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)) Subterm Criterion Processor: simple projection: pi(active#) = 0 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)) Subterm Criterion Processor: simple projection: pi(proper#) = 0 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)) Subterm Criterion Processor: simple projection: pi(s#) = 0 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)) Subterm Criterion Processor: simple projection: pi(from#) = 0 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)) Subterm Criterion Processor: simple projection: pi(cons#) = 1 problem: DPs: cons#(mark(X1),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)) Subterm Criterion Processor: simple projection: pi(cons#) = 0 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)) Subterm Criterion Processor: simple projection: pi(2nd#) = 0 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