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)) CDG 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#(mark(X)) -> proper#(X) top#(ok(X)) -> top#(active(X)) -> top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> active#(X) -> active#(f(f(X))) -> 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))) -> c#(f(g(f(X)))) top#(ok(X)) -> active#(X) -> active#(c(X)) -> d#(X) top#(ok(X)) -> active#(X) -> active#(h(X)) -> d#(X) top#(ok(X)) -> active#(X) -> active#(h(X)) -> c#(d(X)) top#(ok(X)) -> active#(X) -> active#(f(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(f(X)) -> f#(active(X)) top#(ok(X)) -> active#(X) -> active#(h(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(h(X)) -> h#(active(X)) top#(mark(X)) -> proper#(X) -> proper#(f(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(f(X)) -> f#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(c(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(c(X)) -> c#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(g(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(g(X)) -> g#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(d(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(d(X)) -> d#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(h(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(h(X)) -> h#(proper(X)) proper#(h(X)) -> proper#(X) -> proper#(f(X)) -> proper#(X) proper#(h(X)) -> proper#(X) -> proper#(f(X)) -> f#(proper(X)) proper#(h(X)) -> proper#(X) -> proper#(c(X)) -> proper#(X) proper#(h(X)) -> proper#(X) -> proper#(c(X)) -> c#(proper(X)) proper#(h(X)) -> proper#(X) -> proper#(g(X)) -> proper#(X) proper#(h(X)) -> proper#(X) -> proper#(g(X)) -> g#(proper(X)) proper#(h(X)) -> proper#(X) -> proper#(d(X)) -> proper#(X) proper#(h(X)) -> proper#(X) -> proper#(d(X)) -> d#(proper(X)) proper#(h(X)) -> proper#(X) -> proper#(h(X)) -> proper#(X) proper#(h(X)) -> proper#(X) -> proper#(h(X)) -> h#(proper(X)) proper#(d(X)) -> proper#(X) -> proper#(f(X)) -> proper#(X) proper#(d(X)) -> proper#(X) -> proper#(f(X)) -> f#(proper(X)) proper#(d(X)) -> proper#(X) -> proper#(c(X)) -> proper#(X) proper#(d(X)) -> proper#(X) -> proper#(c(X)) -> c#(proper(X)) proper#(d(X)) -> proper#(X) -> proper#(g(X)) -> proper#(X) proper#(d(X)) -> proper#(X) -> proper#(g(X)) -> g#(proper(X)) proper#(d(X)) -> proper#(X) -> proper#(d(X)) -> proper#(X) proper#(d(X)) -> proper#(X) -> proper#(d(X)) -> d#(proper(X)) proper#(d(X)) -> proper#(X) -> proper#(h(X)) -> proper#(X) proper#(d(X)) -> proper#(X) -> proper#(h(X)) -> h#(proper(X)) proper#(c(X)) -> proper#(X) -> proper#(f(X)) -> proper#(X) proper#(c(X)) -> proper#(X) -> proper#(f(X)) -> f#(proper(X)) proper#(c(X)) -> proper#(X) -> proper#(c(X)) -> proper#(X) proper#(c(X)) -> proper#(X) -> proper#(c(X)) -> c#(proper(X)) proper#(c(X)) -> proper#(X) -> proper#(g(X)) -> proper#(X) proper#(c(X)) -> proper#(