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) CDG 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: a#(a(b(x1))) -> a#(a(a(x1))) -> a#(a(b(x1))) -> 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#(a(a(x1))) a#(a(b(x1))) -> a#(a(a(x1))) -> a#(a(b(x1))) -> b#(a(a(a(x1)))) a#(a(b(x1))) -> a#(a(x1)) -> a#(a(b(x1))) -> 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#(a(a(x1))) a#(a(b(x1))) -> a#(a(x1)) -> a#(a(b(x1))) -> b#(a(a(a(x1)))) a#(a(b(x1))) -> a#(x1) -> a#(a(b(x1))) -> 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#(a(a(x1))) a#(a(b(x1))) -> a#(x1) -> a#(a(b(x1))) -> b#(a(a(a(x1)))) 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))) -> 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#(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))) -> b#(x1) b#(a(a(x1))) -> b#(x1) -> b#(a(a(x1))) -> a#(b(x1)) SCC Processor: #sccs: 1 #rules: 6 #arcs: 20/64 DPs: a#(a(b(x1))) -> a#(a(a(x1))) a#(a(b(x1))) -> b#(a(a(a(x1)))) b#(a(a(x1))) -> a#(b(x1)) a#(a(b(x1))) -> a#(a(x1)) a#(a(b(x1))) -> a#(x1) b#(a(a(x1))) -> b#(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: a#(a(b(x1))) -> a#(a(a(x1))) a#(a(b(x1))) -> b#(a(a(a(x1)))) b#(a(a(x1))) -> a#(b(x1)) a#(a(b(x1))) -> a#(a(x1)) a#(a(b(x1))) -> a#(x1) b#(a(a(x1))) -> b#(x1) TRS: a(a(b(x1))) -> b(a(a(a(x1)))) b(a(a(x1))) -> a(b(x1)) Arctic Interpretation Processor: dimension: 2 usable rules: a(a(b(x1))) -> b(a(a(a(x1)))) b(a(a(x1))) -> a(b(x1)) interpretation: [a#](x0) = [-& 0 ]x0 + [2], [b#](x0) = [-& 3 ]x0 + [0], [0 -&] [0] [a](x0) = [0 0 ]x0 + [0], [3 -&] [3 ] [b](x0) = [-& 3 ]x0 + [-&] orientation: a#(a(b(x1))) = [3 3]x1 + [3] >= [0 0]x1 + [2] = a#(a(a(x1))) a#(a(b(x1))) = [3 3]x1 + [3] >= [3 3]x1 + [3] = b#(a(a(a(x1)))) b#(a(a(x1))) = [3 3]x1 + [3] >= [-& 3 ]x1 + [2] = a#(b(x1)) a#(a(b(x1))) = [3 3]x1 + [3] >= [0 0]x1 + [2] = a#(a(x1)) a#(a(b(x1))) = [3 3]x1 + [3] >= [-& 0 ]x1 + [2] = a#(x1) b#(a(a(x1))) = [3 3]x1 + [3] >= [-& 3 ]x1 + [0] = b#(x1) [3 -&] [3] [3 -&] [3] a(a(b(x1))) = [3 3 ]x1 + [3] >= [3 3 ]x1 + [3] = b(a(a(a(x1)))) [3 -&] [3] [3 -&] [3] b(a(a(x1))) = [3 3 ]x1 + [3] >= [3 3 ]x1 + [3] = a(b(x1)) problem: DPs: a#(a(b(x1))) -> b#(a(a(a(x1)))) b#(a(a(x1))) -> a#(b(x1)) b#(a(a(x1))) -> b#(x1) TRS: a(a(b(x1))) -> b(a(a(a(x1)))) b(a(a(x1))) -> a(b(x1)) Restore Modifier: DPs: a#(a(b(x1))) -> b#(a(a(a(x1)))) b#(a(a(x1))) -> a#(b(x1)) b#(a(a(x1))) -> b#(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)))) b#(a(a(x1))) -> a#(b(x1)) b#(a(a(x1))) -> b#(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))) -> 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))) -> b#(x1) -> b#(a(a(x1))) -> a#(b(x1)) b#(a(a(x1))) -> b#(x1) -> b#(a(a(x1))) -> b#(x1) Usable Rule Processor: DPs: a#(a(b(x1))) -> b#(a(a(a(x1)))) b#(a(a(x1))) -> a#(b(x1)) b#(a(a(x1))) -> b#(x1) TRS: a(a(b(x1))) -> b(a(a(a(x1)))) b(a(a(x1))) -> a(b(x1)) Matrix Interpretation Processor: dim=2 interpretation: [a#](x0) = [1 2]x0 + [2], [b#](x0) = [0 2]x0 + [1], [1 0] [0] [a](x0) = [1 1]x0 + [1], [2 0] [1] [b](x0) = [0 1]x0 + [0] orientation: a#(a(b(x1))) = [6 2]x1 + [7] >= [6 2]x1 + [7] = b#(a(a(a(x1)))) b#(a(a(x1))) = [4 2]x1 + [5] >= [2 2]x1 + [3] = a#(b(x1)) b#(a(a(x1))) = [4 2]x1 + [5] >= [0 2]x1 + [1] = b#(x1) [2 0] [1] [2 0] [1] a(a(b(x1))) = [4 1]x1 + [4] >= [3 1]x1 + [3] = b(a(a(a(x1)))) [2 0] [1] [2 0] [1] b(a(a(x1))) = [2 1]x1 + [2] >= [2 1]x1 + [2] = a(b(x1)) problem: DPs: a#(a(b(x1))) -> b#(a(a(a(x1)))) TRS: a(a(b(x1))) -> b(a(a(a(x1)))) b(a(a(x1))) -> a(b(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: Qed