YES Problem: a(s(x1)) -> s(a(x1)) b(a(b(s(x1)))) -> a(b(s(a(x1)))) b(a(b(b(x1)))) -> a(b(a(b(x1)))) a(b(a(a(x1)))) -> b(a(b(a(x1)))) Proof: DP Processor: DPs: a#(s(x1)) -> a#(x1) b#(a(b(s(x1)))) -> a#(x1) b#(a(b(s(x1)))) -> b#(s(a(x1))) b#(a(b(s(x1)))) -> a#(b(s(a(x1)))) b#(a(b(b(x1)))) -> a#(b(x1)) b#(a(b(b(x1)))) -> b#(a(b(x1))) b#(a(b(b(x1)))) -> a#(b(a(b(x1)))) a#(b(a(a(x1)))) -> b#(a(x1)) a#(b(a(a(x1)))) -> a#(b(a(x1))) a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) TRS: a(s(x1)) -> s(a(x1)) b(a(b(s(x1)))) -> a(b(s(a(x1)))) b(a(b(b(x1)))) -> a(b(a(b(x1)))) a(b(a(a(x1)))) -> b(a(b(a(x1)))) TDG Processor: DPs: a#(s(x1)) -> a#(x1) b#(a(b(s(x1)))) -> a#(x1) b#(a(b(s(x1)))) -> b#(s(a(x1))) b#(a(b(s(x1)))) -> a#(b(s(a(x1)))) b#(a(b(b(x1)))) -> a#(b(x1)) b#(a(b(b(x1)))) -> b#(a(b(x1))) b#(a(b(b(x1)))) -> a#(b(a(b(x1)))) a#(b(a(a(x1)))) -> b#(a(x1)) a#(b(a(a(x1)))) -> a#(b(a(x1))) a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) TRS: a(s(x1)) -> s(a(x1)) b(a(b(s(x1)))) -> a(b(s(a(x1)))) b(a(b(b(x1)))) -> a(b(a(b(x1)))) a(b(a(a(x1)))) -> b(a(b(a(x1)))) graph: b#(a(b(b(x1)))) -> b#(a(b(x1))) -> b#(a(b(b(x1)))) -> a#(b(a(b(x1)))) b#(a(b(b(x1)))) -> b#(a(b(x1))) -> b#(a(b(b(x1)))) -> b#(a(b(x1))) b#(a(b(b(x1)))) -> b#(a(b(x1))) -> b#(a(b(b(x1)))) -> a#(b(x1)) b#(a(b(b(x1)))) -> b#(a(b(x1))) -> b#(a(b(s(x1)))) -> a#(b(s(a(x1)))) b#(a(b(b(x1)))) -> b#(a(b(x1))) -> b#(a(b(s(x1)))) -> b#(s(a(x1))) b#(a(b(b(x1)))) -> b#(a(b(x1))) -> b#(a(b(s(x1)))) -> a#(x1) b#(a(b(b(x1)))) -> a#(b(a(b(x1)))) -> a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) b#(a(b(b(x1)))) -> a#(b(a(b(x1)))) -> a#(b(a(a(x1)))) -> a#(b(a(x1))) b#(a(b(b(x1)))) -> a#(b(a(b(x1)))) -> a#(b(a(a(x1)))) -> b#(a(x1)) b#(a(b(b(x1)))) -> a#(b(a(b(x1)))) -> a#(s(x1)) -> a#(x1) b#(a(b(b(x1)))) -> a#(b(x1)) -> a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) b#(a(b(b(x1)))) -> a#(b(x1)) -> a#(b(a(a(x1)))) -> a#(b(a(x1))) b#(a(b(b(x1)))) -> a#(b(x1)) -> a#(b(a(a(x1)))) -> b#(a(x1)) b#(a(b(b(x1)))) -> a#(b(x1)) -> a#(s(x1)) -> a#(x1) b#(a(b(s(x1)))) -> b#(s(a(x1))) -> b#(a(b(b(x1)))) -> a#(b(a(b(x1)))) b#(a(b(s(x1)))) -> b#(s(a(x1))) -> b#(a(b(b(x1)))) -> b#(a(b(x1))) b#(a(b(s(x1)))) -> b#(s(a(x1))) -> b#(a(b(b(x1)))) -> a#(b(x1)) b#(a(b(s(x1)))) -> b#(s(a(x1))) -> b#(a(b(s(x1)))) -> a#(b(s(a(x1)))) b#(a(b(s(x1)))) -> b#(s(a(x1))) -> b#(a(b(s(x1)))) -> b#(s(a(x1))) b#(a(b(s(x1)))) -> b#(s(a(x1))) -> b#(a(b(s(x1)))) -> a#(x1) b#(a(b(s(x1)))) -> a#(b(s(a(x1)))) -> a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) b#(a(b(s(x1)))) -> a#(b(s(a(x1)))) -> a#(b(a(a(x1)))) -> a#(b(a(x1))) b#(a(b(s(x1)))) -> a#(b(s(a(x1)))) -> a#(b(a(a(x1)))) -> b#(a(x1)) b#(a(b(s(x1)))) -> a#(b(s(a(x1)))) -> a#(s(x1)) -> a#(x1) b#(a(b(s(x1)))) -> a#(x1) -> a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) b#(a(b(s(x1)))) -> a#(x1) -> a#(b(a(a(x1)))) -> a#(b(a(x1))) b#(a(b(s(x1)))) -> a#(x1) -> a#(b(a(a(x1)))) -> b#(a(x1)) b#(a(b(s(x1)))) -> a#(x1) -> a#(s(x1)) -> a#(x1) a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) -> b#(a(b(b(x1)))) -> a#(b(a(b(x1)))) a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) -> b#(a(b(b(x1)))) -> b#(a(b(x1))) a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) -> b#(a(b(b(x1)))) -> a#(b(x1)) a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) -> b#(a(b(s(x1)))) -> a#(b(s(a(x1)))) a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) -> b#(a(b(s(x1)))) -> b#(s(a(x1))) a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) -> b#(a(b(s(x1)))) -> a#(x1) a#(b(a(a(x1)))) -> b#(a(x1)) -> b#(a(b(b(x1)))) -> a#(b(a(b(x1)))) a#(b(a(a(x1)))) -> b#(a(x1)) -> b#(a(b(b(x1)))) -> b#(a(b(x1))) a#(b(a(a(x1)))) -> b#(a(x1)) -> b#(a(b(b(x1)))) -> a#(b(x1)) a#(b(a(a(x1)))) -> b#(a(x1)) -> b#(a(b(s(x1)))) -> a#(b(s(a(x1)))) a#(b(a(a(x1)))) -> b#(a(x1)) -> b#(a(b(s(x1)))) -> b#(s(a(x1))) a#(b(a(a(x1)))) -> b#(a(x1)) -> b#(a(b(s(x1)))) -> a#(x1) a#(b(a(a(x1)))) -> a#(b(a(x1))) -> a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) a#(b(a(a(x1)))) -> a#(b(a(x1))) -> a#(b(a(a(x1)))) -> a#(b(a(x1))) a#(b(a(a(x1)))) -> a#(b(a(x1))) -> a#(b(a(a(x1)))) -> b#(a(x1)) a#(b(a(a(x1)))) -> a#(b(a(x1))) -> a#(s(x1)) -> a#(x1) a#(s(x1)) -> a#(x1) -> a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) a#(s(x1)) -> a#(x1) -> a#(b(a(a(x1)))) -> a#(b(a(x1))) a#(s(x1)) -> a#(x1) -> a#(b(a(a(x1)))) -> b#(a(x1)) a#(s(x1)) -> a#(x1) -> a#(s(x1)) -> a#(x1) EDG Processor: DPs: a#(s(x1)) -> a#(x1) b#(a(b(s(x1)))) -> a#(x1) b#(a(b(s(x1)))) -> b#(s(a(x1))) b#(a(b(s(x1)))) -> a#(b(s(a(x1)))) b#(a(b(b(x1)))) -> a#(b(x1)) b#(a(b(b(x1)))) -> b#(a(b(x1))) b#(a(b(b(x1)))) -> a#(b(a(b(x1)))) a#(b(a(a(x1)))) -> b#(a(x1)) a#(b(a(a(x1)))) -> a#(b(a(x1))) a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) TRS: a(s(x1)) -> s(a(x1)) b(a(b(s(x1)))) -> a(b(s(a(x1)))) b(a(b(b(x1)))) -> a(b(a(b(x1)))) a(b(a(a(x1)))) -> b(a(b(a(x1)))) graph: b#(a(b(b(x1)))) -> b#(a(b(x1))) -> b#(a(b(s(x1)))) -> a#(x1) b#(a(b(b(x1)))) -> b#(a(b(x1))) -> b#(a(b(s(x1)))) -> b#(s(a(x1))) b#(a(b(b(x1)))) -> b#(a(b(x1))) -> b#(a(b(s(x1)))) -> a#(b(s(a(x1)))) b#(a(b(b(x1)))) -> b#(a(b(x1))) -> b#(a(b(b(x1)))) -> a#(b(x1)) b#(a(b(b(x1)))) -> b#(a(b(x1))) -> b#(a(b(b(x1)))) -> b#(a(b(x1))) b#(a(b(b(x1)))) -> b#(a(b(x1))) -> b#(a(b(b(x1)))) -> a#(b(a(b(x1)))) b#(a(b(b(x1)))) -> a#(b(a(b(x1)))) -> a#(s(x1)) -> a#(x1) b#(a(b(b(x1)))) -> a#(b(a(b(x1)))) -> a#(b(a(a(x1)))) -> b#(a(x1)) b#(a(b(b(x1)))) -> a#(b(a(b(x1)))) -> a#(b(a(a(x1)))) -> a#(b(a(x1))) b#(a(b(b(x1)))) -> a#(b(a(b(x1)))) -> a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) b#(a(b(b(x1)))) -> a#(b(x1)) -> a#(s(x1)) -> a#(x1) b#(a(b(b(x1)))) -> a#(b(x1)) -> a#(b(a(a(x1)))) -> b#(a(x1)) b#(a(b(b(x1)))) -> a#(b(x1)) -> a#(b(a(a(x1)))) -> a#(b(a(x1))) b#(a(b(b(x1)))) -> a#(b(x1)) -> a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) b#(a(b(s(x1)))) -> a#(x1) -> a#(s(x1)) -> a#(x1) b#(a(b(s(x1)))) -> a#(x1) -> a#(b(a(a(x1)))) -> b#(a(x1)) b#(a(b(s(x1)))) -> a#(x1) -> a#(b(a(a(x1)))) -> a#(b(a(x1))) b#(a(b(s(x1)))) -> a#(x1) -> a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) -> b#(a(b(s(x1)))) -> a#(x1) a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) -> b#(a(b(s(x1)))) -> b#(s(a(x1))) a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) -> b#(a(b(s(x1)))) -> a#(b(s(a(x1)))) a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) -> b#(a(b(b(x1)))) -> a#(b(x1)) a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) -> b#(a(b(b(x1)))) -> b#(a(b(x1))) a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) -> b#(a(b(b(x1)))) -> a#(b(a(b(x1)))) a#(b(a(a(x1)))) -> b#(a(x1)) -> b#(a(b(s(x1)))) -> a#(x1) a#(b(a(a(x1)))) -> b#(a(x1)) -> b#(a(b(s(x1)))) -> b#(s(a(x1))) a#(b(a(a(x1)))) -> b#(a(x1)) -> b#(a(b(s(x1)))) -> a#(b(s(a(x1)))) a#(b(a(a(x1)))) -> b#(a(x1)) -> b#(a(b(b(x1)))) -> a#(b(x1)) a#(b(a(a(x1)))) -> b#(a(x1)) -> b#(a(b(b(x1)))) -> b#(a(b(x1))) a#(b(a(a(x1)))) -> b#(a(x1)) -> b#(a(b(b(x1)))) -> a#(b(a(b(x1)))) a#(b(a(a(x1)))) -> a#(b(a(x1))) -> a#(s(x1)) -> a#(x1) a#(b(a(a(x1)))) -> a#(b(a(x1))) -> a#(b(a(a(x1)))) -> b#(a(x1)) a#(b(a(a(x1)))) -> a#(b(a(x1))) -> a#(b(a(a(x1)))) -> a#(b(a(x1))) a#(b(a(a(x1)))) -> a#(b(a(x1))) -> a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) a#(s(x1)) -> a#(x1) -> a#(s(x1)) -> a#(x1) a#(s(x1)) -> a#(x1) -> a#(b(a(a(x1)))) -> b#(a(x1)) a#(s(x1)) -> a#(x1) -> a#(b(a(a(x1)))) -> a#(b(a(x1))) a#(s(x1)) -> a#(x1) -> a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) SCC Processor: #sccs: 1 #rules: 8 #arcs: 38/100 DPs: b#(a(b(b(x1)))) -> b#(a(b(x1))) b#(a(b(b(x1)))) -> a#(b(a(b(x1)))) a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) b#(a(b(b(x1)))) -> a#(b(x1)) a#(b(a(a(x1)))) -> a#(b(a(x1))) a#(b(a(a(x1)))) -> b#(a(x1)) b#(a(b(s(x1)))) -> a#(x1) a#(s(x1)) -> a#(x1) TRS: a(s(x1)) -> s(a(x1)) b(a(b(s(x1)))) -> a(b(s(a(x1)))) b(a(b(b(x1)))) -> a(b(a(b(x1)))) a(b(a(a(x1)))) -> b(a(b(a(x1)))) KBO Processor: argument filtering: pi(s) = [0] pi(a) = 0 pi(b) = 0 pi(a#) = 0 pi(b#) = 0 weight function: w0 = 1 w(a#) = w(a) = 1 w(b#) = w(b) = w(s) = 0 precedence: b# ~ a# ~ b ~ a ~ s problem: DPs: b#(a(b(b(x1)))) -> b#(a(b(x1))) b#(a(b(b(x1)))) -> a#(b(a(b(x1)))) a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) b#(a(b(b(x1)))) -> a#(b(x1)) a#(b(a(a(x1)))) -> a#(b(a(x1))) a#(b(a(a(x1)))) -> b#(a(x1)) TRS: a(s(x1)) -> s(a(x1)) b(a(b(s(x1)))) -> a(b(s(a(x1)))) b(a(b(b(x1)))) -> a(b(a(b(x1)))) a(b(a(a(x1)))) -> b(a(b(a(x1)))) EDG Processor: DPs: b#(a(b(b(x1)))) -> b#(a(b(x1))) b#(a(b(b(x1)))) -> a#(b(a(b(x1)))) a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) b#(a(b(b(x1)))) -> a#(b(x1)) a#(b(a(a(x1)))) -> a#(b(a(x1))) a#(b(a(a(x1)))) -> b#(a(x1)) TRS: a(s(x1)) -> s(a(x1)) b(a(b(s(x1)))) -> a(b(s(a(x1)))) b(a(b(b(x1)))) -> a(b(a(b(x1)))) a(b(a(a(x1)))) -> b(a(b(a(x1)))) graph: b#(a(b(b(x1)))) -> b#(a(b(x1))) -> b#(a(b(b(x1)))) -> a#(b(a(b(x1)))) b#(a(b(b(x1)))) -> b#(a(b(x1))) -> b#(a(b(b(x1)))) -> b#(a(b(x1))) b#(a(b(b(x1)))) -> b#(a(b(x1))) -> b#(a(b(b(x1)))) -> a#(b(x1)) b#(a(b(b(x1)))) -> a#(b(a(b(x1)))) -> a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) b#(a(b(b(x1)))) -> a#(b(a(b(x1)))) -> a#(b(a(a(x1)))) -> a#(b(a(x1))) b#(a(b(b(x1)))) -> a#(b(a(b(x1)))) -> a#(b(a(a(x1)))) -> b#(a(x1)) b#(a(b(b(x1)))) -> a#(b(x1)) -> a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) b#(a(b(b(x1)))) -> a#(b(x1)) -> a#(b(a(a(x1)))) -> a#(b(a(x1))) b#(a(b(b(x1)))) -> a#(b(x1)) -> a#(b(a(a(x1)))) -> b#(a(x1)) a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) -> b#(a(b(b(x1)))) -> a#(b(a(b(x1)))) a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) -> b#(a(b(b(x1)))) -> b#(a(b(x1))) a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) -> b#(a(b(b(x1)))) -> a#(b(x1)) a#(b(a(a(x1)))) -> b#(a(x1)) -> b#(a(b(b(x1)))) -> a#(b(a(b(x1)))) a#(b(a(a(x1)))) -> b#(a(x1)) -> b#(a(b(b(x1)))) -> b#(a(b(x1))) a#(b(a(a(x1)))) -> b#(a(x1)) -> b#(a(b(b(x1)))) -> a#(b(x1)) a#(b(a(a(x1)))) -> a#(b(a(x1))) -> a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) a#(b(a(a(x1)))) -> a#(b(a(x1))) -> a#(b(a(a(x1)))) -> a#(b(a(x1))) a#(b(a(a(x1)))) -> a#(b(a(x1))) -> a#(b(a(a(x1)))) -> b#(a(x1)) Arctic