x1)) -> d#(x1) -> b#(b(b(x1))) a#(c(c(c(x1)))) -> d#(d(x1)) -> d#(x1) -> b#(b(x1)) a#(c(c(c(x1)))) -> d#(d(x1)) -> d#(x1) -> b#(x1) a#(c(c(c(x1)))) -> d#(d(x1)) -> d#(b(x1)) -> c#(c(x1)) a#(c(c(c(x1)))) -> d#(d(x1)) -> d#(b(x1)) -> c#(x1) a#(c(c(c(x1)))) -> d#(x1) -> d#(x1) -> b#(b(b(b(x1)))) a#(c(c(c(x1)))) -> d#(x1) -> d#(x1) -> b#(b(b(x1))) a#(c(c(c(x1)))) -> d#(x1) -> d#(x1) -> b#(b(x1)) a#(c(c(c(x1)))) -> d#(x1) -> d#(x1) -> b#(x1) a#(c(c(c(x1)))) -> d#(x1) -> d#(b(x1)) -> c#(c(x1)) a#(c(c(c(x1)))) -> d#(x1) -> d#(b(x1)) -> c#(x1) d#(b(x1)) -> c#(c(x1)) -> c#(x1) -> a#(a(a(a(x1)))) d#(b(x1)) -> c#(c(x1)) -> c#(x1) -> a#(a(a(x1))) d#(b(x1)) -> c#(c(x1)) -> c#(x1) -> a#(a(x1)) d#(b(x1)) -> c#(c(x1)) -> c#(x1) -> a#(x1) d#(b(x1)) -> c#(c(x1)) -> c#(c(c(a(x1)))) -> d#(d(x1)) d#(b(x1)) -> c#(c(x1)) -> c#(c(c(a(x1)))) -> d#(x1) d#(b(x1)) -> c#(x1) -> c#(x1) -> a#(a(a(a(x1)))) d#(b(x1)) -> c#(x1) -> c#(x1) -> a#(a(a(x1))) d#(b(x1)) -> c#(x1) -> c#(x1) -> a#(a(x1)) d#(b(x1)) -> c#(x1) -> c#(x1) -> a#(x1) d#(b(x1)) -> c#(x1) -> c#(c(c(a(x1)))) -> d#(d(x1)) d#(b(x1)) -> c#(x1) -> c#(c(c(a(x1)))) -> d#(x1) d#(x1) -> b#(b(b(b(x1)))) -> b#(d(x1)) -> c#(c(x1)) d#(x1) -> b#(b(b(b(x1)))) -> b#(d(x1)) -> c#(x1) d#(x1) -> b#(b(b(x1))) -> b#(d(x1)) -> c#(c(x1)) d#(x1) -> b#(b(b(x1))) -> b#(d(x1)) -> c#(x1) d#(x1) -> b#(b(x1)) -> b#(d(x1)) -> c#(c(x1)) d#(x1) -> b#(b(x1)) -> b#(d(x1)) -> c#(x1) d#(x1) -> b#(x1) -> b#(d(x1)) -> c#(c(x1)) d#(x1) -> b#(x1) -> b#(d(x1)) -> c#(x1) c#(c(c(a(x1)))) -> d#(d(x1)) -> d#(x1) -> b#(b(b(b(x1)))) c#(c(c(a(x1)))) -> d#(d(x1)) -> d#(x1) -> b#(b(b(x1))) c#(c(c(a(x1)))) -> d#(d(x1)) -> d#(x1) -> b#(b(x1)) c#(c(c(a(x1)))) -> d#(d(x1)) -> d#(x1) -> b#(x1) c#(c(c(a(x1)))) -> d#(d(x1)) -> d#(b(x1)) -> c#(c(x1)) c#(c(c(a(x1)))) -> d#(d(x1)) -> d#(b(x1)) -> c#(x1) c#(c(c(a(x1)))) -> d#(x1) -> d#(x1) -> b#(b(b(b(x1)))) c#(c(c(a(x1)))) -> d#(x1) -> d#(x1) -> b#(b(b(x1))) c#(c(c(a(x1)))) -> d#(x1) -> d#(x1) -> b#(b(x1)) c#(c(c(a(x1)))) -> d#(x1) -> d#(x1) -> b#(x1) c#(c(c(a(x1)))) -> d#(x1) -> d#(b(x1)) -> c#(c(x1)) c#(c(c(a(x1)))) -> d#(x1) -> d#(b(x1)) -> c#(x1) c#(x1) -> a#(a(a(a(x1)))) -> a#(c(c(c(x1)))) -> d#(d(x1)) c#(x1) -> a#(a(a(a(x1)))) -> a#(c(c(c(x1)))) -> d#(x1) c#(x1) -> a#(a(a(x1))) -> a#(c(c(c(x1)))) -> d#(d(x1)) c#(x1) -> a#(a(a(x1))) -> a#(c(c(c(x1)))) -> d#(x1) c#(x1) -> a#(a(x1)) -> a#(c(c(c(x1)))) -> d#(d(x1)) c#(x1) -> a#(a(x1)) -> a#(c(c(c(x1)))) -> d#(x1) c#(x1) -> a#(x1) -> a#(c(c(c(x1)))) -> d#(d(x1)) c#(x1) -> a#(x1) -> a#(c(c(c(x1)))) -> d#(x1) Arctic