YES Problem: active(f(f(X))) -> mark(c(f(g(f(X))))) active(c(X)) -> mark(d(X)) active(h(X)) -> mark(c(d(X))) active(f(X)) -> f(active(X)) active(h(X)) -> h(active(X)) f(mark(X)) -> mark(f(X)) h(mark(X)) -> mark(h(X)) proper(f(X)) -> f(proper(X)) proper(c(X)) -> c(proper(X)) proper(g(X)) -> g(proper(X)) proper(d(X)) -> d(proper(X)) proper(h(X)) -> h(proper(X)) f(ok(X)) -> ok(f(X)) c(ok(X)) -> ok(c(X)) g(ok(X)) -> ok(g(X)) d(ok(X)) -> ok(d(X)) h(ok(X)) -> ok(h(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Proof: DP Processor: DPs: active#(f(f(X))) -> g#(f(X)) active#(f(f(X))) -> f#(g(f(X))) active#(f(f(X))) -> c#(f(g(f(X)))) active#(c(X)) -> d#(X) active#(h(X)) -> d#(X) active#(h(X)) -> c#(d(X)) active#(f(X)) -> active#(X) active#(f(X)) -> f#(active(X)) active#(h(X)) -> active#(X) active#(h(X)) -> h#(active(X)) f#(mark(X)) -> f#(X) h#(mark(X)) -> h#(X) proper#(f(X)) -> proper#(X) proper#(f(X)) -> f#(proper(X)) proper#(c(X)) -> proper#(X) proper#(c(X)) -> c#(proper(X)) proper#(g(X)) -> proper#(X) proper#(g(X)) -> g#(proper(X)) proper#(d(X)) -> proper#(X) proper#(d(X)) -> d#(proper(X)) proper#(h(X)) -> proper#(X) proper#(h(X)) -> h#(proper(X)) f#(ok(X)) -> f#(X) c#(ok(X)) -> c#(X) g#(ok(X)) -> g#(X) d#(ok(X)) -> d#(X) h#(ok(X)) -> h#(X) top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) TRS: active(f(f(X))) -> mark(c(f(g(f(X))))) active(c(X)) -> mark(d(X)) active(h(X)) -> mark(c(d(X))) active(f(X)) -> f(active(X)) active(h(X)) -> h(active(X)) f(mark(X)) -> mark(f(X)) h(mark(X)) -> mark(h(X)) proper(f(X)) -> f(proper(X)) proper(c(X)) -> c(proper(X)) proper(g(X)) -> g(proper(X)) proper(d(X)) -> d(proper(X)) proper(h(X)) -> h(proper(X)) f(ok(X)) -> ok(f(X)) c(ok(X)) -> ok(c(X)) g(ok(X)) -> ok(g(X)) d(ok(X)) -> ok(d(X)) h(ok(X)) -> ok(h(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) TDG Processor: DPs: active#(f(f(X))) -> g#(f(X)) active#(f(f(X))) -> f#(g(f(X))) active#(f(f(X))) -> c#(f(g(f(X)))) active#(c(X)) -> d#(X) active#(h(X)) -> d#(X) active#(h(X)) -> c#(d(X)) active#(f(X)) -> active#(X) active#(f(X)) -> f#(active(X)) active#(h(X)) -> active#(X) active#(h(X)) -> h#(active(X)) f#(mark(X)) -> f#(X) h#(mark(X)) -> h#(X) proper#(f(X)) -> proper#(X) proper#(f(X)) -> f#(proper(X)) proper#(c(X)) -> proper#(X) proper#(c(X)) -> c#(proper(X)) proper#(g(X)) -> proper#(X) proper#(g(X)) -> g#(proper(X)) proper#(d(X)) -> proper#(X) proper#(d(X)) -> d#(proper(X)) proper#(h(X)) -> proper#(X) proper#(h(X)) -> h#(proper(X)) f#(ok(X)) -> f#(X) c#(ok(X)) -> c#(X) g#(ok(X)) -> g#(X) d#(ok(X)) -> d#(X) h#(ok(X)) -> h#(X) top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) TRS: active(f(f(X))) -> mark(c(f(g(f(X))))) active(c(X)) -> mark(d(X)) active(h(X)) -> mark(c(d(X))) active(f(X)) -> f(active(X)) active(h(X)) -> h(active(X)) f(mark(X)) -> mark(f(X)) h(mark(X)) -> mark(h(X)) proper(f(X)) -> f(proper(X)) proper(c(X)) -> c(proper(X)) proper(g(X)) -> g(proper(X)) proper(d(X)) -> d(proper(X)) proper(h(X)) -> h(proper(X)) f(ok(X)) -> ok(f(X)) c(ok(X)) -> ok(c(X)) g(ok(X)) -> ok(g(X)) d(ok(X)) -> ok(d(X)) h(ok(X)) -> ok(h(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) graph: top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> top#(active(X)) top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) -> top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> top#(active(X)) -> top#(mark(X)) -> proper#(X) top#(ok(X)) -> active#(X) -> active#(h(X)) -> h#(active(X)) top#(ok(X)) -> active#(X) -> active#(h(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(f(X)) -> f#(active(X)) top#(ok(X)) -> active#(X) -> active#(f(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(h(X)) -> c#(d(X)) top#(ok(X)) -> active#(X) -> active#(h(X)) -> d#(X) top#(ok(X)) -> active#(X) -> active#(c(X)) -> d#(X) top#(ok(X)) -> active#(X) -> active#(f(f(X))) -> c#(f(g(f(X)))) top#(ok(X)) -> active#(X) -> active#(f(f(X))) -> f#(g(f(X))) top#(ok(X)) -> active#(X) -> active#(f(f(X))) -> g#(f(X)) top#(mark(X)) -> top#(proper(X)) -> top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> top#(proper(X)) -> top#(ok(X)) -> active#(X) top#(mark(X)) -> top#(proper(X)) -> top#(mark(X)) -> top#(proper(X)) top#(mark(X)) -> top#(proper(X)) -> top#(mark(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(h(X)) -> h#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(h(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(d(X)) -> d#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(d(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(g(X)) -> g#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(g(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(c(X)) -> c#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(c(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(f(X)) -> f#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(f(X)) -> proper#(X) proper#(h(X)) -> proper#(X) -> proper#(h(X)) -> h#(proper(X)) proper#(h(X)) -> proper#(X) -> proper#(h(X)) -> proper#(X) proper#(h(X)) -> proper#(X) -> proper#(d(X)) -> d#(proper(X)) proper#(h(X)) -> proper#(X) -> proper#(d(X)) -> proper#(X) proper#(h(X)) -> proper#(X) -> proper#(g(X)) -> g#(proper(X)) proper#(h(X)) -> proper#(X) -> proper#(g(X)) -> proper#(X) proper#(h(X)) -> proper#(X) -> proper#(c(X)) -> c#(proper(X)) proper#(h(X)) -> proper#(X) -> proper#(c(X)) -> proper#(X) proper#(h(X)) -> proper#(X) -> proper#(f(X)) -> f#(proper(X)) proper#(h(X)) -> proper#(X) -> proper#(f(X)) -> proper#(X) proper#(h(X)) -> h#(proper(X)) -> h#(ok(X)) -> h#(X) proper#(h(X)) -> h#(proper(X)) -> h#(mark(X)) -> h#(X) proper#(d(X)) -> proper#(X) -> proper#(h(X)) -> h#(proper(X)) proper#(d(X)) -> proper#(X) -> proper#(h(X)) -> proper#(X) proper#(d(X)) -> proper#(X) -> proper#(d(X)) -> d#(proper(X)) proper#(d(X)) -> proper#(X) -> proper#(d(X)) -> proper#(X) proper#(d(X)) -> proper#(X) -> proper#(g(X)) -> g#(proper(X)) proper#(d(X)) -> proper#(X) -> proper#(g(X)) -> proper#(X) proper#(d(X)) -> proper#(X) -> proper#(c(X)) -> c#(proper(X)) proper#(d(X)) -> proper#(X) -> proper#(c(X)) -> proper#(X) proper#(d(X)) -> proper#(X) -> proper#(f(X)) -> f#(proper(X)) proper#(d(X)) -> proper#(X) -> proper#(f(X)) -> proper#(X) proper#(d(X)) -> d#(proper(X)) -> d#(ok(X)) -> d#(X) proper#(c(X)) -> proper#(X) -> proper#(h(X)) -> h#(proper(X)) proper#(c(X)) -> proper#(X) -> proper#(h(X)) -> proper#(X) proper#(c(X)) -> proper#(X) -> proper#(d(X)) -> d#(proper(X)) proper#(c(X)) -> proper#(X) -> proper#(d(X)) -> proper#(X) proper#(c(X)) -> proper#(X) -> proper#(g(X)) -> g#(proper(X)) proper#(c(X)) -> proper#(X) -> proper#(g(X)) -> proper#(X) proper#(c(X)) -> proper#(X) -> proper#(c(X)) -> c#(proper(X)) proper#(c(X)) -> proper#(X) -> proper#(c(X)) -> proper#(X) proper#(c(X)) -> proper#(X) -> proper#(f(X)) -> f#(proper(X)) proper#(c(X)) -> proper#(X) -> proper#(f(X)) -> proper#(X) proper#(c(X)) -> c#(proper(X)) -> c#(ok(X)) -> c#(X) proper#(g(X)) -> proper#(X) -> proper#(h(X)) -> h#(proper(X)) proper#(g(X)) -> proper#(X) -> proper#(h(X)) -> proper#(X) proper#(g(X)) -> proper#(X) -> proper#(d(X)) -> d#(proper(X)) proper#(g(X)) -> proper#(X) -> proper#(d(X)) -> proper#(X) proper#(g(X)) -> proper#(X) -> proper#(g(X)) -> g#(proper(X)) proper#(g(X)) -> proper#(X) -> proper#(g(X)) -> proper#(X) proper#(g(X)) -> proper#(X) -> proper#(c(X)) -> c#(proper(X)) proper#(g(X)) -> proper#(X) -> proper#(c(X)) -> proper#(X) proper#(g(X)) -> proper#(X) -> proper#(f(X)) -> f#(proper(X)) proper#(g(X)) -> proper#(X) -> proper#(f(X)) -> proper#(X) proper#(g(X)) -> g#(proper(X)) -> g#(ok(X)) -> g#(X) proper#(f(X)) -> proper#(X) -> proper#(h(X)) -> h#(proper(X)) proper#(f(X)) -> proper#(X) -> proper#(h(X)) -> proper#(X) proper#(f(X)) -> proper#(X) -> proper#(d(X)) -> d#(proper(X)) proper#(f(X)) -> proper#(X) -> proper#(d(X)) -> proper#(X) proper#(f(X)) -> proper#(X) -> proper#(g(X)) -> g#(proper(X)) proper#(f(X)) -> proper#(X) -> proper#(g(X)) -> proper#(X) proper#(f(X)) -> proper#(X) -> proper#(c(X)) -> c#(proper(X)) proper#(f(X)) -> proper#(X) -> proper#(c(X)) -> proper#(X) proper#(f(X)) -> proper#(X) -> proper#(f(X)) -> f#(proper(X)) proper#(f(X)) -> proper#(X) -> proper#(f(X)) -> proper#(X) proper#(f(X)) -> f#(proper(X)) -> f#(ok(X)) -> f#(X) proper#(f(X)) -> f#(proper(X)) -> f#(mark(X)) -> f#(X) h#(ok(X)) -> h#(X) -> h#(ok(X)) -> h#(X) h#(ok(X)) -> h#(X) -> h#(mark(X)) -> h#(X) h#(mark(X)) -> h#(X) -> h#(ok(X)) -> h#(X) h#(mark(X)) -> h#(X) -> h#(mark(X)) -> h#(X) d#(ok(X)) -> d#(X) -> d#(ok(X)) -> d#(X) c#(ok(X)) -> c#(X) -> c#(ok(X)) -> c#(X) f#(ok(X)) -> f#(X) -> f#(ok(X)) -> f#(X) f#(ok(X)) -> f#(X) -> f#(mark(X)) -> f#(X) f#(mark(X)) -> f#(X) -> f#(ok(X)) -> f#(X) f#(mark(X)) -> f#(X) -> f#(mark(X)) -> f#(X) g#(ok(X)) -> g#(X) -> g#(ok(X)) -> g#(X) active#(h(X)) -> h#(active(X)) -> h#(ok(X)) -> h#(X) active#(h(X)) -> h#(active(X)) -> h#(mark(X)) -> h#(X) active#(h(X)) -> d#(X) -> d#(ok(X)) -> d#(X) active#(h(X)) -> c#(d(X)) -> c#(ok(X)) -> c#(X) active#(h(X)) -> active#(X) -> active#(h(X)) -> h#(active(X)) active#(h(X)) -> active#(X) -> active#(h(X)) -> active#(X) active#(h(X)) -> active#(X) -> active#(f(X)) -> f#(active(X)) active#(h(X)) -> active#(X) -> active#(f(X)) -> active#(X) active#(h(X)) -> active#(X) -> active#(h(X)) -> c#(d(X)) active#(h(X)) -> active#(X) -> active#(h(X)) -> d#(X) active#(h(X)) -> active#(X) -> active#(c(X)) -> d#(X) active#(h(X)) -> active#(X) -> active#(f(f(X))) -> c#(f(g(f(X)))) active#(h(X)) -> active#(X) -> active#(f(f(X))) -> f#(g(f(X))) active#(h(X)) -> active#(X) -> active#(f(f(X))) -> g#(f(X)) active#(c(X)) -> d#(X) -> d#(ok(X)) -> d#(X) active#(f(f(X))) -> c#(f(g(f(X)))) -> c#(ok(X)) -> c#(X) active#(f(f(X))) -> f#(g(f(X))) -> f#(ok(X)) -> f#(X) active#(f(f(X))) -> f#(g(f(X))) -> f#(mark(X)) -> f#(X) active#(f(f(X))) -> g#(f(X)) -> g#(ok(X)) -> g#(X) active#(f(X)) -> f#(active(X)) -> f#(ok(X)) -> f#(X) active#(f(X)) -> f#(active(X)) -> f#(mark(X)) -> f#(X) active#(f(X)) -> active#(X) -> active#(h(X)) -> h#(active(X)) active#(f(X)) -> active#(X) -> active#(h(X)) -> active#(X) active#(f(X)) -> active#(X) -> active#(f(X)) -> f#(active(X)) active#(f(X)) -> active#(X) -> active#(f(X)) -> active#(X) active#(f(X)) -> active#(X) -> active#(h(X)) -> c#(d(X)) active#(f(X)) -> active#(X) -> active#(h(X)) -> d#(X) active#(f(X)) -> active#(X) -> active#(c(X)) -> d#(X) active#(f(X)) -> active#(X) -> active#(f(f(X))) -> c#(f(g(f(X)))) active#(f(X)) -> active#(X) -> active#(f(f(X))) -> f#(g(f(X))) active#(f(X)) -> active#(X) -> active#(f(f(X))) -> g#(f(X)) SCC Processor: #sccs: 8 #rules: 16 #arcs: 127/961 DPs: top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> top#(proper(X)) TRS: active(f(f(X))) -> mark(c(f(g(f(X))))) active(c(X)) -> mark(d(X)) active(h(X)) -> mark(c(d(X))) active(f(X)) -> f(active(X)) active(h(X)) -> h(active(X)) f(mark(X)) -> mark(f(X)) h(mark(X)) -> mark(h(X)) proper(f(X)) -> f(proper(X)) proper(c(X)) -> c(proper(X)) proper(g(X)) -> g(proper(X)) proper(d(X)) -> d(proper(X)) proper(h(X)) -> h(proper(X)) f(ok(X)) -> ok(f(X)) c(ok(X)) -> ok(c(X)) g(ok(X)) -> ok(g(X)) d(ok(X)) -> ok(d(X)) h(ok(X)) -> ok(h(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [top#](x0) = [0 1 1]x0, [0 1 1] [0] [top](x0) = [0 0 1]x0 + [0] [0 1 1] [1], [0] [ok](x0) = [1] [1], [0] [proper](x0) = [0] [0], [0 0 0] [h](x0) = [0 1 0]x0 [0 0 1] , [0 0 0] [d](x0) = [0 0 1]x0 [0 0 1] , [0] [mark](x0) = [0] [1], [0 0 0] [c](x0) = [0 1 