YES Problem: c(c(x1)) -> a(b(x1)) b(x1) -> a(a(x1)) b(b(b(x1))) -> a(c(b(x1))) a(c(a(x1))) -> a(c(c(x1))) Proof: DP Processor: DPs: c#(c(x1)) -> b#(x1) c#(c(x1)) -> a#(b(x1)) b#(x1) -> a#(x1) b#(x1) -> a#(a(x1)) b#(b(b(x1))) -> c#(b(x1)) b#(b(b(x1))) -> a#(c(b(x1))) a#(c(a(x1))) -> c#(x1) a#(c(a(x1))) -> c#(c(x1)) a#(c(a(x1))) -> a#(c(c(x1))) TRS: c(c(x1)) -> a(b(x1)) b(x1) -> a(a(x1)) b(b(b(x1))) -> a(c(b(x1))) a(c(a(x1))) -> a(c(c(x1))) TDG Processor: DPs: c#(c(x1)) -> b#(x1) c#(c(x1)) -> a#(b(x1)) b#(x1) -> a#(x1) b#(x1) -> a#(a(x1)) b#(b(b(x1))) -> c#(b(x1)) b#(b(b(x1))) -> a#(c(b(x1))) a#(c(a(x1))) -> c#(x1) a#(c(a(x1))) -> c#(c(x1)) a#(c(a(x1))) -> a#(c(c(x1))) TRS: c(c(x1)) -> a(b(x1)) b(x1) -> a(a(x1)) b(b(b(x1))) -> a(c(b(x1))) a(c(a(x1))) -> a(c(c(x1))) graph: a#(c(a(x1))) -> a#(c(c(x1))) -> a#(c(a(x1))) -> a#(c(c(x1))) a#(c(a(x1))) -> a#(c(c(x1))) -> a#(c(a(x1))) -> c#(c(x1)) a#(c(a(x1))) -> a#(c(c(x1))) -> a#(c(a(x1))) -> c#(x1) a#(c(a(x1))) -> c#(c(x1)) -> c#(c(x1)) -> a#(b(x1)) a#(c(a(x1))) -> c#(c(x1)) -> c#(c(x1)) -> b#(x1) a#(c(a(x1))) -> c#(x1) -> c#(c(x1)) -> a#(b(x1)) a#(c(a(x1))) -> c#(x1) -> c#(c(x1)) -> b#(x1) b#(b(b(x1))) -> a#(c(b(x1))) -> a#(c(a(x1))) -> a#(c(c(x1))) b#(b(b(x1))) -> a#(c(b(x1))) -> a#(c(a(x1))) -> c#(c(x1)) b#(b(b(x1))) -> a#(c(b(x1))) -> a#(c(a(x1))) -> c#(x1) b#(b(b(x1))) -> c#(b(x1)) -> c#(c(x1)) -> a#(b(x1)) b#(b(b(x1))) -> c#(b(x1)) -> c#(c(x1)) -> b#(x1) b#(x1) -> a#(a(x1)) -> a#(c(a(x1))) -> a#(c(c(x1))) b#(x1) -> a#(a(x1)) -> a#(c(a(x1))) -> c#(c(x1)) b#(x1) -> a#(a(x1)) -> a#(c(a(x1))) -> c#(x1) b#(x1) -> a#(x1) -> a#(c(a(x1))) -> a#(c(c(x1))) b#(x1) -> a#(x1) -> a#(c(a(x1))) -> c#(c(x1)) b#(x1) -> a#(x1) -> a#(c(a(x1))) -> c#(x1) c#(c(x1)) -> a#(b(x1)) -> a#(c(a(x1))) -> a#(c(c(x1))) c#(c(x1)) -> a#(b(x1)) -> a#(c(a(x1))) -> c#(c(x1)) c#(c(x1)) -> a#(b(x1)) -> a#(c(a(x1))) -> c#(x1) c#(c(x1)) -> b#(x1) -> b#(b(b(x1))) -> a#(c(b(x1))) c#(c(x1)) -> b#(x1) -> b#(b(b(x1))) -> c#(b(x1)) c#(c(x1)) -> b#(x1) -> b#(x1) -> a#(a(x1)) c#(c(x1)) -> b#(x1) -> b#(x1) -> a#(x1) EDG