YES Problem: b(d(b(x1))) -> c(d(b(x1))) b(a(c(x1))) -> b(c(x1)) a(d(x1)) -> d(c(x1)) b(b(b(x1))) -> a(b(c(x1))) d(c(x1)) -> b(d(x1)) d(c(x1)) -> d(b(d(x1))) d(a(c(x1))) -> b(b(x1)) Proof: DP Processor: DPs: b#(a(c(x1))) -> b#(c(x1)) a#(d(x1)) -> d#(c(x1)) b#(b(b(x1))) -> b#(c(x1)) b#(b(b(x1))) -> a#(b(c(x1))) d#(c(x1)) -> d#(x1) d#(c(x1)) -> b#(d(x1)) d#(c(x1)) -> d#(b(d(x1))) d#(a(c(x1))) -> b#(x1) d#(a(c(x1))) -> b#(b(x1)) TRS: b(d(b(x1))) -> c(d(b(x1))) b(a(c(x1))) -> b(c(x1)) a(d(x1)) -> d(c(x1)) b(b(b(x1))) -> a(b(c(x1))) d(c(x1)) -> b(d(x1)) d(c(x1)) -> d(b(d(x1))) d(a(c(x1))) -> b(b(x1)) TDG Processor: DPs: b#(a(c(x1))) -> b#(c(x1)) a#(d(x1)) -> d#(c(x1)) b#(b(b(x1))) -> b#(c(x1)) b#(b(b(x1))) -> a#(b(c(x1))) d#(c(x1)) -> d#(x1) d#(c(x1)) -> b#(d(x1)) d#(c(x1)) -> d#(b(d(x1))) d#(a(c(x1))) -> b#(x1) d#(a(c(x1))) -> b#(b(x1)) TRS: b(d(b(x1))) -> c(d(b(x1))) b(a(c(x1))) -> b(c(x1)) a(d(x1)) -> d(c(x1)) b(b(b(x1))) -> a(b(c(x1))) d(c(x1)) -> b(d(x1)) d(c(x1)) -> d(b(d(x1))) d(a(c(x1))) -> b(b(x1)) graph: d#(a(c(x1))) -> b#(b(x1)) -> b#(b(b(x1))) -> a#(b(c(x1))) d#(a(c(x1))) -> b#(b(x1)) -> b#(b(b(x1))) -> b#(c(x1)) d#(a(c(x1))) -> b#(b(x1)) -> b#(a(c(x1))) -> b#(c(x1)) d#(a(c(x1))) -> b#(x1) -> b#(b(b(x1))) -> a#(b(c(x1))) d#(a(c(x1))) -> b#(x1) -> b#(b(b(x1))) -> b#(c(x1)) d#(a(c(x1))) -> b#(x1) -> b#(a(c(x1))) -> b#(c(x1)) d#(c(x1)) -> d#(b(d(x1))) -> d#(a(c(x1))) -> b#(b(x1)) d#(c(x1)) -> d#(b(d(x1))) -> d#(a(c(x1))) -> b#(x1) d#(c(x1)) -> d#(b(d(x1))) -> d#(c(x1)) -> d#(b(d(x1))) d#(c(x1)) -> d#(b(d(x1))) -> d#(c(x1)) -> b#(d(x1)) d#(c(x1)) -> d#(b(d(x1))) -> d#(c(x1)) -> d#(x1) d#(c(x1)) -> d#(x1) -> d#(a(c(x1))) -> b#(b(x1)) d#(c(x1)) -> d#(x1) -> d#(a(c(x1))) -> b#(x1) d#(c(x1)) -> d#(x1) -> d#(c(x1)) -> d#(b(d(x1))) d#(c(x1)) -> d#(x1) -> d#(c(x1)) -> b#(d(x1)) d#(c(x1)) -> d#(x1) -> d#(c(x1)) -> d#(x1) d#(c(x1)) -> b#(d(x1)) -> b#(b(b(x1))) -> a#(b(c(x1))) d#(c(x1)) -> b#(d(x1)) -> b#(b(b(x1))) -> b#(c(x1)) d#(c(x1)) -> b#(d(x1)) -> b#(a(