X) -> proper#(g(X)) -> g#(proper(X)) proper#(c(X)) -> proper#(X) -> proper#(d(X)) -> proper#(X) proper#(c(X)) -> proper#(X) -> proper#(d(X)) -> d#(proper(X)) proper#(c(X)) -> proper#(X) -> proper#(h(X)) -> proper#(X) proper#(c(X)) -> proper#(X) -> proper#(h(X)) -> h#(proper(X)) proper#(g(X)) -> proper#(X) -> proper#(f(X)) -> proper#(X) proper#(g(X)) -> proper#(X) -> proper#(f(X)) -> f#(proper(X)) proper#(g(X)) -> proper#(X) -> proper#(c(X)) -> proper#(X) proper#(g(X)) -> proper#(X) -> proper#(c(X)) -> c#(proper(X)) proper#(g(X)) -> proper#(X) -> proper#(g(X)) -> proper#(X) proper#(g(X)) -> proper#(X) -> proper#(g(X)) -> g#(proper(X)) proper#(g(X)) -> proper#(X) -> proper#(d(X)) -> proper#(X) proper#(g(X)) -> proper#(X) -> proper#(d(X)) -> d#(proper(X)) proper#(g(X)) -> proper#(X) -> proper#(h(X)) -> proper#(X) proper#(g(X)) -> proper#(X) -> proper#(h(X)) -> h#(proper(X)) proper#(f(X)) -> proper#(X) -> proper#(f(X)) -> proper#(X) proper#(f(X)) -> proper#(X) -> proper#(f(X)) -> f#(proper(X)) proper#(f(X)) -> proper#(X) -> proper#(c(X)) -> proper#(X) proper#(f(X)) -> proper#(X) -> proper#(c(X)) -> c#(proper(X)) proper#(f(X)) -> proper#(X) -> proper#(g(X)) -> proper#(X) proper#(f(X)) -> proper#(X) -> proper#(g(X)) -> g#(proper(X)) proper#(f(X)) -> proper#(X) -> proper#(d(X)) -> proper#(X) proper#(f(X)) -> proper#(X) -> proper#(d(X)) -> d#(proper(X)) proper#(f(X)) -> proper#(X) -> proper#(h(X)) -> proper#(X) proper#(f(X)) -> proper#(X) -> proper#(h(X)) -> h#(proper(X)) h#(ok(X)) -> h#(X) -> h#(mark(X)) -> h#(X) h#(ok(X)) -> h#(X) -> h#(ok(X)) -> h#(X) h#(mark(X)) -> h#(X) -> h#(mark(X)) -> h#(X) h#(mark(X)) -> h#(X) -> h#(ok(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#(mark(X)) -> f#(X) f#(ok(X)) -> f#(X) -> f#(ok(X)) -> f#(X) f#(mark(X)) -> f#(X) -> f#(mark(X)) -> f#(X) f#(mark(X)) -> f#(X) -> f#(ok(X)) -> f#(X) g#(ok(X)) -> g#(X) -> g#(ok(X)) -> g#(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#(f(f(X))) -> 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))) -> c#(f(g(f(X)))) active#(h(X)) -> active#(X) -> active#(c(X)) -> d#(X) active#(h(X)) -> active#(X) -> active#(h(X)) -> d#(X) active#(h(X)) -> active#(X) -> active#(h(X)) -> c#(d(X)) active#(h(X)) -> active#(X) -> active#(f(X)) -> active#(X) active#(h(X)) -> active#(X) -> active#(f(X)) -> f#(active(X)) active#(h(X)) -> active#(X) -> active#(h(X)) -> active#(X) active#(h(X)) -> active#(X) -> active#(h(X)) -> h#(active(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))) -> g#(f(X)) -> g#(ok(X)) -> g#(X) active#(f(X)) -> f#(active(X)) -> f#(mark(X)) -> f#(X) active#(f(X)) -> active#(X) -> active#(f(f(X))) -> 