YES Problem: a(a(x1)) -> b(a(b(x1))) b(b(x1)) -> a(c(b(x1))) c(c(x1)) -> c(b(a(x1))) a(b(x1)) -> b(a(x1)) b(c(x1)) -> c(x1) Proof: DP Processor: DPs: a#(a(x1)) -> b#(x1) a#(a(x1)) -> a#(b(x1)) a#(a(x1)) -> b#(a(b(x1))) b#(b(x1)) -> c#(b(x1)) b#(b(x1)) -> a#(c(b(x1))) c#(c(x1)) -> a#(x1) c#(c(x1)) -> b#(a(x1)) c#(c(x1)) -> c#(b(a(x1))) a#(b(x1)) -> a#(x1) a#(b(x1)) -> b#(a(x1)) TRS: a(a(x1)) -> b(a(b(x1))) b(b(x1)) -> a(c(b(x1))) c(c(x1)) -> c(b(a(x1))) a(b(x1)) -> b(a(x1)) b(c(x1)) -> c(x1) TDG Processor: DPs: a#(a(x1)) -> b#(x1) a#(a(x1)) -> a#(b(x1)) a#(a(x1)) -> b#(a(b(x1))) b#(b(x1)) -> c#(b(x1)) b#(b(x1)) -> a#(c(b(x1))) c#(c(x1)) -> a#(x1) c#(c(x1)) -> b#(a(x1)) c#(c(x1)) -> c#(b(a(x1))) a#(b(x1)) -> a#(x1) a#(b(x1)) -> b#(a(x1)) TRS: a(a(x1)) -> b(a(b(x1))) b(b(x1)) -> a(c(b(x1))) c(c(x1)) -> c(b(a(x1))) a(b(x1)) -> b(a(x1)) b(c(x1)) -> c(x1) graph: c#(c(x1)) -> c#(b(a(x1))) -> c#(c(x1)) -> c#(b(a(x1))) c#(c(x1)) -> c#(b(a(x1))) -> c#(c(x1)) -> b#(a(x1)) c#(c(x1)) -> c#(b(a(x1))) -> c#(c(x1)) -> a#(x1) c#(c(x1)) -> b#(a(x1)) -> b#(b(x1)) -> a#(c(b(x1))) c#(c(x1)) -> b#(a(x1)) -> b#(b(x1)) -> c#(b(x1)) c#(c(x1)) -> a#(x1) -> a#(b(x1)) -> b#(a(x1)) c#(c(x1)) -> a#(x1) -> a#(b(x1)) -> a#(x1) c#(c(x1)) -> a#(x1) -> a#(a(x1)) -> b#(a(b(x1))) c#(c(x1)) -> a#(x1) -> a#(a(x1)) -> a#(b(x1)) c#(c(x1)) -> a#(x1) -> a#(a(x1)) -> b#(x1) b#(b(x1)) -> c#(b(x1)) -> c#(c(x1)) -> c#(b(a(x1))) b#(b(x1)) -> c#(b(x1)) -> c#(c(x1)) -> b#(a(x1)) b#(b(x1)) -> c#(b(x1)) -> c#(c(x1)) -> a#(x1) b#(b(x1)) -> a#(c(b(x1))) -> a#(b(x1)) -> b#(a(x1)) b#(b(x1)) -> a#(c(b(x1))) -> a#(b(x1)) -> a#(x1) b#(b(x1)) -> a#(c(b(x1))) -> a#(a(x1)) -> b#(a(b(x1))) b#(b(x1)) -> a#(c(b(x1))) -> a#(a(x1)) -> a#(b(x1)) b#(b(x1)) -> a#(c(b(x1))) -> a#(a(x1)) -> b#(x1) a#(b(x1)) -> b#(a(x1)) -> b#(b(x1)) -> a#(c(b(x1))) a#(b(x1)) -> b#(a(x1)) -> b#(b(x1)) -> c#(b(x1)) a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> b#(a(x1)) a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> a#(x1) a#(b(x1)) -> a#(x1) -> a#(a(x1)) -> b#(a(b(x1))) a#(b(x1)) -> a#(x1) -> a#(a(x1)) -> a#(b(x1)) a#(b(x1)) -> a#(x1) -> a#(a(x1)) -> b#(x1) a#(a(x1)) -> b#(a(b(x1))) -> b#(b(x1)) -> a#(c(b(x1))) a#(a(x1)) -> b#(a(b(x1))) -> b#(b(x1)) -> c#(b(x1)) a#(a(x1)) -> b#(x1) -> b#(b(x1)) -> a#(c(b(x1))) a#(a(x1)) -> b#(x1) -> b#(b(x1)) -> c#(b(x1)) a#(a(x1)) -> a#(b(x1)) -> a#(b(x1)) -> b#(a(x1)) a#(a(x1)) -> a#(b(x1)) -> a#(b(x1)) -> a#(x1) a#(a(x1)) -> a#(b(x1)) -> a#(a(x1)) -> b#(a(b(x1))) a#(a(x1)) -> a#(b(x1)) -> a#(a(x1)) -> a#(b(x1)) a#(a(x1)) -> a#(b(x1)) -> a#(a(x1)) -> b#(x1) CDG Processor: DPs: a#(a(x1)) -> b#(x1) a#(a(x1)) -> a#(b(x1)) a#(a(x1)) -> b#(a(b(x1))) b#(b(x1)) -> c#(b(x1)) b#(b(x1)) -> a#(c(b(x1))) c#(c(x1)) -> a#(x1) c#(c(x1)) -> b#(a(x1)) c#(c(x1)) -> c#(b(a(x1))) a#(b(x1)) -> a#(x1) a#(b(x1)) -> b#(a(x1)) TRS: a(a(x1)) -> b(a(b(x1))) b(b(x1)) -> a(c(b(x1))) c(c(x1)) -> c(b(a(x1))) a(b(x1)) -> b(a(x1)) b(c(x1)) -> c(x1) graph: c#(c(x1)) -> c#(b(a(x1))) -> c#(c(x1)) -> a#(x1) c#(c(x1)) -> c#(b(a(x1))) -> c#(c(x1)) -> b#(a(x1)) c#(c(x1)) -> c#(b(a(x1))) -> c#(c(x1)) -> c#(b(a(x1))) c#(c(x1)) -> b#(a(x1)) -> b#(b(x1)) -> c#(b(x1)) c#(c(x1)) -> b#(a(x1)) -> b#(b(x1)) -> a#(c(b(x1))) c#(c(x1)) -> a#(x1) -> a#(a(x1)) -> b#(x1) c#(c(x1)) -> a#(x1) -> a#(a(x1)) -> a#(b(x1)) c#(c(x1)) -> a#(x1) -> a#(a(x1)) -> b#(a(b(x1))) c#(c(x1)) -> a#(x1) -> a#(b(x1)) -> a#(x1) c#(c(x1)) -> a#(x1) -> a#(b(x1)) -> b#(a(x1)) b#(b(x1)) -> c#(b(x1)) -> c#(c(x1)) -> a#(x1) b#(b(x1)) -> c#(b(x1)) -> c#(c(x1)) -> b#(a(x1)) b#(b(x1)) -> c#(b(x1)) -> c#(c(x1)) -> c#(b(a(x1))) a#(b(x1)) -> b#(a(x1)) -> b#(b(x1)) -> c#(b(x1)) a#(b(x1)) -> b#(a(x1)) -> b#(b(x1)) -> a#(c(b(x1))) a#(b(x1)) -> a#(x1) -> a#(a(x1)) -> b#(x1) a#(b(x1)) -> a#(x1) -> a#(a(x1)) -> a#(b(x1)) a#(b(x1)) -> a#(x1) -> a#(a(x1)) -> b#(a(b(x1))) a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> a#(x1) a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> b#(a(x1)) a#(a(x1)) -> b#(a(b(x1))) -> b#(b(x1)) -> c#(b(x1)) a#(a(x1)) -> b#(a(b(x1))) -> b#(b(x1)) -> a#(c(b(x1))) a#(a(x1)) -> b#(x1) -> b#(b(x1)) -> c#(b(x1)) a#(a(x1)) -> b#(x1) -> b#(b(x1)) -> a#(c(b(x1))) a#(a(x1)) -> a#(b(x1)) -> a#(a(x1)) -> b#(x1) a#(a(x1)) -> a#(b(x1)) -> a#(a(x1)) -> a#(b(x1)) a#(a(x1)) -> a#(b(x1)) -> a#(a(x1)) -> b#(a(b(x1))) a#(a(x1)) -> a#(b(x1)) -> a#(b(x1)) -> a#(x1) a#(a(x1)) -> a#(b(x1)) -> a#(b(x1)) -> b#(a(x1)) SCC