YES Problem: a(x1) -> b(x1) a(a(x1)) -> a(b(a(x1))) a(b(x1)) -> b(b(b(x1))) a(a(a(x1))) -> a(a(b(a(a(x1))))) a(a(b(x1))) -> a(b(b(a(b(x1))))) a(b(a(x1))) -> b(a(b(b(a(x1))))) a(b(b(x1))) -> b(b(b(b(b(x1))))) b(a(x1)) -> b(b(b(x1))) a(b(a(x1))) -> a(b(b(a(b(x1))))) b(a(a(x1))) -> b(a(b(b(a(x1))))) b(b(a(x1))) -> b(b(b(b(b(x1))))) Proof: DP Processor: DPs: a#(x1) -> b#(x1) a#(a(x1)) -> b#(a(x1)) a#(a(x1)) -> a#(b(a(x1))) a#(b(x1)) -> b#(b(x1)) a#(b(x1)) -> b#(b(b(x1))) a#(a(a(x1))) -> b#(a(a(x1))) a#(a(a(x1))) -> a#(b(a(a(x1)))) a#(a(a(x1))) -> a#(a(b(a(a(x1))))) a#(a(b(x1))) -> b#(a(b(x1))) a#(a(b(x1))) -> b#(b(a(b(x1)))) a#(a(b(x1))) -> a#(b(b(a(b(x1))))) a#(b(a(x1))) -> b#(b(a(x1))) a#(b(a(x1))) -> a#(b(b(a(x1)))) a#(b(a(x1))) -> b#(a(b(b(a(x1))))) a#(b(b(x1))) -> b#(b(b(x1))) a#(b(b(x1))) -> b#(b(b(b(x1)))) a#(b(b(x1))) -> b#(b(b(b(b(x1))))) b#(a(x1)) -> b#(x1) b#(a(x1)) -> b#(b(x1)) b#(a(x1)) -> b#(b(b(x1))) a#(b(a(x1))) -> b#(x1) a#(b(a(x1))) -> a#(b(x1)) a#(b(a(x1))) -> b#(a(b(x1))) a#(b(a(x1))) -> b#(b(a(b(x1)))) a#(b(a(x1))) -> a#(b(b(a(b(x1))))) b#(a(a(x1))) -> b#(a(x1)) b#(a(a(x1))) -> b#(b(a(x1))) b#(a(a(x1))) -> a#(b(b(a(x1)))) b#(a(a(x1))) -> b#(a(b(b(a(x1))))) b#(b(a(x1))) -> b#(x1) b#(b(a(x1))) -> b#(b(x1)) b#(b(a(x1))) -> b#(b(b(x1))) b#(b(a(x1))) -> b#(b(b(b(x1)))) b#(b(a(x1))) -> b#(b(b(b(b(x1))))) TRS: a(x1) -> b(x1) a(a(x1)) -> a(b(a(x1))) a(b(x1)) -> b(b(b(x1))) a(a(a(x1))) -> a(a(b(a(a(x1))))) a(a(b(x1))) -> a(b(b(a(b(x1))))) a(b(a(x1))) -> b(a(b(b(a(x1))))) a(b(b(x1))) -> b(b(b(b(b(x1))))) b(a(x1)) -> b(b(b(x1))) a(b(a(x1))) -> a(b(b(a(b(x1))))) b(a(a(x1))) -> b(a(b(b(a(x1))))) b(b(a(x1))) -> b(b(b(b(b(x1))))) TDG Processor: DPs: a#(x1) -> b#(x1) a#(a(x1)) -> b#(a(x1)) a#(a(x1)) -> a#(b(a(x1))) a#(b(x1)) -> b#(b(x1)) a#(b(x1)) -> b#(b(b(x1))) a#(a(a(x1))) -> b#(a(a(x1))) a#(a(a(x1))) -> a#(b(a(a(x1)))) a#(a(a(x1))) -> a#(a(b(a(a(x1))))) a#(a(b(x1))) -> b#(a(b(x1))) a#(a(b(x1))) -> b#(b(a(b(x1)))) a#(a(b(x1))) -> a#(b(b(a(b(x1))))) a#(b(a(x1))) -> b#(b(a(x1))) a#(b(a(x1))) -> a#(b(b(a(x1)))) a#(b(a(x1))) -> b#(a(b(b(a(x1))))) a#(b(b(x1))) -> b#(b(b(x1))) a#(b(b(x1))) -> b#(b(b(b(x1)))) a#(b(b(x1))) -> b#(b(b(b(b(x1))))) b#(a(x1)) -> b#(x1) b#(a(x1)) -> b#(b(x1)) b#(a(x1)) -> b#(b(b(x1))) a#(b(a(x1))) -> b#(x1) a#(b(a(x1))) -> a#(b(x1)) a#(b(a(x1))) -> b#(a(b(x1))) a#(b(a(x1))) -> b#(b(a(b(x1)))) a#(b(a(x1))) -> a#(b(b(a(b(x1))))) b#(a(a(x1))) -> b#(a(x1)) b#(a(a(x1))) -> b#(b(a(x1))) b#(a(a(x1))) -> a#(b(b(a(x1)))) b#(a(a(x1))) -> b#(a(b(b(a(x1))))) b#(b(a(x1))) -> b#(x1) b#(b(a(x1))) -> b#(b(x1)) b#(b(a(x1))) -> b#(b(b(x1))) b#(b(a(x1))) -> b#(b(b(b(x1)))) b#(b(a(x1))) -> b#(b(b(b(b(x1))))) TRS: a(x1) -> b(x1) a(a(x1)) -> a(b(a(x1))) a(b(x1)) -> b(b(b(x1))) a(a(a(x1))) -> a(a(b(a(a(x1))))) a(a(b(x1))) -> a(b(b(a(b(x1))))) a(b(a(x1))) -> b(a(b(b(a(x1))))) a(b(b(x1))) -> b(b(b(b(b(x1))))) b(a(x1)) -> b(b(b(x1))) a(b(a(x1))) -> a(b(b(a(b(x1))))) b(a(a(x1))) -> b(a(b(b(a(x1))))) b(b(a(x1))) -> b(b(b(b(b(x1))))) graph: b#(b(a(x1))) -> b#(b(b(b(b(x1))))) -> b#(b(a(x1))) -> b#(b(b(b(b(x1))))) b#(b(a(x1))) -> b#(b(b(b(b(x1))))) -> b#(b(a(x1))) -> b#(b(b(b(x1)))) b#(b(a(x1))) -> b#(b(b(b(b(x1))))) -> b#(b(a(x1))) -> b#(b(b(x1))) b#(b(a(x1))) -> b#(b(b(b(b(x1))))) -> b#(b(a(x1))) -> b#(b(x1)) b#(b(a(x1))) -> b#(b(b(b(b(x1))))) -> b#(b(a(x1))) -> b#(x1) b#(b(a(x1))) -> b#(b(b(b(b(x1))))) -> b#(a(a(x1))) -> b#(a(b(b(a(x1))))) b#(b(a(x1))) -> b#(b(b(b(b(x1))))) -> b#(a(a(x1))) -> a#(b(b(a(x1)))) b#(b(a(x1))) -> b#(b(b(b(b(x1))))) -> b#(a(a(x1))) -> b#(b(a(x1))) b#(b(a(x1))) -> b#(b(b(b(b(x1))))) -> b#(a(a(x1))) -> b#(a(x1)) b#(b(a(x1))) -> b#(b(b(b(b(x1))))) -> b#(a(x1)) -> b#(b(b(x1))) b#(b(a(x1))) -> b#(b(b(b(b(x1))))) -> b#(a(x1)) -> b#(b(x1)) b#(b(a(x1))) -> b#(b(b(b(b(x1))))) -> b#(a(x1)) -> b#(x1) b#(b(a(x1))) -> b#(b(b(b(x1)))) -> b#(b(a(x1))) -> b#(b(b(b(b(x1))))