Interpretation Processor: dimension: 1 interpretation: [b#](x0) = -2x0 + 0, [a#](x0) = -2x0 + 0, [d#](x0) = 4x0 + 1, [c#](x0) = 1x0 + 0, [b](x0) = 2x0 + -9, [d](x0) = 8x0 + -3, [c](x0) = 5x0 + -6, [a](x0) = 1x0 + -10 orientation: c#(c(c(a(x1)))) = 12x1 + 1 >= 4x1 + 1 = d#(x1) c#(c(c(a(x1)))) = 12x1 + 1 >= 12x1 + 1 = d#(d(x1)) d#(b(x1)) = 6x1 + 1 >= 1x1 + 0 = c#(x1) d#(b(x1)) = 6x1 + 1 >= 6x1 + 0 = c#(c(x1)) c#(x1) = 1x1 + 0 >= -2x1 + 0 = a#(x1) c#(x1) = 1x1 + 0 >= -1x1 + 0 = a#(a(x1)) c#(x1) = 1x1 + 0 >= x1 + 0 = a#(a(a(x1))) c#(x1) = 1x1 + 0 >= 1x1 + 0 = a#(a(a(a(x1)))) d#(x1) = 4x1 + 1 >= -2x1 + 0 = b#(x1) d#(x1) = 4x1 + 1 >= x1 + 0 = b#(b(x1)) d#(x1) = 4x1 + 1 >= 2x1 + 0 = b#(b(b(x1))) d#(x1) = 4x1 + 1 >= 4x1 + 0 = b#(b(b(b(x1)))) b#(d(x1)) = 6x1 + 0 >= 1x1 + 0 = c#(x1) b#(d(x1)) = 6x1 + 0 >= 6x1 + 0 = c#(c(x1)) a#(c(c(c(x1)))) = 13x1 + 2 >= 4x1 + 1 = d#(x1) a#(c(c(c(x1)))) = 13x1 + 2 >= 12x1 + 1 = d#(d(x1)) c(c(c(a(x1)))) = 16x1 + 5 >= 16x1 + 5 = d(d(x1)) d(b(x1)) = 10x1 + -1 >= 10x1 + -1 = c(c(x1)) c(x1) = 5x1 + -6 >= 4x1 + -7 = a(a(a(a(x1)))) d(x1) = 8x1 + -3 >= 8x1 + -3 = b(b(b(b(x1)))) b(d(x1)) = 10x1 + -1 >= 10x1 + -1 = c(c(x1)) a(c(c(c(x1)))) = 16x1 + 5 >= 16x1 + 5 = d(d(x1)) problem: DPs: c#(c(c(a(x1)))) -> d#(x1) c#(c(c(a(x1)))) -> d#(d(x1)) d#(b(x1)) -> c#(c(x1)) c#(x1) -> a#(x1) c#(x1) -> a#(a(x1)) c#(x1) -> a#(a(a(x1))) c#(x1) -> a#(a(a(a(x1)))) d#(x1) -> b#(b(b(b(x1)))) b#(d(x1)) -> c#(x1) b#(d(x1)) -> c#(c(x1)) TRS: c(c(c(a(x1)))) -> d(d(x1)) d(b(x1)) -> c(c(x1)) c(x1) -> a(a(a(a(x1)))) d(x1) -> b(b(b(b(x1)))) b(d(x1)) -> c(c(x1)) a(c(c(c(x1)))) -> d(d(x1)) EDG Processor: DPs: c#(c(c(a(x1)))) -> d#(x1) c#(c(c(a(x1)))) -> d#(d(x1)) d#(b(x1)) -> c#(c(x1)) c#(x1) -> a#(x1) c#(x1) -> a#(a(x1)) c#(x1) -> a#(a(a(x1))) c#(x1) -> a#(a(a(a(x1)))) d#(x1) -> b#(b(b(b(x1)))) b#(d(x1)) -> c#(x1) b#(d(x1)) -> c#(c(x1)) TRS: c(c(c(a(x1)))) -> d(d(x1)) d(b(x1)) -> c(c(x1)) c(x1) -> a(a(a(a(x1)))) d(x1) -> b(b(b(b(x1)))) b(d(x1)) -> c(c(x1)) a(c(c(c(x1)))) -> d(d(x1)) graph: b#(d(x1)) -> c#(c(x1)) -> c#(c(c(a(x1)))) -> d#(x1) b#(d(x1)) -> c#(c(x1)) -> c#(c(c(a(x1)))) -> d#(d(x1)) b#(d(x1)) -> c#(c(x1)) -> c#(x1) -> a#(x1) b#(d(x1)) -> c#(c(x1)) -> c#(x1) -> a#(a(x1)) b#(d(x1)) -> c#(c(x1)) -> c#(x1) -> a#(a(a(x1))) b#(d(x1)) -> c#(c(x1)) -> c#(x1) -> a#(a(a(a(x1)))) b#(d(x1)) -> c#(x1) -> c#(c(c(a(x1)))) -> d#(x1) b#(d(x1)) -> c#(x1) -> c#(c(c(a(x1)))) -> d#(d(x1)) b#(d(x1)) -> c#(x1) -> c#(x1) -> a#(x1) b#(d(x1)) -> c#(x1) -> c#(x1) -> a#(a(x1)) b#(d(x1)) -> c#(x1) -> c#(x1) -> a#(a(a(x1))) b#(d(x1)) -> c#(x1) -> c#(x1) -> a#(a(a(a(x1)))) d#(b(x1)) -> c#(c(x1)) -> c#(c(c(a(x1)))) -> d#(x1) d#(b(x1)) -> c#(c(x1)) -> c#(c(c(a(x1)))) -> d#(d(x1)) d#(b(x1)) -> c#(c(x1)) -> c#(x1) -> a#(x1) d#(b(x1)) -> c#(c(x1)) -> c#(x1) -> a#(a(x1)) d#(b(x1)) -> c#(c(x1)) -> c#(x1) -> a#(a(a(x1))) d#(b(x1)) -> c#(c(x1)) -> c#(x1) -> a#(a(a(a(x1)))) d#(x1) -> b#(b(b(b(x1)))) -> b#(d(x1)) -> c#(x1) d#(x1) -> b#(b(b(b(x1)))) -> b#(d(x1)) -> c#(c(x1)) c#(c(c(a(x1)))) -> d#(d(x1)) -> d#(b(x1)) -> c#(c(x1)) c#(c(c(a(x1)))) -> d#(d(x1)) -> d#(x1) -> b#(b(b(b(x1)))) c#(c(c(a(x1)))) -> d#(x1) -> d#(b(x1)) -> c#(c(x1)) c#(c(c(a(x1)))) -> d#(x1) -> d#(x1) -> b#(b(b(b(x1)))) SCC Processor: #sccs: 1 #rules: 6 #arcs: 24/100 DPs: b#(d(x1)) -> c#(c(x1)) c#(c(c(a(x1)))) -> d#(d(x1)) d#(x1) -> b#(b(b(b(x1)))) b#(d(x1)) -> c#(x1) c#(c(c(a(x1)))) -> d#(x1) d#(b(x1)) -> c#(c(x1)) TRS: c(c(c(a(x1)))) -> d(d(x1)) d(b(x1)) -> c(c(x1)) c(x1) -> a(a(a(a(x1)))) d(x1) -> b(b(b(b(x1)))) b(d(x1)) -> c(c(x1)) a(c(c(c(x1)))) -> d(d(x1)) Arctic Interpretation Processor: dimension: 1 interpretation: [b#](x0) = 1x0, [d#](x0) = 10x0, [c#](x0) = 5x0, [b](x0) = 3x0, [d](x0) = 13x0, [c](x0) = 8x0, [a](x0) = 2x0 orientation: b#(d(x1)) = 14x1 >= 13x1 = c#(c(x1)) c#(c(c(a(x1)))) = 23x1 >= 23x1 = d#(d(x1)) d#(x1) = 10x1 >= 10x1 = b#(b(b(b(x1)))) b#(d(x1)) = 14x1 >= 5x1 = c#(x1) c#(c(c(a(x1)))) = 23x1 >= 10x1 = d#(x1) d#(b(x1)) = 13x1 >= 13x1 = c#(c(x1)) c(c(c(a(x1)))) = 26x1 >= 26x1 = d(d(x1)) d(b(x1)) = 16x1 >= 16x1 = c(c(x1)) c(x1) = 8x1 >= 8x1 = a(a(a(a(x1)))) d(x1) = 13x1 >= 12x1 = b(b(b(b(x1)))) b(d(x1)) = 16x1 >= 16x1 = c(c(x1)) a(c(c(c(x1)))) = 26x1 >= 26x1 = d(d(x1)) problem: DPs: c#(c(c(a(x1)))) -> d#(d(x1)) d#(x1) -> b#(b(b(b(x1)))) d#(b(x1)) -> c#(c(x1)) TRS: c(c(c(a(x1)))) -> d(d(x1)) d(b(x1)) -> c(c(x1)) c(x1) -> a(a(a(a(x1)))) d(x1) -> b(b(b(b(x1)))) b(d(x1)) -> c(c(x1)) a(c(c(c(x1)))) -> d(d(x1)) EDG Processor: DPs: c#(c(c(a(x1)))) -> d#(d(x1)) d#(x1) -> b#(b(b(b(x1)))) d#(b(x1)) -> c#(c(x1)) TRS: c(c(c(a(x1)))) -> d(d(x1)) d(b(x1)) -> c(c(x1)) c(x1) -> a(a(a(a(x1)))) d(x1) -> b(b(b(b(x1)))) b(d(x1)) -> c(c(x1)) a(c(c(c(x1)))) -> d(d(x1)) graph: d#(b(x1)) -> c#(c(x1)) -> c#(c(c(a(x1)))) -> d#(d(x1)) c#(c(c(a(x1)))) -> d#(d(x1)) -> d#(x1) -> b#(b(b(b(x1)))) c#(c(c(a(x1)))) -> d#(d(x1)) -> d#(b(x1)) -> c#(c(x1)) SCC Processor: #sccs: 1 #rules: 2 #arcs: 3/9 DPs: d#(b(x1)) -> c#(c(x1)) c#(c(c(a(x1)))) -> d#(d(x1)) TRS: c(c(c(a(x1)))) -> d(d(x1)) d(b(x1)) -> c(c(x1)) c(x1) -> a(a(a(a(x1)))) d(x1) -> b(b(b(b(x1)))) b(d(x1)) -> c(c(x1)) a(c(c(c(x1)))) -> d(d(x1)) Arctic Interpretation Processor: dimension: 3 interpretation: [d#](x0) = [0 0 0]x0 + [0], [c#](x0) = [0 0 0]x0 + [0], [0 1 1 ] [0] [b](x0) = [-& 0 0 ]x0 + [0] [-& 0 0 ] [0], [1 1 1] [1] [d](x0) = [1 1 1]x0 + [0] [0 0 0] [1], [0 1 0] [0] [c](x0) = [0 0 1]x0 + [0] [0 0 0] [0], [0 0 0] [0] [a](x0) = [0 0 0]x0 + [0] [0 0 0] [0] orientation: d#(b(x1)) = [0 1 1]x1 + [0] >= [0 1 1]x1 + [0] = c#(c(x1)) c#(c(c(a(x1)))) = [2 2 2]x1 + [2] >= [1 1 1]x1 + [1] = d#(d(x1)) [2 2 2] [2] [2 2 2] [2] c(c(c(a(x1)))) = [2 2 2]x1 + [2] >= [2 2 2]x1 + [2] = d(d(x1)) [2 2 2] [2] [1 1 1] [1] [1 2 2] [1] [1 1 2] [1] d(b(x1)) = [1 2 2]x1 + [1] >= [1 1 1]x1 + [1] = c(c(x1)) [0 1 1] [1] [0 1 1] [0] [0 1 0] [0] [0 0 0] [0] c(x1) = [0 0 1]x1 + [0] >= [0 0 0]x1 + [0] = a(a(a(a(x1)))) [0 0 0] [0] [0 0 0] [0] [1 1 1] [1] [0 1 1 ] [1] d(x1) = [1 1 1]x1 + [0] >= [-& 0 0 ]x1 + [0] = b(b(b(b(x1)))) [0 0 0] [1] [-& 0 0 ] [0] [2 2 2] [2] [1 1 2] [1] b(d(x1)) = [1 1 1]x1 + [1] >= [1 1 1]x1 + [1] = c(c(x1)) [1 1 1] [1] [0 1 1] [0] [2 2 2] [2] [2 2 2] [2] a(c(c(c(x1)))) = [2 2 2]x1 + [2] >= [2 2 2]x1 + [2] = d(d(x1)) [2 2 2] [2] [1 1 1] [1] problem: DPs: d#(b(x1)) -> c#(c(x1)) TRS: c(c(c(a(x1)))) -> d(d(x1)) d(b(x1)) -> c(c(x1)) c(x1) -> a(a(a(a(x1)))) d(x1) -> b(b(b(b(x1)))) b(d(x1)) -> c(c(x1)) a(c(c(c(x1)))) -> d(d(x1)) EDG Processor: DPs: d#(b(x1)) -> c#(c(x1)) TRS: c(c(c(a(x1)))) -> d(d(x1)) d(b(x1)) -> c(c(x1)) c(x1) -> a(a(a(a(x1)))) d(x1) -> b(b(b(b(x1)))) b(d(x1)) -> c(c(x1)) a(c(c(c(x1)))) -> d(d(x1)) graph: Qed