Processor: #sccs: 1 #rules: 9 #arcs: 29/100 DPs: c#(c(x1)) -> c#(b(a(x1))) c#(c(x1)) -> b#(a(x1)) b#(b(x1)) -> c#(b(x1)) c#(c(x1)) -> a#(x1) a#(b(x1)) -> b#(a(x1)) a#(b(x1)) -> a#(x1) a#(a(x1)) -> b#(a(b(x1))) a#(a(x1)) -> a#(b(x1)) a#(a(x1)) -> b#(x1) TRS: a(a(x1)) -> b(a(b(x1))) b(b(x1)) -> a(c(b(x1))) c(c(x1)) -> c(b(a(x1))) a(b(x1)) -> b(a(x1)) b(c(x1)) -> c(x1) Arctic Interpretation Processor: dimension: 2 usable rules: a(a(x1)) -> b(a(b(x1))) b(b(x1)) -> a(c(b(x1))) c(c(x1)) -> c(b(a(x1))) a(b(x1)) -> b(a(x1)) b(c(x1)) -> c(x1) interpretation: [c#](x0) = [0 -&]x0 + [0], [b#](x0) = [0 0]x0 + [0], [a#](x0) = [0 0]x0 + [0], [0 0] [2 ] [c](x0) = [0 0]x0 + [-&], [0 -&] [0] [b](x0) = [0 0 ]x0 + [2], [-& -&] [0] [a](x0) = [0 0 ]x0 + [2] orientation: c#(c(x1)) = [0 0]x1 + [2] >= [0] = c#(b(a(x1))) c#(c(x1)) = [0 0]x1 + [2] >= [0 0]x1 + [2] = b#(a(x1)) b#(b(x1)) = [0 0]x1 + [2] >= [0 -&]x1 + [0] = c#(b(x1)) c#(c(x1)) = [0 0]x1 + [2] >= [0 0]x1 + [0] = a#(x1) a#(b(x1)) = [0 0]x1 + [2] >= [0 0]x1 + [2] = b#(a(x1)) a#(b(x1)) = [0 0]x1 + [2] >= [0 0]x1 + [0] = a#(x1) a#(a(x1)) = [0 0]x1 + [2] >= [0 0]x1 + [2] = b#(a(b(x1))) a#(a(x1)) = [0 0]x1 + [2] >= [0 0]x1 + [2] = a#(b(x1)) a#(a(x1)) = [0 0]x1 + [2] >= [0 0]x1 + [0] = b#(x1) [-& -&] [0] [-& -&] [0] a(a(x1)) = [0 0 ]x1 + [2] >= [0 0 ]x1 + [2] = b(a(b(x1))) [0 -&] [0] [-& -&] [0] b(b(x1)) = [0 0 ]x1 + [2] >= [0 0 ]x1 + [2] = a(c(b(x1))) [0 0] [2] [0 0] [2] c(c(x1)) = [0 0]x1 + [2] >= [0 0]x1 + [2] = c(b(a(x1))) [-& -&] [0] [-& -&] [0] a(b(x1)) = [0 0 ]x1 + [2] >= [0 0 ]x1 + [2] = b(a(x1)) [0 0] [2] [0 0] [2 ] b(c(x1)) = [0 0]x1 + [2] >= [0 0]x1 + [-&] = c(x1) problem: DPs: c#(c(x1)) -> b#(a(x1)) b#(b(x1)) -> c#(b(x1)) c#(c(x1)) -> a#(x1) a#(b(x1)) -> b#(a(x1)) a#(b(x1)) -> a#(x1) a#(a(x1)) -> b#(a(b(x1))) a#(a(x1)) -> a#(b(x1)) a#(a(x1)) -> b#(x1) TRS: a(a(x1)) -> b(a(b(x1))) b(b(x1)) -> a(c(b(x1))) c(c(x1)) -> c(b(a(x1))) a(b(x1)) -> b(a(x1)) b(c(x1)) -> c(x1) Restore Modifier: DPs: c#(c(x1)) -> b#(a(x1)) b#(b(x1)) -> c#(b(x1)) c#(c(x1)) -> a#(x1) a#(b(x1)) -> b#(a(x1)) a#(b(x1)) -> a#(x1) a#(a(x1)) -> b#(a(b(x1))) a#(a(x1)) -> a#(b(x1)) a#(a(x1)) -> b#(x1) TRS: a(a(x1)) -> b(a(b(x1))) b(b(x1)) -> a(c(b(x1))) c(c(x1)) -> c(b(a(x1))) a(b(x1)) -> b(a(x1)) b(c(x1)) -> c(x1) EDG Processor: DPs: c#(c(x1)) -> b#(a(x1)) b#(b(x1)) -> c#(b(x1)) c#(c(x1)) -> a#(x1) a#(b(x1)) -> b#(a(x1)) a#(b(x1)) -> a#(x1) a#(a(x1)) -> b#(a(b(x1))) a#(a(x1)) -> a#(b(x1)) a#(a(x1)) -> b#(x1) TRS: a(a(x1)) -> b(a(b(x1))) b(b(x1)) -> a(c(b(x1))) c(c(x1)) -> c(b(a(x1))) a(b(x1)) -> b(a(x1)) b(c(x1)) -> c(x1) graph: c#(c(x1)) -> b#(a(x1)) -> b#(b(x1)) -> c#(b(x1)) c#(c(x1)) -> a#(x1) -> a#(b(x1)) -> b#(a(x1)) c#(c(x1)) -> a#(x1) -> a#(b(x1)) -> a#(x1) c#(c(x1)) -> a#(x1) -> a#(a(x1)) -> b#(a(b(x1))) c#(c(x1)) -> a#(x1) -> a#(a(x1)) -> a#(b(x1)) c#(c(x1)) -> a#(x1) -> a#(a(x1)) -> b#(x1) b#(b(x1)) -> c#(b(x1)) -> c#(c(x1)) -> b#(a(x1)) b#(b(x1)) -> c#(b(x1)) -> c#(c(x1)) -> a#(x1) a#(b(x1)) -> b#(a(x1)) -> b#(b(x1)) -> c#(b(x1)) a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> b#(a(x1)) a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> a#(x1) a#(b(x1)) -> a#(x1) -> a#(a(x1)) -> b#(a(b(x1))) a#(b(x1)) -> a#(x1) -> a#(a(x1)) -> a#(b(x1)) a#(b(x1)) -> a#(x1) -> a#(a(x1)) -> b#(x1) a#(a(x1)) -> b#(a(b(x1))) -> b#(b(x1)) -> c#(b(x1)) a#(a(x1)) -> b#(x1) -> b#(b(x1)) -> c#(b(x1)) a#(a(x1)) -> a#(b(x1)) -> a#(b(x1)) -> b#(a(x1)) a#(a(x1)) -> a#(b(x1)) -> a#(b(x1)) -> a#(x1) a#(a(x1)) -> a#(b(x1)) -> a#(a(x1)) -> b#(a(b(x1))) a#(a(x1)) -> a#(b(x1)) -> a#(a(x1)) -> a#(b(x1)) a#(a(x1)) -> a#(b(x1)) -> a#(a(x1)) -> b#(x1) Arctic Interpretation Processor: dimension: 2 usable rules: a(a(x1)) -> b(a(b(x1))) b(b(x1)) -> a(c(b(x1))) c(c(x1)) -> c(b(a(x1))) a(b(x1)) -> b(a(x1)) b(c(x1)) -> c(x1) interpretation: [c#](x0) = [0 -&]x0 + [0], [b#](x0) = [0 -&]x0 + [0], [a#](x0) = [0 0]x0 + [0], [0 0] [0] [c](x0) = [0 0]x0 + [1], [0 -&] [0] [b](x0) = [0 0 ]x0 + [1], [-& -&] [0] [a](x0) = [0 0 ]x0 + [1] orientation: c#(c(x1)) = [0 0]x1 + [0] >= [0] = b#(a(x1)) b#(b(x1)) = [0 -&]x1 + [0] >= [0 -&]x1 + [0] = c#(b(x1)) c#(c(x1)) = [0 