0]x0 [0 0 1] , [0 0 0] [g](x0) = [0 1 0]x0 [0 0 1] , [0] [active](x0) = [0] [1], [0 0 0] [f](x0) = [0 1 0]x0 [0 0 1] orientation: top#(ok(X)) = [2] >= [1] = top#(active(X)) top#(mark(X)) = [1] >= [0] = top#(proper(X)) [0] [0] active(f(f(X))) = [0] >= [0] = mark(c(f(g(f(X))))) [1] [1] [0] [0] active(c(X)) = [0] >= [0] = mark(d(X)) [1] [1] [0] [0] active(h(X)) = [0] >= [0] = mark(c(d(X))) [1] [1] [0] [0] active(f(X)) = [0] >= [0] = f(active(X)) [1] [1] [0] [0] active(h(X)) = [0] >= [0] = h(active(X)) [1] [1] [0] [0] f(mark(X)) = [0] >= [0] = mark(f(X)) [1] [1] [0] [0] h(mark(X)) = [0] >= [0] = mark(h(X)) [1] [1] [0] [0] proper(f(X)) = [0] >= [0] = f(proper(X)) [0] [0] [0] [0] proper(c(X)) = [0] >= [0] = c(proper(X)) [0] [0] [0] [0] proper(g(X)) = [0] >= [0] = g(proper(X)) [0] [0] [0] [0] proper(d(X)) = [0] >= [0] = d(proper(X)) [0] [0] [0] [0] proper(h(X)) = [0] >= [0] = h(proper(X)) [0] [0] [0] [0] f(ok(X)) = [1] >= [1] = ok(f(X)) [1] [1] [0] [0] c(ok(X)) = [1] >= [1] = ok(c(X)) [1] [1] [0] [0] g(ok(X)) = [1] >= [1] = ok(g(X)) [1] [1] [0] [0] d(ok(X)) = [1] >= [1] = ok(d(X)) [1] [1] [0] [0] h(ok(X)) = [1] >= [1] = ok(h(X)) [1] [1] [1] [0] top(mark(X)) = [1] >= [0] = top(proper(X)) [2] [1] [2] [1] top(ok(X)) = [1] >= [1] = top(active(X)) [3] [2] problem: DPs: TRS: active(f(f(X))) -> mark(c(f(g(f(X))))) active(c(X)) -> mark(d(X)) active(h(X)) -> mark(c(d(X))) active(f(X)) -> f(active(X)) active(h(X)) -> h(active(X)) f(mark(X)) -> mark(f(X)) h(mark(X)) -> mark(h(X)) proper(f(X)) -> f(proper(X)) proper(c(X)) -> c(proper(X)) proper(g(X)) -> g(proper(X)) proper(d(X)) -> d(proper(X)) proper(h(X)) -> h(proper(X)) f(ok(X)) -> ok(f(X)) c(ok(X)) -> ok(c(X)) g(ok(X)) -> ok(g(X)) d(ok(X)) -> ok(d(X)) h(ok(X)) -> ok(h(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: active#(f(X)) -> active#(X) active#(h(X)) -> active#(X) TRS: active(f(f(X))) -> mark(c(f(g(f(X))))) active(c(X)) -> mark(d(X)) active(h(X)) -> mark(c(d(X))) active(f(X)) -> f(active(X)) active(h(X)) -> h(active(X)) f(mark(X)) -> mark(f(X)) h(mark(X)) -> mark(h(X)) proper(f(X)) -> f(proper(X)) proper(c(X)) -> c(proper(X)) proper(g(X)) -> g(proper(X)) proper(d(X)) -> d(proper(X)) proper(h(X)) -> h(proper(X)) f(ok(X)) -> ok(f(X)) c(ok(X)) -> ok(c(X)) g(ok(X)) -> ok(g(X)) d(ok(X)) -> ok(d(X)) h(ok(X)) -> ok(h(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [active#](x0) = [1 1 1]x0 + [1], [0] [top](x0) = [0] [0], [0] [ok](x0) = [0] [0], [1 1 1] [0] [proper](x0) = [1 1 0]x0 + [1] [0 0 0] [1], [0 1 0] [0] [h](x0) = [1 0 1]x0 + [0] [0 0 0] [1], [0] [d](x0) = [1] [0], [mark](x0) = x0 , [0 0 1] [0] [c](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [0 1 1] [0] [g](x0) = [1 0 0]x0 + [1] [0 0 0] [1], [1 0 1] [0] [active](x0) = [0 1 0]x0 + [1] [0 0 0] [1], [0 1 0] [0] [f](x0) = [1 0 1]x0 + [0] [0 0 0] [1] orientation: active#(f(X)) = [1 1 1]X + [2] >= [1 1 1]X + [1] = active#(X) active#(h(X)) = [1 1 1]X + [2] >= [1 1 1]X + [1] = active#(X) [1 0 1] [1] [1] active(f(f(X))) = [0 1 0]X + [2] >= [0] = mark(c(f(g(f(X))))) [0 0 0] [1] [1] [0 0 1] [1] [0] active(c(X)) = [0 0 0]X + [1] >= [1] = mark(d(X)) [0 0 0] [1] [0] [0 1 0] [1] [0] active(h(X)) = [1 0 1]X + [1] >= [0] = mark(c(d(X))) [0 0 0] [1] [1] [0 1 0] [1] [0 1 0] [1] active(f(X)) = [1 0 1]X + [1] >= [1 0 1]X + [1] = f(active(X)) [0 0 0] [1] [0 0 0] [1] [0 1 0] [1] [0 1 0] [1] active(h(X)) = [1 0 1]X + [1] >= [1 0 1]X + [1] = h(active(X)) [0 0 0] [1] [0 0 0] [1] [0 1 0] [0] [0 1 0] [0] f(mark(X)) = [1 0 1]X + [0] >= [1 0 1]X + [0] = mark(f(X)) [0 0 0] [1] [0 0 0] [1] [0 1 0] [0] [0 1 0] [0] h(mark(X)) = [1 0 1]X + [0] >= [1 0 1]X + [0] = mark(h(X)) [0 0 0] [1] [0 0 0] [1] [1 1 1] [1] [1 1 0] [1] proper(f(X)) = [1 1 1]X + [1] >= [1 1 1]X + [1] = f(proper(X)) [0 0 0] [1] [0 0 0] [1] [0 0 1] [1] [1] proper(c(X)) = [0 