) b#(b(a(x1))) -> b#(b(b(b(x1)))) -> b#(b(a(x1))) -> b#(b(b(b(x1)))) b#(b(a(x1))) -> b#(b(b(b(x1)))) -> b#(b(a(x1))) -> b#(b(b(x1))) b#(b(a(x1))) -> b#(b(b(b(x1)))) -> b#(b(a(x1))) -> b#(b(x1)) b#(b(a(x1))) -> b#(b(b(b(x1)))) -> b#(b(a(x1))) -> b#(x1) b#(b(a(x1))) -> b#(b(b(b(x1)))) -> b#(a(a(x1))) -> b#(a(b(b(a(x1))))) b#(b(a(x1))) -> b#(b(b(b(x1)))) -> b#(a(a(x1))) -> a#(b(b(a(x1)))) b#(b(a(x1))) -> b#(b(b(b(x1)))) -> b#(a(a(x1))) -> b#(b(a(x1))) b#(b(a(x1))) -> b#(b(b(b(x1)))) -> b#(a(a(x1))) -> b#(a(x1)) b#(b(a(x1))) -> b#(b(b(b(x1)))) -> b#(a(x1)) -> b#(b(b(x1))) b#(b(a(x1))) -> b#(b(b(b(x1)))) -> b#(a(x1)) -> b#(b(x1)) b#(b(a(x1))) -> b#(b(b(b(x1)))) -> b#(a(x1)) -> b#(x1) b#(b(a(x1))) -> b#(b(b(x1))) -> b#(b(a(x1))) -> b#(b(b(b(b(x1))))) b#(b(a(x1))) -> b#(b(b(x1))) -> b#(b(a(x1))) -> b#(b(b(b(x1)))) b#(b(a(x1))) -> b#(b(b(x1))) -> b#(b(a(x1))) -> b#(b(b(x1))) b#(b(a(x1))) -> b#(b(b(x1))) -> b#(b(a(x1))) -> b#(b(x1)) b#(b(a(x1))) -> b#(b(b(x1))) -> b#(b(a(x1))) -> b#(x1) b#(b(a(x1))) -> b#(b(b(x1))) -> b#(a(a(x1))) -> b#(a(b(b(a(x1))))) b#(b(a(x1))) -> b#(b(b(x1))) -> b#(a(a(x1))) -> a#(b(b(a(x1)))) 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a#(a(x1)) -> a#(b(a(x1))) -> a#(b(x1)) -> b#(b(x1)) a#(a(x1)) -> a#(b(a(x1))) -> a#(a(x1)) -> a#(b(a(x1))) a#(a(x1)) -> a#(b(a(x1))) -> a#(a(x1)) -> b#(a(x1)) a#(a(x1)) -> a#(b(a(x1))) -> a#(x1) -> b#(x1) a#(x1) -> b#(x1) -> b#(b(a(x1))) -> b#(b(b(b(b(x1))))) a#(x1) -> b#(x1) -> b#(b(a(x1))) -> b#(b(b(b(x1)))) a#(x1) -> b#(x1) -> b#(b(a(x1))) -> b#(b(b(x1))) a#(x1) -> b#(x1) -> b#(b(a(x1))) -> b#(b(x1)) a#(x1) -> b#(x1) -> b#(b(a(x1))) -> b#(x1) a#(x1) -> b#(x1) -> b#(a(a(x1))) -> b#(a(b(b(a(x1))))) a#(x1) -> b#(x1) -> b#(a(a(x1))) -> a#(b(b(a(x1)))) a#(x1) -> b#(x1) -> b#(a(a(x1))) -> b#(b(a(x1))) a#(x1) -> b#(x1) -> b#(a(a(x1))) -> b#(a(x1)) a#(x1) -> b#(x1) -> b#(a(x1)) -> b#(b(b(x1))) a#(x1) -> b#(x1) -> b#(a(x1)) -> b#(b(x1)) a#(x1) -> b#(x1) -> b#(a(x1)) -> b#(x1) Matrix