0]x1 + [0] >= [0 0]x1 + [0] = a#(x1) a#(b(x1)) = [0 0]x1 + [1] >= [0] = b#(a(x1)) a#(b(x1)) = [0 0]x1 + [1] >= [0 0]x1 + [0] = a#(x1) a#(a(x1)) = [0 0]x1 + [1] >= [0] = b#(a(b(x1))) a#(a(x1)) = [0 0]x1 + [1] >= [0 0]x1 + [1] = a#(b(x1)) a#(a(x1)) = [0 0]x1 + [1] >= [0 -&]x1 + [0] = b#(x1) [-& -&] [0] [-& -&] [0] a(a(x1)) = [0 0 ]x1 + [1] >= [0 0 ]x1 + [1] = b(a(b(x1))) [0 -&] [0] [-& -&] [0] b(b(x1)) = [0 0 ]x1 + [1] >= [0 0 ]x1 + [1] = a(c(b(x1))) [0 0] [1] [0 0] [1] c(c(x1)) = [0 0]x1 + [1] >= [0 0]x1 + [1] = c(b(a(x1))) [-& -&] [0] [-& -&] [0] a(b(x1)) = [0 0 ]x1 + [1] >= [0 0 ]x1 + [1] = b(a(x1)) [0 0] [0] [0 0] [0] b(c(x1)) = [0 0]x1 + [1] >= [0 0]x1 + [1] = c(x1) problem: DPs: c#(c(x1)) -> b#(a(x1)) b#(b(x1)) -> c#(b(x1)) c#(c(x1)) -> a#(x1) a#(b(x1)) -> a#(x1) a#(a(x1)) -> a#(b(x1)) a#(a(x1)) -> b#(x1) TRS: a(a(x1)) -> b(a(b(x1))) b(b(x1)) -> a(c(b(x1))) c(c(x1)) -> c(b(a(x1))) a(b(x1)) -> b(a(x1)) b(c(x1)) -> c(x1) Restore Modifier: DPs: c#(c(x1)) -> b#(a(x1)) b#(b(x1)) -> c#(b(x1)) c#(c(x1)) -> a#(x1) a#(b(x1)) -> a#(x1) a#(a(x1)) -> a#(b(x1)) a#(a(x1)) -> b#(x1) TRS: a(a(x1)) -> b(a(b(x1))) b(b(x1)) -> a(c(b(x1))) c(c(x1)) -> c(b(a(x1))) a(b(x1)) -> b(a(x1)) b(c(x1)) -> c(x1) EDG Processor: DPs: c#(c(x1)) -> b#(a(x1)) b#(b(x1)) -> c#(b(x1)) c#(c(x1)) -> a#(x1) a#(b(x1)) -> a#(x1) a#(a(x1)) -> a#(b(x1)) a#(a(x1)) -> b#(x1) TRS: a(a(x1)) -> b(a(b(x1))) b(b(x1)) -> a(c(b(x1))) c(c(x1)) -> c(b(a(x1))) a(b(x1)) -> b(a(x1)) b(c(x1)) -> c(x1) graph: c#(c(x1)) -> b#(a(x1)) -> b#(b(x1)) -> c#(b(x1)) c#(c(x1)) -> a#(x1) -> a#(a(x1)) -> b#(x1) c#(c(x1)) -> a#(x1) -> a#(a(x1)) -> a#(b(x1)) c#(c(x1)) -> a#(x1) -> a#(b(x1)) -> a#(x1) b#(b(x1)) -> c#(b(x1)) -> c#(c(x1)) -> a#(x1) b#(b(x1)) -> c#(b(x1)) -> c#(c(x1)) -> b#(a(x1)) a#(b(x1)) -> a#(x1) -> a#(a(x1)) -> b#(x1) a#(b(x1)) -> a#(x1) -> a#(a(x1)) -> a#(b(x1)) a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> a#(x1) a#(a(x1)) -> b#(x1) -> b#(b(x1)) -> c#(b(x1)) a#(a(x1)) -> a#(b(x1)) -> a#(a(x1)) -> b#(x1) a#(a(x1)) -> a#(b(x1)) -> a#(a(x1)) -> a#(b(x1)) a#(a(x1)) -> a#(b(x1)) -> a#(b(x1)) -> a#(x1) Arctic Interpretation