0 1]X + [1] >= [0] = c(proper(X)) [0 0 0] [1] [1] [1 1 1] [2] [1 1 0] [2] proper(g(X)) = [1 1 1]X + [2] >= [1 1 1]X + [1] = g(proper(X)) [0 0 0] [1] [0 0 0] [1] [1] [0] proper(d(X)) = [2] >= [1] = d(proper(X)) [1] [0] [1 1 1] [1] [1 1 0] [1] proper(h(X)) = [1 1 1]X + [1] >= [1 1 1]X + [1] = h(proper(X)) [0 0 0] [1] [0 0 0] [1] [0] [0] f(ok(X)) = [0] >= [0] = ok(f(X)) [1] [0] [0] [0] c(ok(X)) = [0] >= [0] = ok(c(X)) [1] [0] [0] [0] g(ok(X)) = [1] >= [0] = ok(g(X)) [1] [0] [0] [0] d(ok(X)) = [1] >= [0] = ok(d(X)) [0] [0] [0] [0] h(ok(X)) = [0] >= [0] = ok(h(X)) [1] [0] [0] [0] top(mark(X)) = [0] >= [0] = top(proper(X)) [0] [0] [0] [0] top(ok(X)) = [0] >= [0] = top(active(X)) [0] [0] problem: DPs: TRS: active(f(f(X))) -> mark(c(f(g(f(X))))) active(c(X)) -> mark(d(X)) active(h(X)) -> mark(c(d(X))) active(f(X)) -> f(active(X)) active(h(X)) -> h(active(X)) f(mark(X)) -> mark(f(X)) h(mark(X)) -> mark(h(X)) proper(f(X)) -> f(proper(X)) proper(c(X)) -> c(proper(X)) proper(g(X)) -> g(proper(X)) proper(d(X)) -> d(proper(X)) proper(h(X)) -> h(proper(X)) f(ok(X)) -> ok(f(X)) c(ok(X)) -> ok(c(X)) g(ok(X)) -> ok(g(X)) d(ok(X)) -> ok(d(X)) h(ok(X)) -> ok(h(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: proper#(f(X)) -> proper#(X) proper#(c(X)) -> proper#(X) proper#(g(X)) -> proper#(X) proper#(d(X)) -> proper#(X) proper#(h(X)) -> proper#(X) TRS: active(f(f(X))) -> mark(c(f(g(f(X))))) active(c(X)) -> mark(d(X)) active(h(X)) -> mark(c(d(X))) active(f(X)) -> f(active(X)) active(h(X)) -> h(active(X)) f(mark(X)) -> mark(f(X)) h(mark(X)) -> mark(h(X)) proper(f(X)) -> f(proper(X)) proper(c(X)) -> c(proper(X)) proper(g(X)) -> g(proper(X)) proper(d(X)) -> d(proper(X)) proper(h(X)) -> h(proper(X)) f(ok(X)) -> ok(f(X)) c(ok(X)) -> ok(c(X)) g(ok(X)) -> ok(g(X)) d(ok(X)) -> ok(d(X)) h(ok(X)) -> ok(h(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [proper#](x0) = [1 0 1]x0, [0] [top](x0) = [1] [0], [0 1 0] [1] [ok](x0) = [0 1 0]x0 + [0] [0 0 0] [0], [1 0 0] [proper](x0) = [0 0 0]x0 [0 0 1] , [0 0 1] [1] [h](x0) = [0 0 0]x0 + [0] [1 0 0] [0], [1 1 0] [1] [d](x0) = [0 0 0]x0 + [0] [0 0 1] [0], [1] [mark](x0) = [0] [0], [0 0 1] [1] [c](x0) = [0 0 0]x0 + [0] [1 0 0] [0], [1 0 1] [1] [g](x0) = [0 0 0]x0 + [0] [1 0 0] [0], [0 0 1] [1] [active](x0) = [0 0 0]x0 + [0] [1 0 0] [0], [0 0 1] [1] [f](x0) = [0 0 0]x0 + [0] [1 0 0] [0] orientation: proper#(f(X)) = [1 0 1]X + [1] >= [1 0 1]X = proper#(X) proper#(c(X)) = [1 0 1]X + [1] >= [1 0 1]X = proper#(X) proper#(g(X)) = [2 0 1]X + [1] >= [1 0 1]X = proper#(X) proper#(d(X)) = [1 1 1]X + [1] >= [1 0 1]X = proper#(X) proper#(h(X)) = [1 0 1]X + [1] >= [1 0 1]X = proper#(X) [0 0 1] [2] [1] active(f(f(X))) = [0 0 0]X + [0] >= [0] = mark(c(f(g(f(X))))) [1 0 0] [1] [0] [1 0 0] [1] [1] active(c(X)) = [0 0 0]X + [0] >= [0] = mark(d(X)) [0 0 1] [1] [0] [1 0 0] [1] [1] active(h(X)) = [0 0 0]X + [0] >= [0] = mark(c(d(X))) [0 0 1] [1] [0] [1 0 0] [1] [1 0 0] [1] active(f(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = f(active(X)) [0 0 1] [1] [0 0 1] [1] [1 0 0] [1] [1 0 0] [1] active(h(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = h(active(X)) [0 0 1] [1] [0 0 1] [1] [1] [1] f(mark(X)) = [0] >= [0] = mark(f(X)) [1] [0] [1] [1] h(mark(X)) = [0] >= [0] = mark(h(X)) [1] [0] [0 0 1] [1] [0 0 1] [1] proper(f(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = f(proper(X)) [1 0 0] [0] [1 0 0] [0] [0 0 1] [1] [0 0 1] [1] proper(c(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = c(proper(X)) [1 0 0] [0] [1 0 0] [0] [1 0 1] [1] [1 0 1] [1] proper(g(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = g(proper(X)) [1 0 0] [0] [1 0 0] [0] [1 1 0] [1] [1 0 0] [1] proper(d(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = d(proper(X)) [0 0 1] [0] [0 0 1] [0] [0 0 1] [1] [0 0 