Interpretation Processor: dim=3 interpretation: [b#](x0) = [1 1 0]x0, [a#](x0) = [1 1 1]x0 + [1], [0 0 1] [b](x0) = [0 1 0]x0 [0 0 0] , [0 0 2] [1] [a](x0) = [1 1 0]x0 + [0] [0 0 1] [1] orientation: a#(x1) = [1 1 1]x1 + [1] >= [1 1 0]x1 = b#(x1) a#(a(x1)) = [1 1 3]x1 + [3] >= [1 1 2]x1 + [1] = b#(a(x1)) a#(a(x1)) = [1 1 3]x1 + [3] >= [1 1 1]x1 + [2] = a#(b(a(x1))) a#(b(x1)) = [0 1 1]x1 + [1] >= [0 1 1]x1 = b#(b(x1)) a#(b(x1)) = [0 1 1]x1 + [1] >= [0 1 0]x1 = b#(b(b(x1))) a#(a(a(x1))) = [1 1 5]x1 + [7] >= [1 1 4]x1 + [4] = b#(a(a(x1))) a#(a(a(x1))) = [1 1 5]x1 + [7] >= [1 1 3]x1 + [4] = a#(b(a(a(x1)))) a#(a(a(x1))) = [1 1 5]x1 + [7] >= [1 1 3]x1 + [6] = a#(a(b(a(a(x1))))) a#(a(b(x1))) = [0 1 1]x1 + [3] >= [0 1 1]x1 + [1] = b#(a(b(x1))) a#(a(b(x1))) = [0 1 1]x1 + [3] >= [0 1 1]x1 + [1] = b#(b(a(b(x1)))) a#(a(b(x1))) = [0 1 1]x1 + [3] >= [0 1 1]x1 + [1] = a#(b(b(a(b(x1))))) a#(b(a(x1))) = [1 1 1]x1 + [2] >= [1 1 1]x1 + [1] = b#(b(a(x1))) a#(b(a(x1))) = [1 1 1]x1 + [2] >= [1 1 0]x1 + [1] = a#(b(b(a(x1)))) a#(b(a(x1))) = [1 1 1]x1 + [2] >= [1 1 0]x1 + [1] = b#(a(b(b(a(x1))))) a#(b(b(x1))) = [0 1 0]x1 + [1] >= [0 1 0]x1 = b#(b(b(x1))) a#(b(b(x1))) = [0 1 0]x1 + [1] >= [0 1 0]x1 = b#(b(b(b(x1)))) a#(b(b(x1))) = [0 1 0]x1 + [1] >= [0 1 0]x1 = b#(b(b(b(b(x1))))) b#(a(x1)) = [1 1 2]x1 + [1] >= [1 1 0]x1 = b#(x1) b#(a(x1)) = [1 1 2]x1 + [1] >= [0 1 1]x1 = b#(b(x1)) b#(a(x1)) = [1 1 2]x1 + [1] >= [0 1 0]x1 = b#(b(b(x1))) a#(b(a(x1))) = [1 1 1]x1 + [2] >= [1 1 0]x1 = b#(x1) a#(b(a(x1))) = [1 1 1]x1 + [2] >= [0 1 1]x1 + [1] = a#(b(x1)) a#(b(a(x1))) = [1 1 1]x1 + [2] >= [0 1 1]x1 + [1] = b#(a(b(x1))) a#(b(a(x1))) = [1 1 1]x1 + [2] >= [0 1 1]x1 + [1] = b#(b(a(b(x1)))) a#(b(a(x1))) = [1 1 1]x1 + [2] >= [0 1 1]x1 + [1] = a#(b(b(a(b(x1))))) b#(a(a(x1))) = [1 1 4]x1 + [4] >= [1 1 2]x1 + [1] = b#(a(x1)) b#(a(a(x1))) = [1 1 4]x1 + [4] >= [1 1 1]x1 + [1] = b#(b(a(x1))) b#(a(a(x1))) = [1 