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)) KBO Processor: argument filtering: pi(f) = 0 pi(active) = 0 pi(g) = 0 pi(c) = 0 pi(mark) = 0 pi(d) = 0 pi(h) = 0 pi(proper) = 0 pi(ok) = [0] pi(top) = [] pi(top#) = 0 weight function: w0 = 1 w(top) = w(ok) = w(d) = w(g) = 1 w(top#) = w(proper) = w(h) = w(mark) = w(c) = w(active) = w(f) = 0 precedence: top# ~ top ~ ok ~ proper ~ h ~ d ~ mark ~ c ~ g ~ active ~ f problem: DPs: 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)) EDG Processor: DPs: 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)) graph: top#(mark(X)) -> top#(proper(X)) -> top#(mark(X)) -> top#(proper(X)) CDG Processor: DPs: 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)) graph: 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)) Arctic Interpretation Processor: dimension: 1 interpretation: [active#](x0) = x0, [top](x0) = 0, [ok](x0) = 8, [proper](x0) = 8x0, [h](x0) = 11x0 + 10, [d](x0) = 8, [mark](x0) = x0 + 0, [c](x0) = 8, [g](x0) = 9, [active](x0) = 1x0 + 0, [f](x0) = x0 + 12 orientation: active#(f(X)) = X + 12 >= X = active#(X) active#(h(X)) = 11X + 10 >= X = active#(X) active(f(f(X))) = 1X + 13 >= 8 = mark(c(f(g(f(X))))) active(c(X)) = 9 >= 8 = mark(d(X)) active(h(X)) = 12X + 11 >= 8 = mark(c(d(X))) active(f(X)) = 1X + 13 >= 1X + 12 = f(active(X)) active(h(X)) = 12X + 11 >= 12X + 11 = h(active(X)) f(mark(X)) = X + 12 >= X + 12 = mark(f(X)) h(mark(X)) = 11X + 11 >= 11X + 10 = mark(h(X)) proper(f(X)) = 8X + 20 >= 8X + 12 = f(proper(X)) proper(c(X)) = 16 >= 8 = c(proper(X)) proper(g(X)) = 17 >= 9 = g(proper(X)) proper(d(X)) = 16 >= 8 = d(proper(X)) proper(h(X)) = 19X + 18 >= 19X + 10 = h(proper(X)) f(ok(X)) = 12 >= 8 = ok(f(X)) c(ok(X)) = 8 >= 8 = ok(c(X)) g(ok(X)) = 9 >= 8 = ok(g(X)) d(ok(X)) = 8 >= 8 = ok(d(X)) h(ok(X)) = 19 >= 8 = ok(h(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: 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)) EDG Processor: DPs: 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)) graph: active#(f(X)) -> active#(X) -> active#(f(X)) -> active#(X) CDG Processor: DPs: 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)) graph: 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)) KBO Processor: argument filtering: pi(f) = 0 pi(active) = [0] pi(g) = 0 pi(c) = [0] pi(mark) = [] pi(d) = 0 pi(h) = 0 pi(proper) = 0 pi(ok) = 0 pi(top) = [] pi(proper#) = 0 weight function: w0 = 1 w(proper#) = w(top) = w(ok) = w(d) = w(mark) = w(c) = w(active) = 1 w(proper) = w(h) = w(g) = w(f) = 0 precedence: d > proper# ~ top ~ ok ~ proper ~ h ~ mark ~ c ~ g ~ active ~ f problem: DPs: proper#(f(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)) EDG Processor: DPs: proper#(f(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)) graph: proper#(h(X)) -> proper#(X) -> proper#(f(X)) -> proper#(X) proper#(h(X)) -> proper#(X) -> proper#(g(X)) -> proper#(X) proper#(h(X)) -> proper#(X) -> proper#(d(X)) -> proper#(X) proper#(h(X)) -> proper#(X) -> proper#(h(X)) -> proper#(X) proper#(d(X)) -> proper#(X) -> proper#(f(X)) -> proper#(X) proper#(d(X)) -> proper#(X) -> proper#(g(X)) -> proper#(X) proper#(d(X)) -> proper#(X) -> proper#(d(X)) -> proper#(X) proper#(d(X)) -> proper#(X) -> proper#(h(X)) -> proper#(X) proper#(g(X)) -> proper#(X) -> proper#(f(X)) -> proper#(X) proper#(g(X)) -> proper#(X) -> proper#(g(X)) -> proper#(X) proper#(g(X)) -> proper#(X) -> proper#(d(X)) -> proper#(X) proper#(g(X)) -> proper#(X) -> proper#(h(X)) -> proper#(X) proper#(f(X)) -> proper#(X) -> proper#(f(X)) -> proper#(X) proper#(f(X)) -> proper#(X) -> proper#(g(X)) -> proper#(X) proper#(f(X)) -> proper#(X) -> proper#(d(X)) -> proper#(X) proper#(f(X)) -> proper#(X) -> proper#(h(X)) -> proper#(X) CDG Processor: DPs: proper#(f(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)) graph: 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)) Arctic Interpretation Processor: dimension: 1 interpretation: [h#](x0) = x0, [top](x0) = 0, [ok](x0) = 1x0, [proper](x0) = x0, [h](x0) = 12x0, [d](x0) = x0, [mark](x0) = x0, [c](x0) = 1x0, [g](x0) = x0, [active](x0) = 1x0, [f](x0) = x0 orientation: h#(mark(X)) = X >= X = h#(X) h#(ok(X)) = 1X >= X = h#(X) active(f(f(X))) = 1X >= 1X = mark(c(f(g(f(X))))) active(c(X)) = 2X >= X = mark(d(X)) active(h(X)) = 13X >= 1X = mark(c(d(X))) active(f(X)) = 1X >= 1X = f(active(X)) active(h(X)) = 13X >= 13X = h(active(X)) f(mark(X)) = X >= X = mark(f(X)) h(mark(X)) = 12X >= 12X = mark(h(X)) proper(f(X)) = X >= X = f(proper(X)) proper(c(X)) = 1X >= 1X = c(proper(X)) proper(g(X)) = X >= X = g(proper(X)) proper(d(X)) = X >= X = d(proper(X)) proper(h(X)) = 12X >= 12X = h(proper(X)) f(ok(X)) = 1X >= 1X = ok(f(X)) c(ok(X)) = 2X >= 2X = ok(c(X)) g(ok(X)) = 1X >= 1X = ok(g(X)) d(ok(X)) = 1X >= 1X = ok(d(X)) h(ok(X)) = 13X >= 13X = 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)) EDG Processor: 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)) graph: h#(mark(X)) -> h#(X) -> h#(mark(X)) -> h#(X) CDG Processor: 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)) graph: 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)) KBO Processor: argument filtering: pi(f) = 0 pi(active) = [0] pi(g) = 0 pi(c) = [] pi(mark) = [0] pi(d) = [] pi(h) = 0 pi(proper) = 0 pi(ok) = 0 pi(top) = [] pi(f#) = 0 weight function: w0 = 1 w(f#) = w(top) = w(ok) = w(d) = w(c) = w(active) = 1 w(proper) = w(h) = w(mark) = w(g) = w(f) = 0 precedence: f# ~ top ~ ok ~ proper ~ h ~ d ~ mark ~ c ~ g ~ active ~ f problem: DPs: 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)) EDG Processor: DPs: 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)) graph: f#(ok(X)) -> f#(X) -> f#(ok(X)) -> f#(X) CDG Processor: DPs: 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)) graph: Qed