YES Problem: a(b(x1)) -> b(c(a(x1))) b(c(x1)) -> c(b(b(x1))) a(c(x1)) -> c(a(b(x1))) a(a(x1)) -> a(d(d(d(x1)))) d(a(x1)) -> d(d(c(x1))) a(d(d(c(x1)))) -> a(a(a(d(x1)))) e(e(f(f(x1)))) -> f(f(f(e(e(x1))))) e(x1) -> a(x1) b(d(x1)) -> d(d(x1)) Proof: Arctic Interpretation Processor: dimension: 1 interpretation: [e](x0) = 8x0, [f](x0) = x0, [d](x0) = x0, [c](x0) = x0, [a](x0) = x0, [b](x0) = x0 orientation: a(b(x1)) = x1 >= x1 = b(c(a(x1))) b(c(x1)) = x1 >= x1 = c(b(b(x1))) a(c(x1)) = x1 >= x1 = c(a(b(x1))) a(a(x1)) = x1 >= x1 = a(d(d(d(x1)))) d(a(x1)) = x1 >= x1 = d(d(c(x1))) a(d(d(c(x1)))) = x1 >= x1 = a(a(a(d(x1)))) e(e(f(f(x1)))) = 16x1 >= 16x1 = f(f(f(e(e(x1))))) e(x1) = 8x1 >= x1 = a(x1) b(d(x1)) = x1 >= x1 = d(d(x1)) problem: a(b(x1)) -> b(c(a(x1))) b(c(x1)) -> c(b(b(x1))) a(c(x1)) -> c(a(b(x1))) a(a(x1)) -> a(d(d(d(x1)))) d(a(x1)) -> d(d(c(x1))) a(d(d(c(x1)))) -> a(a(a(d(x1)))) e(e(f(f(x1)))) -> f(f(f(e(e(x1))))) b(d(x1)) -> d(d(x1)) String Reversal Processor: b(a(x1)) -> a(c(b(x1))) c(b(x1)) -> b(b(c(x1))) c(a(x1)) -> b(a(c(x1))) a(a(x1)) -> d(d(d(a(x1)))) a(d(x1)) -> c(d(d(x1))) c(d(d(a(x1)))) -> d(a(a(a(x1)))) f(f(e(e(x1)))) -> e(e(f(f(f(x1))))) d(b(x1)) -> d(d(x1)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [0] [e](x0) = [1 0 0]x0 + [1] [1 1 0] [0], [1 0 0] [f](x0) = [0 0 1]x0 [0 0 1] , [1 0 1] [d](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [0] [c](x0) = [0 1 1]x0 + [1] [0 1 1] [0], [1 0 0] [0] [a](x0) = [0 0 0]x0 + [1] [0 0 0] [0], [0] [b](x0) = x0 + [1] [1] orientation: [1 0 0] [0] [1 0 0] [0] b(a(x1)) = [0 0 0]x1 + [2] >= [0 0 0]x1 + [1] = a(c(b(x1))) [0 0 0] [1] [0 0 0] [0] [1 0 0] [0] [1 0 0] [0] c(b(x1)) = [0 1 1]x1 + [3] >= [0 1 1]x1 + [3] = b(b(c(x1))) [0 1 1] [2] [0 1 1] [2] [1 0 0] [0] [1 0 0] [0] c(a(x1)) = [0 0 0]x1 + [2] >= [0 0 0]x1 + [2] = b(a(c(x1))) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] a(a(x1)) = [0 0 0]x1 + [1] >= [0 0 0]x1 = d(d(d(a(x1)))) [0 0 0] [0] [0 0 0] [1 0 1] [0] [1 0 1] [0] a(d(x1)) = [0 0 0]x1 + [1] >= [0 0 0]x1 + [1] = c(d(d(x1))) [0 0 0] [0] [0 0 0] [0] [1 0 0] [0] [1 0 0] c(d(d(a(x1)))) = [0 0 0]x1 + [1] >= [0 0 0]x1 = d(a(a(a(x1)))) [0 0 0] [0] [0 0 0] [1 0 0] [0] [1 0 0] [0] f(f(e(e(x1)))) = [2 0 0]x1 + [1] >= [1 0 0]x1 + [1] = e(e(f(f(f(x1))))) [2 0 0] [1] [2 0 0] [1] [1 0 1] [1] [1 0 1] d(b(x1)) = [0 0 0]x1 + [0] >= [0 0 0]x1 = d(d(x1)) [0 0 0] [0] [0 0 0] problem: b(a(x1)) -> a(c(b(x1))) c(b(x1)) -> b(b(c(x1))) c(a(x1)) -> b(a(c(x1))) a(a(x1)) -> d(d(d(a(x1)))) a(d(x1)) -> c(d(d(x1))) c(d(d(a(x1)))) -> d(a(a(a(x1)))) f(f(e(e(x1)))) -> e(e(f(f(f(x1))))) String Reversal Processor: a(b(x1)) -> b(c(a(x1))) b(c(x1)) -> c(b(b(x1))) a(c(x1)) -> c(a(b(x1))) a(a(x1)) -> a(d(d(d(x1)))) d(a(x1)) -> d(d(c(x1))) a(d(d(c(x1)))) -> a(a(a(d(x1)))) e(e(f(f(x1)))) -> f(f(f(e(e(x1))))) Matrix Interpretation Processor: dim=1 interpretation: [e](x0) = 2x0 + 2, [f](x0) = x0 + 1, [d](x0) = x0, [c](x0) = x0, [a](x0) = x0, [b](x0) = x0 orientation: a(b(x1)) = x1 >= x1 = b(c(a(x1))) b(c(x1)) = x1 >= x1 = c(b(b(x1))) a(c(x1)) = x1 >= x1 = c(a(b(x1))) a(a(x1)) = x1 >= x1 = a(d(d(d(x1)))) d(a(x1)) = x1 >= x1 = d(d(c(x1))) a(d(d(c(x1)))) = x1 >= x1 = a(a(a(d(x1)))) e(e(f(f(x1)))) = 4x1 + 14 >= 4x1 + 9 = f(f(f(e(e(x1))))) problem: a(b(x1)) -> b(c(a(x1))) b(c(x1)) -> c(b(b(x1))) a(c(x1)) -> c(a(b(x1))) a(a(x1)) -> a(d(d(d(x1)))) d(a(x1)) -> d(d(c(x1))) a(d(d(c(x1)))) -> a(a(a(d(x1)))) String Reversal Processor: b(a(x1)) -> a(c(b(x1))) c(b(x1)) -> b(b(c(x1))) c(a(x1)) -> b(a(c(x1))) a(a(x1)) -> d(d(d(a(x1)))) a(d(x1)) -> c(d(d(x1))) c(d(d(a(x1)))) -> d(a(a(a(x1)))) DP Processor: DPs: b#(a(x1)) -> b#(x1) b#(a(x1)) -> c#(b(x1)) b#(a(x1)) -> a#(c(b(x1))) c#(b(x1)) -> c#(x1) c#(b(x1)) -> b#(c(x1)) c#(b(x1)) -> b#(b(c(x1))) c#(a(x1)) -> c#(x1) c#(a(x1)) -> a#(c(x1)) c#(a(x1)) -> b#(a(c(x1))) a#(d(x1)) -> c#(d(d(x1))) c#(d(d(a(x1)))) -> a#(a(x1)) c#(d(d(a(x1)))) -> a#(a(a(x1))) TRS: b(a(x1)) -> a(c(b(x1))) c(b(x1)) -> b(b(c(x1))) c(a(x1)) -> b(a(c(x1))) a(a(x1)) -> d(d(d(a(x1)))) a(d(x1)) -> c(d(d(x1))) c(d(d(a(x1)))) -> d(a(a(a(x1)))) TDG Processor: DPs: b#(a(x1)) -> b#(x1) b#(a(x1)) -> c#(b(x1)) b#(a(x1)) -> a#(c(b(x1))) c#(b(x1)) -> c#(x1) c#(b(x1)) -> b#(c(x1)) c#(b(x1)) -> b#(b(c(x1))) c#(a(x1)) -> c#(x1) c#(a(x1)) -> a#(c(x1)) c#(a(x1)) -> b#(a(c(x1))) a#(d(x1)) -> c#(d(d(x1))) c#(d(d(a(x1)))) -> a#(a(x1)) c#(d(d(a(x1)))) -> a#(a(a(x1))) TRS: b(a(x1)) -> a(c(b(x1))) c(b(x1)) -> b(b(c(x1))) c(a(x1)) -> b(a(c(x1))) a(a(x1)) -> d(d(d(a(x1)))) a(d(x1)) -> c(d(d(x1))) c(d(d(a(x1)))) -> d(a(a(a(x1)))) graph: a#(d(x1)) -> c#(d(d(x1))) -> c#(d(d(a(x1)))) -> a#(a(a(x1))) a#(d(x1)) -> c#(d(d(x1))) -> c#(d(d(a(x1)))) -> a#(a(x1)) a#(d(x1)) -> c#(d(d(x1))) -> c#(a(x1)) -> b#(a(c(x1))) a#(d(x1)) -> c#(d(d(x1))) -> c#(a(x1)) -> a#(c(x1)) a#(d(x1)) -> c#(d(d(x1))) -> c#(a(x1)) -> c#(x1) a#(d(x1)) -> c#(d(d(x1))) -> c#(b(x1)) -> b#(b(c(x1))) a#(d(x1)) -> c#(d(d(x1))) -> c#(b(x1)) -> b#(c(x1)) a#(d(x1)) -> c#(d(d(x1))) -> c#(b(x1)) -> c#(x1) c#(d(d(a(x1)))) -> a#(a(a(x1))) -> a#(d(x1)) -> c#(d(d(x1))) c#(d(d(a(x1)))) -> a#(a(x1)) -> a#(d(x1)) -> c#(d(d(x1))) c#(a(x1)) -> a#(c(x1)) -> a#(d(x1)) -> c#(d(d(x1))) c#(a(x1)) -> c#(x1) -> c#(d(d(a(x1)))) -> a#(a(a(x1))) c#(a(x1)) -> c#(x1) -> c#(d(d(a(x1)))) -> a#(a(x1)) c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> b#(a(c(x1))) c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> a#(c(x1)) c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> c#(x1) c#(a(x1)) -> c#(x1) -> c#(b(x1)) -> b#(b(c(x1))) c#(a(x1)) -> c#(x1) -> c#(b(x1)) -> b#(c(x1)) c#(a(x1)) -> c#(x1) -> c#(b(x1)) -> c#(x1) c#(a(x1)) -> b#(a(c(x1))) -> b#(a(x1)) -> a#(c(b(x1))) c#(a(x1)) -> b#(a(c(x1))) -> b#(a(x1)) -> c#(b(x1)) c#(a(x1)) -> b#(a(c(x1))) -> b#(a(x1)) -> b#(x1) c#(b(x1)) -> c#(x1) -> c#(d(d(a(x1)))) -> a#(a(a(x1))) c#(b(x1)) -> c#(x1) -> c#(d(d(a(x1)))) -> a#(a(x1)) c#(b(x1)) -> c#(x1) -> c#(a(x1)) -> b#(a(c(x1))) c#(b(x1)) -> c#(x1) -> c#(a(x1)) -> a#(c(x1)) c#(b(x1)) -> c#(x1) -> c#(a(x1)) -> c#(x1) c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> b#(b(c(x1))) c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> b#(c(x1)) c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> c#(x1) c#(b(x1)) -> b#(c(x1)) -> b#(a(x1)) -> a#(c(b(x1))) c#(b(x1)) -> b#(c(x1)) -> b#(a(x1)) -> c#(b(x1)) c#(b(x1)) -> b#(c(x1)) -> b#(a(x1)) -> b#(x1) c#(b(x1)) -> b#(b(c(x1))) -> b#(a(x1)) -> a#(c(b(x1))) c#(b(x1)) -> b#(b(c(x1))) -> b#(a(x1)) -> c#(b(x1)) c#(b(x1)) -> b#(b(c(x1))) -> b#(a(x1)) -> b#(x1) b#(a(x1)) -> a#(c(b(x1))) -> a#(d(x1)) -> c#(d(d(x1))) b#(a(x1)) -> c#(b(x1)) -> c#(d(d(a(x1)))) -> a#(a(a(x1))) b#(a(x1)) -> c#(b(x1)) -> c#(d(d(a(x1)))) -> a#(a(x1)) b#(a(x1)) -> c#(b(x1)) -> c#(a(x1)) -> b#(a(c(x1))) b#(a(x1)) -> c#(b(x1)) -> c#(a(x1)) -> a#(c(x1)) b#(a(x1)) -> c#(b(x1)) -> c#(a(x1)) -> c#(x1) b#(a(x1)) -> c#(b(x1)) -> c#(b(x1)) -> b#(b(c(x1))) b#(a(x1)) -> c#(b(x1)) -> c#(b(x1)) -> b#(c(x1)) b#(a(x1)) -> c#(b(x1)) -> c#(b(x1)) -> c#(x1) b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> a#(c(b(x1))) b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> c#(b(x1)) b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> b#(x1) EDG Processor: DPs: b#(a(x1)) -> b#(x1) b#(a(x1)) -> c#(b(x1)) b#(a(x1)) -> a#(c(b(x1))) c#(b(x1)) -> c#(x1) c#(b(x1)) -> b#(c(x1)) c#(b(x1)) -> b#(b(c(x1))) c#(a(x1)) -> c#(x1) c#(a(x1)) -> a#(c(x1)) c#(a(x1)) -> b#(a(c(x1))) a#(d(x1)) -> c#(d(d(x1))) c#(d(d(a(x1)))) -> a#(a(x1)) c#(d(d(a(x1)))) -> a#(a(a(x1))) TRS: b(a(x1)) -> a(c(b(x1))) c(b(x1)) -> b(b(c(x1))) c(a(x1)) -> b(a(c(x1))) a(a(x1)) -> d(d(d(a(x1)))) a(d(x1)) -> c(d(d(x1))) c(d(d(a(x1)))) -> d(a(a(a(x1)))) graph: a#(d(x1)) -> c#(d(d(x1))) -> c#(d(d(a(x1)))) -> a#(a(x1)) a#(d(x1)) -> c#(d(d(x1))) -> c#(d(d(a(x1)))) -> a#(a(a(x1))) c#(d(d(a(x1)))) -> a#(a(a(x1))) -> a#(d(x1)) -> c#(d(d(x1))) c#(d(d(a(x1)))) -> a#(a(x1)) -> a#(d(x1)) -> c#(d(d(x1))) c#(a(x1)) -> a#(c(x1)) -> a#(d(x1)) -> c#(d(d(x1))) c#(a(x1)) -> c#(x1) -> c#(b(x1)) -> c#(x1) c#(a(x1)) -> c#(x1) -> c#(b(x1)) -> b#(c(x1)) c#(a(x1)) -> c#(x1) -> c#(b(x1)) -> b#(b(c(x1))) c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> c#(x1) c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> a#(c(x1)) c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> b#(a(c(x1))) c#(a(x1)) -> c#(x1) -> c#(d(d(a(x1)))) -> a#(a(x1)) c#(a(x1)) -> c#(x1) -> c#(d(d(a(x1)))) -> a#(a(a(x1))) c#(a(x1)) -> b#(a(c(x1))) -> b#(a(x1)) -> b#(x1) c#(a(x1)) -> b#(a(c(x1))) -> b#(a(x1)) -> c#(b(x1)) c#(a(x1)) -> b#(a(c(x1))) -> b#(a(x1)) -> a#(c(b(x1))) c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> c#(x1) c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> b#(c(x1)) c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> b#(b(c(x1))) c#(b(x1)) -> c#(x1) -> c#(a(x1)) -> c#(x1) c#(b(x1)) -> c#(x1) -> c#(a(x1)) -> a#(c(x1)) c#(b(x1)) -> c#(x1) -> c#(a(x1)) -> b#(a(c(x1))) c#(b(x1)) -> c#(x1) -> c#(d(d(a(x1)))) -> a#(a(x1)) c#(b(x1)) -> c#(x1) -> c#(d(d(a(x1)))) -> a#(a(a(x1))) c#(b(x1)) -> b#(c(x1)) -> b#(a(x1)) -> b#(x1) c#(b(x1)) -> b#(c(x1)) -> b#(a(x1)) -> c#(b(x1)) c#(b(x1)) -> b#(c(x1)) -> b#(a(x1)) -> a#(c(b(x1))) c#(b(x1)) -> b#(b(c(x1))) -> b#(a(x1)) -> b#(x1) c#(b(x1)) -> b#(b(c(x1))) -> b#(a(x1)) -> c#(b(x1)) c#(b(x1)) -> b#(b(c(x1))) -> b#(a(x1)) -> a#(c(b(x1))) b#(a(x1)) -> a#(c(b(x1))) -> a#(d(x1)) -> c#(d(d(x1))) b#(a(x1)) -> c#(b(x1)) -> c#(b(x1)) -> c#(x1) b#(a(x1)) -> c#(b(x1)) -> c#(b(x1)) -> b#(c(x1)) b#(a(x1)) -> c#(b(x1)) -> c#(b(x1)) -> b#(b(c(x1))) b#(a(x1)) -> c#(b(x1)) -> c#(a(x1)) -> c#(x1) b#(a(x1)) -> c#(b(x1)) -> c#(a(x1)) -> a#(c(x1)) b#(a(x1)) -> c#(b(x1)) -> c#(a(x1)) -> b#(a(c(x1))) b#(a(x1)) -> c#(b(x1)) -> c#(d(d(a(x1)))) -> a#(a(x1)) b#(a(x1)) -> c#(b(x1)) -> c#(d(d(a(x1)))) -> a#(a(a(x1))) b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> b#(x1) b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> c#(b(x1)) b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> a#(c(b(x1))) SCC Processor: #sccs: 2 #rules: 10 #arcs: 42/144 DPs: c#(a(x1)) -> c#(x1) c#(a(x1)) -> b#(a(c(x1))) b#(a(x1)) -> c#(b(x1)) c#(b(x1)) -> b#(b(c(x1))) b#(a(x1)) -> b#(x1) c#(b(x1)) -> b#(c(x1)) c#(b(x1)) -> c#(x1) TRS: b(a(x1)) -> a(c(b(x1))) c(b(x1)) -> b(b(c(x1))) c(a(x1)) -> b(a(c(x1))) a(a(x1)) -> d(d(d(a(x1)))) a(d(x1)) -> c(d(d(x1))) c(d(d(a(x1)))) -> d(a(a(a(x1)))) Arctic Interpretation Processor: dimension: 1 usable rules: b(a(x1)) -> a(c(b(x1))) c(b(x1)) -> b(b(c(x1))) c(a(x1)) -> b(a(c(x1))) a(a(x1)) -> d(d(d(a(x1)))) a(d(x1)) -> c(d(d(x1))) c(d(d(a(x1)))) -> d(a(a(a(x1)))) interpretation: [c#](x0) = x0 + 0, [b#](x0) = x0 + 0, [d](x0) = 0, [c](x0) = x0 + 0, [a](x0) = 4x0 + 4, [b](x0) = x0 + 0 orientation: c#(a(x1)) = 4x1 + 4 >= x1 + 0 = c#(x1) c#(a(x1)) = 4x1 + 4 >= 4x1 + 4 = b#(a(c(x1))) b#(a(x1)) = 4x1 + 4 >= x1 + 0 = c#(b(x1)) c#(b(x1)) = x1 + 0 >= x1 + 0 = b#(b(c(x1))) b#(a(x1)) = 4x1 + 4 >= x1 + 0 = b#(x1) c#(b(x1)) = x1 + 0 >= x1 + 0 = b#(c(x1)) c#(b(x1)) = x1 + 0 >= x1 + 0 = c#(x1) b(a(x1)) = 4x1 + 4 >= 4x1 + 4 = a(c(b(x1))) c(b(x1)) = x1 + 0 >= x1 + 0 = b(b(c(x1))) c(a(x1)) = 4x1 + 4 >= 4x1 + 4 = b(a(c(x1))) a(a(x1)) = 8x1 + 8 >= 0 = d(d(d(a(x1)))) a(d(x1)) = 4 >= 0 = c(d(d(x1))) c(d(d(a(x1)))) = 0 >= 0 = d(a(a(a(x1)))) problem: DPs: c#(a(x1)) -> b#(a(c(x1))) c#(b(x1)) -> b#(b(c(x1))) c#(b(x1)) -> b#(c(x1)) c#(b(x1)) -> c#(x1) TRS: b(a(x1)) -> a(c(b(x1))) c(b(x1)) -> b(b(c(x1))) c(a(x1)) -> b(a(c(x1))) a(a(x1)) -> d(d(d(a(x1)))) a(d(x1)) -> c(d(d(x1))) c(d(d(a(x1)))) -> d(a(a(a(x1)))) Restore Modifier: DPs: c#(a(x1)) -> b#(a(c(x1))) c#(b(x1)) -> b#(b(c(x1))) c#(b(x1)) -> b#(c(x1)) c#(b(x1)) -> c#(x1) TRS: b(a(x1)) -> a(c(b(x1))) c(b(x1)) -> b(b(c(x1))) c(a(x1)) -> b(a(c(x1))) a(a(x1)) -> d(d(d(a(x1)))) a(d(x1)) -> c(d(d(x1))) c(d(d(a(x1)))) -> d(a(a(a(x1)))) EDG Processor: DPs: c#(a(x1)) -> b#(a(c(x1))) c#(b(x1)) -> b#(b(c(x1))) c#(b(x1)) -> b#(c(x1)) c#(b(x1)) -> c#(x1) TRS: b(a(x1)) -> a(c(b(x1))) c(b(x1)) -> b(b(c(x1))) c(a(x1)) -> b(a(c(x1))) a(a(x1)) -> d(d(d(a(x1)))) a(d(x1)) -> c(d(d(x1))) c(d(d(a(x1)))) -> d(a(a(a(x1)))) graph: c#(b(x1)) -> c#(x1) -> c#(a(x1)) -> b#(a(c(x1))) c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> b#(b(c(x1))) c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> b#(c(x1)) c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> c#(x1) SCC Processor: #sccs: 1 #rules: 1 #arcs: 4/16 DPs: c#(b(x1)) -> c#(x1) TRS: b(a(x1)) -> a(c(b(x1))) c(b(x1)) -> b(b(c(x1))) c(a(x1)) -> b(a(c(x1))) a(a(x1)) -> d(d(d(a(x1)))) a(d(x1)) -> c(d(d(x1))) c(d(d(a(x1)))) -> d(a(a(a(x1)))) Usable Rule Processor: DPs: c#(b(x1)) -> c#(x1) TRS: Arctic Interpretation Processor: dimension: 4 usable rules: interpretation: [c#](x0) = [0 -& -& -&]x0, [1 0 0 0] [0] [1 0 0 0] [0] [b](x0) = [1 0 0 0]x0 + [0] [0 0 0 0] [0] orientation: c#(b(x1)) = [1 0 0 0]x1 + [0] >= [0 -& -& -&]x1 = c#(x1) problem: DPs: TRS: Qed DPs: a#(d(x1)) -> c#(d(d(x1))) c#(d(d(a(x1)))) -> a#(a(a(x1))) c#(d(d(a(x1)))) -> a#(a(x1)) TRS: b(a(x1)) -> a(c(b(x1))) c(b(x1)) -> b(b(c(x1))) c(a(x1)) -> b(a(c(x1))) a(a(x1)) -> d(d(d(a(x1)))) a(d(x1)) -> c(d(d(x1))) c(d(d(a(x1)))) -> d(a(a(a(x1)))) Usable Rule Processor: DPs: a#(d(x1)) -> c#(d(d(x1))) c#(d(d(a(x1)))) -> a#(a(a(x1))) c#(d(d(a(x1)))) -> a#(a(x1)) TRS: a(a(x1)) -> d(d(d(a(x1)))) a(d(x1)) -> c(d(d(x1))) c(d(d(a(x1)))) -> d(a(a(a(x1)))) Matrix Interpretation Processor: dim=4 interpretation: [a#](x0) = [0 0 1 0]x0 + [1], [c#](x0) = [0 1 0 0]x0, [0 0 0 0] [1 0 1 0] [d](x0) = [1 0 0 1]x0 [0 0 0 0] , [0 0 0 0] [1] [0 1 0 0] [0] [c](x0) = [0 1 0 0]x0 + [0] [0 1 0 0] [0], [0 0 0 0] [1] [0 0 1 0] [0] [a](x0) = [0 0 1 0]x0 + [0] [0 0 1 0] [0] orientation: a#(d(x1)) = [1 0 0 1]x1 + [1] >= [1 0 0 1]x1 = c#(d(d(x1))) c#(d(d(a(x1)))) = [0 0 1 0]x1 + [1] >= [0 0 1 0]x1 + [1] = a#(a(a(x1))) c#(d(d(a(x1)))) = [0 0 1 0]x1 + [1] >= [0 0 1 0]x1 + [1] = a#(a(x1)) [0 0 0 0] [1] [0] [0 0 1 0] [0] [0] a(a(x1)) = [0 0 1 0]x1 + [0] >= [0] = d(d(d(a(x1)))) [0 0 1 0] [0] [0] [0 0 0 0] [1] [0 0 0 0] [1] [1 0 0 1] [0] [1 0 0 1] [0] a(d(x1)) = [1 0 0 1]x1 + [0] >= [1 0 0 1]x1 + [0] = c(d(d(x1))) [1 0 0 1] [0] [1 0 0 1] [0] [0 0 0 0] [1] [0 0 0 0] [0] [0 0 1 0] [1] [0 0 1 0] [1] c(d(d(a(x1)))) = [0 0 1 0]x1 + [1] >= [0 0 1 0]x1 + [1] = d(a(a(a(x1)))) [0 0 1 0] [1] [0 0 0 0] [0] problem: DPs: c#(d(d(a(x1)))) -> a#(a(a(x1))) c#(d(d(a(x1)))) -> a#(a(x1)) TRS: a(a(x1)) -> d(d(d(a(x1)))) a(d(x1)) -> c(d(d(x1))) c(d(d(a(x1)))) -> d(a(a(a(x1)))) Restore Modifier: DPs: c#(d(d(a(x1)))) -> a#(a(a(x1))) c#(d(d(a(x1)))) -> a#(a(x1)) TRS: b(a(x1)) -> a(c(b(x1))) c(b(x1)) -> b(b(c(x1))) c(a(x1)) -> b(a(c(x1))) a(a(x1)) -> d(d(d(a(x1)))) a(d(x1)) -> c(d(d(x1))) c(d(d(a(x1)))) -> d(a(a(a(x1)))) EDG Processor: DPs: c#(d(d(a(x1)))) -> a#(a(a(x1))) c#(d(d(a(x1)))) -> a#(a(x1)) TRS: b(a(x1)) -> a(c(b(x1))) c(b(x1)) -> b(b(c(x1))) c(a(x1)) -> b(a(c(x1))) a(a(x1)) -> d(d(d(a(x1)))) a(d(x1)) -> c(d(d(x1))) c(d(d(a(x1)))) -> d(a(a(a(x1)))) graph: Qed