Processor: DPs: c#(c(x1)) -> b#(x1) c#(c(x1)) -> a#(b(x1)) b#(x1) -> a#(x1) b#(x1) -> a#(a(x1)) b#(b(b(x1))) -> c#(b(x1)) b#(b(b(x1))) -> a#(c(b(x1))) a#(c(a(x1))) -> c#(x1) a#(c(a(x1))) -> c#(c(x1)) a#(c(a(x1))) -> a#(c(c(x1))) TRS: c(c(x1)) -> a(b(x1)) b(x1) -> a(a(x1)) b(b(b(x1))) -> a(c(b(x1))) a(c(a(x1))) -> a(c(c(x1))) graph: a#(c(a(x1))) -> a#(c(c(x1))) -> a#(c(a(x1))) -> c#(x1) a#(c(a(x1))) -> a#(c(c(x1))) -> a#(c(a(x1))) -> c#(c(x1)) a#(c(a(x1))) -> a#(c(c(x1))) -> a#(c(a(x1))) -> a#(c(c(x1))) a#(c(a(x1))) -> c#(c(x1)) -> c#(c(x1)) -> b#(x1) a#(c(a(x1))) -> c#(c(x1)) -> c#(c(x1)) -> a#(b(x1)) a#(c(a(x1))) -> c#(x1) -> c#(c(x1)) -> b#(x1) a#(c(a(x1))) -> c#(x1) -> c#(c(x1)) -> a#(b(x1)) b#(b(b(x1))) -> a#(c(b(x1))) -> a#(c(a(x1))) -> c#(x1) b#(b(b(x1))) -> a#(c(b(x1))) -> a#(c(a(x1))) -> c#(c(x1)) b#(b(b(x1))) -> a#(c(b(x1))) -> a#(c(a(x1))) -> a#(c(c(x1))) b#(x1) -> a#(x1) -> a#(c(a(x1))) -> c#(x1) b#(x1) -> a#(x1) -> a#(c(a(x1))) -> c#(c(x1)) b#(x1) -> a#(x1) -> a#(c(a(x1))) -> a#(c(c(x1))) c#(c(x1)) -> b#(x1) -> b#(x1) -> a#(x1) c#(c(x1)) -> b#(x1) -> b#(x1) -> a#(a(x1)) c#(c(x1)) -> b#(x1) -> b#(b(b(x1))) -> c#(b(x1)) c#(c(x1)) -> b#(x1) -> b#(b(b(x1))) -> a#(c(b(x1))) SCC Processor: #sccs: 1 #rules: 6 #arcs: 17/81 DPs: a#(c(a(x1))) -> a#(c(c(x1))) a#(c(a(x1))) -> c#(c(x1)) c#(c(x1)) -> b#(x1) b#(b(b(x1))) -> a#(c(b(x1))) a#(c(a(x1))) -> c#(x1) b#(x1) -> a#(x1) TRS: c(c(x1)) -> a(b(x1)) b(x1) -> a(a(x1)) b(b(b(x1))) -> a(c(b(x1))) a(c(a(x1))) -> a(c(c(x1))) Arctic Interpretation Processor: dimension: 2 interpretation: [a#](x0) = [0 0]x0 + [0], [b#](x0) = [0 0]x0 + [0], [c#](x0) = [0 0]x0 + [0], [0 0] [0 ] [a](x0) = [0 0]x0 + [-&], [2 0] [0] [b](x0) = [0 0]x0 + [3], [0 0] [1] [c](x0) = [2 0]x0 + [3] orientation: a#(c(a(x1))) = [2 2]x1 + [3] >= [2 2]x1 + [3] = a#(c(c(x1))) a#(c(a(x1))) = [2 2]x1 + [3] >= [2 0]x1 + [3] = c#(c(x1)) c#(c(x1)) = [2 0]x1 + [3] >= [0 0]x1 + [0] = b#(x1) b#(b(b(x1))) = [4 2]x1 + [3] >= [4 2]x1 + [3] = a#(c(b(x1))) a#(c(a(x1))) = [2 2]x1 + [3] >= [0 0]x1 + [0] = c#(x1) b#(x1) = [0 0]x1 + [0] >= [0 0]x1 + [0] = a#(x1) [2 0] [3] [2 0] [3] c(c(x1)) = [2 2]x1 + [3] >= [2 0]x1 + [3] = a(b(x1)) [2 0] [0] [0 0] [0] b(x1) = [0 0]x1 + [3] >= [0 0]x1 + [0] = a(a(x1)) [6 4] [5] [4 2] [3] b(b(b(x1))) = [4 2]x1 + [3] >= [4 2]x1 + [3] = a(c(b(x1))) [2 2] [3] [2 2] [3] a(c(a(x1))) = [2 2]x1 + [3] >= [2 2]x1 + [3] = a(c(c(x1))) problem: DPs: a#(c(a(x1))) -> a#(c(c(x1))) a#(c(a(x1))) -> c#(c(x1)) c#(c(x1)) -> b#(x1) b#(b(b(x1))) -> a#(c(b(x1))) b#(x1) -> a#(x1) TRS: c(c(x1)) -> a(b(x1)) b(x1) -> a(a(x1)) b(b(b(x1))) -> a(c(b(x1))) a(c(a(x1))) -> a(c(c(x1))) EDG Processor: DPs: a#(c(a(x1))) -> a#(c(c(x1))) a#(c(a(x1))) -> c#(c(x1)) c#(c(x1)) -> b#(x1) b#(b(b(x1))) -> a#(c(b(x1))) b#(x1) -> a#(x1) TRS: c(c(x1)) -> a(b(x1)) b(x1) -> a(a(x1)) b(b(b(x1))) -> a(c(b(x1))) a(c(a(x1))) -> a(c(c(x1))) graph: a#(c(a(x1))) -> a#(c(c(x1))) -> a#(c(a(x1))) -> a#(c(c(x1))) a#(c(a(x1))) -> a#(c(c(x1))) -> a#(c(a(x1))) -> c#(c(x1)) a#(c(a(x1))) -> c#(c(x1)) -> c#(c(x1)) -> b#(x1) b#(b(b(x1))) -> a#(c(b(x1))) -> a#(c(a(x1))) -> a#(c(c(x1))) b#(b(b(x1))) -> a#(c(b(x1))) -> a#(c(a(x1))) -> c#(c(x1)) b#(x1) -> a#(x1) -> a#(c(a(x1))) -> a#(c(c(x1))) b#(x1) -> a#(x1) -> a#(c(a(x1))) -> c#(c(x1)) c#(c(x1)) -> b#(x1) -> b#(b(b(x1))) -> a#(c(b(x1))) c#(c(x1)) -> b#(x1) -> b#(x1) -> a#(x1) Matrix Interpretation Processor: dim=3 interpretation: [a#](x0) = [0 1 0]x0, [b#](x0) = [0 1 0]x0 + [1], [c#](x0) = [0 0 1]x0 + [1], [0 0 0] [0] [a](x0) = [0 0 0]x0 + [0] [0 1 0] [1], [0 0 0] [0] [b](x0) = [0 0 1]x0 + [0] [0 0 1] [1], [1 0 1] [0] [c](x0) = [0 0 1]x0 + [1] [0 1 0] [1] orientation: a#(c(a(x1))) = [0 1 0]x1 + [2] >= [0 1 0]x1 + [2] = a#(c(c(x1))) a#(c(a(x1))) = [0 1 0]x1 + [2] >= [0 1 0]x1 + [2] = c#(c(x1)) c#(c(x1)) = [0 1 0]x1 + [2] >= [0 1 0]x1 + [1] = b#(x1) b#(b(b(x1))) = [0 0 1]x1 + [2] >= [0 0 1]x1 + [2] = a#(c(b(x1))) b#(x1) = [0 1 0]x1 + [1] >= [0 1 0]x1 = a#(x1) [1 1 1] [1] [0 0 0] [0] c(c(x1)) = [0 1 0]x1 + [2] >= [0 0 0]x1 + [0] = a(b(x1)) [0 0 1] [2] [0 0 1] [1] [0 0 0] [0] [0] b(x1) = [0 0 1]x1 + [0] >= [0] = a(a(x1)) [0 0 1] [1] [1] [0 0 0] [0] [0 0 0] [0] b(b(b(x1))) = [0 0 1]x1 + [2] >= [0 0 0]x1 + [0] = a(c(b(x1))) [0 0 1] [3] [0 0 1] [3] [0 0 0] [0] [0 0 0] [0] a(c(a(x1))) = [0 0 0]x1 + [0] >= [0 0 0]x1 + [0] = a(c(c(x1))) [0 1 0] [3] [0 1 0] [3] problem: DPs: a#(c(a(x1))) -> a#(c(c(x1))) a#(c(a(x1))) -> c#(c(x1)) b#(b(b(x1))) -> a#(c(b(x1))) TRS: c(c(x1)) -> a(b(x1)) b(x1) -> a(a(x1)) b(b(b(x1))) -> a(c(b(x1))) a(c(a(x1))) -> a(c(c(x1))) EDG Processor: DPs: a#(c(a(x1))) -> a#(c(c(x1))) a#(c(a(x1))) -> c#(c(x1)) b#(b(b(x1))) -> a#(c(b(x1))) TRS: c(c(x1)) -> a(b(x1)) b(x1) -> a(a(x1)) b(b(b(x1))) -> a(c(b(x1))) a(c(a(x1))) -> a(c(c(x1))) graph: a#(c(a(x1))) -> a#(c(c(x1))) -> a#(c(a(x1))) -> c#(c(x1)) a#(c(a(x1))) -> a#(c(c(x1))) -> a#(c(a(x1))) -> a#(c(c(x1))) b#(b(b(x1))) -> a#(c(b(x1))) -> a#(c(a(x1))) -> c#(c(x1)) b#(b(b(x1))) -> a#(c(b(x1))) -> a#(c(a(x1))) -> a#(c(c(x1))) SCC Processor: #sccs: 1 #rules: 1 #arcs: 4/9 DPs: a#(c(a(x1))) -> a#(c(c(x1))) TRS: c(c(x1)) -> a(b(x1)) b(x1) -> a(a(x1)) b(b(b(x1))) -> a(c(b(x1))) a(c(a(x1))) -> a(c(c(x1))) Matrix Interpretation Processor: dim=4 interpretation: [a#](x0) = [1 0 0 0]x0, [0 0 0 0] [0] [0 0 0 0] [1] [a](x0) = [0 0 0 1]x0 + [0] [0 0 0 0] [0], [0 1 0 0] [0] [0 0 0 1] [1] [b](x0) = [1 0 1 0]x0 + [0] [0 0 0 0] [0], [0 1 1 0] [0] [0 0 0 1] [0] [c](x0) = [0 0 0 0]x0 + [0] [0 0 1 0] [1] orientation: a#(c(a(x1))) = [0 0 0 1]x1 + [1] >= [0 0 0 1]x1 = a#(c(c(x1))) [0 0 0 1] [0] [0] [0 0 1 0] [1] [1] c(c(x1)) = [0 0 0 0]x1 + [0] >= [0] = a(b(x1)) [0 0 0 0] [1] [0] [0 1 0 0] [0] [0] [0 0 0 1] [1] [1] b(x1) = [1 0 1 0]x1 + [0] >= [0] = a(a(x1)) [0 0 0 0] [0] [0] [0 0 0 0] [1] [0 0 0 0] [0] [0 0 0 0] [1] [0 0 0 0] [1] b(b(b(x1))) = [1 1 1 1]x1 + [1] >= [1 0 1 0]x1 + [1] = a(c(b(x1))) [0 0 0 0] [0] [0 0 0 0] [0] [0 0 0 0] [0] [0] [0 0 0 0] [1] [1] a(c(a(x1))) = [0 0 0 1]x1 + [1] >= [1] = a(c(c(x1))) [0 0 0 0] [0] [0] problem: DPs: TRS: c(c(x1)) -> a(b(x1)) b(x1) -> a(a(x1)) b(b(b(x1))) -> a(c(b(x1))) a(c(a(x1))) -> a(c(c(x1))) Qed