YES Problem: a(a(b(x1))) -> b(a(a(a(x1)))) b(a(b(a(x1)))) -> a(b(b(x1))) Proof: String Reversal Processor: b(a(a(x1))) -> a(a(a(b(x1)))) a(b(a(b(x1)))) -> b(b(a(x1))) DP Processor: DPs: b#(a(a(x1))) -> b#(x1) b#(a(a(x1))) -> a#(b(x1)) b#(a(a(x1))) -> a#(a(b(x1))) b#(a(a(x1))) -> a#(a(a(b(x1)))) a#(b(a(b(x1)))) -> a#(x1) a#(b(a(b(x1)))) -> b#(a(x1)) a#(b(a(b(x1)))) -> b#(b(a(x1))) TRS: b(a(a(x1))) -> a(a(a(b(x1)))) a(b(a(b(x1)))) -> b(b(a(x1))) TDG Processor: DPs: b#(a(a(x1))) -> b#(x1) b#(a(a(x1))) -> a#(b(x1)) b#(a(a(x1))) -> a#(a(b(x1))) b#(a(a(x1))) -> a#(a(a(b(x1)))) a#(b(a(b(x1)))) -> a#(x1) a#(b(a(b(x1)))) -> b#(a(x1)) a#(b(a(b(x1)))) -> b#(b(a(x1))) TRS: b(a(a(x1))) -> a(a(a(b(x1)))) a(b(a(b(x1)))) -> b(b(a(x1))) graph: a#(b(a(b(x1)))) -> a#(x1) -> a#(b(a(b(x1)))) -> b#(b(a(x1))) a#(b(a(b(x1)))) -> a#(x1) -> a#(b(a(b(x1)))) -> b#(a(x1)) a#(b(a(b(x1)))) -> a#(x1) -> a#(b(a(b(x1)))) -> a#(x1) a#(b(a(b(x1)))) -> b#(a(x1)) -> b#(a(a(x1))) -> a#(a(a(b(x1)))) a#(b(a(b(x1)))) -> b#(a(x1)) -> b#(a(a(x1))) -> a#(a(b(x1))) a#(b(a(b(x1)))) -> b#(a(x1)) -> b#(a(a(x1))) -> a#(b(x1)) a#(b(a(b(x1)))) -> b#(a(x1)) -> b#(a(a(x1))) -> b#(x1) a#(b(a(b(x1)))) -> b#(b(a(x1))) -> b#(a(a(x1))) -> a#(a(a(b(x1)))) a#(b(a(b(x1)))) -> b#(b(a(x1))) -> b#(a(a(x1))) -> a#(a(b(x1))) a#(b(a(b(x1)))) -> b#(b(a(x1))) -> b#(a(a(x1))) -> a#(b(x1)) a#(b(a(b(x1)))) -> b#(b(a(x1))) -> b#(a(a(x1))) -> b#(x1) b#(a(a(x1))) -> a#(a(a(b(x1)))) -> a#(b(a(b(x1)))) -> b#(b(a(x1))) b#(a(a(x1))) -> a#(a(a(b(x1)))) -> a#(b(a(b(x1)))) -> b#(a(x1)) b#(a(a(x1))) -> a#(a(a(b(x1)))) -> a#(b(a(b(x1)))) -> a#(x1) b#(a(a(x1))) -> a#(a(b(x1))) -> a#(b(a(b(x1)))) -> b#(b(a(x1))) b#(a(a(x1))) -> a#(a(b(x1))) -> a#(b(a(b(x1)))) -> b#(a(x1)) b#(a(a(x1))) -> a#(a(b(x1))) -> a#(b(a(b(x1)))) -> a#(x1) b#(a(a(x1))) -> a#(b(x1)) -> a#(b(a(b(x1)))) -> b#(b(a(x1))) b#(a(a(x1))) -> a#(b(x1)) -> a#(b(a(b(x1)))) -> b#(a(x1)) b#(a(a(x1))) -> a#(b(x1)) -> a#(b(a(b(x1)))) -> a#(x1) b#(a(a(x1))) -> b#(x1) -> b#(a(a(x1))) -> a#(a(a(b(x1)))) b#(a(a(x1))) -> b#(x1) -> b#(a(a(x1))) -> a#(a(b(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) Arctic