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: DP Processor: DPs: a#(a(b(x1))) -> a#(x1) a#(a(b(x1))) -> b#(a(x1)) 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#(c(x1)) -> b#(x1) TRS: a(a(b(x1))) -> b(a(x1)) b(a(a(x1))) -> a(a(a(b(x1)))) a(c(x1)) -> c(b(x1)) TDG Processor: DPs: a#(a(b(x1))) -> a#(x1) a#(a(b(x1))) -> b#(a(x1)) 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#(c(x1)) -> b#(x1) TRS: a(a(b(x1))) -> b(a(x1)) b(a(a(x1))) -> a(a(a(b(x1)))) a(c(x1)) -> c(b(x1)) graph: 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) b#(a(a(x1))) -> a#(a(a(b(x1)))) -> a#(c(x1)) -> b#(x1) b#(a(a(x1))) -> a#(a(a(b(x1)))) -> a#(a(b(x1))) -> b#(a(x1)) b#(a(a(x1))) -> a#(a(a(b(x1)))) -> a#(a(b(x1))) -> a#(x1) b#(a(a(x1))) -> a#(a(b(x1))) -> a#(c(x1)) -> b#(x1) b#(a(a(x1))) -> a#(a(b(x1))) -> a#(a(b(x1))) -> b#(a(x1)) b#(a(a(x1))) -> a#(a(b(x1))) -> a#(a(b(x1))) -> a#(x1) b#(a(a(x1))) -> a#(b(x1)) -> a#(c(x1)) -> b#(x1) b#(a(a(x1))) -> a#(b(x1)) -> a#(a(b(x1))) -> b#(a(x1)) b#(a(a(x1))) -> a#(b(x1)) -> a#(a(b(x1))) -> a#(x1) a#(c(x1)) -> b#(x1) -> b#(a(a(x1))) -> a#(a(a(b(x1)))) a#(c(x1)) -> b#(x1) -> b#(a(a(x1))) -> a#(a(b(x1))) a#(c(x1)) -> b#(x1) -> b#(a(a(x1))) -> a#(b(x1)) a#(c(x1)) -> b#(x1) -> b#(a(a(x1))) -> b#(x1) a#(a(b(x1))) -> b#(a(x1)) -> b#(a(a(x1))) -> a#(a(a(b(x1)))) a#(a(b(x1))) -> b#(a(x1)) -> b#(a(a(x1))) -> a#(a(b(x1))) a#(a(b(x1))) -> b#(a(x1)) -> b#(a(a(x1))) -> a#(b(x1)) a#(a(b(x1))) -> b#(a(x1)) -> b#(a(a(x1))) -> b#(x1) a#(a(b(x1))) -> a#(x1) -> a#(c(x1)) -> b#(x1) a#(a(b(x1))) -> a#(x1) -> a#(a(b(x1))) -> b#(a(x1)) a#(a(b(x1))) -> a#(x1) -> a#(a(b(x1))) -> a#(x1) Arctic Interpretation Processor: dimension: 1 interpretation: [b#](x0) = x0, [a#](x0) = x0, [c](x0) = 1x0 + 15, [a](x0) = x0 + 8, [b](x0) = x0 + 8 orientation: a#(a(b(x1))) = x1 + 8 >= x1 = a#(x1) a#(a(b(x1))) = x1 + 8 >= x1 + 8 = b#(a(x1)) b#(a(a(x1))) = x1 + 8 >= x1 = b#(x1) b#(a(a(x1))) = x1 + 8 >= x1 + 8 = a#(b(x1)) b#(a(a(x1))) = x1 + 8 >= x1 + 8 = a#(a(b(x1))) b#(a(a(x1))) = x1 + 8 >= x1 + 8 = a#(a(a(b(x1)))) a#(c(x1)) = 1x1 + 15 >= x1 = b#(x1) a(a(b(x1))) = x1 + 8 >= x1 + 8 = b(a(x1)) b(a(a(x1))) = x1 + 8 >= x1 + 8 = a(a(a(b(x1)))) a(c(x1)) = 1x1 + 15 >= 1x1 + 15 = c(b(x1)) problem: DPs: a#(a(b(x1))) -> a#(x1) a#(a(b(x1))) -> b#(a(x1)) 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)))) TRS: a(a(b(x1))) -> b(a(x1)) b(a(a(x1))) -> a(a(a(b(x1)))) a(c(x1)) -> c(b(x1)) EDG Processor: DPs: a#(a(b(x1))) -> a#(x1) a#(a(b(x1))) -> b#(a(x1)) 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)))) TRS: a(a(b(x1))) -> b(a(x1)) b(a(a(x1))) -> a(a(a(b(x1)))) a(c(x1)) -> c(b(x1)) graph: 