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: DP Processor: DPs: a#(a(b(b(x1)))) -> a#(x1) a#(a(b(b(x1)))) -> a#(a(x1)) a#(a(b(b(x1)))) -> a#(a(a(x1))) a#(c(x1)) -> a#(x1) a#(c(x1)) -> c#(a(x1)) c#(b(x1)) -> c#(x1) TRS: a(a(b(b(x1)))) -> b(b(b(a(a(a(x1)))))) a(c(x1)) -> c(a(x1)) c(b(x1)) -> b(c(x1)) TDG Processor: DPs: a#(a(b(b(x1)))) -> a#(x1) a#(a(b(b(x1)))) -> a#(a(x1)) a#(a(b(b(x1)))) -> a#(a(a(x1))) a#(c(x1)) -> a#(x1) a#(c(x1)) -> c#(a(x1)) c#(b(x1)) -> c#(x1) TRS: a(a(b(b(x1)))) -> b(b(b(a(a(a(x1)))))) a(c(x1)) -> c(a(x1)) c(b(x1)) -> b(c(x1)) graph: c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> c#(x1) a#(c(x1)) -> c#(a(x1)) -> c#(b(x1)) -> c#(x1) a#(c(x1)) -> a#(x1) -> a#(c(x1)) -> c#(a(x1)) a#(c(x1)) -> a#(x1) -> a#(c(x1)) -> a#(x1) a#(c(x1)) -> a#(x1) -> a#(a(b(b(x1)))) -> a#(a(a(x1))) a#(c(x1)) -> a#(x1) -> a#(a(b(b(x1)))) -> a#(a(x1)) a#(c(x1)) -> a#(x1) -> a#(a(b(b(x1)))) -> a#(x1) a#(a(b(b(x1)))) -> a#(a(a(x1))) -> a#(c(x1)) -> c#(a(x1)) a#(a(b(b(x1)))) -> a#(a(a(x1))) -> a#(c(x1)) -> a#(x1) a#(a(b(b(x1)))) -> a#(a(a(x1))) -> a#(a(b(b(x1)))) -> a#(a(a(x1))) a#(a(b(b(x1)))) -> a#(a(a(x1))) -> a#(a(b(b(x1)))) -> a#(a(x1)) a#(a(b(b(x1)))) -> a#(a(a(x1))) -> a#(a(b(b(x1)))) -> a#(x1) a#(a(b(b(x1)))) -> a#(a(x1)) -> a#(c(x1)) -> c#(a(x1)) a#(a(b(b(x1)))) -> a#(a(x1)) -> a#(c(x1)) -> a#(x1) a#(a(b(b(x1)))) -> a#(a(x1)) -> a#(a(b(b(x1)))) -> a#(a(a(x1))) a#(a(b(b(x1)))) -> a#(a(x1)) -> a#(a(b(b(x1)))) -> a#(a(x1)) a#(a(b(b(x1)))) -> a#(a(x1)) -> a#(a(b(b(x1)))) -> a#(x1) a#(a(b(b(x1)))) -> a#(x1) -> a#(c(x1)) -> c#(a(x1)) a#(a(b(b(x1)))) -> a#(x1) -> a#(c(x1)) -> a#(x1) a#(a(b(b(x1)))) -> a#(x1) -> a#(a(b(b(x1)))) -> a#(a(a(x1))) a#(a(b(b(x1)))) -> a#(x1) -> a#(a(b(b(x1)))) -> a#(a(x1)) a#(a(b(b(x1)))) -> a#(x1) -> a#(a(b(b(x1)))) -> a#(x1) CDG Processor: DPs: a#(a(b(b(x1)))) -> a#(x1) a#(a(b(b(x1)))) -> a#(a(x1)) a#(a(b(b(x1)))) -> a#(a(a(x1))) a#(c(x1)) -> a#(x1) a#(c(x1)) -> c#(a(x1)) c#(b(x1)) -> c#(x1) TRS: a(a(b(b(x1)))) -> b(b(b(a(a(a(x1)))))) a(c(x1)) -> c(a(x1)) c(b(x1)) -> b(c(x1)) graph: a#(c(x1)) -> c#(a(x1)) -> c#(b(x1)) -> c#(x1) a#(c(x1)) -> a#(x1) -> a#(a(b(b(x1)))) -> a#(x1) a#(c(x1)) -> a#(x1) -> a#(a(b(b(x1)))) -> a#(a(x1)) a#(c(x1)) -> a#(x1) -> a#(a(b(b(x1)))) -> a#(a(a(x1))) a#(c(x1)) -> a#(x1) -> a#(c(x1)) -> a#(x1) a#(c(x1)) -> a#(x1) -> a#(c(x1)) -> c#(a(x1)) a#(a(b(b(x1)))) -> a#(a(a(x1))) -> a#(a(b(b(x1)))) -> a#(x1) a#(a(b(b(x1)))) -> a#(a(a(x1))) -> a#(a(b(b(x1)))) -> a#(a(x1)) a#(a(b(b(x1)))) -> a#(a(a(x1))) -> a#(a(b(b(x1)))) -> a#(a(a(x1))) a#(a(b(b(x1)))) -> a#(a(a(x1))) -> a#(c(x1)) -> a#(x1) a#(a(b(b(x1)))) -> a#(a(a(x1))) -> a#(c(x1)) -> c#(a(x1)) a#(a(b(b(x1)))) -> a#(a(x1)) -> a#(a(b(b(x1)))) -> a#(x1) a#(a(b(b(x1)))) -> a#(a(x1)) -> a#(a(b(b(x1)))) -> a#(a(x1)) a#(a(b(b(x1)))) -> a#(a(x1)) -> a#(a(b(b(x1)))) -> a#(a(a(x1))) a#(a(b(b(x1)))) -> a#(a(x1)) -> a#(c(x1)) -> a#(x1) a#(a(b(b(x1)))) -> a#(a(x1)) -> a#(c(x1)) -> c#(a(x1)) a#(a(b(b(x1)))) -> a#(x1) -> a#(a(b(b(x1)))) -> a#(x1) a#(a(b(b(x1)))) -> a#(x1) -> a#(a(b(b(x1)))) -> a#(a(x1)) a#(a(b(b(x1)))) -> a#(x1) -> a#(a(b(b(x1)))) -> a#(a(a(x1))) a#(a(b(b(x1)))) -> a#(x1) -> a#(c(x1)) -> a#(x1) a#(a(b(b(x1)))) -> a#(x1) -> a#(c(x1)) -> c#(a(x1)) SCC Processor: #sccs: 1 #rules: 4 #arcs: 21/36 DPs: a#(c(x1)) -> a#(x1) a#(a(b(b(x1)))) -> a#(a(a(x1))) a#(a(b(b(x1)))) -> a#(a(x1)) a#(a(b(b(x1)))) -> a#(x1) TRS: a(a(b(b(x1)))) -> b(b(b(a(a(a(x1)))))) a(c(x1)) -> c(a(x1)) c(b(x1)) -> b(c(x1)) Arctic Interpretation Processor: dimension: 1 interpretation: [a#](x0) = x0 + 0, [c](x0) = 10x0 + 3, [a](x0) = x0, [b](x0) = x0 orientation: a#(c(x1)) = 10x1 + 3 >= x1 + 0 = a#(x1) a#(a(b(b(x1)))) = x1 + 0 >= x1 + 0 = a#(a(a(x1))) a#(a(b(b(x1)))) = x1 + 0 >= x1 + 0 = a#(a(x1)) a#(a(b(b(x1)))) = x1 + 0 >= x1 + 0 = a#(x1) a(a(b(b(x1)))) = x1 >= x1 = b(b(b(a(a(a(x1)))))) a(c(x1)) = 10x1 + 3 >= 10x1 + 3 = c(a(x1)) c(b(x1)) = 10x1 + 3 >= 10x1 + 3 = b(c(x1)) problem: DPs: a#(a(b(b(x1)))) -> a#(a(a(x1))) a#(a(b(b(x1)))) -> a#(a(x1)) a#(a(b(b(x1)))) -> a#(x1) TRS: a(a(b(b(x1)))) -> b(b(b(a(a(a(x1)))))) a(c(x1)) -> c(a(x1)) c(b(x1)) -> b(c(x1)) EDG Processor: DPs: a#(a(b(b(x1)))) -> a#(a(a(x1))) a#(a(b(b(x1)))) -> a#(a(x1)) a#(a(b(b(x1)))) -> a#(x1) TRS: a(a(b(b(x1)))) -> b(b(b(a(a(a(x1)))))) a(c(x1)) -> c(a(x1)) c(b(x1)) -> b(c(x1)) graph: a#(a(b(b(x1)))) -> a#(a(a(x1))) -> a#(a(b(b(x1)))) -> a#(a(a(x1))) a#(a(b(b(x1)))) -> a#(a(a(x1))) -> a#(a(b(b(x1)))) -> a#(a(x1)) a#(a(b(b(x1)))) -> a#(a(a(x1))) -> a#(a(b(b(x1)))) -> a#(x1) a#(a(b(b(x1)))) -> a#(a(x1)) -> a#(a(b(b(x1)))) -> a#(a(a(x1))) a#(a(b(b(x1)))) -> a#(a(x1)) -> a#(a(b(b(x1)))) -> a#(a(x1)) a#(a(b(b(x1)))) -> a#(a(x1)) -> a#(a(b(b(x1)))) -> a#(x1) a#(a(b(b(x1)))) -> a#(x1) -> a#(a(b(b(x1)))) -> a#(a(a(x1))) a#(a(b(b(x1)))) -> a#(x1) -> a#(a(b(b(x1)))) -> a#(a(x1)) a#(a(b(b(x1)))) -> a#(x1) -> a#(a(b(b(x1)))) -> a#(x1) Matrix Interpretation Processor: dim=3 interpretation: [a#](x0) = [1 0 0]x0, [c](x0) = x0 , [0 1 0] [a](x0) = [1 0 0]x0 [2 0 0] , [0 0 0] [0] [b](x0) = [0 0 1]x0 + [0] [2 1 0] [1] orientation: a#(a(b(b(x1)))) = [2 1 0]x1 + [1] >= [1 0 0]x1 = a#(a(a(x1))) a#(a(b(b(x1)))) = [2 1 0]x1 + [1] >= [0 1 0]x1 = a#(a(x1)) a#(a(b(b(x1)))) = [2 1 0]x1 + [1] >= [1 0 0]x1 = a#(x1) [0 0 0] [0] [0 0 0] [0] a(a(b(b(x1)))) = [2 1 0]x1 + [1] >= [2 0 0]x1 + [1] = b(b(b(a(a(a(x1)))))) [4 2 0] [2] [1 2 0] [2] [0 1 0] [0 1 0] a(c(x1)) = [1 0 0]x1 >= [1 0 0]x1 = c(a(x1)) [2 0 0] [2 0 0] [0 0 0] [0] [0 0 0] [0] c(b(x1)) = [0 0 1]x1 + [0] >= [0 0 1]x1 + [0] = b(c(x1)) [2 1 0] [1] [2 1 0] [1] problem: DPs: TRS: a(a(b(b(x1)))) -> b(b(b(a(a(a(x1)))))) a(c(x1)) -> c(a(x1)) c(b(x1)) -> b(c(x1)) Qed