1 4]x1 + [4] >= [1 1 0]x1 + [1] = a#(b(b(a(x1)))) b#(a(a(x1))) = [1 1 4]x1 + [4] >= [1 1 0]x1 + [1] = b#(a(b(b(a(x1))))) b#(b(a(x1))) = [1 1 1]x1 + [1] >= [1 1 0]x1 = b#(x1) b#(b(a(x1))) = [1 1 1]x1 + [1] >= [0 1 1]x1 = b#(b(x1)) b#(b(a(x1))) = [1 1 1]x1 + [1] >= [0 1 0]x1 = b#(b(b(x1))) b#(b(a(x1))) = [1 1 1]x1 + [1] >= [0 1 0]x1 = b#(b(b(b(x1)))) b#(b(a(x1))) = [1 1 1]x1 + [1] >= [0 1 0]x1 = b#(b(b(b(b(x1))))) [0 0 2] [1] [0 0 1] a(x1) = [1 1 0]x1 + [0] >= [0 1 0]x1 = b(x1) [0 0 1] [1] [0 0 0] [0 0 2] [3] [0 0 0] [1] a(a(x1)) = [1 1 2]x1 + [1] >= [1 1 1]x1 + [1] = a(b(a(x1))) [0 0 1] [2] [0 0 0] [1] [0 0 0] [1] [0 0 0] a(b(x1)) = [0 1 1]x1 + [0] >= [0 1 0]x1 = b(b(b(x1))) [0 0 0] [1] [0 0 0] [0 0 2] [5] [0 0 0] [3] a(a(a(x1))) = [1 1 4]x1 + [4] >= [1 1 3]x1 + [4] = a(a(b(a(a(x1))))) [0 0 1] [3] [0 0 0] [2] [0 0 0] [3] [0 0 0] [1] a(a(b(x1))) = [0 1 1]x1 + [1] >= [0 1 1]x1 + [0] = a(b(b(a(b(x1))))) [0 0 0] [2] [0 0 0] [1] [0 0 0] [1] [0 0 0] [1] a(b(a(x1))) = [1 1 1]x1 + [1] >= [1 1 0]x1 + [0] = b(a(b(b(a(x1))))) [0 0 0] [1] [0 0 0] [0] [0 0 0] [1] [0 0 0] a(b(b(x1))) = [0 1 0]x1 + [0] >= [0 1 0]x1 = b(b(b(b(b(x1))))) [0 0 0] [1] [0 0 0] [0 0 1] [1] [0 0 0] b(a(x1)) = [1 1 0]x1 + [0] >= [0 1 0]x1 = b(b(b(x1))) [0 0 0] [0] [0 0 0] [0 0 0] [1] [0 0 0] [1] a(b(a(x1))) = [1 1 1]x1 + [1] >= [0 1 1]x1 + [0] = a(b(b(a(b(x1))))) [0 0 0] [1] [0 0 0] [1] [0 0 1] [2] [0 0 0] [1] b(a(a(x1))) = [1 1 2]x1 + [1] >= [1 1 0]x1 + [0] = b(a(b(b(a(x1))))) [0 0 0] [0] [0 0 0] [0] [0 0 0] [0 0 0] b(b(a(x1))) = [1 1 0]x1 >= [0 1 0]x1 = b(b(b(b(b(x1))))) [0 0 0] [0 0 0] problem: DPs: TRS: a(x1) -> b(x1) a(a(x1)) -> a(b(a(x1))) a(b(x1)) -> b(b(b(x1))) a(a(a(x1))) -> a(a(b(a(a(x1))))) a(a(b(x1))) -> a(b(b(a(b(x1))))) a(b(a(x1))) -> b(a(b(b(a(x1))))) a(b(b(x1))) -> b(b(b(b(b(x1))))) b(a(x1)) -> b(b(b(x1))) a(b(a(x1))) -> a(b(b(a(b(x1))))) b(a(a(x1))) -> b(a(b(b(a(x1))))) b(b(a(x1))) -> b(b(b(b(b(x1))))) Qed