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)))) b#(a(a(x1))) -> a#(a(a(b(x1)))) -> a#(a(b(x1))) -> a#(x1) b#(a(a(x1))) -> a#(a(a(b(x1)))) -> a#(a(b(x1))) -> b#(a(x1)) b#(a(a(x1))) -> a#(a(b(x1))) -> a#(a(b(x1))) -> a#(x1) b#(a(a(x1))) -> a#(a(b(x1))) -> a#(a(b(x1))) -> b#(a(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))) -> b#(a(x1)) a#(a(b(x1))) -> b#(a(x1)) -> b#(a(a(x1))) -> b#(x1) a#(a(b(x1))) -> b#(a(x1)) -> b#(a(a(x1))) -> a#(b(x1)) a#(a(b(x1))) -> b#(a(x1)) -> b#(a(a(x1))) -> a#(a(b(x1))) a#(a(b(x1))) -> b#(a(x1)) -> b#(a(a(x1))) -> a#(a(a(b(x1)))) a#(a(b(x1))) -> a#(x1) -> a#(a(b(x1))) -> a#(x1) a#(a(b(x1))) -> a#(x1) -> a#(a(b(x1))) -> b#(a(x1)) Arctic Interpretation Processor: dimension: 1 interpretation: [b#](x0) = 1x0 + -15, [a#](x0) = x0, [c](x0) = 0, [a](x0) = x0, [b](x0) = 1x0 + -15 orientation: a#(a(b(x1))) = 1x1 + -15 >= x1 = a#(x1) a#(a(b(x1))) = 1x1 + -15 >= 1x1 + -15 = b#(a(x1)) b#(a(a(x1))) = 1x1 + -15 >= 1x1 + -15 = b#(x1) b#(a(a(x1))) = 1x1 + -15 >= 1x1 + -15 = a#(b(x1)) b#(a(a(x1))) = 1x1 + -15 >= 1x1 + -15 = a#(a(b(x1))) b#(a(a(x1))) = 1x1 + -15 >= 1x1 + -15 = a#(a(a(b(x1)))) a(a(b(x1))) = 1x1 + -15 >= 1x1 + -15 = b(a(x1)) b(a(a(x1))) = 1x1 + -15 >= 1x1 + -15 = a(a(a(b(x1)))) a(c(x1)) = 0 >= 0 = c(b(x1)) problem: DPs: a#(a(b(x1))) -> b#(a(x1)) 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)))) TRS: a(a(b(x1))) -> b(a(x1)) b(a(a(x1))) -> a(a(a(b(x1)))) a(c(x1)) -> c(b(x1)) EDG Processor: DPs: a#(a(b(x1))) -> b#(a(x1)) 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)))) TRS: a(a(b(x1))) -> b(a(x1)) b(a(a(x1))) -> a(a(a(b(x1)))) a(c(x1)) -> c(b(x1)) graph: 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) b#(a(a(x1))) -> a#(a(a(b(x1)))) -> a#(a(b(x1))) -> b#(a(x1)) b#(a(a(x1))) -> a#(a(b(x1))) -> a#(a(b(x1))) -> b#(a(x1)) b#(a(a(x1))) -> a#(b(x1)) -> a#(a(b(x1))) -> b#(a(x1)) a#(a(b(x1))) -> b#(a(x1)) -> b#(a(a(x1))) -> a#(a(a(b(x1)))) a#(a(b(x1))) -> b#(a(x1)) -> b#(a(a(x1))) -> a#(a(b(x1))) a#(a(b(x1))) -> b#(a(x1)) -> b#(a(a(x1))) -> a#(b(x1)) a#(a(b(x1))) -> b#(a(x1)) -> b#(a(a(x1))) -> b#(x1) Matrix Interpretation Processor: dim=4 interpretation: [b#](x0) = [0 1 1 0]x0 + [1], [a#](x0) = [0 0 1 1]x0, [0 0 0 0] [0] [0 0 0 1] [1] [c](x0) = [0 0 0 1]x0 + [1] [0 0 0 0] [0], [0 0 0 1] [0] [0 0 1 0] [0] [a](x0) = [1 1 0 0]x0 + [0] [0 0 0 0] [1], [0 0 0 1] [1 1 1 0] [b](x0) = [1 1 1 0]x0 [0 0 0 1] orientation: a#(a(b(x1))) = [1 1 1 1]x1 + [1] >= [1 1 1 0]x1 + [1] = b#(a(x1)) b#(a(a(x1))) = [1 1 1 1]x1 + [1] >= [0 1 1 0]x1 + [1] = b#(x1) b#(a(a(x1))) = [1 1 1 1]x1 + [1] >= [1 1 1 1]x1 = a#(b(x1)) b#(a(a(x1))) = [1 1 1 1]x1 + [1] >= [1 1 1 1]x1 + [1] = a#(a(b(x1))) b#(a(a(x1))) = [1 1 1 1]x1 + [1] >= [1 1 1 1]x1 + [1] = a#(a(a(b(x1)))) [0 0 0 0] [1] [0 0 0 0] [1] [1 1 1 1] [0] [1 1 1 1] [0] a(a(b(x1))) = [1 1 1 1]x1 + [0] >= [1 1 1 1]x1 + [0] = b(a(x1)) [0 0 0 0] [1] [0 0 0 0] [1] [0 0 0 0] [1] [0 0 0 0] [1] [1 1 1 1] [1] [1 1 1 1] [0] b(a(a(x1))) = [1 1 1 1]x1 + [1] >= [1 1 1 1]x1 + [1] = a(a(a(b(x1)))) [0 0 0 0] [1] [0 0 0 0] [1] [0 0 0 0] [0] [0 0 0 0] [0] [0 0 0 1] [1] [0 0 0 1] [1] a(c(x1)) = [0 0 0 1]x1 + [1] >= [0 0 0 1]x1 + [1] = c(b(x1)) [0 0 0 0] [1] [0 0 0 0] [0] problem: DPs: a#(a(b(x1))) -> b#(a(x1)) b#(a(a(x1))) -> b#(x1) b#(a(a(x1))) -> a#(a(b(x1))) b#(a(a(x1))) -> a#(a(a(b(x1)))) TRS: a(a(b(x1))) -> b(a(x1)) b(a(a(x1))) -> a(a(a(b(x1)))) a(c(x1)) -> c(b(x1)) EDG Processor: DPs: a#(a(b(x1))) -> b#(a(x1)) b#(a(a(x1))) -> b#(x1) b#(a(a(x1))) -> a#(a(b(x1))) b#(a(a(x1))) -> a#(a(a(b(x1)))) TRS: a(a(b(x1))) -> b(a(x1)) b(a(a(x1))) -> a(a(a(b(x1)))) a(c(x1)) -> c(b(x1)) graph: b#(a(a(x1))) -> b#(x1) -> b#(a(a(x1))) -> 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)))) b#(a(a(x1))) -> a#(a(a(b(x1)))) -> a#(a(b(x1))) -> b#(a(x1)) b#(a(a(x1))) -> a#(a(b(x1))) -> a#(a(b(x1))) -> b#(a(x1)) a#(a(b(x1))) -> b#(a(x1)) -> b#(a(a(x1))) -> b#(x1) a#(a(b(x1))) -> b#(a(x1)) -> b#(a(a(x1))) -> a#(a(b(x1))) a#(a(b(x1))) -> b#(a(x1)) -> b#(a(a(x1))) -> a#(a(a(b(x1)))) Matrix Interpretation Processor: dim=4 interpretation: [b#](x0) = [1 1 1 0]x0, [a#](x0) = [1 0 0 1]x0, [1] [0] [c](x0) = [1] [0], [0 1 1 0] [0] [0 0 0 1] [0] [a](x0) = [1 0 0 0]x0 + [0] [0 0 0 0] [1], [1 1 1 0] [0 0 0 1] [b](x0) = [1 1 1 0]x0 [0 0 0 1] orientation: a#(a(b(x1))) = [1 1 1 1]x1 + [1] >= [1 1 1 1]x1 = b#(a(x1)) b#(a(a(x1))) = [1 1 1 1]x1 + [1] >= [1 1 1 0]x1 = b#(x1) b#(a(a(x1))) = [1 1 1 1]x1 + [1] >= [1 1 1 1]x1 + [1] = a#(a(b(x1))) b#(a(a(x1))) = [1 1 1 1]x1 + [1] >= [1 1 1 1]x1 + [1] = a#(a(a(b(x1)))) [1 1 1 1] [0] [1 1 1 1] [0] [0 0 0 0] [1] [0 0 0 0] [1] a(a(b(x1))) = [1 1 1 1]x1 + [0] >= [1 1 1 1]x1 + [0] = b(a(x1)) [0 0 0 0] [1] [0 0 0 0] [1] [1 1 1 1] [1] [1 1 1 1] [1] [0 0 0 0] [1] [0 0 0 0] [1] b(a(a(x1))) = [1 1 1 1]x1 + [1] >= [1 1 1 1]x1 + [0] = a(a(a(b(x1)))) [0 0 0 0] [1] [0 0 0 0] [1] [1] [1] [0] [0] a(c(x1)) = [1] >= [1] = c(b(x1)) [1] [0] problem: DPs: b#(a(a(x1))) -> a#(a(b(x1))) b#(a(a(x1))) -> a#(a(a(b(x1)))) TRS: a(a(b(x1))) -> b(a(x1)) b(a(a(x1))) -> a(a(a(b(x1)))) a(c(x1)) -> c(b(x1)) EDG Processor: DPs: b#(a(a(x1))) -> a#(a(b(x1))) b#(a(a(x1))) -> a#(a(a(b(x1)))) TRS: a(a(b(x1))) -> b(a(x1)) b(a(a(x1))) -> a(a(a(b(x1)))) a(c(x1)) -> c(b(x1)) graph: Qed