YES Problem: a(a(x1)) -> b(x1) a(a(a(x1))) -> a(b(a(x1))) a(b(a(x1))) -> b(b(b(x1))) a(a(a(a(x1)))) -> a(a(b(a(a(x1))))) a(a(b(a(x1)))) -> a(b(b(a(b(x1))))) a(b(a(a(x1)))) -> b(a(b(b(a(x1))))) a(b(b(a(x1)))) -> b(b(b(b(b(x1))))) a(a(a(a(a(x1))))) -> a(a(a(b(a(a(a(x1))))))) a(a(a(b(a(x1))))) -> a(a(b(b(a(a(b(x1))))))) a(a(b(a(a(x1))))) -> a(b(a(b(a(b(a(x1))))))) a(a(b(b(a(x1))))) -> a(b(b(b(a(b(b(x1))))))) a(b(a(a(a(x1))))) -> b(a(a(b(b(a(a(x1))))))) a(b(a(b(a(x1))))) -> b(a(b(b(b(a(b(x1))))))) a(b(b(a(a(x1))))) -> b(b(a(b(b(b(a(x1))))))) a(b(b(b(a(x1))))) -> b(b(b(b(b(b(b(x1))))))) Proof: DP Processor: DPs: a#(a(a(x1))) -> a#(b(a(x1))) a#(a(a(a(x1)))) -> a#(b(a(a(x1)))) a#(a(a(a(x1)))) -> a#(a(b(a(a(x1))))) a#(a(b(a(x1)))) -> a#(b(x1)) a#(a(b(a(x1)))) -> a#(b(b(a(b(x1))))) a#(b(a(a(x1)))) -> a#(b(b(a(x1)))) a#(a(a(a(a(x1))))) -> a#(b(a(a(a(x1))))) a#(a(a(a(a(x1))))) -> a#(a(b(a(a(a(x1)))))) a#(a(a(a(a(x1))))) -> a#(a(a(b(a(a(a(x1))))))) a#(a(a(b(a(x1))))) -> a#(b(x1)) a#(a(a(b(a(x1))))) -> a#(a(b(x1))) a#(a(a(b(a(x1))))) -> a#(b(b(a(a(b(x1)))))) a#(a(a(b(a(x1))))) -> a#(a(b(b(a(a(b(x1))))))) a#(a(b(a(a(x1))))) -> a#(b(a(x1))) a#(a(b(a(a(x1))))) -> a#(b(a(b(a(x1))))) a#(a(b(a(a(x1))))) -> a#(b(a(b(a(b(a(x1))))))) a#(a(b(b(a(x1))))) -> a#(b(b(x1))) a#(a(b(b(a(x1))))) -> a#(b(b(b(a(b(b(x1))))))) a#(b(a(a(a(x1))))) -> a#(b(b(a(a(x1))))) a#(b(a(a(a(x1))))) -> a#(a(b(b(a(a(x1)))))) a#(b(a(b(a(x1))))) -> a#(b(x1)) a#(b(a(b(a(x1))))) -> a#(b(b(b(a(b(x1)))))) a#(b(b(a(a(x1))))) -> a#(b(b(b(a(x1))))) TRS: a(a(x1)) -> b(x1) a(a(a(x1))) -> a(b(a(x1))) a(b(a(x1))) -> b(b(b(x1))) a(a(a(a(x1)))) -> a(a(b(a(a(x1))))) a(a(b(a(x1)))) -> a(b(b(a(b(x1))))) a(b(a(a(x1)))) -> b(a(b(b(a(x1))))) a(b(b(a(x1)))) -> b(b(b(b(b(x1))))) a(a(a(a(a(x1))))) -> a(a(a(b(a(a(a(x1))))))) a(a(a(b(a(x1))))) -> a(a(b(b(a(a(b(x1))))))) a(a(b(a(a(x1))))) -> a(b(a(b(a(b(a(x1))))))) a(a(b(b(a(x1))))) -> a(b(b(b(a(b(b(x1))))))) a(b(a(a(a(x1))))) -> b(a(a(b(b(a(a(x1))))))) a(b(a(b(a(x1))))) -> b(a(b(b(b(a(b(x1))))))) a(b(b(a(a(x1))))) -> b(b(a(b(b(b(a(x1))))))) a(b(b(b(a(x1))))) -> b(b(b(b(b(b(b(x1))))))) Arctic Interpretation Processor: dimension: 1 interpretation: [a#](x0) = 4x0 + 8, [b](x0) = -3x0 + 0, [a](x0) = 2x0 + 9 orientation: a#(a(a(x1))) = 8x1 + 15 >= 3x1 + 10 = a#(b(a(x1))) a#(a(a(a(x1)))) = 10x1 + 17 >= 5x1 + 12 = a#(b(a(a(x1)))) a#(a(a(a(x1)))) = 10x1 + 17 >= 7x1 + 14 = a#(a(b(a(a(x1))))) a#(a(b(a(x1)))) = 5x1 + 13 >= 1x1 + 8 = a#(b(x1)) a#(a(b(a(x1)))) = 5x1 + 13 >= -3x1 + 8 = a#(b(b(a(b(x1))))) a#(b(a(a(x1)))) = 5x1 + 12 >= x1 + 8 = a#(b(b(a(x1)))) a#(a(a(a(a(x1))))) = 