c(x1))) -> b#(c(x1)) a#(d(x1)) -> d#(c(x1)) -> d#(a(c(x1))) -> b#(b(x1)) a#(d(x1)) -> d#(c(x1)) -> d#(a(c(x1))) -> b#(x1) a#(d(x1)) -> d#(c(x1)) -> d#(c(x1)) -> d#(b(d(x1))) a#(d(x1)) -> d#(c(x1)) -> d#(c(x1)) -> b#(d(x1)) a#(d(x1)) -> d#(c(x1)) -> d#(c(x1)) -> d#(x1) b#(a(c(x1))) -> b#(c(x1)) -> b#(b(b(x1))) -> a#(b(c(x1))) b#(a(c(x1))) -> b#(c(x1)) -> b#(b(b(x1))) -> b#(c(x1)) b#(a(c(x1))) -> b#(c(x1)) -> b#(a(c(x1))) -> b#(c(x1)) b#(b(b(x1))) -> a#(b(c(x1))) -> a#(d(x1)) -> d#(c(x1)) b#(b(b(x1))) -> b#(c(x1)) -> b#(b(b(x1))) -> a#(b(c(x1))) b#(b(b(x1))) -> b#(c(x1)) -> b#(b(b(x1))) -> b#(c(x1)) b#(b(b(x1))) -> b#(c(x1)) -> b#(a(c(x1))) -> b#(c(x1)) EDG Processor: DPs: b#(a(c(x1))) -> b#(c(x1)) a#(d(x1)) -> d#(c(x1)) b#(b(b(x1))) -> b#(c(x1)) b#(b(b(x1))) -> a#(b(c(x1))) d#(c(x1)) -> d#(x1) d#(c(x1)) -> b#(d(x1)) d#(c(x1)) -> d#(b(d(x1))) d#(a(c(x1))) -> b#(x1) d#(a(c(x1))) -> b#(b(x1)) TRS: b(d(b(x1))) -> c(d(b(x1))) b(a(c(x1))) -> b(c(x1)) a(d(x1)) -> d(c(x1)) b(b(b(x1))) -> a(b(c(x1))) d(c(x1)) -> b(d(x1)) d(c(x1)) -> d(b(d(x1))) d(a(c(x1))) -> b(b(x1)) graph: d#(a(c(x1))) -> b#(b(x1)) -> b#(a(c(x1))) -> b#(c(x1)) d#(a(c(x1))) -> b#(b(x1)) -> b#(b(b(x1))) -> b#(c(x1)) d#(a(c(x1))) -> b#(b(x1)) -> b#(b(b(x1))) -> a#(b(c(x1))) d#(a(c(x1))) -> b#(x1) -> b#(a(c(x1))) -> b#(c(x1)) d#(a(c(x1))) -> b#(x1) -> b#(b(b(x1))) -> b#(c(x1)) d#(a(c(x1))) -> b#(x1) -> b#(b(b(x1))) -> a#(b(c(x1))) d#(c(x1)) -> d#(b(d(x1))) -> d#(c(x1)) -> d#(x1) d#(c(x1)) -> d#(b(d(x1))) -> d#(c(x1)) -> b#(d(x1)) d#(c(x1)) -> d#(b(d(x1))) -> d#(c(x1)) -> d#(b(d(x1))) d#(c(x1)) -> d#(b(d(x1))) -> d#(a(c(x1))) -> b#(x1) d#(c(x1)) -> d#(b(d(x1))) -> d#(a(c(x1))) -> b#(b(x1)) d#(c(x1)) -> d#(x1) -> d#(c(x1)) -> d#(x1) d#(c(x1)) -> d#(x1) -> d#(c(x1)) -> b#(d(x1)) d#(c(x1)) -> d#(x1) -> d#(c(x1)) -> d#(b(d(x1))) d#(c(x1)) -> d#(x1) -> d#(a(c(x1))) -> b#(x1) d#(c(x1)) -> d#(x1) -> d#(a(c(x1))) -> b#(b(x1)) d#(c(x1)) -> b#(d(x1)) -> b#(a(c(x1))) -> b#(c(x1)) d#(c(x1)) -> b#(d(x1)) -> b#(b(b(x1))) -> b#(c(x1)) d#(c(x1)) -> b#(d(x1)) -> b#(b(b(x1))) -> a#(b(c(x1))) a#(d(x1)) -> d#(c(x1)) -> d#(c(x1)) -> d#(x1) a#(d(x1)) -> d#(c(x1)) -> d#(c(x1)) -> b#(d(x1)) a#(d(x1)) -> d#(c(x1)) -> d#(c(x1)) -> d#(b(d(x1))) CDG