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