12x1 + 19 >= 7x1 + 14 = a#(b(a(a(a(x1))))) a#(a(a(a(a(x1))))) = 12x1 + 19 >= 9x1 + 16 = a#(a(b(a(a(a(x1)))))) a#(a(a(a(a(x1))))) = 12x1 + 19 >= 11x1 + 18 = a#(a(a(b(a(a(a(x1))))))) a#(a(a(b(a(x1))))) = 7x1 + 15 >= 1x1 + 8 = a#(b(x1)) a#(a(a(b(a(x1))))) = 7x1 + 15 >= 3x1 + 13 = a#(a(b(x1))) a#(a(a(b(a(x1))))) = 7x1 + 15 >= -1x1 + 9 = a#(b(b(a(a(b(x1)))))) a#(a(a(b(a(x1))))) = 7x1 + 15 >= 1x1 + 13 = a#(a(b(b(a(a(b(x1))))))) a#(a(b(a(a(x1))))) = 7x1 + 14 >= 3x1 + 10 = a#(b(a(x1))) a#(a(b(a(a(x1))))) = 7x1 + 14 >= 2x1 + 10 = a#(b(a(b(a(x1))))) a#(a(b(a(a(x1))))) = 7x1 + 14 >= 1x1 + 10 = a#(b(a(b(a(b(a(x1))))))) a#(a(b(b(a(x1))))) = 2x1 + 13 >= -2x1 + 8 = a#(b(b(x1))) a#(a(b(b(a(x1))))) = 2x1 + 13 >= -9x1 + 8 = a#(b(b(b(a(b(b(x1))))))) a#(b(a(a(a(x1))))) = 7x1 + 14 >= 2x1 + 9 = a#(b(b(a(a(x1))))) a#(b(a(a(a(x1))))) = 7x1 + 14 >= 4x1 + 13 = a#(a(b(b(a(a(x1)))))) a#(b(a(b(a(x1))))) = 2x1 + 10 >= 1x1 + 8 = a#(b(x1)) a#(b(a(b(a(x1))))) = 2x1 + 10 >= -6x1 + 8 = a#(b(b(b(a(b(x1)))))) a#(b(b(a(a(x1))))) = 2x1 + 9 >= -3x1 + 8 = a#(b(b(b(a(x1))))) a(a(x1)) = 4x1 + 11 >= -3x1 + 0 = b(x1) a(a(a(x1))) = 6x1 + 13 >= 1x1 + 9 = a(b(a(x1))) a(b(a(x1))) = 1x1 + 9 >= -9x1 + 0 = b(b(b(x1))) a(a(a(a(x1)))) = 8x1 + 15 >= 5x1 + 12 = a(a(b(a(a(x1))))) a(a(b(a(x1)))) = 3x1 + 11 >= -5x1 + 9 = a(b(b(a(b(x1))))) a(b(a(a(x1)))) = 3x1 + 10 >= -5x1 + 6 = b(a(b(b(a(x1))))) a(b(b(a(x1)))) = -2x1 + 9 >= -15x1 + 0 = b(b(b(b(b(x1))))) a(a(a(a(a(x1))))) = 10x1 + 17 >= 9x1 + 16 = a(a(a(b(a(a(a(x1))))))) a(a(a(b(a(x1))))) = 5x1 + 13 >= -1x1 + 11 = a(a(b(b(a(a(b(x1))))))) a(a(b(a(a(x1))))) = 5x1 + 12 >= -1x1 + 9 = a(b(a(b(a(b(a(x1))))))) a(a(b(b(a(x1))))) = x1 + 11 >= -11x1 + 9 = a(b(b(b(a(b(b(x1))))))) a(b(a(a(a(x1))))) = 5x1 + 12 >= -1x1 + 8 = b(a(a(b(b(a(a(x1))))))) a(b(a(b(a(x1))))) = x1 + 9 >= -11x1 + 6 = b(a(b(b(b(a(b(x1))))))) a(b(b(a(a(x1))))) = x1 + 9 >= -11x1 + 3 = b(b(a(b(b(b(a(x1))))))) a(b(b(b(a(x1))))) = -5x1 + 9 >= -21x1 + 0 = b(b(b(b(b(b(b(x1))))))) problem: DPs: TRS: a(a(x1)) -> b(x1) a(a(a(x1))) -> a(b(a(x1))) a(b(a(x1))) -> b(b(b(x1))) a(a(a(a(x1)))) -> a(a(b(a(a(x1))))) a(a(b(a(x1)))) -> a(b(b(a(b(x1))))) a(b(a(a(x1)))) -> b(a(b(b(a(x1))))) a(b(b(a(x1)))) -> b(b(b(b(b(x1))))) a(a(a(a(a(x1))))) -> a(a(a(b(a(a(a(x1))))))) a(a(a(b(a(x1))))) -> a(a(b(b(a(a(b(x1))))))) a(a(b(a(a(x1))))) -> a(b(a(b(a(b(a(x1))))))) a(a(b(b(a(x1))))) -> a(b(b(b(a(b(b(x1))))))) a(b(a(a(a(x1))))) -> b(a(a(b(b(a(a(x1))))))) a(b(a(b(a(x1))))) -> b(a(b(b(b(a(b(x1))))))) a(b(b(a(a(x1))))) -> b(b(a(b(b(b(a(x1))))))) a(b(b(b(a(x1))))) -> b(b(b(b(b(b(b(x1))))))) Qed