YES Problem: a(a(b(b(x1)))) -> b(b(b(a(a(a(x1)))))) a(c(x1)) -> c(a(x1)) c(b(x1)) -> b(c(x1)) Proof: String Reversal Processor: b(b(a(a(x1)))) -> a(a(a(b(b(b(x1)))))) c(a(x1)) -> a(c(x1)) b(c(x1)) -> c(b(x1)) DP Processor: DPs: b#(b(a(a(x1)))) -> b#(x1) b#(b(a(a(x1)))) -> b#(b(x1)) b#(b(a(a(x1)))) -> b#(b(b(x1))) c#(a(x1)) -> c#(x1) b#(c(x1)) -> b#(x1) b#(c(x1)) -> c#(b(x1)) TRS: b(b(a(a(x1)))) -> a(a(a(b(b(b(x1)))))) c(a(x1)) -> a(c(x1)) b(c(x1)) -> c(b(x1)) TDG Processor: DPs: b#(b(a(a(x1)))) -> b#(x1) b#(b(a(a(x1)))) -> b#(b(x1)) b#(b(a(a(x1)))) -> b#(b(b(x1))) c#(a(x1)) -> c#(x1) b#(c(x1)) -> b#(x1) b#(c(x1)) -> c#(b(x1)) TRS: b(b(a(a(x1)))) -> a(a(a(b(b(b(x1)))))) c(a(x1)) -> a(c(x1)) b(c(x1)) -> c(b(x1)) graph: c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> c#(x1) b#(c(x1)) -> c#(b(x1)) -> c#(a(x1)) -> c#(x1) b#(c(x1)) -> b#(x1) -> b#(c(x1)) -> c#(b(x1)) b#(c(x1)) -> b#(x1) -> b#(c(x1)) -> b#(x1) b#(c(x1)) -> b#(x1) -> b#(b(a(a(x1)))) -> b#(b(b(x1))) b#(c(x1)) -> b#(x1) -> b#(b(a(a(x1)))) -> b#(b(x1)) b#(c(x1)) -> b#(x1) -> b#(b(a(a(x1)))) -> b#(x1) b#(b(a(a(x1)))) -> b#(b(b(x1))) -> b#(c(x1)) -> c#(b(x1)) b#(b(a(a(x1)))) -> b#(b(b(x1))) -> b#(c(x1)) -> b#(x1) b#(b(a(a(x1)))) -> b#(b(b(x1))) -> b#(b(a(a(x1)))) -> b#(b(b(x1))) b#(b(a(a(x1)))) -> b#(b(b(x1))) -> b#(b(a(a(x1)))) -> b#(b(x1)) b#(b(a(a(x1)))) -> b#(b(b(x1))) -> b#(b(a(a(x1)))) -> b#(x1) b#(b(a(a(x1)))) -> b#(b(x1)) -> b#(c(x1)) -> c#(b(x1)) b#(b(a(a(x1)))) -> b#(b(x1)) -> b#(c(x1)) -> b#(x1) b#(b(a(a(x1)))) -> b#(b(x1)) -> b#(b(a(a(x1)))) -> b#(b(b(x1))) b#(b(a(a(x1)))) -> b#(b(x1)) -> b#(b(a(a(x1)))) -> b#(b(x1)) b#(b(a(a(x1)))) -> b#(b(x1)) -> b#(b(a(a(x1)))) -> b#(x1) b#(b(a(a(x1)))) -> b#(x1) -> b#(c(x1)) -> c#(b(x1)) b#(b(a(a(x1)))) -> b#(x1) -> b#(c(x1)) -> b#(x1) b#(b(a(a(x1)))) -> b#(x1) -> b#(b(a(a(x1)))) -> b#(b(b(x1))) b#(b(a(a(x1)))) -> b#(x1) -> b#(b(a(a(x1)))) -> b#(b(x1)) b#(b(a(a(x1)))) -> b#(x1) -> b#(b(a(a(x1)))) -> b#(x1) CDG Processor: DPs: b#(b(a(a(x1)))) -> b#(x1) b#(b(a(a(x1)))) -> b#(b(x1)) b#(b(a(a(x1)))) -> b#(b(b(x1))) c#(a(x1)) -> c#(x1) b#(c(x1)) -> b#(x1) b#(c(x1)) -> c#(b(x1)) TRS: b(b(a(a(x1)))) -> a(a(a(b(b(b(x1)))))) c(a(x1)) -> a(c(x1)) b(c(x1)) -> c(b(x1)) graph: b#(c(x1)) -> c#(b(x1)) -> c#(a(x1)) -> c#(x1) b#(c(x1)) -> b#(x1) -> b#(b(a(a(x1)))) -> b#(x1) b#(c(x1)) -> b#(x1) -> b#(b(a(a(x1)))) -> b#(b(x1)) b#(c(x1)) -> b#(x1) -> b#(b(a(a(x1)))) -> b#(b(b(x1))) b#(c(x1)) -> b#(x1) -> b#(c(x1)) -> b#(x1) b#(c(x1)) -> b#(x1) -> b#(c(x1)) -> c#(b(x1)) b#(b(a(a(x1)))) -> b#(b(b(x1))) -> b#(b(a(a(x1)))) -> b#(x1) b#(b(a(a(x1)))) -> b#(b(b(x1))) -> b#(b(a(a(x1)))) -> b#(b(x1)) b#(b(a(a(x1)))) -> b#(b(b(x1))) -> b#(b(a(a(x1)))) -> b#(b(b(x1))) b#(b(a(a(x1)))) -> b#(b(b(x1))) -> b#(c(x1)) -> b#(x1) b#(b(a(a(x1)))) -> b#(b(b(x1))) -> b#(c(x1)) -> c#(b(x1)) b#(b(a(a(x1)))) -> b#(b(x1)) -> b#(b(a(a(x1)))) -> b#(x1) b#(b(a(a(x1)))) -> b#(b(x1)) -> b#(b(a(a(x1)))) -> b#(b(x1)) b#(b(a(a(x1)))) -> b#(b(x1)) -> b#(b(a(a(x1)))) -> b#(b(b(x1))) b#(b(a(a(x1)))) -> b#(b(x1)) -> b#(c(x1)) -> b#(x1) b#(b(a(a(x1)))) -> b#(b(x1)) -> b#(c(x1)) -> c#(b(x1)) b#(b(a(a(x1)))) -> b#(x1) -> b#(b(a(a(x1)))) -> b#(x1) b#(b(a(a(x1)))) -> b#(x1) -> b#(b(a(a(x1)))) -> b#(b(x1)) b#(b(a(a(x1)))) -> b#(x1) -> b#(b(a(a(x1)))) -> b#(b(b(x1))) b#(b(a(a(x1)))) -> b#(x1) -> b#(c(x1)) -> b#(x1) b#(b(a(a(x1)))) -> b#(x1) -> b#(c(x1)) -> c#(b(x1)) SCC Processor: #sccs: 1 #rules: 4 #arcs: 21/36 DPs: b#(c(x1)) -> b#(x1) b#(b(a(a(x1)))) -> b#(b(b(x1))) b#(b(a(a(x1)))) -> b#(b(x1)) b#(b(a(a(x1)))) -> b#(x1) TRS: b(b(a(a(x1)))) -> a(a(a(b(b(b(x1)))))) c(a(x1)) -> a(c(x1)) b(c(x1)) -> c(b(x1)) Arctic Interpretation Processor: dimension: 1 usable rules: b(b(a(a(x1)))) -> a(a(a(b(b(b(x1)))))) c(a(x1)) -> a(c(x1)) b(c(x1)) -> c(b(x1)) interpretation: [b#](x0) = x0 + 0, [c](x0) = 2x0 + 1, [a](x0) = x0 + 1, [b](x0) = x0 orientation: b#(c(x1)) = 2x1 + 1 >= x1 + 0 = b#(x1) b#(b(a(a(x1)))) = x1 + 1 >= x1 + 0 = b#(b(b(x1))) b#(b(a(a(x1)))) = x1 + 1 >= x1 + 0 = b#(b(x1)) b#(b(a(a(x1)))) = x1 + 1 >= x1 + 0 = b#(x1) b(b(a(a(x1)))) = x1 + 1 >= x1 + 1 = a(a(a(b(b(b(x1)))))) c(a(x1)) = 2x1 + 3 >= 2x1 + 1 = a(c(x1)) b(c(x1)) = 2x1 + 1 >= 2x1 + 1 = c(b(x1)) problem: DPs: b#(b(a(a(x1)))) -> b#(b(b(x1))) b#(b(a(a(x1)))) -> b#(b(x1)) b#(b(a(a(x1)))) -> b#(x1) TRS: b(b(a(a(x1)))) -> a(a(a(b(b(b(x1)))))) c(a(x1)) -> a(c(x1)) b(c(x1)) -> c(b(x1)) Restore Modifier: DPs: b#(b(a(a(x1)))) -> b#(b(b(x1))) b#(b(a(a(x1)))) -> b#(b(x1)) b#(b(a(a(x1)))) -> b#(x1) TRS: b(b(a(a(x1)))) -> a(a(a(b(b(b(x1)))))) c(a(x1)) -> a(c(x1)) b(c(x1)) -> c(b(x1)) EDG Processor: DPs: b#(b(a(a(x1)))) -> b#(b(b(x1))) b#(b(a(a(x1)))) -> b#(b(x1)) b#(b(a(a(x1)))) -> b#(x1) TRS: b(b(a(a(x1)))) -> a(a(a(b(b(b(x1)))))) c(a(x1)) -> a(c(x1)) b(c(x1)) -> c(b(x1)) graph: b#(b(a(a(x1)))) -> b#(b(b(x1))) -> b#(b(a(a(x1)))) -> b#(b(b(x1))) b#(b(a(a(x1)))) -> b#(b(b(x1))) -> b#(b(a(a(x1)))) -> b#(b(x1)) b#(b(a(a(x1)))) -> b#(b(b(x1))) -> b#(b(a(a(x1)))) -> b#(x1) b#(b(a(a(x1)))) -> b#(b(x1)) -> b#(b(a(a(x1)))) -> b#(b(b(x1))) b#(b(a(a(x1)))) -> b#(b(x1)) -> b#(b(a(a(x1)))) -> b#(b(x1)) b#(b(a(a(x1)))) -> b#(b(x1)) -> b#(b(a(a(x1)))) -> b#(x1) b#(b(a(a(x1)))) -> b#(x1) -> b#(b(a(a(x1)))) -> b#(b(b(x1))) b#(b(a(a(x1)))) -> b#(x1) -> b#(b(a(a(x1)))) -> b#(b(x1)) b#(b(a(a(x1)))) -> b#(x1) -> b#(b(a(a(x1)))) -> b#(x1) Matrix Interpretation Processor: dim=5 interpretation: [b#](x0) = [0 0 1 0 0]x0, [c](x0) = x0 , [0 0 0 1 1] [0] [0 0 0 0 0] [0] [a](x0) = [0 0 0 1 0]x0 + [0] [0 1 1 0 0] [1] [0 1 0 0 0] [0], [0 1 0 0 0] [0 0 1 0 0] [b](x0) = [0 0 1 0 0]x0 [0 0 1 0 1] [1 0 0 0 0] orientation: b#(b(a(a(x1)))) = [0 1 1 0 0]x1 + [1] >= [0 0 1 0 0]x1 = b#(b(b(x1))) b#(b(a(a(x1)))) = [0 1 1 0 0]x1 + [1] >= [0 0 1 0 0]x1 = b#(b(x1)) b#(b(a(a(x1)))) = [0 1 1 0 0]x1 + [1] >= [0 0 1 0 0]x1 = b#(x1) [0 1 1 0 0] [1] [0 1 1 0 0] [1] [0 1 1 0 0] [1] [0 0 0 0 0] [0] b(b(a(a(x1)))) = [0 1 1 0 0]x1 + [1] >= [0 1 1 0 0]x1 + [1] = a(a(a(b(b(b(x1)))))) [0 3 2 0 0] [2] [0 0 2 0 0] [2] [0 0 0 0 0] [0] [0 0 0 0 0] [0] [0 0 0 1 1] [0] [0 0 0 1 1] [0] [0 0 0 0 0] [0] [0 0 0 0 0] [0] c(a(x1)) = [0 0 0 1 0]x1 + [0] >= [0 0 0 1 0]x1 + [0] = a(c(x1)) [0 1 1 0 0] [1] [0 1 1 0 0] [1] [0 1 0 0 0] [0] [0 1 0 0 0] [0] [0 1 0 0 0] [0 1 0 0 0] [0 0 1 0 0] [0 0 1 0 0] b(c(x1)) = [0 0 1 0 0]x1 >= [0 0 1 0 0]x1 = c(b(x1)) [0 0 1 0 1] [0 0 1 0 1] [1 0 0 0 0] [1 0 0 0 0] problem: DPs: TRS: b(b(a(a(x1)))) -> a(a(a(b(b(b(x1)))))) c(a(x1)) -> a(c(x1)) b(c(x1)) -> c(b(x1)) Qed