Processor: dimension: 2 usable rules: a(a(x1)) -> b(a(b(x1))) b(b(x1)) -> a(c(b(x1))) c(c(x1)) -> c(b(a(x1))) a(b(x1)) -> b(a(x1)) b(c(x1)) -> c(x1) interpretation: [c#](x0) = [-& 3 ]x0 + [0], [b#](x0) = [0 3]x0 + [0], [a#](x0) = [3 0]x0 + [0], [0 0] [-&] [c](x0) = [0 0]x0 + [0 ], [0 0 ] [0 ] [b](x0) = [-& 0 ]x0 + [-&], [0 0 ] [0 ] [a](x0) = [-& -&]x0 + [-&] orientation: c#(c(x1)) = [3 3]x1 + [3] >= [0 0]x1 + [0] = b#(a(x1)) b#(b(x1)) = [0 3]x1 + [0] >= [-& 3 ]x1 + [0] = c#(b(x1)) c#(c(x1)) = [3 3]x1 + [3] >= [3 0]x1 + [0] = a#(x1) a#(b(x1)) = [3 3]x1 + [3] >= [3 0]x1 + [0] = a#(x1) a#(a(x1)) = [3 3]x1 + [3] >= [3 3]x1 + [3] = a#(b(x1)) a#(a(x1)) = [3 3]x1 + [3] >= [0 3]x1 + [0] = b#(x1) [0 0 ] [0 ] [0 0 ] [0 ] a(a(x1)) = [-& -&]x1 + [-&] >= [-& -&]x1 + [-&] = b(a(b(x1))) [0 0 ] [0 ] [0 0 ] [0 ] b(b(x1)) = [-& 0 ]x1 + [-&] >= [-& -&]x1 + [-&] = a(c(b(x1))) [0 0] [0] [0 0] [0] c(c(x1)) = [0 0]x1 + [0] >= [0 0]x1 + [0] = c(b(a(x1))) [0 0 ] [0 ] [0 0 ] [0 ] a(b(x1)) = [-& -&]x1 + [-&] >= [-& -&]x1 + [-&] = b(a(x1)) [0 0] [0] [0 0] [-&] b(c(x1)) = [0 0]x1 + [0] >= [0 0]x1 + [0 ] = c(x1) problem: DPs: b#(b(x1)) -> c#(b(x1)) c#(c(x1)) -> a#(x1) a#(b(x1)) -> a#(x1) a#(a(x1)) -> a#(b(x1)) a#(a(x1)) -> b#(x1) TRS: a(a(x1)) -> b(a(b(x1))) b(b(x1)) -> a(c(b(x1))) c(c(x1)) -> c(b(a(x1))) a(b(x1)) -> b(a(x1)) b(c(x1)) -> c(x1) Restore Modifier: DPs: b#(b(x1)) -> c#(b(x1)) c#(c(x1)) -> a#(x1) a#(b(x1)) -> a#(x1) a#(a(x1)) -> a#(b(x1)) a#(a(x1)) -> b#(x1) TRS: a(a(x1)) -> b(a(b(x1))) b(b(x1)) -> a(c(b(x1))) c(c(x1)) -> c(b(a(x1))) a(b(x1)) -> b(a(x1)) b(c(x1)) -> c(x1) EDG Processor: DPs: b#(b(x1)) -> c#(b(x1)) c#(c(x1)) -> a#(x1) a#(b(x1)) -> a#(x1) a#(a(x1)) -> a#(b(x1)) a#(a(x1)) -> b#(x1) TRS: a(a(x1)) -> b(a(b(x1))) b(b(x1)) -> a(c(b(x1))) c(c(x1)) -> c(b(a(x1))) a(b(x1)) -> b(a(x1)) b(c(x1)) -> c(x1) graph: c#(c(x1)) -> a#(x1) -> a#(b(x1)) -> a#(x1) c#(c(x1)) -> a#(x1) -> a#(a(x1)) -> a#(b(x1)) c#(c(x1)) -> a#(x1) -> a#(a(x1)) -> b#(x1) b#(b(x1)) -> c#(b(x1)) -> c#(c(x1)) -> a#(x1) a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> a#(x1) a#(b(x1)) -> a#(x1) -> a#(a(x1)) -> a#(b(x1)) a#(b(x1)) -> a#(x1) -> a#(a(x1)) -> b#(x1) a#(a(x1)) -> b#(x1) -> b#(b(x1)) -> c#(b(x1)) a#(a(x1)) -> a#(b(x1)) -> a#(b(x1)) -> a#(x1) a#(a(x1)) -> a#(b(x1)) -> a#(a(x1)) -> a#(b(x1)) a#(a(x1)) -> a#(b(x1)) -> a#(a(x1)) -> b#(x1) Arctic