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))))))) EDG 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))))))) graph: a#(b(a(b(a(x1))))) -> a#(b(x1)) -> a#(b(a(a(x1)))) -> a#(b(b(a(x1)))) a#(b(a(b(a(x1))))) -> a#(b(x1)) -> a#(b(a(a(a(x1))))) -> a#(b(b(a(a(x1))))) a#(b(a(b(a(x1))))) -> a#(b(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(x1)) a#(b(a(b(a(x1))))) -> a#(b(x1)) -> a#(b(a(b(a(x1))))) -> a#(b(b(b(a(b(x1)))))) 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-> a#(b(a(b(a(b(a(x1))))))) a#(a(a(a(x1)))) -> a#(a(b(a(a(x1))))) -> a#(a(b(b(a(x1))))) -> a#(b(b(x1))) a#(a(a(a(x1)))) -> a#(a(b(a(a(x1))))) -> a#(a(b(b(a(x1))))) -> a#(b(b(b(a(b(b(x1))))))) a#(a(a(a(x1)))) -> a#(a(b(a(a(x1))))) -> a#(b(a(a(a(x1))))) -> a#(b(b(a(a(x1))))) a#(a(a(a(x1)))) -> a#(a(b(a(a(x1))))) -> a#(b(a(a(a(x1))))) -> a#(a(b(b(a(a(x1)))))) a#(a(a(a(x1)))) -> a#(a(b(a(a(x1))))) -> a#(b(a(b(a(x1))))) -> a#(b(x1)) a#(a(a(a(x1)))) -> a#(a(b(a(a(x1))))) -> a#(b(a(b(a(x1))))) -> a#(b(b(b(a(b(x1)))))) a#(a(a(a(x1)))) -> a#(a(b(a(a(x1))))) -> a#(b(b(a(a(x1))))) -> a#(b(b(b(a(x1))))) a#(a(a(x1))) -> a#(b(a(x1))) -> a#(b(a(a(x1)))) -> a#(b(b(a(x1)))) a#(a(a(x1))) -> a#(b(a(x1))) -> a#(b(a(a(a(x1))))) -> a#(b(b(a(a(x1))))) a#(a(a(x1))) -> a#(b(a(x1))) -> a#(b(a(a(a(x1))))) -> a#(a(b(b(a(a(x1)))))) a#(a(a(x1))) -> a#(b(a(x1))) -> a#(b(a(b(a(x1))))) -> a#(b(x1)) a#(a(a(x1))) -> a#(b(a(x1))) -> a#(b(a(b(a(x1))))) -> a#(b(b(b(a(b(x1)))))) a#(a(a(x1))) -> a#(b(a(x1))) -> a#(b(b(a(a(x1))))) -> a#(b(b(b(a(x1))))) SCC Processor: #sccs: 1 #rules: 15 #arcs: 197/529 DPs: a#(b(a(b(a(x1))))) -> a#(b(x1)) a#(b(a(a(a(x1))))) -> a#(a(b(b(a(a(x1)))))) a#(a(b(a(a(x1))))) -> a#(b(a(b(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(x1))) a#(a(a(b(a(x1))))) -> a#(a(b(b(a(a(b(x1))))))) a#(a(a(b(a(x1))))) -> a#(a(b(x1))) a#(a(a(b(a(x1))))) -> a#(b(x1)) a#(a(a(a(a(x1))))) -> a#(a(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#(b(a(a(a(x1))))) a#(a(b(a(x1)))) -> a#(b(x1)) a#(a(a(a(x1)))) -> a#(a(b(a(a(x1))))) a#(a(a(a(x1)))) -> a#(b(a(a(x1)))) a#(a(a(x1))) -> a#(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 + 8 orientation: a#(b(a(b(a(x1))))) = 2x1 + 9 >= 1x1 + 8 = a#(b(x1)) a#(b(a(a(a(x1))))) = 7x1 + 13 >= 4x1 + 12 = a#(a(b(b(a(a(x1)))))) a#(a(b(a(a(x1))))) = 7x1 + 13 >= 1x1 + 9 = a#(b(a(b(a(b(a(x1))))))) a#(a(b(a(a(x1))))) = 7x1 + 13 >= 2x1 + 9 = a#(b(a(b(a(x1))))) a#(a(b(a(a(x1))))) = 7x1 + 13 >= 3x1 + 9 = a#(b(a(x1))) a#(a(a(b(a(x1))))) = 7x1 + 14 >= 1x1 + 12 = a#(a(b(b(a(a(b(x1))))))) a#(a(a(b(a(x1))))) = 7x1 + 14 >= 3x1 + 12 = a#(a(b(x1))) a#(a(a(b(a(x1))))) = 7x1 + 14 >= 1x1 + 8 = a#(b(x1)) a#(a(a(a(a(x1))))) = 12x1 + 18 >= 11x1 + 17 = a#(a(a(b(a(a(a(x1))))))) a#(a(a(a(a(x1))))) = 12x1 + 18 >= 9x1 + 15 = a#(a(b(a(a(a(x1)))))) a#(a(a(a(a(x1))))) = 12x1 + 18 >= 7x1 + 13 = a#(b(a(a(a(x1))))) a#(a(b(a(x1)))) = 5x1 + 12 >= 1x1 + 8 = a#(b(x1)) a#(a(a(a(x1)))) = 10x1 + 16 >= 7x1 + 13 = a#(a(b(a(a(x1))))) a#(a(a(a(x1)))) = 10x1 + 16 >= 5x1 + 11 = a#(b(a(a(x1)))) a#(a(a(x1))) = 8x1 + 14 >= 3x1 + 9 = a#(b(a(x1))) a(a(x1)) = 4x1 + 10 >= -3x1 + 0 = b(x1) a(a(a(x1))) = 6x1 + 12 >= 1x1 + 8 = a(b(a(x1))) a(b(a(x1))) = 1x1 + 8 >= -9x1 + 0 = b(b(b(x1))) a(a(a(a(x1)))) = 8x1 + 14 >= 5x1 + 11 = a(a(b(a(a(x1))))) a(a(b(a(x1)))) = 3x1 + 10 >= -5x1 + 8 = a(b(b(a(b(x1))))) a(b(a(a(x1)))) = 3x1 + 9 >= -5x1 + 5 = b(a(b(b(a(x1))))) a(b(b(a(x1)))) = -2x1 + 8 >= -15x1 + 0 = b(b(b(b(b(x1))))) a(a(a(a(a(x1))))) = 10x1 + 16 >= 9x1 + 15 = a(a(a(b(a(a(a(x1))))))) a(a(a(b(a(x1))))) = 5x1 + 12 >= -1x1 + 10 = a(a(b(b(a(a(b(x1))))))) a(a(b(a(a(x1))))) = 5x1 + 11 >= -1x1 + 8 = a(b(a(b(a(b(a(x1))))))) a(a(b(b(a(x1))))) = x1 + 10 >= -11x1 + 8 = a(b(b(b(a(b(b(x1))))))) a(b(a(a(a(x1))))) = 5x1 + 11 >= -1x1 + 7 = b(a(a(b(b(a(a(x1))))))) a(b(a(b(a(x1))))) = x1 + 8 >= -11x1 + 5 = b(a(b(b(b(a(b(x1))))))) a(b(b(a(a(x1))))) = x1 + 8 >= -11x1 + 2 = b(b(a(b(b(b(a(x1))))))) a(b(b(b(a(x1))))) = -5x1 + 8 >= -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