YES Problem: a(a(b(x1))) -> b(a(x1)) b(a(a(x1))) -> a(a(a(b(x1)))) a(c(x1)) -> c(b(x1)) Proof: String Reversal Processor: b(a(a(x1))) -> a(b(x1)) a(a(b(x1))) -> b(a(a(a(x1)))) c(a(x1)) -> b(c(x1)) DP Processor: DPs: b#(a(a(x1))) -> b#(x1) b#(a(a(x1))) -> a#(b(x1)) a#(a(b(x1))) -> a#(x1) a#(a(b(x1))) -> a#(a(x1)) a#(a(b(x1))) -> a#(a(a(x1))) a#(a(b(x1))) -> b#(a(a(a(x1)))) c#(a(x1)) -> c#(x1) c#(a(x1)) -> b#(c(x1)) TRS: b(a(a(x1))) -> a(b(x1)) a(a(b(x1))) -> b(a(a(a(x1)))) c(a(x1)) -> b(c(x1)) TDG Processor: DPs: b#(a(a(x1))) -> b#(x1) b#(a(a(x1))) -> a#(b(x1)) a#(a(b(x1))) -> a#(x1) a#(a(b(x1))) -> a#(a(x1)) a#(a(b(x1))) -> a#(a(a(x1))) a#(a(b(x1))) -> b#(a(a(a(x1)))) c#(a(x1)) -> c#(x1) c#(a(x1)) -> b#(c(x1)) TRS: b(a(a(x1))) -> a(b(x1)) a(a(b(x1))) -> b(a(a(a(x1)))) c(a(x1)) -> b(c(x1)) graph: c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> b#(c(x1)) c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> c#(x1) c#(a(x1)) -> b#(c(x1)) -> b#(a(a(x1))) -> a#(b(x1)) c#(a(x1)) -> b#(c(x1)) -> b#(a(a(x1))) -> b#(x1) a#(a(b(x1))) -> a#(a(a(x1))) -> a#(a(b(x1))) -> b#(a(a(a(x1)))) a#(a(b(x1))) -> a#(a(a(x1))) -> a#(a(b(x1))) -> a#(a(a(x1))) a#(a(b(x1))) -> a#(a(a(x1))) -> a#(a(b(x1))) -> a#(a(x1)) a#(a(b(x1))) -> a#(a(a(x1))) -> a#(a(b(x1))) -> a#(x1) a#(a(b(x1))) -> a#(a(x1)) -> a#(a(b(x1))) -> b#(a(a(a(x1)))) a#(a(b(x1))) -> a#(a(x1)) -> a#(a(b(x1))) -> a#(a(a(x1))) a#(a(b(x1))) -> a#(a(x1)) -> a#(a(b(x1))) -> a#(a(x1)) a#(a(b(x1))) -> a#(a(x1)) -> a#(a(b(x1))) -> a#(x1) a#(a(b(x1))) -> a#(x1) -> a#(a(b(x1))) -> b#(a(a(a(x1)))) a#(a(b(x1))) -> a#(x1) -> a#(a(b(x1))) -> a#(a(a(x1))) a#(a(b(x1))) -> a#(x1) -> a#(a(b(x1))) -> a#(a(x1)) a#(a(b(x1))) -> a#(x1) -> a#(a(b(x1))) -> a#(x1) a#(a(b(x1))) -> b#(a(a(a(x1)))) -> b#(a(a(x1))) -> a#(b(x1)) a#(a(b(x1))) -> b#(a(a(a(x1)))) -> b#(a(a(x1))) -> b#(x1) b#(a(a(x1))) -> a#(b(x1)) -> a#(a(b(x1))) -> b#(a(a(a(x1)))) b#(a(a(x1))) -> a#(b(x1)) -> a#(a(b(x1))) -> a#(a(a(x1))) b#(a(a(x1))) -> a#(b(x1)) -> a#(a(b(x1))) -> a#(a(x1)) b#(a(a(x1))) -> a#(b(x1)) -> a#(a(b(x1))) -> a#(x1) b#(a(a(x1))) -> b#(x1) -> b#(a(a(x1))) -> a#(b(x1)) b#(a(a(x1))) -> b#(x1) -> b#(a(a(x1))) -> b#(x1) SCC Processor: #sccs: 2 #rules: 7 #arcs: 24/64 DPs: c#(a(x1)) -> c#(x1) TRS: b(a(a(x1))) -> a(b(x1)) a(a(b(x1))) -> b(a(a(a(x1)))) c(a(x1)) -> b(c(x1)) Usable Rule Processor: DPs: c#(a(x1)) -> c#(x1) TRS: Arctic Interpretation Processor: dimension: 4 usable rules: interpretation: [1 0 0 0] [0] [0 0 0 0] [0] [a](x0) = [1 0 0 0]x0 + [0] [0 0 0 0] [0], [c#](x0) = [0 -& -& -&]x0 orientation: c#(a(x1)) = [1 0 0 0]x1 + [0] >= [0 -& -& -&]x1 = c#(x1) problem: DPs: TRS: Qed DPs: b#(a(a(x1))) -> b#(x1) b#(a(a(x1))) -> a#(b(x1)) a#(a(b(x1))) -> a#(x1) a#(a(b(x1))) -> a#(a(x1)) a#(a(b(x1))) -> a#(a(a(x1))) a#(a(b(x1))) -> b#(a(a(a(x1)))) TRS: b(a(a(x1))) -> a(b(x1)) a(a(b(x1))) -> b(a(a(a(x1)))) c(a(x1)) -> b(c(x1)) Usable Rule Processor: DPs: b#(a(a(x1))) -> b#(x1) b#(a(a(x1))) -> a#(b(x1)) a#(a(b(x1))) -> a#(x1) a#(a(b(x1))) -> a#(a(x1)) a#(a(b(x1))) -> a#(a(a(x1))) a#(a(b(x1))) -> b#(a(a(a(x1)))) TRS: b(a(a(x1))) -> a(b(x1)) a(a(b(x1))) -> b(a(a(a(x1)))) Arctic Interpretation Processor: dimension: 1 usable rules: b(a(a(x1))) -> a(b(x1)) a(a(b(x1))) -> b(a(a(a(x1)))) interpretation: [a#](x0) = 1x0 + 0, [a](x0) = x0 + 3, [b#](x0) = 5x0 + 8, [b](x0) = 4x0 + 7 orientation: b#(a(a(x1))) = 5x1 + 8 >= 5x1 + 8 = b#(x1) b#(a(a(x1))) = 5x1 + 8 >= 5x1 + 8 = a#(b(x1)) a#(a(b(x1))) = 5x1 + 8 >= 1x1 + 0 = a#(x1) a#(a(b(x1))) = 5x1 + 8 >= 1x1 + 4 = a#(a(x1)) a#(a(b(x1))) = 5x1 + 8 >= 1x1 + 4 = a#(a(a(x1))) a#(a(b(x1))) = 5x1 + 8 >= 5x1 + 8 = b#(a(a(a(x1)))) b(a(a(x1))) = 4x1 + 7 >= 4x1 + 7 = a(b(x1)) a(a(b(x1))) = 4x1 + 7 >= 4x1 + 7 = b(a(a(a(x1)))) problem: DPs: b#(a(a(x1))) -> b#(x1) b#(a(a(x1))) -> a#(b(x1)) a#(a(b(x1))) -> b#(a(a(a(x1)))) TRS: b(a(a(x1))) -> a(b(x1)) a(a(b(x1))) -> b(a(a(a(x1)))) Restore Modifier: DPs: b#(a(a(x1))) -> b#(x1) b#(a(a(x1))) -> a#(b(x1)) a#(a(b(x1))) -> b#(a(a(a(x1)))) TRS: b(a(a(x1))) -> a(b(x1)) a(a(b(x1))) -> b(a(a(a(x1)))) c(a(x1)) -> b(c(x1)) EDG Processor: DPs: b#(a(a(x1))) -> b#(x1) b#(a(a(x1))) -> a#(b(x1)) a#(a(b(x1))) -> b#(a(a(a(x1)))) TRS: b(a(a(x1))) -> a(b(x1)) a(a(b(x1))) -> b(a(a(a(x1)))) c(a(x1)) -> b(c(x1)) graph: a#(a(b(x1))) -> b#(a(a(a(x1)))) -> b#(a(a(x1))) -> b#(x1) a#(a(b(x1))) -> b#(a(a(a(x1)))) -> b#(a(a(x1))) -> a#(b(x1)) b#(a(a(x1))) -> a#(b(x1)) -> a#(a(b(x1))) -> b#(a(a(a(x1)))) b#(a(a(x1))) -> b#(x1) -> b#(a(a(x1))) -> b#(x1) b#(a(a(x1))) -> b#(x1) -> b#(a(a(x1))) -> a#(b(x1)) Usable Rule Processor: DPs: b#(a(a(x1))) -> b#(x1) b#(a(a(x1))) -> a#(b(x1)) a#(a(b(x1))) -> b#(a(a(a(x1)))) TRS: b(a(a(x1))) -> a(b(x1)) a(a(b(x1))) -> b(a(a(a(x1)))) Matrix Interpretation Processor: dim=2 interpretation: [a#](x0) = [1 1]x0 + [1], [1 0] [0] [a](x0) = [1 1]x0 + [1], [b#](x0) = [0 2]x0, [3 0] [2] [b](x0) = [0 2]x0 + [0] orientation: b#(a(a(x1))) = [4 2]x1 + [4] >= [0 2]x1 = b#(x1) b#(a(a(x1))) = [4 2]x1 + [4] >= [3 2]x1 + [3] = a#(b(x1)) a#(a(b(x1))) = [6 2]x1 + [6] >= [6 2]x1 + [6] = b#(a(a(a(x1)))) [3 0] [2] [3 0] [2] b(a(a(x1))) = [4 2]x1 + [4] >= [3 2]x1 + [3] = a(b(x1)) [3 0] [2] [3 0] [2] a(a(b(x1))) = [6 2]x1 + [6] >= [6 2]x1 + [6] = b(a(a(a(x1)))) problem: DPs: a#(a(b(x1))) -> b#(a(a(a(x1)))) TRS: b(a(a(x1))) -> a(b(x1)) a(a(b(x1))) -> b(a(a(a(x1)))) Restore Modifier: DPs: a#(a(b(x1))) -> b#(a(a(a(x1)))) TRS: b(a(a(x1))) -> a(b(x1)) a(a(b(x1))) -> b(a(a(a(x1)))) c(a(x1)) -> b(c(x1)) EDG Processor: DPs: a#(a(b(x1))) -> b#(a(a(a(x1)))) TRS: b(a(a(x1))) -> a(b(x1)) a(a(b(x1))) -> b(a(a(a(x1)))) c(a(x1)) -> b(c(x1)) graph: SCC Processor: #sccs: 0 #rules: 0 #arcs: 0/1