1] [1] proper(h(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = h(proper(X)) [1 0 0] [0] [1 0 0] [0] [0 0 0] [1] [1] f(ok(X)) = [0 0 0]X + [0] >= [0] = ok(f(X)) [0 1 0] [1] [0] [0 0 0] [1] [1] c(ok(X)) = [0 0 0]X + [0] >= [0] = ok(c(X)) [0 1 0] [1] [0] [0 1 0] [2] [1] g(ok(X)) = [0 0 0]X + [0] >= [0] = ok(g(X)) [0 1 0] [1] [0] [0 2 0] [2] [1] d(ok(X)) = [0 0 0]X + [0] >= [0] = ok(d(X)) [0 0 0] [0] [0] [0 0 0] [1] [1] h(ok(X)) = [0 0 0]X + [0] >= [0] = ok(h(X)) [0 1 0] [1] [0] [0] [0] top(mark(X)) = [1] >= [1] = top(proper(X)) [0] [0] [0] [0] top(ok(X)) = [1] >= [1] = top(active(X)) [0] [0] problem: DPs: TRS: active(f(f(X))) -> mark(c(f(g(f(X))))) active(c(X)) -> mark(d(X)) active(h(X)) -> mark(c(d(X))) active(f(X)) -> f(active(X)) active(h(X)) -> h(active(X)) f(mark(X)) -> mark(f(X)) h(mark(X)) -> mark(h(X)) proper(f(X)) -> f(proper(X)) proper(c(X)) -> c(proper(X)) proper(g(X)) -> g(proper(X)) proper(d(X)) -> d(proper(X)) proper(h(X)) -> h(proper(X)) f(ok(X)) -> ok(f(X)) c(ok(X)) -> ok(c(X)) g(ok(X)) -> ok(g(X)) d(ok(X)) -> ok(d(X)) h(ok(X)) -> ok(h(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: h#(mark(X)) -> h#(X) h#(ok(X)) -> h#(X) TRS: active(f(f(X))) -> mark(c(f(g(f(X))))) active(c(X)) -> mark(d(X)) active(h(X)) -> mark(c(d(X))) active(f(X)) -> f(active(X)) active(h(X)) -> h(active(X)) f(mark(X)) -> mark(f(X)) h(mark(X)) -> mark(h(X)) proper(f(X)) -> f(proper(X)) proper(c(X)) -> c(proper(X)) proper(g(X)) -> g(proper(X)) proper(d(X)) -> d(proper(X)) proper(h(X)) -> h(proper(X)) f(ok(X)) -> ok(f(X)) c(ok(X)) -> ok(c(X)) g(ok(X)) -> ok(g(X)) d(ok(X)) -> ok(d(X)) h(ok(X)) -> ok(h(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [h#](x0) = [0 1 0]x0, [0 0 0] [0] [top](x0) = [0 1 0]x0 + [1] [0 0 0] [0], [0 0 0] [0] [ok](x0) = [0 1 0]x0 + [1] [0 0 0] [0], [1 0 0] [proper](x0) = [0 0 0]x0 [0 1 1] , [0 0 0] [1] [h](x0) = [0 1 0]x0 + [0] [0 0 0] [0], [0 1 0] [1] [d](x0) = [0 1 0]x0 + [0] [0 1 0] [1], [0 0 0] [0] [mark](x0) = [0 1 0]x0 + [1] [0 0 0] [0], [0 0 0] [1] [c](x0) = [0 1 0]x0 + [0] [0 1 0] [0], [0 0 0] [g](x0) = [0 1 0]x0 [0 1 0] , [1 0 0] [1] [active](x0) = [0 1 0]x0 + [1] [0 0 0] [0], [0 0 0] [f](x0) = [0 1 0]x0 [0 0 0] orientation: h#(mark(X)) = [0 1 0]X + [1] >= [0 1 0]X = h#(X) h#(ok(X)) = [0 1 0]X + [1] >= [0 1 0]X = h#(X) [0 0 0] [1] [0 0 0] [0] active(f(f(X))) = [0 1 0]X + [1] >= [0 1 0]X + [1] = mark(c(f(g(f(X))))) [0 0 0] [0] [0 0 0] [0] [0 0 0] [2] [0 0 0] [0] active(c(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = mark(d(X)) [0 0 0] [0] [0 0 0] [0] [0 0 0] [2] [0 0 0] [0] active(h(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = mark(c(d(X))) [0 0 0] [0] [0 0 0] [0] [0 0 0] [1] [0 0 0] [0] active(f(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = f(active(X)) [0 0 0] [0] [0 0 0] [0] [0 0 0] [2] [0 0 0] [1] active(h(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = h(active(X)) [0 0 0] [0] [0 0 0] [0] [0 0 0] [0] [0 0 0] [0] f(mark(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = mark(f(X)) [0 0 0] [0] [0 0 0] [0] [0 0 0] [1] [0 0 0] [0] h(mark(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = mark(h(X)) [0 0 0] [0] [0 0 0] [0] [0 0 0] [0] proper(f(X)) = [0 0 0]X >= [0] = f(proper(X)) [0 1 0] [0] [0 0 0] [1] [1] proper(c(X)) = [0 0 0]X + [0] >= [0] = c(proper(X)) [0 2 0] [0] [0] [0 0 0] [0] proper(g(X)) = [0 0 0]X >= [0] = g(proper(X)) [0 2 0] [0] [0 1 0] [1] [1] proper(d(X)) = [0 0 0]X + [0] >= [0] = d(proper(X)) [0 2 0] [1] [1] [0 0 0] [1] [1] proper(h(X)) = [0 0 0]X + [0] >= [0] = h(proper(X)) [0 1 0] [0] [0] [0 0 0] [0] [0 0 0] [0] f(ok(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = ok(f(X)) [0 0 0] [0] [0 0 0] [0] [0 0 0] [1] [0 0 0] [0] c(ok(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = ok(c(X)) [0 1 0] [1] [0 0 0] [0] [0 0 0] [0] [0 0 