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 interpretation: [d#](x0) = 5x0, [a#](x0) = 5x0, [c#](x0) = 10x0, [b#](x0) = x0, [d](x0) = 5x0, [c](x0) = 5x0, [b](x0) = x0, [a](x0) = 5x0 orientation: b#(a(a(x1))) = 10x1 >= 10x1 = c#(x1) b#(a(a(x1))) = 10x1 >= 5x1 = b#(c(x1)) b#(a(a(x1))) = 10x1 >= 10x1 = a#(b(c(x1))) c#(a(x1)) = 15x1 >= 10x1 = c#(x1) c#(a(x1)) = 15x1 >= 10x1 = a#(c(x1)) b#(c(a(x1))) = 10x1 >= 10x1 = c#(x1) b#(c(a(x1))) = 10x1 >= 5x1 = b#(c(x1)) b#(c(a(x1))) = 10x1 >= 10x1 = a#(b(c(x1))) c#(b(x1)) = 10x1 >= 5x1 = d#(x1) d#(x1) = 5x1 >= 5x1 = a#(x1) d#(x1) = 5x1 >= 5x1 = b#(a(x1)) a#(d(x1)) = 10x1 >= 5x1 = a#(x1) a#(d(x1)) = 10x1 >= 10x1 = d#(a(x1)) a#(a(x1)) = 10x1 >= 5x1 = b#(a(x1)) a#(a(x1)) = 10x1 >= 10x1 = a#(b(a(x1))) b(a(a(x1))) = 10x1 >= 10x1 = a(b(c(x1))) c(a(x1)) = 10x1 >= 10x1 = a(c(x1)) b(c(a(x1))) = 10x1 >= 10x1 = a(b(c(x1))) c(b(x1)) = 5x1 >= 5x1 = d(x1) d(x1) = 5x1 >= 5x1 = b(a(x1)) a(d(x1)) = 10x1 >= 10x1 = d(a(x1)) a(a(x1)) = 10x1 >= 10x1 = 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))) 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)) CDG 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#(d(x1)) -> d#(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))) -> 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#(a(x1)) -> a#(b(a(x1))) -> a#(a(x1)) -> a#(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#(a(x1)) -> a#(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#(a(x1)) -> a#(b(a(x1))) SCC Processor: #sccs: 2 #rules: 5 #arcs: 11/64 DPs: d#(x1) -> a#(x1) a#(d(x1)) -> d#(a(x1)) d#(x1) -> b#(a(x1)) b#(a(a(x1))) -> a#(b(c(x1))) 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: 2 interpretation: [d#](x0) = [3 3]x0 + [3], [a#](x0) = [2 3]x0 + [0], [b#](x0) = [3 3]x0 + [0], [0 0] [0] [d](x0) = [0 0]x0 + [0], [0 0] [0] [c](x0) = [0 0]x0 + [0], [0 0 ] [0 ] [b](x0) = [-& -&]x0 + [-&], [0 0 ] [0 ] [a](x0) = [-& 0 ]x0 + [-&] orientation: d#(x1) = [3 3]x1 + [3] >= [2 3]x1 + [0] = a#(x1) a#(d(x1)) = [3 3]x1 + [3] >= [3 3]x1 + [3] = d#(a(x1)) d#(x1) = [3 3]x1 + [3] >= [3 3]x1 + [3] = b#(a(x1)) b#(a(a(x1))) = [3 3]x1 + [3] >= [2 2]x1 + [2] = a#(b(c(x1))) [0 0 ] [0 ] [0 0 ] [0 ] b(a(a(x1))) = [-& -&]x1 + [-&] >= [-& -&]x1 + [-&] = a(b(c(x1))) [0 0] [0] [0 0] [0] c(a(x1)) = [0 0]x1 + [0] >= [0 0]x1 + [0] = a(c(x1)) [0 0 ] [0 ] [0 0 ] [0 ] b(c(a(x1))) = [-& -&]x1 + [-&] >= [-& -&]x1 + [-&] = a(b(c(x1))) [0 0] [0] [0 0] [0] c(b(x1)) = [0 0]x1 + [0] >= [0 0]x1 + [0] = d(x1) [0 0] [0] [0 0 ] [0 ] d(x1) = [0 0]x1 + [0] >= [-& -&]x1 + [-&] = b(a(x1)) [0 0] [0] [0 0] [0] a(d(x1)) = [0 0]x1 + [0] >= [0 0]x1 + [0] = d(a(x1)) [0 0 ] [0 ] [0 0 ] [0 ] a(a(x1)) = [-& 0 ]x1 + [-&] >= [-& -&]x1 + [-&] = a(b(a(x1))) problem: DPs: d#(x1) -> a#(x1) a#(d(x1)) -> d#(a(x1)) d#(x1) -> 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)) 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)) a#(d(x1)) -> d#(a(x1)) -> d#(x1) -> b#(a(x1)) a#(d(x1)) -> d#(a(x1)) -> d#(x1) -> a#(x1) SCC Processor: #sccs: 1 #rules: 2 #arcs: 3/9 DPs: d#(x1) -> 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: 2 interpretation: [d#](x0) = [-& 1 ]x0 + [2], [a#](x0) = [-& 1 ]x0 + [2], [2 0 ] [3] [d](x0) = [-& 1 ]x0 + [3], [2 0] [3] [c](x0) = [1 0]x0 + [3], [0 0 ] [0] [b](x0) = [-& -&]x0 + [0], [2 0 ] [3] [a](x0) = [-& 0 ]x0 + [0] orientation: d#(x1) = [-& 1 ]x1 + [2] >= [-& 1 ]x1 + [2] = a#(x1) a#(d(x1)) = [-& 2 ]x1 + [4] >= [-& 1 ]x1 + [2] = d#(a(x1)) [4 2 ] [5] [4 2 ] [5] b(a(a(x1))) = [-& -&]x1 + [0] >= [-& -&]x1 + [0] = a(b(c(x1))) [4 2] [5] [4 2] [5] c(a(x1)) = [3 1]x1 + [4] >= [1 0]x1 + [3] = a(c(x1)) [4 2 ] [5] [4 2 ] [5] b(c(a(x1))) = [-& -&]x1 + [0] >= [-& -&]x1 + [0] = a(b(c(x1))) [2 2] [3] [2 0 ] [3] c(b(x1)) = [1 1]x1 + [3] >= [-& 1 ]x1 + [3] = d(x1) [2 0 ] [3] [2 0 ] [3] d(x1) = [-& 1 ]x1 + [3] >= [-& -&]x1 + [0] = b(a(x1)) [4 2 ] [5] [4 2 ] [5] a(d(x1)) = [-& 1 ]x1 + [3] >= [-& 1 ]x1 + [3] = d(a(x1)) [4 2 ] [5] [4 2 ] [5] a(a(x1)) = [-& 0 ]x1 + [0] >= [-& -&]x1 + [0] = a(b(a(x1))) problem: DPs: d#(x1) -> 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) 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: Qed 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 interpretation: [a#](x0) = x0 + 0, [d](x0) = 14x0 + 9, [c](x0) = 15x0 + 0, [b](x0) = -1x0 + 1, [a](x0) = 15x0 + 2 orientation: a#(a(x1)) = 15x1 + 2 >= 14x1 + 1 = a#(b(a(x1))) b(a(a(x1))) = 29x1 + 16 >= 29x1 + 16 = a(b(c(x1))) c(a(x1)) = 30x1 + 17 >= 30x1 + 15 = a(c(x1)) b(c(a(x1))) = 29x1 + 16 >= 29x1 + 16 = a(b(c(x1))) c(b(x1)) = 14x1 + 16 >= 14x1 + 9 = d(x1) d(x1) = 14x1 + 9 >= 14x1 + 1 = b(a(x1)) a(d(x1)) = 29x1 + 24 >= 29x1 + 16 = d(a(x1)) a(a(x1)) = 30x1 + 17 >= 29x1 + 16 = 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