Interpretation Processor: dimension: 1 interpretation: [b#](x0) = x0, [a#](x0) = x0, [b](x0) = 1x0, [a](x0) = 1x0, [s](x0) = 15x0 + 13 orientation: b#(a(b(b(x1)))) = 3x1 >= 2x1 = b#(a(b(x1))) b#(a(b(b(x1)))) = 3x1 >= 3x1 = a#(b(a(b(x1)))) a#(b(a(a(x1)))) = 3x1 >= 3x1 = b#(a(b(a(x1)))) b#(a(b(b(x1)))) = 3x1 >= 1x1 = a#(b(x1)) a#(b(a(a(x1)))) = 3x1 >= 2x1 = a#(b(a(x1))) a#(b(a(a(x1)))) = 3x1 >= 1x1 = b#(a(x1)) a(s(x1)) = 16x1 + 14 >= 16x1 + 13 = s(a(x1)) b(a(b(s(x1)))) = 18x1 + 16 >= 18x1 + 15 = a(b(s(a(x1)))) b(a(b(b(x1)))) = 4x1 >= 4x1 = a(b(a(b(x1)))) a(b(a(a(x1)))) = 4x1 >= 4x1 = b(a(b(a(x1)))) problem: DPs: b#(a(b(b(x1)))) -> a#(b(a(b(x1)))) a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) TRS: a(s(x1)) -> s(a(x1)) b(a(b(s(x1)))) -> a(b(s(a(x1)))) b(a(b(b(x1)))) -> a(b(a(b(x1)))) a(b(a(a(x1)))) -> b(a(b(a(x1)))) EDG Processor: DPs: b#(a(b(b(x1)))) -> a#(b(a(b(x1)))) a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) TRS: a(s(x1)) -> s(a(x1)) b(a(b(s(x1)))) -> a(b(s(a(x1)))) b(a(b(b(x1)))) -> a(b(a(b(x1)))) a(b(a(a(x1)))) -> b(a(b(a(x1)))) graph: b#(a(b(b(x1)))) -> a#(b(a(b(x1)))) -> a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) a#(b(a(a(x1)))) -> b#(a(b(a(x1)))) -> b#(a(b(b(x1)))) -> a#(b(a(b(x1)))) Matrix Interpretation Processor: dim=3 interpretation: [b#](x0) = [0 1 0]x0, [a#](x0) = [0 1 0]x0, [0 0 0] [1] [b](x0) = [1 1 0]x0 + [0] [1 1 0] [0], [1 0 0] [0] [a](x0) = [0 0 1]x0 + [0] [0 0 1] [1], [1] [s](x0) = [1] [1] orientation: b#(a(b(b(x1)))) = [1 1 0]x1 + [1] >= [1 1 0]x1 + [1] = a#(b(a(b(x1)))) a#(b(a(a(x1)))) = [1 0 1]x1 + [1] >= [1 0 1]x1 = b#(a(b(a(x1)))) [1] [1] a(s(x1)) = [1] >= [1] = s(a(x1)) [2] [1] [1] [1] b(a(b(s(x1)))) = [3] >= [2] = a(b(s(a(x1)))) [3] [3] [0 0 0] [1] [0 0 0] [1] b(a(b(b(x1)))) = [1 1 0]x1 + [2] >= [1 1 0]x1 + [1] = a(b(a(b(x1)))) [1 1 0] [2] [1 1 0] [2] [0 0 0] [1] [0 0 0] [1] a(b(a(a(x1)))) = [1 0 1]x1 + [1] >= [1 0 1]x1 + [1] = b(a(b(a(x1)))) [1 0 1] [2] [1 0 1] [1] problem: DPs: b#(a(b(b(x1)))) -> a#(b(a(b(x1)))) TRS: a(s(x1)) -> s(a(x1)) b(a(b(s(x1)))) -> a(b(s(a(x1)))) b(a(b(b(x1)))) -> a(b(a(b(x1)))) a(b(a(a(x1)))) -> b(a(b(a(x1)))) EDG Processor: DPs: b#(a(b(b(x1)))) -> a#(b(a(b(x1)))) TRS: a(s(x1)) -> s(a(x1)) b(a(b(s(x1)))) -> a(b(s(a(x1)))) b(a(b(b(x1)))) -> a(b(a(b(x1)))) a(b(a(a(x1)))) -> b(a(b(a(x1)))) graph: Qed