YES Problem: a(b(b(a(x1)))) -> b(a(a(b(x1)))) b(a(b(x1))) -> a(b(b(b(x1)))) Proof: DP Processor: DPs: a#(b(b(a(x1)))) -> b#(x1) a#(b(b(a(x1)))) -> a#(b(x1)) a#(b(b(a(x1)))) -> a#(a(b(x1))) a#(b(b(a(x1)))) -> b#(a(a(b(x1)))) b#(a(b(x1))) -> b#(b(x1)) b#(a(b(x1))) -> b#(b(b(x1))) b#(a(b(x1))) -> a#(b(b(b(x1)))) TRS: a(b(b(a(x1)))) -> b(a(a(b(x1)))) b(a(b(x1))) -> a(b(b(b(x1)))) TDG Processor: DPs: a#(b(b(a(x1)))) -> b#(x1) a#(b(b(a(x1)))) -> a#(b(x1)) a#(b(b(a(x1)))) -> a#(a(b(x1))) a#(b(b(a(x1)))) -> b#(a(a(b(x1)))) b#(a(b(x1))) -> b#(b(x1)) b#(a(b(x1))) -> b#(b(b(x1))) b#(a(b(x1))) -> a#(b(b(b(x1)))) TRS: a(b(b(a(x1)))) -> b(a(a(b(x1)))) b(a(b(x1))) -> a(b(b(b(x1)))) graph: b#(a(b(x1))) -> b#(b(b(x1))) -> b#(a(b(x1))) -> a#(b(b(b(x1)))) b#(a(b(x1))) -> b#(b(b(x1))) -> b#(a(b(x1))) -> b#(b(b(x1))) b#(a(b(x1))) -> b#(b(b(x1))) -> b#(a(b(x1))) -> b#(b(x1)) b#(a(b(x1))) -> b#(b(x1)) -> b#(a(b(x1))) -> a#(b(b(b(x1)))) b#(a(b(x1))) -> b#(b(x1)) -> b#(a(b(x1))) -> b#(b(b(x1))) b#(a(b(x1))) -> b#(b(x1)) -> b#(a(b(x1))) -> b#(b(x1)) b#(a(b(x1))) -> a#(b(b(b(x1)))) -> a#(b(b(a(x1)))) -> b#(a(a(b(x1)))) b#(a(b(x1))) -> a#(b(b(b(x1)))) -> a#(b(b(a(x1)))) -> a#(a(b(x1))) b#(a(b(x1))) -> a#(b(b(b(x1)))) -> a#(b(b(a(x1)))) -> a#(b(x1)) b#(a(b(x1))) -> a#(b(b(b(x1)))) -> a#(b(b(a(x1)))) -> b#(x1) a#(b(b(a(x1)))) -> b#(a(a(b(x1)))) -> b#(a(b(x1))) -> a#(b(b(b(x1)))) a#(b(b(a(x1)))) -> b#(a(a(b(x1)))) -> b#(a(b(x1))) -> b#(b(b(x1))) a#(b(b(a(x1)))) -> b#(a(a(b(x1)))) -> b#(a(b(x1))) -> b#(b(x1)) a#(b(b(a(x1)))) -> b#(x1) -> b#(a(b(x1))) -> a#(b(b(b(x1)))) a#(b(b(a(x1)))) -> b#(x1) -> b#(a(b(x1))) -> b#(b(b(x1))) a#(b(b(a(x1)))) -> b#(x1) -> b#(a(b(x1))) -> b#(b(x1)) a#(b(b(a(x1)))) -> a#(b(x1)) -> a#(b(b(a(x1)))) -> b#(a(a(b(x1)))) a#(b(b(a(x1)))) -> a#(b(x1)) -> a#(b(b(a(x1)))) -> a#(a(b(x1))) a#(b(b(a(x1)))) -> a#(b(x1)) -> a#(b(b(a(x1)))) -> a#(b(x1)) a#(b(b(a(x1)))) -> a#(b(x1)) -> a#(b(b(a(x1)))) -> b#(x1) a#(b(b(a(x1)))) -> a#(a(b(x1))) -> a#(b(b(a(x1)))) -> b#(a(a(b(x1)))) a#(b(b(a(x1)))) -> a#(a(b(x1))) -> a#(b(b(a(x1)))) -> a#(a(b(x1))) a#(b(b(a(x1)))) -> a#(a(b(x1))) -> a#(b(b(a(x1)))) -> a#(b(x1)) a#(b(b(a(x1)))) -> a#(a(b(x1))) -> a#(b(b(a(x1)))) -> b#(x1) Arctic Interpretation Processor: dimension: 1 interpretation: [b#](x0) = x0 + 0, [a#](x0) = 1x0 + 1, [b](x0) = x0, [a](x0) = 1x0 + 1 orientation: a#(b(b(a(x1)))) = 2x1 + 2 >= x1 + 0 = b#(x1) a#(b(b(a(x1)))) = 2x1 + 2 >= 1x1 + 1 = a#(b(x1)) a#(b(b(a(x1)))) = 2x1 + 2 >= 2x1 + 2 = a#(a(b(x1))) a#(b(b(a(x1)))) = 2x1 + 2 >= 2x1 + 2 = b#(a(a(b(x1)))) b#(a(b(x1))) = 1x1 + 1 >= x1 + 0 = b#(b(x1)) b#(a(b(x1))) = 1x1 + 1 >= x1 + 0 = b#(b(b(x1))) b#(a(b(x1))) = 1x1 + 1 >= 1x1 + 1 = a#(b(b(b(x1)))) a(b(b(a(x1)))) = 2x1 + 2 >= 2x1 + 2 = b(a(a(b(x1)))) b(a(b(x1))) = 1x1 + 1 >= 1x1 + 1 = a(b(b(b(x1)))) problem: DPs: a#(b(b(a(x1)))) -> a#(a(b(x1))) a#(b(b(a(x1)))) -> b#(a(a(b(x1)))) b#(a(b(x1))) -> a#(b(b(b(x1)))) TRS: a(b(b(a(x1)))) -> b(a(a(b(x1)))) b(a(b(x1))) -> a(b(b(b(x1)))) EDG Processor: DPs: a#(b(b(a(x1)))) -> a#(a(b(x1))) a#(b(b(a(x1)))) -> b#(a(a(b(x1)))) b#(a(b(x1))) -> a#(b(b(b(x1)))) TRS: a(b(b(a(x1)))) -> b(a(a(b(x1)))) b(a(b(x1))) -> a(b(b(b(x1)))) graph: b#(a(b(x1))) -> a#(b(b(b(x1)))) -> a#(b(b(a(x1)))) -> a#(a(b(x1))) b#(a(b(x1))) -> a#(b(b(b(x1)))) -> a#(b(b(a(x1)))) -> b#(a(a(b(x1)))) a#(b(b(a(x1)))) -> b#(a(a(b(x1)))) -> b#(a(b(x1))) -> a#(b(b(b(x1)))) a#(b(b(a(x1)))) -> a#(a(b(x1))) -> a#(b(b(a(x1)))) -> a#(a(b(x1))) a#(b(b(a(x1)))) -> a#(a(b(x1))) -> a#(b(b(a(x1)))) -> b#(a(a(b(x1)))) Matrix Interpretation Processor: dim=4 interpretation: [b#](x0) = [1], [a#](x0) = [1 0 0 0]x0, [0 0 1 0] [0] [0 0 0 0] [1] [b](x0) = [0 0 0 0]x0 + [1] [0 0 0 1] [0], [0 0 0 0] [0] [0 0 0 0] [1] [a](x0) = [1 0 0 0]x0 + [0] [0 0 0 0] [0] orientation: a#(b(b(a(x1)))) = [1] >= [0] = a#(a(b(x1))) a#(b(b(a(x1)))) = [1] >= [1] = b#(a(a(b(x1)))) b#(a(b(x1))) = [1] >= [1] = a#(b(b(b(x1)))) [0] [0] [1] [1] a(b(b(a(x1)))) = [1] >= [1] = b(a(a(b(x1)))) [0] [0] [0 0 1 0] [0] [0] [0 0 0 0] [1] [1] b(a(b(x1))) = [0 0 0 0]x1 + [1] >= [1] = a(b(b(b(x1)))) [0 0 0 0] [0] [0] problem: DPs: a#(b(b(a(x1)))) -> b#(a(a(b(x1)))) b#(a(b(x1))) -> a#(b(b(b(x1)))) TRS: a(b(b(a(x1)))) -> b(a(a(b(x1)))) b(a(b(x1))) -> a(b(b(b(x1)))) EDG Processor: DPs: a#(b(b(a(x1)))) -> b#(a(a(b(x1)))) b#(a(b(x1))) -> a#(b(b(b(x1)))) TRS: a(b(b(a(x1)))) -> b(a(a(b(x1)))) b(a(b(x1))) -> a(b(b(b(x1)))) graph: b#(a(b(x1))) -> a#(b(b(b(x1)))) -> a#(b(b(a(x1)))) -> b#(a(a(b(x1)))) a#(b(b(a(x1)))) -> b#(a(a(b(x1)))) -> b#(a(b(x1))) -> a#(b(b(b(x1)))) Matrix Interpretation Processor: dim=4 interpretation: [b#](x0) = [0 1 0 0]x0, [a#](x0) = [0 0 0 1]x0, [0 1 1 0] [0 1 0 0] [b](x0) = [0 0 1 0]x0 [1 0 0 0] , [0 0 0 1] [0] [1 0 0 0] [0] [a](x0) = [0 0 0 0]x0 + [1] [0 0 0 0] [0] orientation: a#(b(b(a(x1)))) = [1 0 0 0]x1 + [1] >= [1 0 0 0]x1 = b#(a(a(b(x1)))) b#(a(b(x1))) = [0 1 1 0]x1 >= [0 1 1 0]x1 = a#(b(b(b(x1)))) [1 0 0 0] [1] [1 0 0 0] [1] [1 0 0 0] [1] [1 0 0 0] [0] a(b(b(a(x1)))) = [0 0 0 0]x1 + [1] >= [0 0 0 0]x1 + [1] = b(a(a(b(x1)))) [0 0 0 0] [0] [0 0 0 0] [0] [0 1 1 0] [1] [0 1 1 0] [0] [0 1 1 0] [0] [0 1 1 0] [0] b(a(b(x1))) = [0 0 0 0]x1 + [1] >= [0 0 0 0]x1 + [1] = a(b(b(b(x1)))) [1 0 0 0] [0] [0 0 0 0] [0] problem: DPs: b#(a(b(x1))) -> a#(b(b(b(x1)))) TRS: a(b(b(a(x1)))) -> b(a(a(b(x1)))) b(a(b(x1))) -> a(b(b(b(x1)))) EDG Processor: DPs: b#(a(b(x1))) -> a#(b(b(b(x1)))) TRS: a(b(b(a(x1)))) -> b(a(a(b(x1)))) b(a(b(x1))) -> a(b(b(b(x1)))) graph: Qed