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))) -> c#(f(g(f(X)))) active#(f(X)) -> active#(X) -> active#(c(X)) -> d#(X) active#(f(X)) -> active#(X) -> active#(h(X)) -> d#(X) active#(f(X)) -> active#(X) -> active#(h(X)) -> c#(d(X)) active#(f(X)) -> active#(X) -> active#(f(X)) -> active#(X) active#(f(X)) -> active#(X) -> active#(f(X)) -> f#(active(X)) active#(f(X)) -> active#(X) -> active#(h(X)) -> active#(X) active#(f(X)) -> active#(X) -> active#(h(X)) -> h#(active(X)) SCC Processor: #sccs: 7 #rules: 14 #arcs: 111/961 DPs: active#(h(X)) -> active#(X) active#(f(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: dimension: 1 interpretation: [active#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [h](x0) = x0 + 1, [d](x0) = x0, [mark](x0) = x0, [c](x0) = x0, [g](x0) = x0 + 1, [active](x0) = x0 + 1, [f](x0) = x0 + 1 orientation: active#(h(X)) = X + 2 >= X + 1 = active#(X) active#(f(X)) = X + 2 >= X + 1 = active#(X) active(f(f(X))) = X + 3 >= X + 3 = mark(c(f(g(f(X))))) active(c(X)) = X + 1 >= X = mark(d(X)) active(h(X)) = X + 2 >= X = mark(c(d(X))) active(f(X)) = X + 2 >= X + 2 = f(active(X)) active(h(X)) = X + 2 >= X + 2 = h(active(X)) f(mark(X)) = X + 1 >= X + 1 = mark(f(X)) h(mark(X)) = X + 1 >= X + 1 = mark(h(X)) proper(f(X)) = X + 2 >= X + 2 = f(proper(X)) proper(c(X)) = X + 1 >= X + 1 = c(proper(X)) proper(g(X)) = X + 2 >= X + 2 = g(proper(X)) proper(d(X)) = X + 1 >= X + 1 = d(proper(X)) proper(h(X)) = X + 2 >= X + 2 = h(proper(X)) f(ok(X)) = X + 2 >= X + 2 = ok(f(X)) c(ok(X)) = X + 1 >= X + 1 = ok(c(X)) g(ok(X)) = X + 2 >= X + 2 = ok(g(X)) d(ok(X)) = X + 1 >= X + 1 = ok(d(X)) h(ok(X)) = X + 2 >= X + 2 = ok(h(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = 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 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)) Matrix Interpretation Processor: dimension: 1 interpretation: [g#](x0) = x0 + 1, [top](x0) = 1, [ok](x0) = x0 + 1, [proper](x0) = 1, [h](x0) = x0, [d](x0) = x0, [mark](x0) = 1, [c](x0) = x0, [g](x0) = x0, [active](x0) = 1, [f](x0) = x0 orientation: g#(ok(X)) = X + 2 >= X + 1 = g#(X) active(f(f(X))) = 1 >= 1 = mark(c(f(g(f(X))))) active(c(X)) = 1 >= 1 = mark(d(X)) active(h(X)) = 1 >= 1 = mark(c(d(X))) active(f(X)) = 1 >= 1 = f(active(X)) active(h(X)) = 1 >= 1 = h(active(X)) f(mark(X)) = 1 >= 1 = mark(f(X)) h(mark(X)) = 1 >= 1 = mark(h(X)) proper(f(X)) = 1 >= 1 = f(proper(X)) proper(c(X)) = 1 >= 1 = c(proper(X)) proper(g(X)) = 1 >= 1 = g(proper(X)) proper(d(X)) = 1 >= 1 = d(proper(X)) proper(h(X)) = 1 >= 1 = h(proper(X)) f(ok(X)) = X + 1 >= X + 1 = ok(f(X)) c(ok(X)) = X + 1 >= X + 1 = ok(c(X)) g(ok(X)) = X + 1 >= X + 1 = ok(g(X)) d(ok(X)) = X + 1 >= X + 1 = ok(d(X)) h(ok(X)) = X + 1 >= X + 1 = ok(h(X)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = 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 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)) Matrix Interpretation Processor: dimension: 1 interpretation: [d#](x0) = x0 + 1, [top](x0) = 1, [ok](x0) = x0 + 1, [proper](x0) = 1, [h](x0) = x0, [d](x0) = x0, [mark](x0) = 1, [c](x0) = x0, [g](x0) = x0, [active](x0) = 1, [f](x0) = x0 orientation: d#(ok(X)) = X + 2 >= X + 1 = d#(X) active(f(f(X))) = 1 >= 1 = mark(c(f(g(f(X))))) active(c(X)) = 1 >= 1 = mark(d(X)) active(h(X)) = 1 >= 1 = mark(c(d(X))) active(f(X)) = 1 >= 1 = f(active(X)) active(h(X)) = 1 >= 1 = h(active(X)) f(mark(X)) = 1 >= 1 = mark(f(X)) h(mark(X)) = 1 >= 1 = mark(h(X)) proper(f(X)) = 1 >= 1 = f(proper(X)) proper(c(X)) = 1 >= 1 = c(proper(X)) proper(g(X)) = 1 >= 1 = g(proper(X)) proper(d(X)) = 1 >= 1 = d(proper(X)) proper(h(X)) = 1 >= 1 = h(proper(X)) f(ok(X)) = X + 1 >= X + 1 = ok(f(X)) c(ok(X)) = X + 1 >= X + 1 = ok(c(X)) g(ok(X)) = X + 1 >= X + 1 = ok(g(X)) d(ok(X)) = X + 1 >= X + 1 = ok(d(X)) h(ok(X)) = X + 1 >= X + 1 = ok(h(X)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = 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 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)) Matrix Interpretation Processor: dimension: 1 interpretation: [c#](x0) = x0 + 1, [top](x0) = 1, [ok](x0) = x0 + 1, [proper](x0) = 1, [h](x0) = x0, [d](x0) = x0, [mark](x0) = 1, [c](x0) = x0, [g](x0) = x0, [active](x0) = 1, [f](x0) = x0 orientation: c#(ok(X)) = X + 2 >= X + 1 = c#(X) active(f(f(X))) = 1 >= 1 = mark(c(f(g(f(X))))) active(c(X)) = 1 >= 1 = mark(d(X)) active(h(X)) = 1 >= 1 = mark(c(d(X))) active(f(X)) = 1 >= 1 = f(active(X)) active(h(X)) = 1 >= 1 = h(active(X)) f(mark(X)) = 1 >= 1 = mark(f(X)) h(mark(X)) = 1 >= 1 = mark(h(X)) proper(f(X)) = 1 >= 1 = f(proper(X)) proper(c(X)) = 1 >= 1 = c(proper(X)) proper(g(X)) = 1 >= 1 = g(proper(X)) proper(d(X)) = 1 >= 1 = d(proper(X)) proper(h(X)) = 1 >= 1 = h(proper(X)) f(ok(X)) = X + 1 >= X + 1 = ok(f(X)) c(ok(X)) = X + 1 >= X + 1 = ok(c(X)) g(ok(X)) = X + 1 >= X + 1 = ok(g(X)) d(ok(X)) = X + 1 >= X + 1 = ok(d(X)) h(ok(X)) = X + 1 >= X + 1 = ok(h(X)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = 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 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: dimension: 1 interpretation: [f#](x0) = x0, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [h](x0) = x0, [d](x0) = x0, [mark](x0) = x0, [c](x0) = x0 + 1, [g](x0) = x0, [active](x0) = x0 + 1, [f](x0) = x0 + 1 orientation: f#(mark(X)) = X >= X = f#(X) f#(ok(X)) = X + 1 >= X = f#(X) active(f(f(X))) = X + 3 >= X + 3 = mark(c(f(g(f(X))))) active(c(X)) = X + 2 >= X = mark(d(X)) active(h(X)) = X + 1 >= X + 1 = mark(c(d(X))) active(f(X)) = X + 2 >= X + 2 = f(active(X)) active(h(X)) = X + 1 >= X + 1 = h(active(X)) f(mark(X)) = X + 1 >= X + 1 = mark(f(X)) h(mark(X)) = X >= X = mark(h(X)) proper(f(X)) = X + 2 >= X + 2 = f(proper(X)) proper(c(X)) = X + 2 >= X + 2 = c(proper(X)) proper(g(X)) = X + 1 >= X + 1 = g(proper(X)) proper(d(X)) = X + 1 >= X + 1 = d(proper(X)) proper(h(X)) = X + 1 >= X + 1 = h(proper(X)) f(ok(X)) = X + 2 >= X + 2 = ok(f(X)) c(ok(X)) = X + 2 >= X + 2 = ok(c(X)) g(ok(X)) = X + 1 >= X + 1 = ok(g(X)) d(ok(X)) = X + 1 >= X + 1 = ok(d(X)) h(ok(X)) = X + 1 >= X + 1 = ok(h(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: f#(mark(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: dimension: 1 interpretation: [f#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = 0, [h](x0) = x0, [d](x0) = 0, [mark](x0) = x0 + 1, [c](x0) = x0, [g](x0) = 0, [active](x0) = x0 + 1, [f](x0) = x0 orientation: f#(mark(X)) = X + 2 >= X + 1 = f#(X) active(f(f(X))) = X + 1 >= 1 = mark(c(f(g(f(X))))) active(c(X)) = X + 1 >= 1 = mark(d(X)) active(h(X)) = X + 1 >= 1 = mark(c(d(X))) active(f(X)) = X + 1 >= X + 1 = f(active(X)) active(h(X)) = X + 1 >= X + 1 = h(active(X)) f(mark(X)) = X + 1 >= X + 1 = mark(f(X)) h(mark(X)) = X + 1 >= X + 1 = mark(h(X)) proper(f(X)) = 0 >= 0 = f(proper(X)) proper(c(X)) = 0 >= 0 = c(proper(X)) proper(g(X)) = 0 >= 0 = g(proper(X)) proper(d(X)) = 0 >= 0 = d(proper(X)) proper(h(X)) = 0 >= 0 = h(proper(X)) f(ok(X)) = 0 >= 0 = ok(f(X)) c(ok(X)) = 0 >= 0 = ok(c(X)) g(ok(X)) = 0 >= 0 = ok(g(X)) d(ok(X)) = 0 >= 0 = ok(d(X)) h(ok(X)) = 0 >= 0 = ok(h(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = 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 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: dimension: 1 interpretation: [h#](x0) = x0, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [h](x0) = x0, [d](x0) = x0, [mark](x0) = x0, [c](x0) = x0 + 1, [g](x0) = x0, [active](x0) = x0 + 1, [f](x0) = x0 + 1 orientation: h#(mark(X)) = X >= X = h#(X) h#(ok(X)) = X + 1 >= X = h#(X) active(f(f(X))) = X + 3 >= X + 3 = mark(c(f(g(f(X))))) active(c(X)) = X + 2 >= X = mark(d(X)) active(h(X)) = X + 1 >= X + 1 = mark(c(d(X))) active(f(X)) = X + 2 >= X + 2 = f(active(X)) active(h(X)) = X + 1 >= X + 1 = h(active(X)) f(mark(X)) = X + 1 >= X + 1 = mark(f(X)) h(mark(X)) = X >= X = mark(h(X)) proper(f(X)) = X + 2 >= X + 2 = f(proper(X)) proper(c(X)) = X + 2 >= X + 2 = c(proper(X)) proper(g(X)) = X + 1 >= X + 1 = g(proper(X)) proper(d(X)) = X + 1 >= X + 1 = d(proper(X)) proper(h(X)) = X + 1 >= X + 1 = h(proper(X)) f(ok(X)) = X + 2 >= X + 2 = ok(f(X)) c(ok(X)) = X + 2 >= X + 2 = ok(c(X)) g(ok(X)) = X + 1 >= X + 1 = ok(g(X)) d(ok(X)) = X + 1 >= X + 1 = ok(d(X)) h(ok(X)) = X + 1 >= X + 1 = ok(h(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: h#(mark(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: dimension: 1 interpretation: [h#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = 0, [h](x0) = x0, [d](x0) = 0, [mark](x0) = x0 + 1, [c](x0) = x0, [g](x0) = 0, [active](x0) = x0 + 1, [f](x0) = x0 orientation: h#(mark(X)) = X + 2 >= X + 1 = h#(X) active(f(f(X))) = X + 1 >= 1 = mark(c(f(g(f(X))))) active(c(X)) = X + 1 >= 1 = mark(d(X)) active(h(X)) = X + 1 >= 1 = mark(c(d(X))) active(f(X)) = X + 1 >= X + 1 = f(active(X)) active(h(X)) = X + 1 >= X + 1 = h(active(X)) f(mark(X)) = X + 1 >= X + 1 = mark(f(X)) h(mark(X)) = X + 1 >= X + 1 = mark(h(X)) proper(f(X)) = 0 >= 0 = f(proper(X)) proper(c(X)) = 0 >= 0 = c(proper(X)) proper(g(X)) = 0 >= 0 = g(proper(X)) proper(d(X)) = 0 >= 0 = d(proper(X)) proper(h(X)) = 0 >= 0 = h(proper(X)) f(ok(X)) = 0 >= 0 = ok(f(X)) c(ok(X)) = 0 >= 0 = ok(c(X)) g(ok(X)) = 0 >= 0 = ok(g(X)) d(ok(X)) = 0 >= 0 = ok(d(X)) h(ok(X)) = 0 >= 0 = ok(h(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = 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 DPs: proper#(h(X)) -> proper#(X) proper#(d(X)) -> proper#(X) proper#(g(X)) -> proper#(X) proper#(c(X)) -> proper#(X) proper#(f(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: dimension: 1 interpretation: [proper#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = x0 + 1, [h](x0) = x0 + 1, [d](x0) = x0 + 1, [mark](x0) = 0, [c](x0) = x0 + 1, [g](x0) = x0 + 1, [active](x0) = x0, [f](x0) = x0 orientation: proper#(h(X)) = X + 2 >= X + 1 = proper#(X) proper#(d(X)) = X + 2 >= X + 1 = proper#(X) proper#(g(X)) = X + 2 >= X + 1 = proper#(X) proper#(c(X)) = X + 2 >= X + 1 = proper#(X) proper#(f(X)) = X + 1 >= X + 1 = proper#(X) active(f(f(X))) = X >= 0 = mark(c(f(g(f(X))))) active(c(X)) = X + 1 >= 0 = mark(d(X)) active(h(X)) = X + 1 >= 0 = mark(c(d(X))) active(f(X)) = X >= X = f(active(X)) active(h(X)) = X + 1 >= X + 1 = h(active(X)) f(mark(X)) = 0 >= 0 = mark(f(X)) h(mark(X)) = 1 >= 0 = mark(h(X)) proper(f(X)) = X + 1 >= X + 1 = f(proper(X)) proper(c(X)) = X + 2 >= X + 2 = c(proper(X)) proper(g(X)) = X + 2 >= X + 2 = g(proper(X)) proper(d(X)) = X + 2 >= X + 2 = d(proper(X)) proper(h(X)) = X + 2 >= X + 2 = h(proper(X)) f(ok(X)) = 0 >= 0 = ok(f(X)) c(ok(X)) = 1 >= 0 = ok(c(X)) g(ok(X)) = 1 >= 0 = ok(g(X)) d(ok(X)) = 1 >= 0 = ok(d(X)) h(ok(X)) = 1 >= 0 = ok(h(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: proper#(f(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: dimension: 1 interpretation: [proper#](x0) = x0 + 1, [top](x0) = 1, [ok](x0) = x0, [proper](x0) = x0, [h](x0) = 0, [d](x0) = 1, [mark](x0) = x0, [c](x0) = 1, [g](x0) = 0, [active](x0) = x0 + 1, [f](x0) = x0 + 1 orientation: proper#(f(X)) = X + 2 >= X + 1 = proper#(X) active(f(f(X))) = X + 3 >= 1 = mark(c(f(g(f(X))))) active(c(X)) = 2 >= 1 = mark(d(X)) active(h(X)) = 1 >= 1 = mark(c(d(X))) active(f(X)) = X + 2 >= X + 2 = f(active(X)) active(h(X)) = 1 >= 0 = h(active(X)) f(mark(X)) = X + 1 >= X + 1 = mark(f(X)) h(mark(X)) = 0 >= 0 = mark(h(X)) proper(f(X)) = X + 1 >= X + 1 = f(proper(X)) proper(c(X)) = 1 >= 1 = c(proper(X)) proper(g(X)) = 0 >= 0 = g(proper(X)) proper(d(X)) = 1 >= 1 = d(proper(X)) proper(h(X)) = 0 >= 0 = h(proper(X)) f(ok(X)) = X + 1 >= X + 1 = ok(f(X)) c(ok(X)) = 1 >= 1 = ok(c(X)) g(ok(X)) = 0 >= 0 = ok(g(X)) d(ok(X)) = 1 >= 1 = ok(d(X)) h(ok(X)) = 0 >= 0 = ok(h(X)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = 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