Processor: DPs: b#(a(c(x1))) -> b#(c(x1)) a#(d(x1)) -> d#(c(x1)) b#(b(b(x1))) -> b#(c(x1)) b#(b(b(x1))) -> a#(b(c(x1))) d#(c(x1)) -> d#(x1) d#(c(x1)) -> b#(d(x1)) d#(c(x1)) -> d#(b(d(x1))) d#(a(c(x1))) -> b#(x1) d#(a(c(x1))) -> b#(b(x1)) TRS: b(d(b(x1))) -> c(d(b(x1))) b(a(c(x1))) -> b(c(x1)) a(d(x1)) -> d(c(x1)) b(b(b(x1))) -> a(b(c(x1))) d(c(x1)) -> b(d(x1)) d(c(x1)) -> d(b(d(x1))) d(a(c(x1))) -> b(b(x1)) graph: d#(a(c(x1))) -> b#(b(x1)) -> b#(b(b(x1))) -> a#(b(c(x1))) d#(a(c(x1))) -> b#(b(x1)) -> b#(b(b(x1))) -> b#(c(x1)) d#(a(c(x1))) -> b#(b(x1)) -> b#(a(c(x1))) -> b#(c(x1)) d#(c(x1)) -> d#(b(d(x1))) -> d#(a(c(x1))) -> b#(b(x1)) d#(c(x1)) -> d#(b(d(x1))) -> d#(a(c(x1))) -> b#(x1) d#(c(x1)) -> d#(b(d(x1))) -> d#(c(x1)) -> d#(b(d(x1))) d#(c(x1)) -> d#(b(d(x1))) -> d#(c(x1)) -> b#(d(x1)) d#(c(x1)) -> d#(b(d(x1))) -> d#(c(x1)) -> d#(x1) d#(c(x1)) -> d#(x1) -> d#(a(c(x1))) -> b#(b(x1)) d#(c(x1)) -> d#(x1) -> d#(a(c(x1))) -> b#(x1) d#(c(x1)) -> d#(x1) -> d#(c(x1)) -> d#(b(d(x1))) d#(c(x1)) -> d#(x1) -> d#(c(x1)) -> b#(d(x1)) d#(c(x1)) -> d#(x1) -> d#(c(x1)) -> d#(x1) d#(c(x1)) -> b#(d(x1)) -> b#(b(b(x1))) -> a#(b(c(x1))) d#(c(x1)) -> b#(d(x1)) -> b#(b(b(x1))) -> b#(c(x1)) a#(d(x1)) -> d#(c(x1)) -> d#(c(x1)) -> d#(b(d(x1))) a#(d(x1)) -> d#(c(x1)) -> d#(c(x1)) -> b#(d(x1)) a#(d(x1)) -> d#(c(x1)) -> d#(c(x1)) -> d#(x1) SCC Processor: #sccs: 1 #rules: 2 #arcs: 18/81 DPs: d#(c(x1)) -> d#(b(d(x1))) d#(c(x1)) -> d#(x1) TRS: b(d(b(x1))) -> c(d(b(x1))) b(a(c(x1))) -> b(c(x1)) a(d(x1)) -> d(c(x1)) b(b(b(x1))) -> a(b(c(x1))) d(c(x1)) -> b(d(x1)) d(c(x1)) -> d(b(d(x1))) d(a(c(x1))) -> b(b(x1)) Arctic Interpretation Processor: dimension: 1 interpretation: [d#](x0) = x0, [a](x0) = 2x0, [c](x0) = 2x0, [d](x0) = x0, [b](x0) = 2x0 orientation: d#(c(x1)) = 2x1 >= 2x1 = d#(b(d(x1))) d#(c(x1)) = 2x1 >= x1 = d#(x1) b(d(b(x1))) = 4x1 >= 4x1 = c(d(b(x1))) b(a(c(x1))) = 6x1 >= 4x1 = b(c(x1)) a(d(x1)) = 2x1 >= 2x1 = d(c(x1)) b(b(b(x1))) = 6x1 >= 6x1 = a(b(c(x1))) d(c(x1)) = 2x1 >= 2x1 = b(d(x1)) d(c(x1)) = 2x1 >= 2x1 = d(b(d(x1))) d(a(c(x1))) = 4x1 >= 4x1 = b(b(x1)) problem: DPs: d#(c(x1)) -> d#(b(d(x1))) TRS: b(d(b(x1))) -> c(d(b(x1))) b(a(c(x1))) -> b(c(x1)) a(d(x1)) -> d(c(x1)) b(b(b(x1))) -> a(b(c(x1))) d(c(x1)) -> b(d(x1)) d(c(x1)) -> d(b(d(x1))) d(a(c(x1))) -> b(b(x1)) EDG Processor: DPs: d#(c(x1)) -> d#(b(d(x1))) TRS: b(d(b(x1))) -> c(d(b(x1))) b(a(c(x1))) -> b(c(x1)) a(d(x1)) -> d(c(x1)) b(b(b(x1))) -> a(b(c(x1))) d(c(x1)) -> b(d(x1)) d(c(x1)) -> d(b(d(x1))) d(a(c(x1))) -> b(b(x1)) graph: d#(c(x1)) -> d#(b(d(x1))) -> d#(c(x1)) -> d#(b(d(x1))) Matrix Interpretation Processor: dim=3 interpretation: [d#](x0) = [0 1 0]x0, [0 1 0] [1] [a](x0) = [0 1 0]x0 + [1] [0 1 0] [1], [0 0 0] [0] [c](x0) = [0 0 1]x0 + [1] [0 0 1] [1], [0 0 1] [d](x0) = [0 0 1]x0 [0 1 0] , [0 0 0] [1] [b](x0) = [1 0 0]x0 + [0] [1 0 0] [1] orientation: d#(c(x1)) = [0 0 1]x1 + [1] >= [0 0 1]x1 = d#(b(d(x1))) [0 0 0] [1] [0 0 0] [0] b(d(b(x1))) = [1 0 0]x1 + [1] >= [1 0 0]x1 + [1] = c(d(b(x1))) [1 0 0] [2] [1 0 0] [1] [0 0 0] [1] [1] b(a(c(x1))) = [0 0 1]x1 + [2] >= [0] = b(c(x1)) [0 0 1] [3] [1] [0 0 1] [1] [0 0 1] [1] a(d(x1)) = [0 0 1]x1 + [1] >= [0 0 1]x1 + [1] = d(c(x1)) [0 0 1] [1] [0 0 1] [1] [1] [1] b(b(b(x1))) = [1] >= [1] = a(b(c(x1))) [2] [1] [0 0 1] [1] [0 0 0] [1] d(c(x1)) = [0 0 1]x1 + [1] >= [0 0 1]x1 + [0] = b(d(x1)) [0 0 1] [1] [0 0 1] [1] [0 0 1] [1] [0 0 1] [1] d(c(x1)) = [0 0 1]x1 + [1] >= [0 0 1]x1 + [1] = d(b(d(x1))) [0 0 1] [1] [0 0 1] [0] [0 0 1] [2] [1] d(a(c(x1))) = [0 0 1]x1 + [2] >= [1] = b(b(x1)) [0 0 1] [2] [2] problem: DPs: TRS: b(d(b(x1))) -> c(d(b(x1))) b(a(c(x1))) -> b(c(x1)) a(d(x1)) -> d(c(x1)) b(b(b(x1))) -> a(b(c(x1))) d(c(x1)) -> b(d(x1)) d(c(x1)) -> d(b(d(x1))) d(a(c(x1))) -> b(b(x1)) Qed