Interpretation Processor: dimension: 2 usable rules: b(a(a(x1))) -> a(a(a(b(x1)))) a(b(a(b(x1)))) -> b(b(a(x1))) interpretation: [a#](x0) = [-& 0 ]x0 + [0], [b#](x0) = [0 0]x0 + [1], [0 0 ] [0] [a](x0) = [-& 0 ]x0 + [0], [0 1] [0] [b](x0) = [0 0]x0 + [1] orientation: b#(a(a(x1))) = [0 0]x1 + [1] >= [0 0]x1 + [1] = b#(x1) b#(a(a(x1))) = [0 0]x1 + [1] >= [0 0]x1 + [1] = a#(b(x1)) b#(a(a(x1))) = [0 0]x1 + [1] >= [0 0]x1 + [1] = a#(a(b(x1))) b#(a(a(x1))) = [0 0]x1 + [1] >= [0 0]x1 + [1] = a#(a(a(b(x1)))) a#(b(a(b(x1)))) = [0 1]x1 + [1] >= [-& 0 ]x1 + [0] = a#(x1) a#(b(a(b(x1)))) = [0 1]x1 + [1] >= [0 0]x1 + [1] = b#(a(x1)) a#(b(a(b(x1)))) = [0 1]x1 + [1] >= [0 1]x1 + [1] = b#(b(a(x1))) [0 1] [1] [0 1] [1] b(a(a(x1))) = [0 0]x1 + [1] >= [0 0]x1 + [1] = a(a(a(b(x1)))) [1 1] [2] [1 1] [2] a(b(a(b(x1)))) = [0 1]x1 + [1] >= [0 1]x1 + [1] = b(b(a(x1))) problem: DPs: b#(a(a(x1))) -> b#(x1) b#(a(a(x1))) -> a#(b(x1)) b#(a(a(x1))) -> a#(a(b(x1))) b#(a(a(x1))) -> a#(a(a(b(x1)))) a#(b(a(b(x1)))) -> b#(a(x1)) a#(b(a(b(x1)))) -> b#(b(a(x1))) TRS: b(a(a(x1))) -> a(a(a(b(x1)))) a(b(a(b(x1)))) -> b(b(a(x1))) Restore Modifier: DPs: b#(a(a(x1))) -> b#(x1) b#(a(a(x1))) -> a#(b(x1)) b#(a(a(x1))) -> a#(a(b(x1))) b#(a(a(x1))) -> a#(a(a(b(x1)))) a#(b(a(b(x1)))) -> b#(a(x1)) a#(b(a(b(x1)))) -> b#(b(a(x1))) TRS: b(a(a(x1))) -> a(a(a(b(x1)))) a(b(a(b(x1)))) -> b(b(a(x1))) EDG Processor: DPs: b#(a(a(x1))) -> b#(x1) b#(a(a(x1))) -> a#(b(x1)) b#(a(a(x1))) -> a#(a(b(x1))) b#(a(a(x1))) -> a#(a(a(b(x1)))) a#(b(a(b(x1)))) -> b#(a(x1)) a#(b(a(b(x1)))) -> b#(b(a(x1))) TRS: b(a(a(x1))) -> a(a(a(b(x1)))) a(b(a(b(x1)))) -> b(b(a(x1))) graph: a#(b(a(b(x1)))) -> b#(a(x1)) -> b#(a(a(x1))) -> b#(x1) a#(b(a(b(x1)))) -> b#(a(x1)) -> b#(a(a(x1))) -> a#(b(x1)) a#(b(a(b(x1)))) -> b#(a(x1)) -> b#(a(a(x1))) -> a#(a(b(x1))) a#(b(a(b(x1)))) -> b#(a(x1)) -> b#(a(a(x1))) -> a#(a(a(b(x1)))) a#(b(a(b(x1)))) -> b#(b(a(x1))) -> b#(a(a(x1))) -> b#(x1) a#(b(a(b(x1)))) -> b#(b(a(x1))) -> b#(a(a(x1))) -> a#(b(x1)) a#(b(a(b(x1)))) -> b#(b(a(x1))) -> b#(a(a(x1))) -> a#(a(b(x1))) a#(b(a(b(x1)))) -> b#(b(a(x1))) -> b#(a(a(x1))) -> a#(a(a(b(x1)))) b#(a(a(x1))) -> a#(a(a(b(x1)))) -> a#(b(a(b(x1)))) -> b#(a(x1)) b#(a(a(x1))) -> a#(a(a(b(x1)))) -> a#(b(a(b(x1)))) -> b#(b(a(x1))) b#(a(a(x1))) -> a#(a(b(x1))) -> a#(b(a(b(x1)))) -> b#(a(x1)) b#(a(a(x1))) -> a#(a(b(x1))) -> a#(b(a(b(x1)))) -> b#(b(a(x1))) b#(a(a(x1))) -> a#(b(x1)) -> a#(b(a(b(x1)))) -> b#(a(x1)) b#(a(a(x1))) -> a#(b(x1)) -> a#(b(a(b(x1)))) -> b#(b(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)) b#(a(a(x1))) -> b#(x1) -> b#(a(a(x1))) -> a#(a(b(x1))) b#(a(a(x1))) -> b#(x1) -> b#(a(a(x1))) -> a#(a(a(b(x1)))) Arctic Interpretation Processor: dimension: 2 usable rules: b(a(a(x1))) -> a(a(a(b(x1)))) a(b(a(b(x1)))) -> b(b(a(x1))) interpretation: [a#](x0) = [-& 0 ]x0 + [0], [b#](x0) = [0 1]x0, [0 0 ] [-&] [a](x0) = [-& 0 ]x0 + [0 ], [0 1] [-&] [b](x0) = [0 1]x0 + [1 ] orientation: b#(a(a(x1))) = [0 1]x1 + [1] >= [0 1]x1 = b#(x1) b#(a(a(x1))) = [0 1]x1 + [1] >= [0 1]x1 + [1] = a#(b(x1)) b#(a(a(x1))) = [0 1]x1 + [1] >= [0 1]x1 + [1] = a#(a(b(x1))) b#(a(a(x1))) = [0 1]x1 + [1] >= [0 1]x1 + [1] = a#(a(a(b(x1)))) a#(b(a(b(x1)))) = [1 2]x1 + [2] >= [0 1]x1 + [1] = b#(a(x1)) a#(b(a(b(x1)))) = [1 2]x1 + [2] >= [1 2]x1 + [2] = b#(b(a(x1))) [0 1] [1] [0 1] [1] b(a(a(x1))) = [0 1]x1 + [1] >= [0 1]x1 + [1] = a(a(a(b(x1)))) [1 2] [2] [1 2] [2] a(b(a(b(x1)))) = [1 2]x1 + [2] >= [1 2]x1 + [2] = b(b(a(x1))) problem: DPs: b#(a(a(x1))) -> b#(x1) b#(a(a(x1))) -> a#(b(x1)) b#(a(a(x1))) -> a#(a(b(x1))) b#(a(a(x1))) -> a#(a(a(b(x1)))) a#(b(a(b(x1)))) -> b#(b(a(x1))) TRS: b(a(a(x1))) -> a(a(a(b(x1)))) a(b(a(b(x1)))) -> b(b(a(x1))) Restore Modifier: DPs: b#(a(a(x1))) -> b#(x1) b#(a(a(x1))) -> a#(b(x1)) b#(a(a(x1))) -> a#(a(b(x1))) b#(a(a(x1))) -> a#(a(a(b(x1)))) a#(b(a(b(x1)))) -> b#(b(a(x1))) TRS: b(a(a(x1))) -> a(a(a(b(x1)))) a(b(a(b(x1)))) -> b(b(a(x1))) EDG Processor: DPs: b#(a(a(x1))) -> b#(x1) b#(a(a(x1))) -> a#(b(x1)) b#(a(a(x1))) -> a#(a(b(x1))) b#(a(a(x1))) -> a#(a(a(b(x1)))) a#(b(a(b(x1)))) -> b#(b(a(x1))) TRS: b(a(a(x1))) -> a(a(a(b(x1)))) a(b(a(b(x1)))) -> b(b(a(x1))) graph: a#(b(a(b(x1)))) -> b#(b(a(x1))) -> b#(a(a(x1))) -> a#(a(a(b(x1)))) a#(b(a(b(x1)))) -> b#(b(a(x1))) -> b#(a(a(x1))) -> a#(a(b(x1))) a#(b(a(b(x1)))) -> b#(b(a(x1))) -> b#(a(a(x1))) -> a#(b(x1)) a#(b(a(b(x1)))) -> b#(b(a(x1))) -> b#(a(a(x1))) -> b#(x1) b#(a(a(x1))) -> a#(a(a(b(x1)))) -> a#(b(a(b(x1)))) -> b#(b(a(x1))) b#(a(a(x1))) -> a#(a(b(x1))) -> a#(b(a(b(x1)))) -> b#(b(a(x1))) b#(a(a(x1))) -> a#(b(x1)) -> a#(b(a(b(x1)))) -> b#(b(a(x1))) b#(a(a(x1))) -> b#(x1) -> b#(a(a(x1))) -> a#(a(a(b(x1)))) b#(a(a(x1))) -> b#(x1) -> b#(a(a(x1))) -> a#(a(b(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) Matrix Interpretation Processor: dim=1 usable rules: b(a(a(x1))) -> a(a(a(b(x1)))) a(b(a(b(x1)))) -> b(b(a(x1))) interpretation: [a#](x0) = 4x0 + 2, [b#](x0) = 6x0 + 1/2, [a](x0) = x0 + 1/2, [b](x0) = 3/2x0 orientation: b#(a(a(x1))) = 6x1 + 13/2 >= 6x1 + 1/2 = b#(x1) b#(a(a(x1))) = 6x1 + 13/2 >= 6x1 + 2 = a#(b(x1)) b#(a(a(x1))) = 6x1 + 13/2 >= 6x1 + 4 = a#(a(b(x1))) b#(a(a(x1))) = 6x1 + 13/2 >= 6x1 + 6 = a#(a(a(b(x1)))) a#(b(a(b(x1)))) = 9x1 + 5 >= 9x1 + 5 = b#(b(a(x1))) b(a(a(x1))) = 3/2x1 + 3/2 >= 3/2x1 + 3/2 = a(a(a(b(x1)))) a(b(a(b(x1)))) = 9/4x1 + 5/4 >= 9/4x1 + 9/8 = b(b(a(x1))) problem: DPs: a#(b(a(b(x1)))) -> b#(b(a(x1))) TRS: b(a(a(x1))) -> a(a(a(b(x1)))) a(b(a(b(x1)))) -> b(b(a(x1))) Restore Modifier: DPs: a#(b(a(b(x1)))) -> b#(b(a(x1))) TRS: b(a(a(x1))) -> a(a(a(b(x1)))) a(b(a(b(x1)))) -> b(b(a(x1))) EDG Processor: DPs: a#(b(a(b(x1)))) -> b#(b(a(x1))) TRS: b(a(a(x1))) -> a(a(a(b(x1)))) a(b(a(b(x1)))) -> b(b(a(x1))) graph: Qed