Interpretation Processor: dimension: 3 usable rules: a(a(x1)) -> b(a(b(x1))) b(b(x1)) -> a(c(b(x1))) c(c(x1)) -> c(b(a(x1))) a(b(x1)) -> b(a(x1)) b(c(x1)) -> c(x1) interpretation: [c#](x0) = [0 0 0]x0 + [0], [b#](x0) = [1 0 1]x0 + [1], [a#](x0) = [0 0 0]x0 + [1], [-& 0 0 ] [0] [c](x0) = [0 1 0 ]x0 + [1] [-& 0 0 ] [0], [0 0 1 ] [-&] [b](x0) = [-& 0 -&]x0 + [0 ] [-& 0 -&] [0 ], [1 0 1 ] [0] [a](x0) = [-& -& -&]x0 + [0] [0 -& 0 ] [0] orientation: b#(b(x1)) = [1 1 2]x1 + [1] >= [0 0 1]x1 + [0] = c#(b(x1)) c#(c(x1)) = [0 1 0]x1 + [1] >= [0 0 0]x1 + [1] = a#(x1) a#(b(x1)) = [0 0 1]x1 + [1] >= [0 0 0]x1 + [1] = a#(x1) a#(a(x1)) = [1 0 1]x1 + [1] >= [0 0 1]x1 + [1] = a#(b(x1)) a#(a(x1)) = [1 0 1]x1 + [1] >= [1 0 1]x1 + [1] = b#(x1) [2 1 2 ] [1] [1 1 2 ] [1] a(a(x1)) = [-& -& -&]x1 + [0] >= [-& -& -&]x1 + [0] = b(a(b(x1))) [1 0 1 ] [0] [-& -& -&] [0] [0 1 1 ] [1] [0 1 1 ] [1] b(b(x1)) = [-& 0 -&]x1 + [0] >= [-& -& -&]x1 + [0] = a(c(b(x1))) [-& 0 -&] [0] [-& 0 -&] [0] [0 1 0] [1] [-& -& -&] [0] c(c(x1)) = [1 2 1]x1 + [2] >= [1 0 1 ]x1 + [1] = c(b(a(x1))) [0 1 0] [1] [-& -& -&] [0] [1 1 2 ] [1] [1 0 1 ] [1] a(b(x1)) = [-& -& -&]x1 + [0] >= [-& -& -&]x1 + [0] = b(a(x1)) [0 0 1 ] [0] [-& -& -&] [0] [0 1 1] [1] [-& 0 0 ] [0] b(c(x1)) = [0 1 0]x1 + [1] >= [0 1 0 ]x1 + [1] = c(x1) [0 1 0] [1] [-& 0 0 ] [0] problem: DPs: c#(c(x1)) -> a#(x1) a#(b(x1)) -> a#(x1) a#(a(x1)) -> a#(b(x1)) a#(a(x1)) -> b#(x1) TRS: a(a(x1)) -> b(a(b(x1))) b(b(x1)) -> a(c(b(x1))) c(c(x1)) -> c(b(a(x1))) a(b(x1)) -> b(a(x1)) b(c(x1)) -> c(x1) Restore Modifier: DPs: c#(c(x1)) -> a#(x1) a#(b(x1)) -> a#(x1) a#(a(x1)) -> a#(b(x1)) a#(a(x1)) -> b#(x1) TRS: a(a(x1)) -> b(a(b(x1))) b(b(x1)) -> a(c(b(x1))) c(c(x1)) -> c(b(a(x1))) a(b(x1)) -> b(a(x1)) b(c(x1)) -> c(x1) EDG Processor: DPs: c#(c(x1)) -> a#(x1) a#(b(x1)) -> a#(x1) a#(a(x1)) -> a#(b(x1)) a#(a(x1)) -> b#(x1) TRS: a(a(x1)) -> b(a(b(x1))) b(b(x1)) -> a(c(b(x1))) c(c(x1)) -> c(b(a(x1))) a(b(x1)) -> b(a(x1)) b(c(x1)) -> c(x1) graph: c#(c(x1)) -> a#(x1) -> a#(a(x1)) -> b#(x1) c#(c(x1)) -> a#(x1) -> a#(a(x1)) -> a#(b(x1)) c#(c(x1)) -> a#(x1) -> a#(b(x1)) -> a#(x1) a#(b(x1)) -> a#(x1) -> a#(a(x1)) -> b#(x1) a#(b(x1)) -> a#(x1) -> a#(a(x1)) -> a#(b(x1)) a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> a#(x1) a#(a(x1)) -> a#(b(x1)) -> a#(a(x1)) -> b#(x1) a#(a(x1)) -> a#(b(x1)) -> a#(a(x1)) -> a#(b(x1)) a#(a(x1)) -> a#(b(x1)) -> a#(b(x1)) -> a#(x1) SCC Processor: #sccs: 1 #rules: 2 #arcs: 9/16 DPs: a#(b(x1)) -> a#(x1) a#(a(x1)) -> a#(b(x1)) TRS: a(a(x1)) -> b(a(b(x1))) b(b(x1)) -> a(c(b(x1))) c(c(x1)) -> c(b(a(x1))) a(b(x1)) -> b(a(x1)) b(c(x1)) -> c(x1) Arctic Interpretation Processor: dimension: 2 usable rules: a(a(x1)) -> b(a(b(x1))) b(b(x1)) -> a(c(b(x1))) c(c(x1)) -> c(b(a(x1))) a(b(x1)) -> b(a(x1)) b(c(x1)) -> c(x1) interpretation: [a#](x0) = [0 0]x0, [0 ] [c](x0) = [-&], [-& 0 ] [1] [b](x0) = [0 1 ]x0 + [2], [0 1] [1] [a](x0) = [1 2]x0 + [3] orientation: a#(b(x1)) = [0 1]x1 + [2] >= [0 0]x1 = a#(x1) a#(a(x1)) = [1 2]x1 + [3] >= [0 1]x1 + [2] = a#(b(x1)) [2 3] [4] [2 3] [4] a(a(x1)) = [3 4]x1 + [5] >= [3 4]x1 + [5] = b(a(b(x1))) [0 1] [2] [1] b(b(x1)) = [1 2]x1 + [3] >= [3] = a(c(b(x1))) [0 ] [0 ] c(c(x1)) = [-&] >= [-&] = c(b(a(x1))) [1 2] [3] [1 2] [3] a(b(x1)) = [2 3]x1 + [4] >= [2 3]x1 + [4] = b(a(x1)) [1] [0 ] b(c(x1)) = [2] >= [-&] = c(x1) problem: DPs: a#(b(x1)) -> a#(x1) TRS: a(a(x1)) -> b(a(b(x1))) b(b(x1)) -> a(c(b(x1))) c(c(x1)) -> c(b(a(x1))) a(b(x1)) -> b(a(x1)) b(c(x1)) -> c(x1) Restore Modifier: DPs: a#(b(x1)) -> a#(x1) TRS: a(a(x1)) -> b(a(b(x1))) b(b(x1)) -> a(c(b(x1))) c(c(x1)) -> c(b(a(x1))) a(b(x1)) -> b(a(x1)) b(c(x1)) -> c(x1) EDG Processor: DPs: a#(b(x1)) -> a#(x1) TRS: a(a(x1)) -> b(a(b(x1))) b(b(x1)) -> a(c(b(x1))) c(c(x1)) -> c(b(a(x1))) a(b(x1)) -> b(a(x1)) b(c(x1)) -> c(x1) graph: a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> a#(x1) CDG Processor: DPs: a#(b(x1)) -> a#(x1) TRS: a(a(x1)) -> b(a(b(x1))) b(b(x1)) -> a(c(b(x1))) c(c(x1)) -> c(b(a(x1))) a(b(x1)) -> b(a(x1)) b(c(x1)) -> c(x1) graph: Qed