MAYBE Problem: active(from(X)) -> mark(cons(X,from(s(X)))) active(after(0(),XS)) -> mark(XS) active(after(s(N),cons(X,XS))) -> mark(after(N,XS)) active(from(X)) -> from(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(after(X1,X2)) -> after(active(X1),X2) active(after(X1,X2)) -> after(X1,active(X2)) from(mark(X)) -> mark(from(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) after(mark(X1),X2) -> mark(after(X1,X2)) after(X1,mark(X2)) -> mark(after(X1,X2)) proper(from(X)) -> from(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(after(X1,X2)) -> after(proper(X1),proper(X2)) proper(0()) -> ok(0()) from(ok(X)) -> ok(from(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) after(ok(X1),ok(X2)) -> ok(after(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Proof: DP Processor: DPs: active#(from(X)) -> s#(X) active#(from(X)) -> from#(s(X)) active#(from(X)) -> cons#(X,from(s(X))) active#(after(s(N),cons(X,XS))) -> after#(N,XS) active#(from(X)) -> active#(X) active#(from(X)) -> from#(active(X)) active#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(s(X)) -> active#(X) active#(s(X)) -> s#(active(X)) active#(after(X1,X2)) -> active#(X1) active#(after(X1,X2)) -> after#(active(X1),X2) active#(after(X1,X2)) -> active#(X2) active#(after(X1,X2)) -> after#(X1,active(X2)) from#(mark(X)) -> from#(X) cons#(mark(X1),X2) -> cons#(X1,X2) s#(mark(X)) -> s#(X) after#(mark(X1),X2) -> after#(X1,X2) after#(X1,mark(X2)) -> after#(X1,X2) proper#(from(X)) -> proper#(X) proper#(from(X)) -> from#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(after(X1,X2)) -> proper#(X2) proper#(after(X1,X2)) -> proper#(X1) proper#(after(X1,X2)) -> after#(proper(X1),proper(X2)) from#(ok(X)) -> from#(X) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) s#(ok(X)) -> s#(X) after#(ok(X1),ok(X2)) -> after#(X1,X2) top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) TRS: active(from(X)) -> mark(cons(X,from(s(X)))) active(after(0(),XS)) -> mark(XS) active(after(s(N),cons(X,XS))) -> mark(after(N,XS)) active(from(X)) -> from(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(after(X1,X2)) -> after(active(X1),X2) active(after(X1,X2)) -> after(X1,active(X2)) from(mark(X)) -> mark(from(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) after(mark(X1),X2) -> mark(after(X1,X2)) after(X1,mark(X2)) -> mark(after(X1,X2)) proper(from(X)) -> from(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(after(X1,X2)) -> after(proper(X1),proper(X2)) proper(0()) -> ok(0()) from(ok(X)) -> ok(from(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) after(ok(X1),ok(X2)) -> ok(after(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) TDG Processor: DPs: active#(from(X)) -> s#(X) active#(from(X)) -> from#(s(X)) active#(from(X)) -> cons#(X,from(s(X))) active#(after(s(N),cons(X,XS))) -> after#(N,XS) active#(from(X)) -> active#(X) active#(from(X)) -> from#(active(X)) active#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(s(X)) -> active#(X) active#(s(X)) -> s#(active(X)) active#(after(X1,X2)) -> active#(X1) active#(after(X1,X2)) -> after#(active(X1),X2) active#(after(X1,X2)) -> active#(X2) active#(after(X1,X2)) -> after#(X1,active(X2)) from#(mark(X)) -> from#(X) cons#(mark(X1),X2) -> cons#(X1,X2) s#(mark(X)) -> s#(X) after#(mark(X1),X2) -> after#(X1,X2) after#(X1,mark(X2)) -> after#(X1,X2) proper#(from(X)) -> proper#(X) proper#(from(X)) -> from#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(after(X1,X2)) -> proper#(X2) proper#(after(X1,X2)) -> proper#(X1) proper#(after(X1,X2)) -> after#(proper(X1),proper(X2)) from#(ok(X)) -> from#(X) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) s#(ok(X)) -> s#(X) after#(ok(X1),ok(X2)) -> after#(X1,X2) top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) TRS: active(from(X)) -> mark(cons(X,from(s(X)))) active(after(0(),XS)) -> mark(XS) active(after(s(N),cons(X,XS))) -> mark(after(N,XS)) active(from(X)) -> from(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(after(X1,X2)) -> after(active(X1),X2) active(after(X1,X2)) -> after(X1,active(X2)) from(mark(X)) -> mark(from(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) after(mark(X1),X2) -> mark(after(X1,X2)) after(X1,mark(X2)) -> mark(after(X1,X2)) proper(from(X)) -> from(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(after(X1,X2)) -> after(proper(X1),proper(X2)) proper(0()) -> ok(0()) from(ok(X)) -> ok(from(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) after(ok(X1),ok(X2)) -> ok(after(X1,X2)) 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#(after(X1,X2)) -> after#(X1,active(X2)) top#(ok(X)) -> active#(X) -> active#(after(X1,X2)) -> active#(X2) top#(ok(X)) -> active#(X) -> active#(after(X1,X2)) -> after#(active(X1),X2) top#(ok(X)) -> active#(X) -> active#(after(X1,X2)) -> active#(X1) top#(ok(X)) -> active#(X) -> active#(s(X)) -> s#(active(X)) top#(ok(X)) -> active#(X) -> active#(s(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) top#(ok(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) top#(ok(X)) -> active#(X) -> active#(from(X)) -> from#(active(X)) top#(ok(X)) -> active#(X) -> active#(from(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(after(s(N),cons(X,XS))) -> after#(N,XS) top#(ok(X)) -> active#(X) -> active#(from(X)) -> cons#(X,from(s(X))) top#(ok(X)) -> active#(X) -> active#(from(X)) -> from#(s(X)) top#(ok(X)) -> active#(X) -> active#(from(X)) -> s#(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#(after(X1,X2)) -> after#(proper(X1),proper(X2)) top#(mark(X)) -> proper#(X) -> proper#(after(X1,X2)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(after(X1,X2)) -> proper#(X2) top#(mark(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) top#(mark(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) top#(mark(X)) -> proper#(X) -> proper#(from(X)) -> from#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(from(X)) -> proper#(X) proper#(after(X1,X2)) -> proper#(X2) -> proper#(after(X1,X2)) -> after#(proper(X1),proper(X2)) proper#(after(X1,X2)) -> proper#(X2) -> proper#(after(X1,X2)) -> proper#(X1) proper#(after(X1,X2)) -> proper#(X2) -> proper#(after(X1,X2)) -> proper#(X2) proper#(after(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> s#(proper(X)) proper#(after(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> proper#(X) proper#(after(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(after(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(after(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(after(X1,X2)) -> proper#(X2) -> proper#(from(X)) -> from#(proper(X)) proper#(after(X1,X2)) -> proper#(X2) -> proper#(from(X)) -> proper#(X) proper#(after(X1,X2)) -> proper#(X1) -> proper#(after(X1,X2)) -> after#(proper(X1),proper(X2)) proper#(after(X1,X2)) -> proper#(X1) -> proper#(after(X1,X2)) -> proper#(X1) proper#(after(X1,X2)) -> proper#(X1) -> proper#(after(X1,X2)) -> proper#(X2) proper#(after(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> s#(proper(X)) proper#(after(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> proper#(X) proper#(after(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(after(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(after(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(after(X1,X2)) -> proper#(X1) -> proper#(from(X)) -> from#(proper(X)) proper#(after(X1,X2)) -> proper#(X1) -> proper#(from(X)) -> proper#(X) proper#(after(X1,X2)) -> after#(proper(X1),proper(X2)) -> after#(ok(X1),ok(X2)) -> after#(X1,X2) proper#(after(X1,X2)) -> after#(proper(X1),proper(X2)) -> after#(X1,mark(X2)) -> after#(X1,X2) proper#(after(X1,X2)) -> after#(proper(X1),proper(X2)) -> after#(mark(X1),X2) -> after#(X1,X2) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(after(X1,X2)) -> after#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(after(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(after(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> s#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(from(X)) -> from#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(from(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(after(X1,X2)) -> after#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(after(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(after(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> s#(proper(X)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(from(X)) -> from#(proper(X)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(from(X)) -> proper#(X) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) -> cons#(mark(X1),X2) -> cons#(X1,X2) proper#(s(X)) -> proper#(X) -> proper#(after(X1,X2)) -> after#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) -> proper#(after(X1,X2)) -> proper#(X1) proper#(s(X)) -> proper#(X) -> proper#(after(X1,X2)) -> proper#(X2) proper#(s(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) proper#(s(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(s(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(s(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(s(X)) -> proper#(X) -> proper#(from(X)) -> from#(proper(X)) proper#(s(X)) -> proper#(X) -> proper#(from(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) -> s#(ok(X)) -> s#(X) proper#(s(X)) -> s#(proper(X)) -> s#(mark(X)) -> s#(X) proper#(from(X)) -> proper#(X) -> proper#(after(X1,X2)) -> after#(proper(X1),proper(X2)) proper#(from(X)) -> proper#(X) -> proper#(after(X1,X2)) -> proper#(X1) proper#(from(X)) -> proper#(X) -> proper#(after(X1,X2)) -> proper#(X2) proper#(from(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) proper#(from(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(from(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(from(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(from(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(from(X)) -> proper#(X) -> proper#(from(X)) -> from#(proper(X)) proper#(from(X)) -> proper#(X) -> proper#(from(X)) -> proper#(X) proper#(from(X)) -> from#(proper(X)) -> from#(ok(X)) -> from#(X) proper#(from(X)) -> from#(proper(X)) -> from#(mark(X)) -> from#(X) after#(ok(X1),ok(X2)) -> after#(X1,X2) -> after#(ok(X1),ok(X2)) -> after#(X1,X2) after#(ok(X1),ok(X2)) -> after#(X1,X2) -> after#(X1,mark(X2)) -> after#(X1,X2) after#(ok(X1),ok(X2)) -> after#(X1,X2) -> after#(mark(X1),X2) -> after#(X1,X2) after#(mark(X1),X2) -> after#(X1,X2) -> after#(ok(X1),ok(X2)) -> after#(X1,X2) after#(mark(X1),X2) -> after#(X1,X2) -> after#(X1,mark(X2)) -> after#(X1,X2) after#(mark(X1),X2) -> after#(X1,X2) -> after#(mark(X1),X2) -> after#(X1,X2) after#(X1,mark(X2)) -> after#(X1,X2) -> after#(ok(X1),ok(X2)) -> after#(X1,X2) after#(X1,mark(X2)) -> after#(X1,X2) -> after#(X1,mark(X2)) -> after#(X1,X2) after#(X1,mark(X2)) -> after#(X1,X2) -> after#(mark(X1),X2) -> after#(X1,X2) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) -> cons#(mark(X1),X2) -> cons#(X1,X2) cons#(mark(X1),X2) -> cons#(X1,X2) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) cons#(mark(X1),X2) -> cons#(X1,X2) -> cons#(mark(X1),X2) -> cons#(X1,X2) from#(ok(X)) -> from#(X) -> from#(ok(X)) -> from#(X) from#(ok(X)) -> from#(X) -> from#(mark(X)) -> from#(X) from#(mark(X)) -> from#(X) -> from#(ok(X)) -> from#(X) from#(mark(X)) -> from#(X) -> from#(mark(X)) -> from#(X) s#(ok(X)) -> s#(X) -> s#(ok(X)) -> s#(X) s#(ok(X)) -> s#(X) -> s#(mark(X)) -> s#(X) s#(mark(X)) -> s#(X) -> s#(ok(X)) -> s#(X) s#(mark(X)) -> s#(X) -> s#(mark(X)) -> s#(X) active#(after(s(N),cons(X,XS))) -> after#(N,XS) -> after#(ok(X1),ok(X2)) -> after#(X1,X2) active#(after(s(N),cons(X,XS))) -> after#(N,XS) -> after#(X1,mark(X2)) -> after#(X1,X2) active#(after(s(N),cons(X,XS))) -> after#(N,XS) -> after#(mark(X1),X2) -> after#(X1,X2) active#(after(X1,X2)) -> after#(active(X1),X2) -> after#(ok(X1),ok(X2)) -> after#(X1,X2) active#(after(X1,X2)) -> after#(active(X1),X2) -> after#(X1,mark(X2)) -> after#(X1,X2) active#(after(X1,X2)) -> after#(active(X1),X2) -> after#(mark(X1),X2) -> after#(X1,X2) active#(after(X1,X2)) -> after#(X1,active(X2)) -> after#(ok(X1),ok(X2)) -> after#(X1,X2) active#(after(X1,X2)) -> after#(X1,active(X2)) -> after#(X1,mark(X2)) -> after#(X1,X2) active#(after(X1,X2)) -> after#(X1,active(X2)) -> after#(mark(X1),X2) -> after#(X1,X2) active#(after(X1,X2)) -> active#(X2) -> active#(after(X1,X2)) -> after#(X1,active(X2)) active#(after(X1,X2)) -> active#(X2) -> active#(after(X1,X2)) -> active#(X2) active#(after(X1,X2)) -> active#(X2) -> active#(after(X1,X2)) -> after#(active(X1),X2) active#(after(X1,X2)) -> active#(X2) -> active#(after(X1,X2)) -> active#(X1) active#(after(X1,X2)) -> active#(X2) -> active#(s(X)) -> s#(active(X)) active#(after(X1,X2)) -> active#(X2) -> active#(s(X)) -> active#(X) active#(after(X1,X2)) -> active#(X2) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(after(X1,X2)) -> active#(X2) -> active#(cons(X1,X2)) -> active#(X1) active#(after(X1,X2)) -> active#(X2) -> active#(from(X)) -> from#(active(X)) active#(after(X1,X2)) -> active#(X2) -> active#(from(X)) -> active#(X) active#(after(X1,X2)) -> active#(X2) -> active#(after(s(N),cons(X,XS))) -> after#(N,XS) active#(after(X1,X2)) -> active#(X2) -> active#(from(X)) -> cons#(X,from(s(X))) active#(after(X1,X2)) -> active#(X2) -> active#(from(X)) -> from#(s(X)) active#(after(X1,X2)) -> active#(X2) -> active#(from(X)) -> s#(X) active#(after(X1,X2)) -> active#(X1) -> active#(after(X1,X2)) -> after#(X1,active(X2)) active#(after(X1,X2)) -> active#(X1) -> active#(after(X1,X2)) -> active#(X2) active#(after(X1,X2)) -> active#(X1) -> active#(after(X1,X2)) -> after#(active(X1),X2) active#(after(X1,X2)) -> active#(X1) -> active#(after(X1,X2)) -> active#(X1) active#(after(X1,X2)) -> active#(X1) -> active#(s(X)) -> s#(active(X)) active#(after(X1,X2)) -> active#(X1) -> active#(s(X)) -> active#(X) active#(after(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(after(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> active#(X1) active#(after(X1,X2)) -> active#(X1) -> active#(from(X)) -> from#(active(X)) active#(after(X1,X2)) -> active#(X1) -> active#(from(X)) -> active#(X) active#(after(X1,X2)) -> active#(X1) -> active#(after(s(N),cons(X,XS))) -> after#(N,XS) active#(after(X1,X2)) -> active#(X1) -> active#(from(X)) -> cons#(X,from(s(X))) active#(after(X1,X2)) -> active#(X1) -> active#(from(X)) -> from#(s(X)) active#(after(X1,X2)) -> active#(X1) -> active#(from(X)) -> s#(X) active#(cons(X1,X2)) -> cons#(active(X1),X2) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) active#(cons(X1,X2)) -> cons#(active(X1),X2) -> cons#(mark(X1),X2) -> cons#(X1,X2) active#(cons(X1,X2)) -> active#(X1) -> active#(after(X1,X2)) -> after#(X1,active(X2)) active#(cons(X1,X2)) -> active#(X1) -> active#(after(X1,X2)) -> active#(X2) active#(cons(X1,X2)) -> active#(X1) -> active#(after(X1,X2)) -> after#(active(X1),X2) active#(cons(X1,X2)) -> active#(X1) -> active#(after(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> active#(X1) -> active#(s(X)) -> s#(active(X)) active#(cons(X1,X2)) -> active#(X1) -> active#(s(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(cons(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> active#(X1) -> active#(from(X)) -> from#(active(X)) active#(cons(X1,X2)) -> active#(X1) -> active#(from(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) -> active#(after(s(N),cons(X,XS))) -> after#(N,XS) active#(cons(X1,X2)) -> active#(X1) -> active#(from(X)) -> cons#(X,from(s(X))) active#(cons(X1,X2)) -> active#(X1) -> active#(from(X)) -> from#(s(X)) active#(cons(X1,X2)) -> active#(X1) -> active#(from(X)) -> s#(X) active#(s(X)) -> s#(active(X)) -> s#(ok(X)) -> s#(X) active#(s(X)) -> s#(active(X)) -> s#(mark(X)) -> s#(X) active#(s(X)) -> active#(X) -> active#(after(X1,X2)) -> after#(X1,active(X2)) active#(s(X)) -> active#(X) -> active#(after(X1,X2)) -> active#(X2) active#(s(X)) -> active#(X) -> active#(after(X1,X2)) -> after#(active(X1),X2) active#(s(X)) -> active#(X) -> active#(after(X1,X2)) -> active#(X1) active#(s(X)) -> active#(X) -> active#(s(X)) -> s#(active(X)) active#(s(X)) -> active#(X) -> active#(s(X)) -> active#(X) active#(s(X)) -> active#(X) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(s(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) active#(s(X)) -> active#(X) -> active#(from(X)) -> from#(active(X)) active#(s(X)) -> active#(X) -> active#(from(X)) -> active#(X) active#(s(X)) -> active#(X) -> active#(after(s(N),cons(X,XS))) -> after#(N,XS) active#(s(X)) -> active#(X) -> active#(from(X)) -> cons#(X,from(s(X))) active#(s(X)) -> active#(X) -> active#(from(X)) -> from#(s(X)) active#(s(X)) -> active#(X) -> active#(from(X)) -> s#(X) active#(from(X)) -> cons#(X,from(s(X))) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) active#(from(X)) -> cons#(X,from(s(X))) -> cons#(mark(X1),X2) -> cons#(X1,X2) active#(from(X)) -> from#(s(X)) -> from#(ok(X)) -> from#(X) active#(from(X)) -> from#(s(X)) -> from#(mark(X)) -> from#(X) active#(from(X)) -> from#(active(X)) -> from#(ok(X)) -> from#(X) active#(from(X)) -> from#(active(X)) -> from#(mark(X)) -> from#(X) active#(from(X)) -> s#(X) -> s#(ok(X)) -> s#(X) active#(from(X)) -> s#(X) -> s#(mark(X)) -> s#(X) active#(from(X)) -> active#(X) -> active#(after(X1,X2)) -> after#(X1,active(X2)) active#(from(X)) -> active#(X) -> active#(after(X1,X2)) -> active#(X2) active#(from(X)) -> active#(X) -> active#(after(X1,X2)) -> after#(active(X1),X2) active#(from(X)) -> active#(X) -> active#(after(X1,X2)) -> active#(X1) active#(from(X)) -> active#(X) -> active#(s(X)) -> s#(active(X)) active#(from(X)) -> active#(X) -> active#(s(X)) -> active#(X) active#(from(X)) -> active#(X) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(from(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) active#(from(X)) -> active#(X) -> active#(from(X)) -> from#(active(X)) active#(from(X)) -> active#(X) -> active#(from(X)) -> active#(X) active#(from(X)) -> active#(X) -> active#(after(s(N),cons(X,XS))) -> after#(N,XS) active#(from(X)) -> active#(X) -> active#(from(X)) -> cons#(X,from(s(X))) active#(from(X)) -> active#(X) -> active#(from(X)) -> from#(s(X)) active#(from(X)) -> active#(X) -> active#(from(X)) -> s#(X) SCC Processor: #sccs: 7 #rules: 22 #arcs: 213/1369 DPs: top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> top#(proper(X)) TRS: active(from(X)) -> mark(cons(X,from(s(X)))) active(after(0(),XS)) -> mark(XS) active(after(s(N),cons(X,XS))) -> mark(after(N,XS)) active(from(X)) -> from(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(after(X1,X2)) -> after(active(X1),X2) active(after(X1,X2)) -> after(X1,active(X2)) from(mark(X)) -> mark(from(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) after(mark(X1),X2) -> mark(after(X1,X2)) after(X1,mark(X2)) -> mark(after(X1,X2)) proper(from(X)) -> from(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(after(X1,X2)) -> after(proper(X1),proper(X2)) proper(0()) -> ok(0()) from(ok(X)) -> ok(from(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) after(ok(X1),ok(X2)) -> ok(after(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: active#(from(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) active#(s(X)) -> active#(X) active#(after(X1,X2)) -> active#(X1) active#(after(X1,X2)) -> active#(X2) TRS: active(from(X)) -> mark(cons(X,from(s(X)))) active(after(0(),XS)) -> mark(XS) active(after(s(N),cons(X,XS))) -> mark(after(N,XS)) active(from(X)) -> from(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(after(X1,X2)) -> after(active(X1),X2) active(after(X1,X2)) -> after(X1,active(X2)) from(mark(X)) -> mark(from(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) after(mark(X1),X2) -> mark(after(X1,X2)) after(X1,mark(X2)) -> mark(after(X1,X2)) proper(from(X)) -> from(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(after(X1,X2)) -> after(proper(X1),proper(X2)) proper(0()) -> ok(0()) from(ok(X)) -> ok(from(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) after(ok(X1),ok(X2)) -> ok(after(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [active#](x0) = x0 + 1, [top](x0) = 1, [ok](x0) = x0, [proper](x0) = x0, [after](x0, x1) = x0 + x1 + 1, [0] = 0, [mark](x0) = 1, [cons](x0, x1) = x0, [s](x0) = x0 + 1, [active](x0) = x0, [from](x0) = x0 + 1 orientation: active#(from(X)) = X + 2 >= X + 1 = active#(X) active#(cons(X1,X2)) = X1 + 1 >= X1 + 1 = active#(X1) active#(s(X)) = X + 2 >= X + 1 = active#(X) active#(after(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = active#(X1) active#(after(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = active#(X2) active(from(X)) = X + 1 >= 1 = mark(cons(X,from(s(X)))) active(after(0(),XS)) = XS + 1 >= 1 = mark(XS) active(after(s(N),cons(X,XS))) = N + X + 2 >= 1 = mark(after(N,XS)) active(from(X)) = X + 1 >= X + 1 = from(active(X)) active(cons(X1,X2)) = X1 >= X1 = cons(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(after(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = after(active(X1),X2) active(after(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = after(X1,active(X2)) from(mark(X)) = 2 >= 1 = mark(from(X)) cons(mark(X1),X2) = 1 >= 1 = mark(cons(X1,X2)) s(mark(X)) = 2 >= 1 = mark(s(X)) after(mark(X1),X2) = X2 + 2 >= 1 = mark(after(X1,X2)) after(X1,mark(X2)) = X1 + 2 >= 1 = mark(after(X1,X2)) proper(from(X)) = X + 1 >= X + 1 = from(proper(X)) proper(cons(X1,X2)) = X1 >= X1 = cons(proper(X1),proper(X2)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(after(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = after(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) from(ok(X)) = X + 1 >= X + 1 = ok(from(X)) cons(ok(X1),ok(X2)) = X1 >= X1 = ok(cons(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) after(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(after(X1,X2)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: active#(cons(X1,X2)) -> active#(X1) TRS: active(from(X)) -> mark(cons(X,from(s(X)))) active(after(0(),XS)) -> mark(XS) active(after(s(N),cons(X,XS))) -> mark(after(N,XS)) active(from(X)) -> from(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(after(X1,X2)) -> after(active(X1),X2) active(after(X1,X2)) -> after(X1,active(X2)) from(mark(X)) -> mark(from(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) after(mark(X1),X2) -> mark(after(X1,X2)) after(X1,mark(X2)) -> mark(after(X1,X2)) proper(from(X)) -> from(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(after(X1,X2)) -> after(proper(X1),proper(X2)) proper(0()) -> ok(0()) from(ok(X)) -> ok(from(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) after(ok(X1),ok(X2)) -> ok(after(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [active#](x0) = x0 + 1, [top](x0) = 1, [ok](x0) = x0, [proper](x0) = x0, [after](x0, x1) = 0, [0] = 0, [mark](x0) = 0, [cons](x0, x1) = x0 + 1, [s](x0) = 0, [active](x0) = x0, [from](x0) = 0 orientation: active#(cons(X1,X2)) = X1 + 2 >= X1 + 1 = active#(X1) active(from(X)) = 0 >= 0 = mark(cons(X,from(s(X)))) active(after(0(),XS)) = 0 >= 0 = mark(XS) active(after(s(N),cons(X,XS))) = 0 >= 0 = mark(after(N,XS)) active(from(X)) = 0 >= 0 = from(active(X)) active(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(active(X1),X2) active(s(X)) = 0 >= 0 = s(active(X)) active(after(X1,X2)) = 0 >= 0 = after(active(X1),X2) active(after(X1,X2)) = 0 >= 0 = after(X1,active(X2)) from(mark(X)) = 0 >= 0 = mark(from(X)) cons(mark(X1),X2) = 1 >= 0 = mark(cons(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) after(mark(X1),X2) = 0 >= 0 = mark(after(X1,X2)) after(X1,mark(X2)) = 0 >= 0 = mark(after(X1,X2)) proper(from(X)) = 0 >= 0 = from(proper(X)) proper(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(proper(X1),proper(X2)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(after(X1,X2)) = 0 >= 0 = after(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) from(ok(X)) = 0 >= 0 = ok(from(X)) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) s(ok(X)) = 0 >= 0 = ok(s(X)) after(ok(X1),ok(X2)) = 0 >= 0 = ok(after(X1,X2)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: TRS: active(from(X)) -> mark(cons(X,from(s(X)))) active(after(0(),XS)) -> mark(XS) active(after(s(N),cons(X,XS))) -> mark(after(N,XS)) active(from(X)) -> from(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(after(X1,X2)) -> after(active(X1),X2) active(after(X1,X2)) -> after(X1,active(X2)) from(mark(X)) -> mark(from(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) after(mark(X1),X2) -> mark(after(X1,X2)) after(X1,mark(X2)) -> mark(after(X1,X2)) proper(from(X)) -> from(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(after(X1,X2)) -> after(proper(X1),proper(X2)) proper(0()) -> ok(0()) from(ok(X)) -> ok(from(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) after(ok(X1),ok(X2)) -> ok(after(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: proper#(from(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(s(X)) -> proper#(X) proper#(after(X1,X2)) -> proper#(X2) proper#(after(X1,X2)) -> proper#(X1) TRS: active(from(X)) -> mark(cons(X,from(s(X)))) active(after(0(),XS)) -> mark(XS) active(after(s(N),cons(X,XS))) -> mark(after(N,XS)) active(from(X)) -> from(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(after(X1,X2)) -> after(active(X1),X2) active(after(X1,X2)) -> after(X1,active(X2)) from(mark(X)) -> mark(from(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) after(mark(X1),X2) -> mark(after(X1,X2)) after(X1,mark(X2)) -> mark(after(X1,X2)) proper(from(X)) -> from(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(after(X1,X2)) -> after(proper(X1),proper(X2)) proper(0()) -> ok(0()) from(ok(X)) -> ok(from(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) after(ok(X1),ok(X2)) -> ok(after(X1,X2)) 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) = x0, [proper](x0) = x0, [after](x0, x1) = x0 + x1, [0] = 1, [mark](x0) = 1, [cons](x0, x1) = x0 + x1 + 1, [s](x0) = x0 + 1, [active](x0) = x0, [from](x0) = x0 + 1 orientation: proper#(from(X)) = X + 2 >= X + 1 = proper#(X) proper#(cons(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) proper#(cons(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(s(X)) = X + 2 >= X + 1 = proper#(X) proper#(after(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = proper#(X2) proper#(after(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = proper#(X1) active(from(X)) = X + 1 >= 1 = mark(cons(X,from(s(X)))) active(after(0(),XS)) = XS + 1 >= 1 = mark(XS) active(after(s(N),cons(X,XS))) = N + X + XS + 2 >= 1 = mark(after(N,XS)) active(from(X)) = X + 1 >= X + 1 = from(active(X)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(after(X1,X2)) = X1 + X2 >= X1 + X2 = after(active(X1),X2) active(after(X1,X2)) = X1 + X2 >= X1 + X2 = after(X1,active(X2)) from(mark(X)) = 2 >= 1 = mark(from(X)) cons(mark(X1),X2) = X2 + 2 >= 1 = mark(cons(X1,X2)) s(mark(X)) = 2 >= 1 = mark(s(X)) after(mark(X1),X2) = X2 + 1 >= 1 = mark(after(X1,X2)) after(X1,mark(X2)) = X1 + 1 >= 1 = mark(after(X1,X2)) proper(from(X)) = X + 1 >= X + 1 = from(proper(X)) proper(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(proper(X1),proper(X2)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(after(X1,X2)) = X1 + X2 >= X1 + X2 = after(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) from(ok(X)) = X + 1 >= X + 1 = ok(from(X)) cons(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(cons(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) after(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(after(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: proper#(after(X1,X2)) -> proper#(X2) proper#(after(X1,X2)) -> proper#(X1) TRS: active(from(X)) -> mark(cons(X,from(s(X)))) active(after(0(),XS)) -> mark(XS) active(after(s(N),cons(X,XS))) -> mark(after(N,XS)) active(from(X)) -> from(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(after(X1,X2)) -> after(active(X1),X2) active(after(X1,X2)) -> after(X1,active(X2)) from(mark(X)) -> mark(from(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) after(mark(X1),X2) -> mark(after(X1,X2)) after(X1,mark(X2)) -> mark(after(X1,X2)) proper(from(X)) -> from(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(after(X1,X2)) -> after(proper(X1),proper(X2)) proper(0()) -> ok(0()) from(ok(X)) -> ok(from(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) after(ok(X1),ok(X2)) -> ok(after(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [proper#](x0) = x0, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = x0, [after](x0, x1) = x0 + x1 + 1, [0] = 1, [mark](x0) = 0, [cons](x0, x1) = 1, [s](x0) = 1, [active](x0) = x0, [from](x0) = 1 orientation: proper#(after(X1,X2)) = X1 + X2 + 1 >= X2 = proper#(X2) proper#(after(X1,X2)) = X1 + X2 + 1 >= X1 = proper#(X1) active(from(X)) = 1 >= 0 = mark(cons(X,from(s(X)))) active(after(0(),XS)) = XS + 2 >= 0 = mark(XS) active(after(s(N),cons(X,XS))) = 3 >= 0 = mark(after(N,XS)) active(from(X)) = 1 >= 1 = from(active(X)) active(cons(X1,X2)) = 1 >= 1 = cons(active(X1),X2) active(s(X)) = 1 >= 1 = s(active(X)) active(after(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = after(active(X1),X2) active(after(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = after(X1,active(X2)) from(mark(X)) = 1 >= 0 = mark(from(X)) cons(mark(X1),X2) = 1 >= 0 = mark(cons(X1,X2)) s(mark(X)) = 1 >= 0 = mark(s(X)) after(mark(X1),X2) = X2 + 1 >= 0 = mark(after(X1,X2)) after(X1,mark(X2)) = X1 + 1 >= 0 = mark(after(X1,X2)) proper(from(X)) = 1 >= 1 = from(proper(X)) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(after(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = after(proper(X1),proper(X2)) proper(0()) = 1 >= 0 = ok(0()) from(ok(X)) = 1 >= 0 = ok(from(X)) cons(ok(X1),ok(X2)) = 1 >= 0 = ok(cons(X1,X2)) s(ok(X)) = 1 >= 0 = ok(s(X)) after(ok(X1),ok(X2)) = 1 >= 0 = ok(after(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(from(X)) -> mark(cons(X,from(s(X)))) active(after(0(),XS)) -> mark(XS) active(after(s(N),cons(X,XS))) -> mark(after(N,XS)) active(from(X)) -> from(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(after(X1,X2)) -> after(active(X1),X2) active(after(X1,X2)) -> after(X1,active(X2)) from(mark(X)) -> mark(from(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) after(mark(X1),X2) -> mark(after(X1,X2)) after(X1,mark(X2)) -> mark(after(X1,X2)) proper(from(X)) -> from(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(after(X1,X2)) -> after(proper(X1),proper(X2)) proper(0()) -> ok(0()) from(ok(X)) -> ok(from(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) after(ok(X1),ok(X2)) -> ok(after(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: after#(mark(X1),X2) -> after#(X1,X2) after#(X1,mark(X2)) -> after#(X1,X2) after#(ok(X1),ok(X2)) -> after#(X1,X2) TRS: active(from(X)) -> mark(cons(X,from(s(X)))) active(after(0(),XS)) -> mark(XS) active(after(s(N),cons(X,XS))) -> mark(after(N,XS)) active(from(X)) -> from(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(after(X1,X2)) -> after(active(X1),X2) active(after(X1,X2)) -> after(X1,active(X2)) from(mark(X)) -> mark(from(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) after(mark(X1),X2) -> mark(after(X1,X2)) after(X1,mark(X2)) -> mark(after(X1,X2)) proper(from(X)) -> from(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(after(X1,X2)) -> after(proper(X1),proper(X2)) proper(0()) -> ok(0()) from(ok(X)) -> ok(from(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) after(ok(X1),ok(X2)) -> ok(after(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [after#](x0, x1) = x1 + 1, [top](x0) = 1, [ok](x0) = x0 + 1, [proper](x0) = 1, [after](x0, x1) = x1, [0] = 0, [mark](x0) = x0, [cons](x0, x1) = x1, [s](x0) = x0, [active](x0) = x0 + 1, [from](x0) = x0 orientation: after#(mark(X1),X2) = X2 + 1 >= X2 + 1 = after#(X1,X2) after#(X1,mark(X2)) = X2 + 1 >= X2 + 1 = after#(X1,X2) after#(ok(X1),ok(X2)) = X2 + 2 >= X2 + 1 = after#(X1,X2) active(from(X)) = X + 1 >= X = mark(cons(X,from(s(X)))) active(after(0(),XS)) = XS + 1 >= XS = mark(XS) active(after(s(N),cons(X,XS))) = XS + 1 >= XS = mark(after(N,XS)) active(from(X)) = X + 1 >= X + 1 = from(active(X)) active(cons(X1,X2)) = X2 + 1 >= X2 = cons(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(after(X1,X2)) = X2 + 1 >= X2 = after(active(X1),X2) active(after(X1,X2)) = X2 + 1 >= X2 + 1 = after(X1,active(X2)) from(mark(X)) = X >= X = mark(from(X)) cons(mark(X1),X2) = X2 >= X2 = mark(cons(X1,X2)) s(mark(X)) = X >= X = mark(s(X)) after(mark(X1),X2) = X2 >= X2 = mark(after(X1,X2)) after(X1,mark(X2)) = X2 >= X2 = mark(after(X1,X2)) proper(from(X)) = 1 >= 1 = from(proper(X)) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(after(X1,X2)) = 1 >= 1 = after(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) from(ok(X)) = X + 1 >= X + 1 = ok(from(X)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) after(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(after(X1,X2)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: after#(mark(X1),X2) -> after#(X1,X2) after#(X1,mark(X2)) -> after#(X1,X2) TRS: active(from(X)) -> mark(cons(X,from(s(X)))) active(after(0(),XS)) -> mark(XS) active(after(s(N),cons(X,XS))) -> mark(after(N,XS)) active(from(X)) -> from(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(after(X1,X2)) -> after(active(X1),X2) active(after(X1,X2)) -> after(X1,active(X2)) from(mark(X)) -> mark(from(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) after(mark(X1),X2) -> mark(after(X1,X2)) after(X1,mark(X2)) -> mark(after(X1,X2)) proper(from(X)) -> from(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(after(X1,X2)) -> after(proper(X1),proper(X2)) proper(0()) -> ok(0()) from(ok(X)) -> ok(from(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) after(ok(X1),ok(X2)) -> ok(after(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: s#(mark(X)) -> s#(X) s#(ok(X)) -> s#(X) TRS: active(from(X)) -> mark(cons(X,from(s(X)))) active(after(0(),XS)) -> mark(XS) active(after(s(N),cons(X,XS))) -> mark(after(N,XS)) active(from(X)) -> from(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(after(X1,X2)) -> after(active(X1),X2) active(after(X1,X2)) -> after(X1,active(X2)) from(mark(X)) -> mark(from(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) after(mark(X1),X2) -> mark(after(X1,X2)) after(X1,mark(X2)) -> mark(after(X1,X2)) proper(from(X)) -> from(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(after(X1,X2)) -> after(proper(X1),proper(X2)) proper(0()) -> ok(0()) from(ok(X)) -> ok(from(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) after(ok(X1),ok(X2)) -> ok(after(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [s#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = 1, [after](x0, x1) = x1, [0] = 0, [mark](x0) = x0, [cons](x0, x1) = x1, [s](x0) = x0, [active](x0) = x0, [from](x0) = x0 orientation: s#(mark(X)) = X + 1 >= X + 1 = s#(X) s#(ok(X)) = X + 2 >= X + 1 = s#(X) active(from(X)) = X >= X = mark(cons(X,from(s(X)))) active(after(0(),XS)) = XS >= XS = mark(XS) active(after(s(N),cons(X,XS))) = XS >= XS = mark(after(N,XS)) active(from(X)) = X >= X = from(active(X)) active(cons(X1,X2)) = X2 >= X2 = cons(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(after(X1,X2)) = X2 >= X2 = after(active(X1),X2) active(after(X1,X2)) = X2 >= X2 = after(X1,active(X2)) from(mark(X)) = X >= X = mark(from(X)) cons(mark(X1),X2) = X2 >= X2 = mark(cons(X1,X2)) s(mark(X)) = X >= X = mark(s(X)) after(mark(X1),X2) = X2 >= X2 = mark(after(X1,X2)) after(X1,mark(X2)) = X2 >= X2 = mark(after(X1,X2)) proper(from(X)) = 1 >= 1 = from(proper(X)) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(after(X1,X2)) = 1 >= 1 = after(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) from(ok(X)) = X + 1 >= X + 1 = ok(from(X)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) after(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(after(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: s#(mark(X)) -> s#(X) TRS: active(from(X)) -> mark(cons(X,from(s(X)))) active(after(0(),XS)) -> mark(XS) active(after(s(N),cons(X,XS))) -> mark(after(N,XS)) active(from(X)) -> from(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(after(X1,X2)) -> after(active(X1),X2) active(after(X1,X2)) -> after(X1,active(X2)) from(mark(X)) -> mark(from(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) after(mark(X1),X2) -> mark(after(X1,X2)) after(X1,mark(X2)) -> mark(after(X1,X2)) proper(from(X)) -> from(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(after(X1,X2)) -> after(proper(X1),proper(X2)) proper(0()) -> ok(0()) from(ok(X)) -> ok(from(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) after(ok(X1),ok(X2)) -> ok(after(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: cons#(mark(X1),X2) -> cons#(X1,X2) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) TRS: active(from(X)) -> mark(cons(X,from(s(X)))) active(after(0(),XS)) -> mark(XS) active(after(s(N),cons(X,XS))) -> mark(after(N,XS)) active(from(X)) -> from(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(after(X1,X2)) -> after(active(X1),X2) active(after(X1,X2)) -> after(X1,active(X2)) from(mark(X)) -> mark(from(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) after(mark(X1),X2) -> mark(after(X1,X2)) after(X1,mark(X2)) -> mark(after(X1,X2)) proper(from(X)) -> from(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(after(X1,X2)) -> after(proper(X1),proper(X2)) proper(0()) -> ok(0()) from(ok(X)) -> ok(from(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) after(ok(X1),ok(X2)) -> ok(after(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [cons#](x0, x1) = x1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [after](x0, x1) = x0, [0] = 1, [mark](x0) = 0, [cons](x0, x1) = x0, [s](x0) = x0, [active](x0) = x0, [from](x0) = x0 orientation: cons#(mark(X1),X2) = X2 >= X2 = cons#(X1,X2) cons#(ok(X1),ok(X2)) = X2 + 1 >= X2 = cons#(X1,X2) active(from(X)) = X >= 0 = mark(cons(X,from(s(X)))) active(after(0(),XS)) = 1 >= 0 = mark(XS) active(after(s(N),cons(X,XS))) = N >= 0 = mark(after(N,XS)) active(from(X)) = X >= X = from(active(X)) active(cons(X1,X2)) = X1 >= X1 = cons(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(after(X1,X2)) = X1 >= X1 = after(active(X1),X2) active(after(X1,X2)) = X1 >= X1 = after(X1,active(X2)) from(mark(X)) = 0 >= 0 = mark(from(X)) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) after(mark(X1),X2) = 0 >= 0 = mark(after(X1,X2)) after(X1,mark(X2)) = X1 >= 0 = mark(after(X1,X2)) proper(from(X)) = X + 1 >= X + 1 = from(proper(X)) proper(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(proper(X1),proper(X2)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(after(X1,X2)) = X1 + 1 >= X1 + 1 = after(proper(X1),proper(X2)) proper(0()) = 2 >= 2 = ok(0()) from(ok(X)) = X + 1 >= X + 1 = ok(from(X)) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) after(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(after(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: cons#(mark(X1),X2) -> cons#(X1,X2) TRS: active(from(X)) -> mark(cons(X,from(s(X)))) active(after(0(),XS)) -> mark(XS) active(after(s(N),cons(X,XS))) -> mark(after(N,XS)) active(from(X)) -> from(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(after(X1,X2)) -> after(active(X1),X2) active(after(X1,X2)) -> after(X1,active(X2)) from(mark(X)) -> mark(from(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) after(mark(X1),X2) -> mark(after(X1,X2)) after(X1,mark(X2)) -> mark(after(X1,X2)) proper(from(X)) -> from(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(after(X1,X2)) -> after(proper(X1),proper(X2)) proper(0()) -> ok(0()) from(ok(X)) -> ok(from(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) after(ok(X1),ok(X2)) -> ok(after(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: from#(mark(X)) -> from#(X) from#(ok(X)) -> from#(X) TRS: active(from(X)) -> mark(cons(X,from(s(X)))) active(after(0(),XS)) -> mark(XS) active(after(s(N),cons(X,XS))) -> mark(after(N,XS)) active(from(X)) -> from(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(after(X1,X2)) -> after(active(X1),X2) active(after(X1,X2)) -> after(X1,active(X2)) from(mark(X)) -> mark(from(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) after(mark(X1),X2) -> mark(after(X1,X2)) after(X1,mark(X2)) -> mark(after(X1,X2)) proper(from(X)) -> from(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(after(X1,X2)) -> after(proper(X1),proper(X2)) proper(0()) -> ok(0()) from(ok(X)) -> ok(from(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) after(ok(X1),ok(X2)) -> ok(after(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [from#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = 1, [after](x0, x1) = x1, [0] = 0, [mark](x0) = x0, [cons](x0, x1) = x1, [s](x0) = x0, [active](x0) = x0, [from](x0) = x0 orientation: from#(mark(X)) = X + 1 >= X + 1 = from#(X) from#(ok(X)) = X + 2 >= X + 1 = from#(X) active(from(X)) = X >= X = mark(cons(X,from(s(X)))) active(after(0(),XS)) = XS >= XS = mark(XS) active(after(s(N),cons(X,XS))) = XS >= XS = mark(after(N,XS)) active(from(X)) = X >= X = from(active(X)) active(cons(X1,X2)) = X2 >= X2 = cons(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(after(X1,X2)) = X2 >= X2 = after(active(X1),X2) active(after(X1,X2)) = X2 >= X2 = after(X1,active(X2)) from(mark(X)) = X >= X = mark(from(X)) cons(mark(X1),X2) = X2 >= X2 = mark(cons(X1,X2)) s(mark(X)) = X >= X = mark(s(X)) after(mark(X1),X2) = X2 >= X2 = mark(after(X1,X2)) after(X1,mark(X2)) = X2 >= X2 = mark(after(X1,X2)) proper(from(X)) = 1 >= 1 = from(proper(X)) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(after(X1,X2)) = 1 >= 1 = after(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) from(ok(X)) = X + 1 >= X + 1 = ok(from(X)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) after(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(after(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: from#(mark(X)) -> from#(X) TRS: active(from(X)) -> mark(cons(X,from(s(X)))) active(after(0(),XS)) -> mark(XS) active(after(s(N),cons(X,XS))) -> mark(after(N,XS)) active(from(X)) -> from(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(after(X1,X2)) -> after(active(X1),X2) active(after(X1,X2)) -> after(X1,active(X2)) from(mark(X)) -> mark(from(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) after(mark(X1),X2) -> mark(after(X1,X2)) after(X1,mark(X2)) -> mark(after(X1,X2)) proper(from(X)) -> from(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(after(X1,X2)) -> after(proper(X1),proper(X2)) proper(0()) -> ok(0()) from(ok(X)) -> ok(from(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) after(ok(X1),ok(X2)) -> ok(after(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open