0] [0] g(ok(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = ok(g(X)) [0 1 0] [1] [0 0 0] [0] [0 1 0] [2] [0 0 0] [0] d(ok(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = ok(d(X)) [0 1 0] [2] [0 0 0] [0] [0 0 0] [1] [0 0 0] [0] h(ok(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = ok(h(X)) [0 0 0] [0] [0 0 0] [0] [0 0 0] [0] [0] top(mark(X)) = [0 1 0]X + [2] >= [1] = top(proper(X)) [0 0 0] [0] [0] [0 0 0] [0] [0 0 0] [0] top(ok(X)) = [0 1 0]X + [2] >= [0 1 0]X + [2] = top(active(X)) [0 0 0] [0] [0 0 0] [0] problem: DPs: TRS: active(f(f(X))) -> mark(c(f(g(f(X))))) active(c(X)) -> mark(d(X)) active(h(X)) -> mark(c(d(X))) active(f(X)) -> f(active(X)) active(h(X)) -> h(active(X)) f(mark(X)) -> mark(f(X)) h(mark(X)) -> mark(h(X)) proper(f(X)) -> f(proper(X)) proper(c(X)) -> c(proper(X)) proper(g(X)) -> g(proper(X)) proper(d(X)) -> d(proper(X)) proper(h(X)) -> h(proper(X)) f(ok(X)) -> ok(f(X)) c(ok(X)) -> ok(c(X)) g(ok(X)) -> ok(g(X)) d(ok(X)) -> ok(d(X)) h(ok(X)) -> ok(h(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: d#(ok(X)) -> d#(X) TRS: active(f(f(X))) -> mark(c(f(g(f(X))))) active(c(X)) -> mark(d(X)) active(h(X)) -> mark(c(d(X))) active(f(X)) -> f(active(X)) active(h(X)) -> h(active(X)) f(mark(X)) -> mark(f(X)) h(mark(X)) -> mark(h(X)) proper(f(X)) -> f(proper(X)) proper(c(X)) -> c(proper(X)) proper(g(X)) -> g(proper(X)) proper(d(X)) -> d(proper(X)) proper(h(X)) -> h(proper(X)) f(ok(X)) -> ok(f(X)) c(ok(X)) -> ok(c(X)) g(ok(X)) -> ok(g(X)) d(ok(X)) -> ok(d(X)) h(ok(X)) -> ok(h(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) KBO Processor: argument filtering: pi(f) = 0 pi(active) = [] pi(g) = [0] pi(c) = 0 pi(mark) = [] pi(d) = [0] pi(h) = 0 pi(proper) = 0 pi(ok) = [0] pi(top) = [] pi(d#) = 0 weight function: w0 = 1 w(d#) = w(top) = w(ok) = w(proper) = w(h) = w(d) = w(mark) = w(g) = w( active) = w(f) = 1 w(c) = 0 precedence: d# ~ top ~ proper ~ d ~ g ~ active > ok ~ h ~ mark ~ c ~ f problem: DPs: TRS: active(f(f(X))) -> mark(c(f(g(f(X))))) active(c(X)) -> mark(d(X)) active(h(X)) -> mark(c(d(X))) active(f(X)) -> f(active(X)) active(h(X)) -> h(active(X)) f(mark(X)) -> mark(f(X)) h(mark(X)) -> mark(h(X)) proper(f(X)) -> f(proper(X)) proper(c(X)) -> c(proper(X)) proper(g(X)) -> g(proper(X)) proper(d(X)) -> d(proper(X)) proper(h(X)) -> h(proper(X)) f(ok(X)) -> ok(f(X)) c(ok(X)) -> ok(c(X)) g(ok(X)) -> ok(g(X)) d(ok(X)) -> ok(d(X)) h(ok(X)) -> ok(h(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: g#(ok(X)) -> g#(X) TRS: active(f(f(X))) -> mark(c(f(g(f(X))))) active(c(X)) -> mark(d(X)) active(h(X)) -> mark(c(d(X))) active(f(X)) -> f(active(X)) active(h(X)) -> h(active(X)) f(mark(X)) -> mark(f(X)) h(mark(X)) -> mark(h(X)) proper(f(X)) -> f(proper(X)) proper(c(X)) -> c(proper(X)) proper(g(X)) -> g(proper(X)) proper(d(X)) -> d(proper(X)) proper(h(X)) -> h(proper(X)) f(ok(X)) -> ok(f(X)) c(ok(X)) -> ok(c(X)) g(ok(X)) -> ok(g(X)) d(ok(X)) -> ok(d(X)) h(ok(X)) -> ok(h(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) KBO Processor: argument filtering: pi(f) = 0 pi(active) = [] pi(g) = [0] pi(c) = 0 pi(mark) = [] pi(d) = [0] pi(h) = 0 pi(proper) = 0 pi(ok) = [0] pi(top) = [] pi(g#) = 0 weight function: w0 = 1 w(g#) = w(top) = w(ok) = w(proper) = w(h) = w(d) = w(mark) = w(g) = w( active) = w(f) = 1 w(c) = 0 precedence: g# ~ top ~ proper ~ d ~ g ~ active > ok ~ h ~ mark ~ c ~ f problem: DPs: TRS: active(f(f(X))) -> mark(c(f(g(f(X))))) active(c(X)) -> mark(d(X)) active(h(X)) -> mark(c(d(X))) active(f(X)) -> f(active(X)) active(h(X)) -> h(active(X)) f(mark(X)) -> mark(f(X)) h(mark(X)) -> mark(h(X)) proper(f(X)) -> f(proper(X)) proper(c(X)) -> c(proper(X)) proper(g(X)) -> g(proper(X)) proper(d(X)) -> d(proper(X)) proper(h(X)) -> h(proper(X)) f(ok(X)) -> ok(f(X)) c(ok(X)) -> ok(c(X)) g(ok(X)) -> ok(g(X)) d(ok(X)) -> ok(d(X)) h(ok(X)) -> ok(h(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: c#(ok(X)) -> c#(X) TRS: active(f(f(X))) -> mark(c(f(g(f(X))))) active(c(X)) -> mark(d(X)) active(h(X)) -> mark(c(d(X))) active(f(X)) -> f(active(X)) active(h(X)) -> h(active(X)) f(mark(X)) -> mark(f(X)) h(mark(X)) -> mark(h(X)) proper(f(X)) -> f(proper(X)) proper(c(X)) -> c(proper(X)) proper(g(X)) -> g(proper(X)) proper(d(X)) -> d(proper(X)) proper(h(X)) -> h(proper(X)) f(ok(X)) -> ok(f(X)) c(ok(X)) -> ok(c(X)) g(ok(X)) -> ok(g(X)) d(ok(X)) -> ok(d(X)) h(ok(X)) -> ok(h(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) KBO Processor: argument filtering: pi(f) = 0 pi(active) = [] pi(g) = [0] pi(c) = 0 pi(mark) = [] pi(d) = [0] pi(h) = 0 pi(proper) = 0 pi(ok) = [0] pi(top) = [] pi(c#) = 0 weight function: w0 = 1 w(c#) = w(top) = w(ok) = w(proper) = w(h) = w(d) = w(mark) = w(g) = w( active) = w(f) = 1 w(c) = 0 precedence: c# ~ top ~ proper ~ d ~ g ~ active > ok ~ h ~ mark ~ c ~ f problem: DPs: TRS: active(f(f(X))) -> mark(c(f(g(f(X))))) active(c(X)) -> mark(d(X)) active(h(X)) -> mark(c(d(X))) active(f(X)) -> f(active(X)) active(h(X)) -> h(active(X)) f(mark(X)) -> mark(f(X)) h(mark(X)) -> mark(h(X)) proper(f(X)) -> f(proper(X)) proper(c(X)) -> c(proper(X)) proper(g(X)) -> g(proper(X)) proper(d(X)) -> d(proper(X)) proper(h(X)) -> h(proper(X)) f(ok(X)) -> ok(f(X)) c(ok(X)) -> ok(c(X)) g(ok(X)) -> ok(g(X)) d(ok(X)) -> ok(d(X)) h(ok(X)) -> ok(h(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: f#(mark(X)) -> f#(X) f#(ok(X)) -> f#(X) TRS: active(f(f(X))) -> mark(c(f(g(f(X))))) active(c(X)) -> mark(d(X)) active(h(X)) -> mark(c(d(X))) active(f(X)) -> f(active(X)) active(h(X)) -> h(active(X)) f(mark(X)) -> mark(f(X)) h(mark(X)) -> mark(h(X)) proper(f(X)) -> f(proper(X)) proper(c(X)) -> c(proper(X)) proper(g(X)) -> g(proper(X)) proper(d(X)) -> d(proper(X)) proper(h(X)) -> h(proper(X)) f(ok(X)) -> ok(f(X)) c(ok(X)) -> ok(c(X)) g(ok(X)) -> ok(g(X)) d(ok(X)) -> ok(d(X)) h(ok(X)) -> ok(h(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [f#](x0) = 4x0, [top](x0) = 4, [ok](x0) = x0 + 1/2, [proper](x0) = 4x0, [h](x0) = x0 + 4, [d](x0) = 5/2x0, [mark](x0) = x0 + 1/2, [c](x0) = x0 + 1/2, [g](x0) = 2x0, [active](x0) = 5/2x0 + 1/2, [f](x0) = 5/2x0 + 1/2 orientation: f#(mark(X)) = 4X + 2 >= 4X = f#(X) f#(ok(X)) = 4X + 2 >= 4X = f#(X) active(f(f(X))) = 125/8X + 39/8 >= 25/2X + 4 = mark(c(f(g(f(X))))) active(c(X)) = 5/2X + 7/4 >= 5/2X + 1/2 = mark(d(X)) active(h(X)) = 5/2X + 21/2 >= 5/2X + 1 = mark(c(d(X))) active(f(X)) = 25/4X + 7/4 >= 25/4X + 7/4 = f(active(X)) active(h(X)) = 5/2X + 21/2 >= 5/2X + 9/2 = h(active(X)) f(mark(X)) = 5/2X + 7/4 >= 5/2X + 1 = mark(f(X)) h(mark(X)) = X + 9/2 >= X + 9/2 = mark(h(X)) proper(f(X)) = 10X + 2 >= 10X + 1/2 = f(proper(X)) proper(c(X)) = 4X + 2 >= 4X + 1/2 = c(proper(X)) proper(g(X)) = 8X >= 8X = g(proper(X)) proper(d(X)) = 10X >= 10X = d(proper(X)) proper(h(X)) = 4X + 16 >= 4X + 4 = h(proper(X)) f(ok(X)) = 5/2X + 7/4 >= 5/2X + 1 = ok(f(X)) c(ok(X)) = X + 1 >= X + 1 = ok(c(X)) g(ok(X)) = 2X + 1 >= 2X + 1/2 = ok(g(X)) d(ok(X)) = 5/2X + 5/4 >= 5/2X + 1/2 = ok(d(X)) h(ok(X)) = X + 9/2 >= X + 9/2 = ok(h(X)) top(mark(X)) = 4 >= 4 = top(proper(X)) top(ok(X)) = 4 >= 4 = top(active(X)) problem: DPs: TRS: active(f(f(X))) -> mark(c(f(g(f(X))))) active(c(X)) -> mark(d(X)) active(h(X)) -> mark(c(d(X))) active(f(X)) -> f(active(X)) active(h(X)) -> h(active(X)) f(mark(X)) -> mark(f(X)) h(mark(X)) -> mark(h(X)) proper(f(X)) -> f(proper(X)) proper(c(X)) -> c(proper(X)) proper(g(X)) -> g(proper(X)) proper(d(X)) -> d(proper(X)) proper(h(X)) -> h(proper(X)) f(ok(X)) -> ok(f(X)) c(ok(X)) -> ok(c(X)) g(ok(X)) -> ok(g(X)) d(ok(X)) -> ok(d(X)) h(ok(X)) -> ok(h(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed