YES Problem: active(2nd(cons(X,cons(Y,Z)))) -> mark(Y) active(from(X)) -> mark(cons(X,from(s(X)))) active(2nd(X)) -> 2nd(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(from(X)) -> from(active(X)) active(s(X)) -> s(active(X)) 2nd(mark(X)) -> mark(2nd(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) from(mark(X)) -> mark(from(X)) s(mark(X)) -> mark(s(X)) proper(2nd(X)) -> 2nd(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) proper(s(X)) -> s(proper(X)) 2nd(ok(X)) -> ok(2nd(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) s(ok(X)) -> ok(s(X)) 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#(2nd(X)) -> active#(X) active#(2nd(X)) -> 2nd#(active(X)) active#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(from(X)) -> active#(X) active#(from(X)) -> from#(active(X)) active#(s(X)) -> active#(X) active#(s(X)) -> s#(active(X)) 2nd#(mark(X)) -> 2nd#(X) cons#(mark(X1),X2) -> cons#(X1,X2) from#(mark(X)) -> from#(X) s#(mark(X)) -> s#(X) proper#(2nd(X)) -> proper#(X) proper#(2nd(X)) -> 2nd#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(from(X)) -> proper#(X) proper#(from(X)) -> from#(proper(X)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) 2nd#(ok(X)) -> 2nd#(X) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) from#(ok(X)) -> from#(X) s#(ok(X)) -> s#(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(2nd(cons(X,cons(Y,Z)))) -> mark(Y) active(from(X)) -> mark(cons(X,from(s(X)))) active(2nd(X)) -> 2nd(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(from(X)) -> from(active(X)) active(s(X)) -> s(active(X)) 2nd(mark(X)) -> mark(2nd(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) from(mark(X)) -> mark(from(X)) s(mark(X)) -> mark(s(X)) proper(2nd(X)) -> 2nd(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) proper(s(X)) -> s(proper(X)) 2nd(ok(X)) -> ok(2nd(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) s(ok(X)) -> ok(s(X)) 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#(2nd(X)) -> active#(X) active#(2nd(X)) -> 2nd#(active(X)) active#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(from(X)) -> active#(X) active#(from(X)) -> from#(active(X)) active#(s(X)) -> active#(X) active#(s(X)) -> s#(active(X)) 2nd#(mark(X)) -> 2nd#(X) cons#(mark(X1),X2) -> cons#(X1,X2) from#(mark(X)) -> from#(X) s#(mark(X)) -> s#(X) proper#(2nd(X)) -> proper#(X) proper#(2nd(X)) -> 2nd#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(from(X)) -> proper#(X) proper#(from(X)) -> from#(proper(X)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) 2nd#(ok(X)) -> 2nd#(X) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) from#(ok(X)) -> from#(X) s#(ok(X)) -> s#(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(2nd(cons(X,cons(Y,Z)))) -> mark(Y) active(from(X)) -> mark(cons(X,from(s(X)))) active(2nd(X)) -> 2nd(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(from(X)) -> from(active(X)) active(s(X)) -> s(active(X)) 2nd(mark(X)) -> mark(2nd(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) from(mark(X)) -> mark(from(X)) s(mark(X)) -> mark(s(X)) proper(2nd(X)) -> 2nd(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) proper(s(X)) -> s(proper(X)) 2nd(ok(X)) -> ok(2nd(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) s(ok(X)) -> ok(s(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#(s(X)) -> s#(active(X)) top#(ok(X)) -> active#(X) -> active#(s(X)) -> active#(X) 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#(cons(X1,X2)) -> cons#(active(X1),X2) top#(ok(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) top#(ok(X)) -> active#(X) -> active#(2nd(X)) -> 2nd#(active(X)) top#(ok(X)) -> active#(X) -> active#(2nd(X)) -> active#(X) 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#(s(X)) -> s#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(from(X)) -> from#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(from(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#(2nd(X)) -> 2nd#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(2nd(X)) -> proper#(X) 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#(from(X)) -> from#(proper(X)) proper#(s(X)) -> proper#(X) -> proper#(from(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#(2nd(X)) -> 2nd#(proper(X)) proper#(s(X)) -> proper#(X) -> proper#(2nd(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#(s(X)) -> s#(proper(X)) proper#(from(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(from(X)) -> proper#(X) -> proper#(from(X)) -> from#(proper(X)) proper#(from(X)) -> proper#(X) -> proper#(from(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#(2nd(X)) -> 2nd#(proper(X)) proper#(from(X)) -> proper#(X) -> proper#(2nd(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) proper#(2nd(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) proper#(2nd(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(2nd(X)) -> proper#(X) -> proper#(from(X)) -> from#(proper(X)) proper#(2nd(X)) -> proper#(X) -> proper#(from(X)) -> proper#(X) proper#(2nd(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(2nd(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(2nd(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(2nd(X)) -> proper#(X) -> proper#(2nd(X)) -> 2nd#(proper(X)) proper#(2nd(X)) -> proper#(X) -> proper#(2nd(X)) -> proper#(X) proper#(2nd(X)) -> 2nd#(proper(X)) -> 2nd#(ok(X)) -> 2nd#(X) proper#(2nd(X)) -> 2nd#(proper(X)) -> 2nd#(mark(X)) -> 2nd#(X) 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#(from(X)) -> from#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(from(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#(2nd(X)) -> 2nd#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(2nd(X)) -> proper#(X) 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#(from(X)) -> from#(proper(X)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(from(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#(2nd(X)) -> 2nd#(proper(X)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(2nd(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) 2nd#(ok(X)) -> 2nd#(X) -> 2nd#(ok(X)) -> 2nd#(X) 2nd#(ok(X)) -> 2nd#(X) -> 2nd#(mark(X)) -> 2nd#(X) 2nd#(mark(X)) -> 2nd#(X) -> 2nd#(ok(X)) -> 2nd#(X) 2nd#(mark(X)) -> 2nd#(X) -> 2nd#(mark(X)) -> 2nd#(X) 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#(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#(s(X)) -> s#(active(X)) active#(s(X)) -> active#(X) -> active#(s(X)) -> active#(X) 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#(cons(X1,X2)) -> cons#(active(X1),X2) active#(s(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) active#(s(X)) -> active#(X) -> active#(2nd(X)) -> 2nd#(active(X)) active#(s(X)) -> active#(X) -> active#(2nd(X)) -> active#(X) 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#(s(X)) -> s#(active(X)) active#(from(X)) -> active#(X) -> active#(s(X)) -> active#(X) 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#(cons(X1,X2)) -> cons#(active(X1),X2) active#(from(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) active#(from(X)) -> active#(X) -> active#(2nd(X)) -> 2nd#(active(X)) active#(from(X)) -> active#(X) -> active#(2nd(X)) -> active#(X) 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) active#(2nd(X)) -> 2nd#(active(X)) -> 2nd#(ok(X)) -> 2nd#(X) active#(2nd(X)) -> 2nd#(active(X)) -> 2nd#(mark(X)) -> 2nd#(X) active#(2nd(X)) -> active#(X) -> active#(s(X)) -> s#(active(X)) active#(2nd(X)) -> active#(X) -> active#(s(X)) -> active#(X) active#(2nd(X)) -> active#(X) -> active#(from(X)) -> from#(active(X)) active#(2nd(X)) -> active#(X) -> active#(from(X)) -> active#(X) active#(2nd(X)) -> active#(X) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(2nd(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) active#(2nd(X)) -> active#(X) -> active#(2nd(X)) -> 2nd#(active(X)) active#(2nd(X)) -> active#(X) -> active#(2nd(X)) -> active#(X) active#(2nd(X)) -> active#(X) -> active#(from(X)) -> cons#(X,from(s(X))) active#(2nd(X)) -> active#(X) -> active#(from(X)) -> from#(s(X)) active#(2nd(X)) -> active#(X) -> 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#(s(X)) -> s#(active(X)) active#(cons(X1,X2)) -> active#(X1) -> active#(s(X)) -> active#(X) 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#(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#(2nd(X)) -> 2nd#(active(X)) active#(cons(X1,X2)) -> active#(X1) -> active#(2nd(X)) -> active#(X) 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) SCC Processor: #sccs: 7 #rules: 19 #arcs: 155/1024 DPs: top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> top#(proper(X)) TRS: active(2nd(cons(X,cons(Y,Z)))) -> mark(Y) active(from(X)) -> mark(cons(X,from(s(X)))) active(2nd(X)) -> 2nd(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(from(X)) -> from(active(X)) active(s(X)) -> s(active(X)) 2nd(mark(X)) -> mark(2nd(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) from(mark(X)) -> mark(from(X)) s(mark(X)) -> mark(s(X)) proper(2nd(X)) -> 2nd(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) proper(s(X)) -> s(proper(X)) 2nd(ok(X)) -> ok(2nd(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) s(ok(X)) -> ok(s(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) KBO Processor: argument filtering: pi(cons) = 0 pi(2nd) = 0 pi(active) = [] pi(mark) = [] pi(from) = 0 pi(s) = 0 pi(proper) = [] pi(ok) = [0] pi(top) = [] pi(top#) = 0 weight function: w0 = 1 w(top#) = w(top) = w(ok) = w(proper) = w(mark) = w(active) = 1 w(s) = w(from) = w(2nd) = w(cons) = 0 precedence: active > mark > top# ~ top ~ ok ~ proper ~ s ~ from ~ 2nd ~ cons problem: DPs: TRS: active(2nd(cons(X,cons(Y,Z)))) -> mark(Y) active(from(X)) -> mark(cons(X,from(s(X)))) active(2nd(X)) -> 2nd(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(from(X)) -> from(active(X)) active(s(X)) -> s(active(X)) 2nd(mark(X)) -> mark(2nd(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) from(mark(X)) -> mark(from(X)) s(mark(X)) -> mark(s(X)) proper(2nd(X)) -> 2nd(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) proper(s(X)) -> s(proper(X)) 2nd(ok(X)) -> ok(2nd(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) s(ok(X)) -> ok(s(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: active#(2nd(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) active#(from(X)) -> active#(X) active#(s(X)) -> active#(X) TRS: active(2nd(cons(X,cons(Y,Z)))) -> mark(Y) active(from(X)) -> mark(cons(X,from(s(X)))) active(2nd(X)) -> 2nd(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(from(X)) -> from(active(X)) active(s(X)) -> s(active(X)) 2nd(mark(X)) -> mark(2nd(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) from(mark(X)) -> mark(from(X)) s(mark(X)) -> mark(s(X)) proper(2nd(X)) -> 2nd(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) proper(s(X)) -> s(proper(X)) 2nd(ok(X)) -> ok(2nd(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) s(ok(X)) -> ok(s(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [active#](x0) = x0 + 4, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = 4x0 + 2, [s](x0) = 5x0 + 7, [from](x0) = 2x0 + 6, [mark](x0) = 4, [active](x0) = x0, [2nd](x0) = 6x0 + 6, [cons](x0, x1) = 2x0 + 3 orientation: active#(2nd(X)) = 6X + 10 >= X + 4 = active#(X) active#(cons(X1,X2)) = 2X1 + 7 >= X1 + 4 = active#(X1) active#(from(X)) = 2X + 10 >= X + 4 = active#(X) active#(s(X)) = 5X + 11 >= X + 4 = active#(X) active(2nd(cons(X,cons(Y,Z)))) = 12X + 24 >= 4 = mark(Y) active(from(X)) = 2X + 6 >= 4 = mark(cons(X,from(s(X)))) active(2nd(X)) = 6X + 6 >= 6X + 6 = 2nd(active(X)) active(cons(X1,X2)) = 2X1 + 3 >= 2X1 + 3 = cons(active(X1),X2) active(from(X)) = 2X + 6 >= 2X + 6 = from(active(X)) active(s(X)) = 5X + 7 >= 5X + 7 = s(active(X)) 2nd(mark(X)) = 30 >= 4 = mark(2nd(X)) cons(mark(X1),X2) = 11 >= 4 = mark(cons(X1,X2)) from(mark(X)) = 14 >= 4 = mark(from(X)) s(mark(X)) = 27 >= 4 = mark(s(X)) proper(2nd(X)) = 24X + 26 >= 24X + 18 = 2nd(proper(X)) proper(cons(X1,X2)) = 8X1 + 14 >= 8X1 + 7 = cons(proper(X1),proper(X2)) proper(from(X)) = 8X + 26 >= 8X + 10 = from(proper(X)) proper(s(X)) = 20X + 30 >= 20X + 17 = s(proper(X)) 2nd(ok(X)) = 6X + 6 >= 6X + 6 = ok(2nd(X)) cons(ok(X1),ok(X2)) = 2X1 + 3 >= 2X1 + 3 = ok(cons(X1,X2)) from(ok(X)) = 2X + 6 >= 2X + 6 = ok(from(X)) s(ok(X)) = 5X + 7 >= 5X + 7 = ok(s(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(2nd(cons(X,cons(Y,Z)))) -> mark(Y) active(from(X)) -> mark(cons(X,from(s(X)))) active(2nd(X)) -> 2nd(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(from(X)) -> from(active(X)) active(s(X)) -> s(active(X)) 2nd(mark(X)) -> mark(2nd(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) from(mark(X)) -> mark(from(X)) s(mark(X)) -> mark(s(X)) proper(2nd(X)) -> 2nd(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) proper(s(X)) -> s(proper(X)) 2nd(ok(X)) -> ok(2nd(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) s(ok(X)) -> ok(s(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: proper#(2nd(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(from(X)) -> proper#(X) proper#(s(X)) -> proper#(X) TRS: active(2nd(cons(X,cons(Y,Z)))) -> mark(Y) active(from(X)) -> mark(cons(X,from(s(X)))) active(2nd(X)) -> 2nd(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(from(X)) -> from(active(X)) active(s(X)) -> s(active(X)) 2nd(mark(X)) -> mark(2nd(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) from(mark(X)) -> mark(from(X)) s(mark(X)) -> mark(s(X)) proper(2nd(X)) -> 2nd(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) proper(s(X)) -> s(proper(X)) 2nd(ok(X)) -> ok(2nd(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) s(ok(X)) -> ok(s(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [proper#](x0) = 2x0, [top](x0) = 0, [ok](x0) = 2, [proper](x0) = 5x0 + 4, [s](x0) = 4x0 + 4, [from](x0) = 3x0 + 2, [mark](x0) = 0, [active](x0) = 4x0, [2nd](x0) = x0 + 1, [cons](x0, x1) = 2x0 + x1 + 2 orientation: proper#(2nd(X)) = 2X + 2 >= 2X = proper#(X) proper#(cons(X1,X2)) = 4X1 + 2X2 + 4 >= 2X2 = proper#(X2) proper#(cons(X1,X2)) = 4X1 + 2X2 + 4 >= 2X1 = proper#(X1) proper#(from(X)) = 6X + 4 >= 2X = proper#(X) proper#(s(X)) = 8X + 8 >= 2X = proper#(X) active(2nd(cons(X,cons(Y,Z)))) = 8X + 8Y + 4Z + 20 >= 0 = mark(Y) active(from(X)) = 12X + 8 >= 0 = mark(cons(X,from(s(X)))) active(2nd(X)) = 4X + 4 >= 4X + 1 = 2nd(active(X)) active(cons(X1,X2)) = 8X1 + 4X2 + 8 >= 8X1 + X2 + 2 = cons(active(X1),X2) active(from(X)) = 12X + 8 >= 12X + 2 = from(active(X)) active(s(X)) = 16X + 16 >= 16X + 4 = s(active(X)) 2nd(mark(X)) = 1 >= 0 = mark(2nd(X)) cons(mark(X1),X2) = X2 + 2 >= 0 = mark(cons(X1,X2)) from(mark(X)) = 2 >= 0 = mark(from(X)) s(mark(X)) = 4 >= 0 = mark(s(X)) proper(2nd(X)) = 5X + 9 >= 5X + 5 = 2nd(proper(X)) proper(cons(X1,X2)) = 10X1 + 5X2 + 14 >= 10X1 + 5X2 + 14 = cons(proper(X1),proper(X2)) proper(from(X)) = 15X + 14 >= 15X + 14 = from(proper(X)) proper(s(X)) = 20X + 24 >= 20X + 20 = s(proper(X)) 2nd(ok(X)) = 3 >= 2 = ok(2nd(X)) cons(ok(X1),ok(X2)) = 8 >= 2 = ok(cons(X1,X2)) from(ok(X)) = 8 >= 2 = ok(from(X)) s(ok(X)) = 12 >= 2 = ok(s(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(2nd(cons(X,cons(Y,Z)))) -> mark(Y) active(from(X)) -> mark(cons(X,from(s(X)))) active(2nd(X)) -> 2nd(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(from(X)) -> from(active(X)) active(s(X)) -> s(active(X)) 2nd(mark(X)) -> mark(2nd(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) from(mark(X)) -> mark(from(X)) s(mark(X)) -> mark(s(X)) proper(2nd(X)) -> 2nd(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) proper(s(X)) -> s(proper(X)) 2nd(ok(X)) -> ok(2nd(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) s(ok(X)) -> ok(s(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: s#(mark(X)) -> s#(X) s#(ok(X)) -> s#(X) TRS: active(2nd(cons(X,cons(Y,Z)))) -> mark(Y) active(from(X)) -> mark(cons(X,from(s(X)))) active(2nd(X)) -> 2nd(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(from(X)) -> from(active(X)) active(s(X)) -> s(active(X)) 2nd(mark(X)) -> mark(2nd(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) from(mark(X)) -> mark(from(X)) s(mark(X)) -> mark(s(X)) proper(2nd(X)) -> 2nd(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) proper(s(X)) -> s(proper(X)) 2nd(ok(X)) -> ok(2nd(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) s(ok(X)) -> ok(s(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [s#](x0) = x0, [top](x0) = 6, [ok](x0) = 2x0 + 1, [proper](x0) = 4x0, [s](x0) = x0, [from](x0) = 5x0 + 4, [mark](x0) = x0 + 1, [active](x0) = 2x0, [2nd](x0) = 2x0 + 1, [cons](x0, x1) = 2x0 + x1 orientation: s#(mark(X)) = X + 1 >= X = s#(X) s#(ok(X)) = 2X + 1 >= X = s#(X) active(2nd(cons(X,cons(Y,Z)))) = 8X + 8Y + 4Z + 2 >= Y + 1 = mark(Y) active(from(X)) = 10X + 8 >= 7X + 5 = mark(cons(X,from(s(X)))) active(2nd(X)) = 4X + 2 >= 4X + 1 = 2nd(active(X)) active(cons(X1,X2)) = 4X1 + 2X2 >= 4X1 + X2 = cons(active(X1),X2) active(from(X)) = 10X + 8 >= 10X + 4 = from(active(X)) active(s(X)) = 2X >= 2X = s(active(X)) 2nd(mark(X)) = 2X + 3 >= 2X + 2 = mark(2nd(X)) cons(mark(X1),X2) = 2X1 + X2 + 2 >= 2X1 + X2 + 1 = mark(cons(X1,X2)) from(mark(X)) = 5X + 9 >= 5X + 5 = mark(from(X)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) proper(2nd(X)) = 8X + 4 >= 8X + 1 = 2nd(proper(X)) proper(cons(X1,X2)) = 8X1 + 4X2 >= 8X1 + 4X2 = cons(proper(X1),proper(X2)) proper(from(X)) = 20X + 16 >= 20X + 4 = from(proper(X)) proper(s(X)) = 4X >= 4X = s(proper(X)) 2nd(ok(X)) = 4X + 3 >= 4X + 3 = ok(2nd(X)) cons(ok(X1),ok(X2)) = 4X1 + 2X2 + 3 >= 4X1 + 2X2 + 1 = ok(cons(X1,X2)) from(ok(X)) = 10X + 9 >= 10X + 9 = ok(from(X)) s(ok(X)) = 2X + 1 >= 2X + 1 = ok(s(X)) top(mark(X)) = 6 >= 6 = top(proper(X)) top(ok(X)) = 6 >= 6 = top(active(X)) problem: DPs: TRS: active(2nd(cons(X,cons(Y,Z)))) -> mark(Y) active(from(X)) -> mark(cons(X,from(s(X)))) active(2nd(X)) -> 2nd(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(from(X)) -> from(active(X)) active(s(X)) -> s(active(X)) 2nd(mark(X)) -> mark(2nd(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) from(mark(X)) -> mark(from(X)) s(mark(X)) -> mark(s(X)) proper(2nd(X)) -> 2nd(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) proper(s(X)) -> s(proper(X)) 2nd(ok(X)) -> ok(2nd(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) s(ok(X)) -> ok(s(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: from#(mark(X)) -> from#(X) from#(ok(X)) -> from#(X) TRS: active(2nd(cons(X,cons(Y,Z)))) -> mark(Y) active(from(X)) -> mark(cons(X,from(s(X)))) active(2nd(X)) -> 2nd(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(from(X)) -> from(active(X)) active(s(X)) -> s(active(X)) 2nd(mark(X)) -> mark(2nd(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) from(mark(X)) -> mark(from(X)) s(mark(X)) -> mark(s(X)) proper(2nd(X)) -> 2nd(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) proper(s(X)) -> s(proper(X)) 2nd(ok(X)) -> ok(2nd(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) s(ok(X)) -> ok(s(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [from#](x0) = x0, [top](x0) = 6, [ok](x0) = 2x0 + 1, [proper](x0) = 4x0, [s](x0) = x0, [from](x0) = 5x0 + 4, [mark](x0) = x0 + 1, [active](x0) = 2x0, [2nd](x0) = 2x0 + 1, [cons](x0, x1) = 2x0 + x1 orientation: from#(mark(X)) = X + 1 >= X = from#(X) from#(ok(X)) = 2X + 1 >= X = from#(X) active(2nd(cons(X,cons(Y,Z)))) = 8X + 8Y + 4Z + 2 >= Y + 1 = mark(Y) active(from(X)) = 10X + 8 >= 7X + 5 = mark(cons(X,from(s(X)))) active(2nd(X)) = 4X + 2 >= 4X + 1 = 2nd(active(X)) active(cons(X1,X2)) = 4X1 + 2X2 >= 4X1 + X2 = cons(active(X1),X2) active(from(X)) = 10X + 8 >= 10X + 4 = from(active(X)) active(s(X)) = 2X >= 2X = s(active(X)) 2nd(mark(X)) = 2X + 3 >= 2X + 2 = mark(2nd(X)) cons(mark(X1),X2) = 2X1 + X2 + 2 >= 2X1 + X2 + 1 = mark(cons(X1,X2)) from(mark(X)) = 5X + 9 >= 5X + 5 = mark(from(X)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) proper(2nd(X)) = 8X + 4 >= 8X + 1 = 2nd(proper(X)) proper(cons(X1,X2)) = 8X1 + 4X2 >= 8X1 + 4X2 = cons(proper(X1),proper(X2)) proper(from(X)) = 20X + 16 >= 20X + 4 = from(proper(X)) proper(s(X)) = 4X >= 4X = s(proper(X)) 2nd(ok(X)) = 4X + 3 >= 4X + 3 = ok(2nd(X)) cons(ok(X1),ok(X2)) = 4X1 + 2X2 + 3 >= 4X1 + 2X2 + 1 = ok(cons(X1,X2)) from(ok(X)) = 10X + 9 >= 10X + 9 = ok(from(X)) s(ok(X)) = 2X + 1 >= 2X + 1 = ok(s(X)) top(mark(X)) = 6 >= 6 = top(proper(X)) top(ok(X)) = 6 >= 6 = top(active(X)) problem: DPs: TRS: active(2nd(cons(X,cons(Y,Z)))) -> mark(Y) active(from(X)) -> mark(cons(X,from(s(X)))) active(2nd(X)) -> 2nd(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(from(X)) -> from(active(X)) active(s(X)) -> s(active(X)) 2nd(mark(X)) -> mark(2nd(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) from(mark(X)) -> mark(from(X)) s(mark(X)) -> mark(s(X)) proper(2nd(X)) -> 2nd(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) proper(s(X)) -> s(proper(X)) 2nd(ok(X)) -> ok(2nd(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) s(ok(X)) -> ok(s(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: cons#(mark(X1),X2) -> cons#(X1,X2) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) TRS: active(2nd(cons(X,cons(Y,Z)))) -> mark(Y) active(from(X)) -> mark(cons(X,from(s(X)))) active(2nd(X)) -> 2nd(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(from(X)) -> from(active(X)) active(s(X)) -> s(active(X)) 2nd(mark(X)) -> mark(2nd(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) from(mark(X)) -> mark(from(X)) s(mark(X)) -> mark(s(X)) proper(2nd(X)) -> 2nd(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) proper(s(X)) -> s(proper(X)) 2nd(ok(X)) -> ok(2nd(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) s(ok(X)) -> ok(s(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [cons#](x0, x1) = x0 + 4x1, [top](x0) = 0, [ok](x0) = x0 + 2, [proper](x0) = 4x0, [s](x0) = x0, [from](x0) = x0 + 1, [mark](x0) = x0 + 2, [active](x0) = 3x0 + 1, [2nd](x0) = 2x0 + 2, [cons](x0, x1) = x0 + 2x1 orientation: cons#(mark(X1),X2) = X1 + 4X2 + 2 >= X1 + 4X2 = cons#(X1,X2) cons#(ok(X1),ok(X2)) = X1 + 4X2 + 10 >= X1 + 4X2 = cons#(X1,X2) active(2nd(cons(X,cons(Y,Z)))) = 6X + 12Y + 24Z + 7 >= Y + 2 = mark(Y) active(from(X)) = 3X + 4 >= 3X + 4 = mark(cons(X,from(s(X)))) active(2nd(X)) = 6X + 7 >= 6X + 4 = 2nd(active(X)) active(cons(X1,X2)) = 3X1 + 6X2 + 1 >= 3X1 + 2X2 + 1 = cons(active(X1),X2) active(from(X)) = 3X + 4 >= 3X + 2 = from(active(X)) active(s(X)) = 3X + 1 >= 3X + 1 = s(active(X)) 2nd(mark(X)) = 2X + 6 >= 2X + 4 = mark(2nd(X)) cons(mark(X1),X2) = X1 + 2X2 + 2 >= X1 + 2X2 + 2 = mark(cons(X1,X2)) from(mark(X)) = X + 3 >= X + 3 = mark(from(X)) s(mark(X)) = X + 2 >= X + 2 = mark(s(X)) proper(2nd(X)) = 8X + 8 >= 8X + 2 = 2nd(proper(X)) proper(cons(X1,X2)) = 4X1 + 8X2 >= 4X1 + 8X2 = cons(proper(X1),proper(X2)) proper(from(X)) = 4X + 4 >= 4X + 1 = from(proper(X)) proper(s(X)) = 4X >= 4X = s(proper(X)) 2nd(ok(X)) = 2X + 6 >= 2X + 4 = ok(2nd(X)) cons(ok(X1),ok(X2)) = X1 + 2X2 + 6 >= X1 + 2X2 + 2 = ok(cons(X1,X2)) from(ok(X)) = X + 3 >= X + 3 = ok(from(X)) s(ok(X)) = X + 2 >= X + 2 = ok(s(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(2nd(cons(X,cons(Y,Z)))) -> mark(Y) active(from(X)) -> mark(cons(X,from(s(X)))) active(2nd(X)) -> 2nd(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(from(X)) -> from(active(X)) active(s(X)) -> s(active(X)) 2nd(mark(X)) -> mark(2nd(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) from(mark(X)) -> mark(from(X)) s(mark(X)) -> mark(s(X)) proper(2nd(X)) -> 2nd(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) proper(s(X)) -> s(proper(X)) 2nd(ok(X)) -> ok(2nd(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) s(ok(X)) -> ok(s(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: 2nd#(mark(X)) -> 2nd#(X) 2nd#(ok(X)) -> 2nd#(X) TRS: active(2nd(cons(X,cons(Y,Z)))) -> mark(Y) active(from(X)) -> mark(cons(X,from(s(X)))) active(2nd(X)) -> 2nd(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(from(X)) -> from(active(X)) active(s(X)) -> s(active(X)) 2nd(mark(X)) -> mark(2nd(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) from(mark(X)) -> mark(from(X)) s(mark(X)) -> mark(s(X)) proper(2nd(X)) -> 2nd(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) proper(s(X)) -> s(proper(X)) 2nd(ok(X)) -> ok(2nd(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) s(ok(X)) -> ok(s(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [2nd#](x0) = x0, [top](x0) = 6, [ok](x0) = 2x0 + 1, [proper](x0) = 4x0, [s](x0) = x0, [from](x0) = 5x0 + 4, [mark](x0) = x0 + 1, [active](x0) = 2x0, [2nd](x0) = 2x0 + 1, [cons](x0, x1) = 2x0 + x1 orientation: 2nd#(mark(X)) = X + 1 >= X = 2nd#(X) 2nd#(ok(X)) = 2X + 1 >= X = 2nd#(X) active(2nd(cons(X,cons(Y,Z)))) = 8X + 8Y + 4Z + 2 >= Y + 1 = mark(Y) active(from(X)) = 10X + 8 >= 7X + 5 = mark(cons(X,from(s(X)))) active(2nd(X)) = 4X + 2 >= 4X + 1 = 2nd(active(X)) active(cons(X1,X2)) = 4X1 + 2X2 >= 4X1 + X2 = cons(active(X1),X2) active(from(X)) = 10X + 8 >= 10X + 4 = from(active(X)) active(s(X)) = 2X >= 2X = s(active(X)) 2nd(mark(X)) = 2X + 3 >= 2X + 2 = mark(2nd(X)) cons(mark(X1),X2) = 2X1 + X2 + 2 >= 2X1 + X2 + 1 = mark(cons(X1,X2)) from(mark(X)) = 5X + 9 >= 5X + 5 = mark(from(X)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) proper(2nd(X)) = 8X + 4 >= 8X + 1 = 2nd(proper(X)) proper(cons(X1,X2)) = 8X1 + 4X2 >= 8X1 + 4X2 = cons(proper(X1),proper(X2)) proper(from(X)) = 20X + 16 >= 20X + 4 = from(proper(X)) proper(s(X)) = 4X >= 4X = s(proper(X)) 2nd(ok(X)) = 4X + 3 >= 4X + 3 = ok(2nd(X)) cons(ok(X1),ok(X2)) = 4X1 + 2X2 + 3 >= 4X1 + 2X2 + 1 = ok(cons(X1,X2)) from(ok(X)) = 10X + 9 >= 10X + 9 = ok(from(X)) s(ok(X)) = 2X + 1 >= 2X + 1 = ok(s(X)) top(mark(X)) = 6 >= 6 = top(proper(X)) top(ok(X)) = 6 >= 6 = top(active(X)) problem: DPs: TRS: active(2nd(cons(X,cons(Y,Z)))) -> mark(Y) active(from(X)) -> mark(cons(X,from(s(X)))) active(2nd(X)) -> 2nd(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(from(X)) -> from(active(X)) active(s(X)) -> s(active(X)) 2nd(mark(X)) -> mark(2nd(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) from(mark(X)) -> mark(from(X)) s(mark(X)) -> mark(s(X)) proper(2nd(X)) -> 2nd(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) proper(s(X)) -> s(proper(X)) 2nd(ok(X)) -> ok(2nd(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) s(ok(X)) -> ok(s(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed