MAYBE Problem: active(incr(nil())) -> mark(nil()) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(adx(nil())) -> mark(nil()) active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(head(cons(X,L))) -> mark(X) active(tail(cons(X,L))) -> mark(L) active(incr(X)) -> incr(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(adx(X)) -> adx(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) incr(mark(X)) -> mark(incr(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) adx(mark(X)) -> mark(adx(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(incr(X)) -> incr(proper(X)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(adx(X)) -> adx(proper(X)) proper(nats()) -> ok(nats()) proper(zeros()) -> ok(zeros()) proper(0()) -> ok(0()) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) incr(ok(X)) -> ok(incr(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) adx(ok(X)) -> ok(adx(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Proof: DP Processor: DPs: active#(incr(cons(X,L))) -> incr#(L) active#(incr(cons(X,L))) -> s#(X) active#(incr(cons(X,L))) -> cons#(s(X),incr(L)) active#(adx(cons(X,L))) -> adx#(L) active#(adx(cons(X,L))) -> cons#(X,adx(L)) active#(adx(cons(X,L))) -> incr#(cons(X,adx(L))) active#(nats()) -> adx#(zeros()) active#(zeros()) -> cons#(0(),zeros()) active#(incr(X)) -> active#(X) active#(incr(X)) -> incr#(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#(adx(X)) -> active#(X) active#(adx(X)) -> adx#(active(X)) active#(head(X)) -> active#(X) active#(head(X)) -> head#(active(X)) active#(tail(X)) -> active#(X) active#(tail(X)) -> tail#(active(X)) incr#(mark(X)) -> incr#(X) cons#(mark(X1),X2) -> cons#(X1,X2) s#(mark(X)) -> s#(X) adx#(mark(X)) -> adx#(X) head#(mark(X)) -> head#(X) tail#(mark(X)) -> tail#(X) proper#(incr(X)) -> proper#(X) proper#(incr(X)) -> incr#(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#(adx(X)) -> proper#(X) proper#(adx(X)) -> adx#(proper(X)) proper#(head(X)) -> proper#(X) proper#(head(X)) -> head#(proper(X)) proper#(tail(X)) -> proper#(X) proper#(tail(X)) -> tail#(proper(X)) incr#(ok(X)) -> incr#(X) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) s#(ok(X)) -> s#(X) adx#(ok(X)) -> adx#(X) head#(ok(X)) -> head#(X) tail#(ok(X)) -> tail#(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(incr(nil())) -> mark(nil()) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(adx(nil())) -> mark(nil()) active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(head(cons(X,L))) -> mark(X) active(tail(cons(X,L))) -> mark(L) active(incr(X)) -> incr(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(adx(X)) -> adx(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) incr(mark(X)) -> mark(incr(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) adx(mark(X)) -> mark(adx(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(incr(X)) -> incr(proper(X)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(adx(X)) -> adx(proper(X)) proper(nats()) -> ok(nats()) proper(zeros()) -> ok(zeros()) proper(0()) -> ok(0()) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) incr(ok(X)) -> ok(incr(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) adx(ok(X)) -> ok(adx(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) TDG Processor: DPs: active#(incr(cons(X,L))) -> incr#(L) active#(incr(cons(X,L))) -> s#(X) active#(incr(cons(X,L))) -> cons#(s(X),incr(L)) active#(adx(cons(X,L))) -> adx#(L) active#(adx(cons(X,L))) -> cons#(X,adx(L)) active#(adx(cons(X,L))) -> incr#(cons(X,adx(L))) active#(nats()) -> adx#(zeros()) active#(zeros()) -> cons#(0(),zeros()) active#(incr(X)) -> active#(X) active#(incr(X)) -> incr#(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#(adx(X)) -> active#(X) active#(adx(X)) -> adx#(active(X)) active#(head(X)) -> active#(X) active#(head(X)) -> head#(active(X)) active#(tail(X)) -> active#(X) active#(tail(X)) -> tail#(active(X)) incr#(mark(X)) -> incr#(X) cons#(mark(X1),X2) -> cons#(X1,X2) s#(mark(X)) -> s#(X) adx#(mark(X)) -> adx#(X) head#(mark(X)) -> head#(X) tail#(mark(X)) -> tail#(X) proper#(incr(X)) -> proper#(X) proper#(incr(X)) -> incr#(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#(adx(X)) -> proper#(X) proper#(adx(X)) -> adx#(proper(X)) proper#(head(X)) -> proper#(X) proper#(head(X)) -> head#(proper(X)) proper#(tail(X)) -> proper#(X) proper#(tail(X)) -> tail#(proper(X)) incr#(ok(X)) -> incr#(X) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) s#(ok(X)) -> s#(X) adx#(ok(X)) -> adx#(X) head#(ok(X)) -> head#(X) tail#(ok(X)) -> tail#(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(incr(nil())) -> mark(nil()) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(adx(nil())) -> mark(nil()) active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(head(cons(X,L))) -> mark(X) active(tail(cons(X,L))) -> mark(L) active(incr(X)) -> incr(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(adx(X)) -> adx(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) incr(mark(X)) -> mark(incr(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) adx(mark(X)) -> mark(adx(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(incr(X)) -> incr(proper(X)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(adx(X)) -> adx(proper(X)) proper(nats()) -> ok(nats()) proper(zeros()) -> ok(zeros()) proper(0()) -> ok(0()) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) incr(ok(X)) -> ok(incr(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) adx(ok(X)) -> ok(adx(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(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#(tail(X)) -> tail#(active(X)) top#(ok(X)) -> active#(X) -> active#(tail(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(head(X)) -> head#(active(X)) top#(ok(X)) -> active#(X) -> active#(head(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(adx(X)) -> adx#(active(X)) top#(ok(X)) -> active#(X) -> active#(adx(X)) -> active#(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#(cons(X1,X2)) -> cons#(active(X1),X2) top#(ok(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) top#(ok(X)) -> active#(X) -> active#(incr(X)) -> incr#(active(X)) top#(ok(X)) -> active#(X) -> active#(incr(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(zeros()) -> cons#(0(),zeros()) top#(ok(X)) -> active#(X) -> active#(nats()) -> adx#(zeros()) top#(ok(X)) -> active#(X) -> active#(adx(cons(X,L))) -> incr#(cons(X,adx(L))) top#(ok(X)) -> active#(X) -> active#(adx(cons(X,L))) -> cons#(X,adx(L)) top#(ok(X)) -> active#(X) -> active#(adx(cons(X,L))) -> adx#(L) top#(ok(X)) -> active#(X) -> active#(incr(cons(X,L))) -> cons#(s(X),incr(L)) top#(ok(X)) -> active#(X) -> active#(incr(cons(X,L))) -> s#(X) top#(ok(X)) -> active#(X) -> active#(incr(cons(X,L))) -> incr#(L) 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#(tail(X)) -> tail#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(tail(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(head(X)) -> head#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(head(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(adx(X)) -> adx#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(adx(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#(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#(incr(X)) -> incr#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(incr(X)) -> proper#(X) proper#(tail(X)) -> proper#(X) -> proper#(tail(X)) -> tail#(proper(X)) proper#(tail(X)) -> proper#(X) -> proper#(tail(X)) -> proper#(X) proper#(tail(X)) -> proper#(X) -> proper#(head(X)) -> head#(proper(X)) proper#(tail(X)) -> proper#(X) -> proper#(head(X)) -> proper#(X) proper#(tail(X)) -> proper#(X) -> proper#(adx(X)) -> adx#(proper(X)) proper#(tail(X)) -> proper#(X) -> proper#(adx(X)) -> proper#(X) proper#(tail(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) proper#(tail(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(tail(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(tail(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(tail(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(tail(X)) -> proper#(X) -> proper#(incr(X)) -> incr#(proper(X)) proper#(tail(X)) -> proper#(X) -> proper#(incr(X)) -> proper#(X) proper#(tail(X)) -> tail#(proper(X)) -> tail#(ok(X)) -> tail#(X) proper#(tail(X)) -> tail#(proper(X)) -> tail#(mark(X)) -> tail#(X) proper#(head(X)) -> proper#(X) -> proper#(tail(X)) -> tail#(proper(X)) proper#(head(X)) -> proper#(X) -> proper#(tail(X)) -> proper#(X) proper#(head(X)) -> proper#(X) -> proper#(head(X)) -> head#(proper(X)) proper#(head(X)) -> proper#(X) -> proper#(head(X)) -> proper#(X) proper#(head(X)) -> proper#(X) -> proper#(adx(X)) -> adx#(proper(X)) proper#(head(X)) -> proper#(X) -> proper#(adx(X)) -> proper#(X) proper#(head(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) proper#(head(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(head(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(head(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(head(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(head(X)) -> proper#(X) -> proper#(incr(X)) -> incr#(proper(X)) proper#(head(X)) -> proper#(X) -> proper#(incr(X)) -> proper#(X) proper#(head(X)) -> head#(proper(X)) -> head#(ok(X)) -> head#(X) proper#(head(X)) -> head#(proper(X)) -> head#(mark(X)) -> head#(X) proper#(adx(X)) -> proper#(X) -> proper#(tail(X)) -> tail#(proper(X)) proper#(adx(X)) -> proper#(X) -> proper#(tail(X)) -> proper#(X) proper#(adx(X)) -> proper#(X) -> proper#(head(X)) -> head#(proper(X)) proper#(adx(X)) -> proper#(X) -> proper#(head(X)) -> proper#(X) proper#(adx(X)) -> proper#(X) -> proper#(adx(X)) -> adx#(proper(X)) proper#(adx(X)) -> proper#(X) -> proper#(adx(X)) -> proper#(X) proper#(adx(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) proper#(adx(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(adx(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(adx(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(adx(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(adx(X)) -> proper#(X) -> proper#(incr(X)) -> incr#(proper(X)) proper#(adx(X)) -> proper#(X) -> proper#(incr(X)) -> proper#(X) proper#(adx(X)) -> adx#(proper(X)) -> adx#(ok(X)) -> adx#(X) proper#(adx(X)) -> adx#(proper(X)) -> adx#(mark(X)) -> adx#(X) proper#(s(X)) -> proper#(X) -> proper#(tail(X)) -> tail#(proper(X)) proper#(s(X)) -> proper#(X) -> proper#(tail(X)) -> proper#(X) proper#(s(X)) -> proper#(X) -> proper#(head(X)) -> head#(proper(X)) proper#(s(X)) -> proper#(X) -> proper#(head(X)) -> proper#(X) proper#(s(X)) -> proper#(X) -> proper#(adx(X)) -> adx#(proper(X)) proper#(s(X)) -> proper#(X) -> proper#(adx(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#(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#(incr(X)) -> incr#(proper(X)) proper#(s(X)) -> proper#(X) -> proper#(incr(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#(cons(X1,X2)) -> proper#(X2) -> proper#(tail(X)) -> tail#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(tail(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(head(X)) -> head#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(head(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(adx(X)) -> adx#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(adx(X)) -> proper#(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#(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#(incr(X)) -> incr#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(incr(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(tail(X)) -> tail#(proper(X)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(tail(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(head(X)) -> head#(proper(X)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(head(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(adx(X)) -> adx#(proper(X)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(adx(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#(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#(incr(X)) -> incr#(proper(X)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(incr(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#(incr(X)) -> proper#(X) -> proper#(tail(X)) -> tail#(proper(X)) proper#(incr(X)) -> proper#(X) -> proper#(tail(X)) -> proper#(X) proper#(incr(X)) -> proper#(X) -> proper#(head(X)) -> head#(proper(X)) proper#(incr(X)) -> proper#(X) -> proper#(head(X)) -> proper#(X) proper#(incr(X)) -> proper#(X) -> proper#(adx(X)) -> adx#(proper(X)) proper#(incr(X)) -> proper#(X) -> proper#(adx(X)) -> proper#(X) proper#(incr(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) proper#(incr(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(incr(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(incr(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(incr(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(incr(X)) -> proper#(X) -> proper#(incr(X)) -> incr#(proper(X)) proper#(incr(X)) -> proper#(X) -> proper#(incr(X)) -> proper#(X) proper#(incr(X)) -> incr#(proper(X)) -> incr#(ok(X)) -> incr#(X) proper#(incr(X)) -> incr#(proper(X)) -> incr#(mark(X)) -> incr#(X) tail#(ok(X)) -> tail#(X) -> tail#(ok(X)) -> tail#(X) tail#(ok(X)) -> tail#(X) -> tail#(mark(X)) -> tail#(X) tail#(mark(X)) -> tail#(X) -> tail#(ok(X)) -> tail#(X) tail#(mark(X)) -> tail#(X) -> tail#(mark(X)) -> tail#(X) head#(ok(X)) -> head#(X) -> head#(ok(X)) -> head#(X) head#(ok(X)) -> head#(X) -> head#(mark(X)) -> head#(X) head#(mark(X)) -> head#(X) -> head#(ok(X)) -> head#(X) head#(mark(X)) -> head#(X) -> head#(mark(X)) -> head#(X) adx#(ok(X)) -> adx#(X) -> adx#(ok(X)) -> adx#(X) adx#(ok(X)) -> adx#(X) -> adx#(mark(X)) -> adx#(X) adx#(mark(X)) -> adx#(X) -> adx#(ok(X)) -> adx#(X) adx#(mark(X)) -> adx#(X) -> adx#(mark(X)) -> adx#(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) 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) incr#(ok(X)) -> incr#(X) -> incr#(ok(X)) -> incr#(X) incr#(ok(X)) -> incr#(X) -> incr#(mark(X)) -> incr#(X) incr#(mark(X)) -> incr#(X) -> incr#(ok(X)) -> incr#(X) incr#(mark(X)) -> incr#(X) -> incr#(mark(X)) -> incr#(X) active#(tail(X)) -> tail#(active(X)) -> tail#(ok(X)) -> tail#(X) active#(tail(X)) -> tail#(active(X)) -> tail#(mark(X)) -> tail#(X) active#(tail(X)) -> active#(X) -> active#(tail(X)) -> tail#(active(X)) active#(tail(X)) -> active#(X) -> active#(tail(X)) -> active#(X) active#(tail(X)) -> active#(X) -> active#(head(X)) -> head#(active(X)) active#(tail(X)) -> active#(X) -> active#(head(X)) -> active#(X) active#(tail(X)) -> active#(X) -> active#(adx(X)) -> adx#(active(X)) active#(tail(X)) -> active#(X) -> active#(adx(X)) -> active#(X) active#(tail(X)) -> active#(X) -> active#(s(X)) -> s#(active(X)) active#(tail(X)) -> active#(X) -> active#(s(X)) -> active#(X) active#(tail(X)) -> active#(X) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(tail(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) active#(tail(X)) -> active#(X) -> active#(incr(X)) -> incr#(active(X)) active#(tail(X)) -> active#(X) -> active#(incr(X)) -> active#(X) active#(tail(X)) -> active#(X) -> active#(zeros()) -> cons#(0(),zeros()) active#(tail(X)) -> active#(X) -> active#(nats()) -> adx#(zeros()) active#(tail(X)) -> active#(X) -> active#(adx(cons(X,L))) -> incr#(cons(X,adx(L))) active#(tail(X)) -> active#(X) -> active#(adx(cons(X,L))) -> cons#(X,adx(L)) active#(tail(X)) -> active#(X) -> active#(adx(cons(X,L))) -> adx#(L) active#(tail(X)) -> active#(X) -> active#(incr(cons(X,L))) -> cons#(s(X),incr(L)) active#(tail(X)) -> active#(X) -> active#(incr(cons(X,L))) -> s#(X) active#(tail(X)) -> active#(X) -> active#(incr(cons(X,L))) -> incr#(L) active#(head(X)) -> head#(active(X)) -> head#(ok(X)) -> head#(X) active#(head(X)) -> head#(active(X)) -> head#(mark(X)) -> head#(X) active#(head(X)) -> active#(X) -> active#(tail(X)) -> tail#(active(X)) active#(head(X)) -> active#(X) -> active#(tail(X)) -> active#(X) active#(head(X)) -> active#(X) -> active#(head(X)) -> head#(active(X)) active#(head(X)) -> active#(X) -> active#(head(X)) -> active#(X) active#(head(X)) -> active#(X) -> active#(adx(X)) -> adx#(active(X)) active#(head(X)) -> active#(X) -> active#(adx(X)) -> active#(X) active#(head(X)) -> active#(X) -> active#(s(X)) -> s#(active(X)) active#(head(X)) -> active#(X) -> active#(s(X)) -> active#(X) active#(head(X)) -> active#(X) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(head(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) active#(head(X)) -> active#(X) -> active#(incr(X)) -> incr#(active(X)) active#(head(X)) -> active#(X) -> active#(incr(X)) -> active#(X) active#(head(X)) -> active#(X) -> active#(zeros()) -> cons#(0(),zeros()) active#(head(X)) -> active#(X) -> active#(nats()) -> adx#(zeros()) active#(head(X)) -> active#(X) -> active#(adx(cons(X,L))) -> incr#(cons(X,adx(L))) active#(head(X)) -> active#(X) -> active#(adx(cons(X,L))) -> cons#(X,adx(L)) active#(head(X)) -> active#(X) -> active#(adx(cons(X,L))) -> adx#(L) active#(head(X)) -> active#(X) -> active#(incr(cons(X,L))) -> cons#(s(X),incr(L)) active#(head(X)) -> active#(X) -> active#(incr(cons(X,L))) -> s#(X) active#(head(X)) -> active#(X) -> active#(incr(cons(X,L))) -> incr#(L) active#(zeros()) -> cons#(0(),zeros()) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) active#(zeros()) -> cons#(0(),zeros()) -> cons#(mark(X1),X2) -> cons#(X1,X2) active#(nats()) -> adx#(zeros()) -> adx#(ok(X)) -> adx#(X) active#(nats()) -> adx#(zeros()) -> adx#(mark(X)) -> adx#(X) active#(adx(cons(X,L))) -> adx#(L) -> adx#(ok(X)) -> adx#(X) active#(adx(cons(X,L))) -> adx#(L) -> adx#(mark(X)) -> adx#(X) active#(adx(cons(X,L))) -> cons#(X,adx(L)) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) active#(adx(cons(X,L))) -> cons#(X,adx(L)) -> cons#(mark(X1),X2) -> cons#(X1,X2) active#(adx(cons(X,L))) -> incr#(cons(X,adx(L))) -> incr#(ok(X)) -> incr#(X) active#(adx(cons(X,L))) -> incr#(cons(X,adx(L))) -> incr#(mark(X)) -> incr#(X) active#(adx(X)) -> adx#(active(X)) -> adx#(ok(X)) -> adx#(X) active#(adx(X)) -> adx#(active(X)) -> adx#(mark(X)) -> adx#(X) active#(adx(X)) -> active#(X) -> active#(tail(X)) -> tail#(active(X)) active#(adx(X)) -> active#(X) -> active#(tail(X)) -> active#(X) active#(adx(X)) -> active#(X) -> active#(head(X)) -> head#(active(X)) active#(adx(X)) -> active#(X) -> active#(head(X)) -> active#(X) active#(adx(X)) -> active#(X) -> active#(adx(X)) -> adx#(active(X)) active#(adx(X)) -> active#(X) -> active#(adx(X)) -> active#(X) active#(adx(X)) -> active#(X) -> active#(s(X)) -> s#(active(X)) active#(adx(X)) -> active#(X) -> active#(s(X)) -> active#(X) active#(adx(X)) -> active#(X) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(adx(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) active#(adx(X)) -> active#(X) -> active#(incr(X)) -> incr#(active(X)) active#(adx(X)) -> active#(X) -> active#(incr(X)) -> active#(X) active#(adx(X)) -> active#(X) -> active#(zeros()) -> cons#(0(),zeros()) active#(adx(X)) -> active#(X) -> active#(nats()) -> adx#(zeros()) active#(adx(X)) -> active#(X) -> active#(adx(cons(X,L))) -> incr#(cons(X,adx(L))) active#(adx(X)) -> active#(X) -> active#(adx(cons(X,L))) -> cons#(X,adx(L)) active#(adx(X)) -> active#(X) -> active#(adx(cons(X,L))) -> adx#(L) active#(adx(X)) -> active#(X) -> active#(incr(cons(X,L))) -> cons#(s(X),incr(L)) active#(adx(X)) -> active#(X) -> active#(incr(cons(X,L))) -> s#(X) active#(adx(X)) -> active#(X) -> active#(incr(cons(X,L))) -> incr#(L) 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#(tail(X)) -> tail#(active(X)) active#(s(X)) -> active#(X) -> active#(tail(X)) -> active#(X) active#(s(X)) -> active#(X) -> active#(head(X)) -> head#(active(X)) active#(s(X)) -> active#(X) -> active#(head(X)) -> active#(X) active#(s(X)) -> active#(X) -> active#(adx(X)) -> adx#(active(X)) active#(s(X)) -> active#(X) -> active#(adx(X)) -> active#(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#(cons(X1,X2)) -> cons#(active(X1),X2) active#(s(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) active#(s(X)) -> active#(X) -> active#(incr(X)) -> incr#(active(X)) active#(s(X)) -> active#(X) -> active#(incr(X)) -> active#(X) active#(s(X)) -> active#(X) -> active#(zeros()) -> cons#(0(),zeros()) active#(s(X)) -> active#(X) -> active#(nats()) -> adx#(zeros()) active#(s(X)) -> active#(X) -> active#(adx(cons(X,L))) -> incr#(cons(X,adx(L))) active#(s(X)) -> active#(X) -> active#(adx(cons(X,L))) -> cons#(X,adx(L)) active#(s(X)) -> active#(X) -> active#(adx(cons(X,L))) -> adx#(L) active#(s(X)) -> active#(X) -> active#(incr(cons(X,L))) -> cons#(s(X),incr(L)) active#(s(X)) -> active#(X) -> active#(incr(cons(X,L))) -> s#(X) active#(s(X)) -> active#(X) -> active#(incr(cons(X,L))) -> incr#(L) 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#(tail(X)) -> tail#(active(X)) active#(cons(X1,X2)) -> active#(X1) -> active#(tail(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) -> active#(head(X)) -> head#(active(X)) active#(cons(X1,X2)) -> active#(X1) -> active#(head(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) -> active#(adx(X)) -> adx#(active(X)) active#(cons(X1,X2)) -> active#(X1) -> active#(adx(X)) -> active#(X) 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#(incr(X)) -> incr#(active(X)) active#(cons(X1,X2)) -> active#(X1) -> active#(incr(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) -> active#(zeros()) -> cons#(0(),zeros()) active#(cons(X1,X2)) -> active#(X1) -> active#(nats()) -> adx#(zeros()) active#(cons(X1,X2)) -> active#(X1) -> active#(adx(cons(X,L))) -> incr#(cons(X,adx(L))) active#(cons(X1,X2)) -> active#(X1) -> active#(adx(cons(X,L))) -> cons#(X,adx(L)) active#(cons(X1,X2)) -> active#(X1) -> active#(adx(cons(X,L))) -> adx#(L) active#(cons(X1,X2)) -> active#(X1) -> active#(incr(cons(X,L))) -> cons#(s(X),incr(L)) active#(cons(X1,X2)) -> active#(X1) -> active#(incr(cons(X,L))) -> s#(X) active#(cons(X1,X2)) -> active#(X1) -> active#(incr(cons(X,L))) -> incr#(L) active#(incr(cons(X,L))) -> cons#(s(X),incr(L)) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) active#(incr(cons(X,L))) -> cons#(s(X),incr(L)) -> cons#(mark(X1),X2) -> cons#(X1,X2) active#(incr(cons(X,L))) -> s#(X) -> s#(ok(X)) -> s#(X) active#(incr(cons(X,L))) -> s#(X) -> s#(mark(X)) -> s#(X) active#(incr(cons(X,L))) -> incr#(L) -> incr#(ok(X)) -> incr#(X) active#(incr(cons(X,L))) -> incr#(L) -> incr#(mark(X)) -> incr#(X) active#(incr(X)) -> incr#(active(X)) -> incr#(ok(X)) -> incr#(X) active#(incr(X)) -> incr#(active(X)) -> incr#(mark(X)) -> incr#(X) active#(incr(X)) -> active#(X) -> active#(tail(X)) -> tail#(active(X)) active#(incr(X)) -> active#(X) -> active#(tail(X)) -> active#(X) active#(incr(X)) -> active#(X) -> active#(head(X)) -> head#(active(X)) active#(incr(X)) -> active#(X) -> active#(head(X)) -> active#(X) active#(incr(X)) -> active#(X) -> active#(adx(X)) -> adx#(active(X)) active#(incr(X)) -> active#(X) -> active#(adx(X)) -> active#(X) active#(incr(X)) -> active#(X) -> active#(s(X)) -> s#(active(X)) active#(incr(X)) -> active#(X) -> active#(s(X)) -> active#(X) active#(incr(X)) -> active#(X) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(incr(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) active#(incr(X)) -> active#(X) -> active#(incr(X)) -> incr#(active(X)) active#(incr(X)) -> active#(X) -> active#(incr(X)) -> active#(X) active#(incr(X)) -> active#(X) -> active#(zeros()) -> cons#(0(),zeros()) active#(incr(X)) -> active#(X) -> active#(nats()) -> adx#(zeros()) active#(incr(X)) -> active#(X) -> active#(adx(cons(X,L))) -> incr#(cons(X,adx(L))) active#(incr(X)) -> active#(X) -> active#(adx(cons(X,L))) -> cons#(X,adx(L)) active#(incr(X)) -> active#(X) -> active#(adx(cons(X,L))) -> adx#(L) active#(incr(X)) -> active#(X) -> active#(incr(cons(X,L))) -> cons#(s(X),incr(L)) active#(incr(X)) -> active#(X) -> active#(incr(cons(X,L))) -> s#(X) active#(incr(X)) -> active#(X) -> active#(incr(cons(X,L))) -> incr#(L) SCC Processor: #sccs: 9 #rules: 27 #arcs: 316/2401 DPs: top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> top#(proper(X)) TRS: active(incr(nil())) -> mark(nil()) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(adx(nil())) -> mark(nil()) active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(head(cons(X,L))) -> mark(X) active(tail(cons(X,L))) -> mark(L) active(incr(X)) -> incr(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(adx(X)) -> adx(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) incr(mark(X)) -> mark(incr(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) adx(mark(X)) -> mark(adx(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(incr(X)) -> incr(proper(X)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(adx(X)) -> adx(proper(X)) proper(nats()) -> ok(nats()) proper(zeros()) -> ok(zeros()) proper(0()) -> ok(0()) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) incr(ok(X)) -> ok(incr(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) adx(ok(X)) -> ok(adx(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: active#(incr(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) active#(s(X)) -> active#(X) active#(adx(X)) -> active#(X) active#(head(X)) -> active#(X) active#(tail(X)) -> active#(X) TRS: active(incr(nil())) -> mark(nil()) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(adx(nil())) -> mark(nil()) active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(head(cons(X,L))) -> mark(X) active(tail(cons(X,L))) -> mark(L) active(incr(X)) -> incr(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(adx(X)) -> adx(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) incr(mark(X)) -> mark(incr(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) adx(mark(X)) -> mark(adx(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(incr(X)) -> incr(proper(X)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(adx(X)) -> adx(proper(X)) proper(nats()) -> ok(nats()) proper(zeros()) -> ok(zeros()) proper(0()) -> ok(0()) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) incr(ok(X)) -> ok(incr(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) adx(ok(X)) -> ok(adx(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(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(incr(nil())) -> mark(nil()) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(adx(nil())) -> mark(nil()) active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(head(cons(X,L))) -> mark(X) active(tail(cons(X,L))) -> mark(L) active(incr(X)) -> incr(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(adx(X)) -> adx(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) incr(mark(X)) -> mark(incr(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) adx(mark(X)) -> mark(adx(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(incr(X)) -> incr(proper(X)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(adx(X)) -> adx(proper(X)) proper(nats()) -> ok(nats()) proper(zeros()) -> ok(zeros()) proper(0()) -> ok(0()) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) incr(ok(X)) -> ok(incr(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) adx(ok(X)) -> ok(adx(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: proper#(incr(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(s(X)) -> proper#(X) proper#(adx(X)) -> proper#(X) proper#(head(X)) -> proper#(X) proper#(tail(X)) -> proper#(X) TRS: active(incr(nil())) -> mark(nil()) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(adx(nil())) -> mark(nil()) active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(head(cons(X,L))) -> mark(X) active(tail(cons(X,L))) -> mark(L) active(incr(X)) -> incr(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(adx(X)) -> adx(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) incr(mark(X)) -> mark(incr(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) adx(mark(X)) -> mark(adx(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(incr(X)) -> incr(proper(X)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(adx(X)) -> adx(proper(X)) proper(nats()) -> ok(nats()) proper(zeros()) -> ok(zeros()) proper(0()) -> ok(0()) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) incr(ok(X)) -> ok(incr(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) adx(ok(X)) -> ok(adx(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(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(incr(nil())) -> mark(nil()) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(adx(nil())) -> mark(nil()) active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(head(cons(X,L))) -> mark(X) active(tail(cons(X,L))) -> mark(L) active(incr(X)) -> incr(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(adx(X)) -> adx(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) incr(mark(X)) -> mark(incr(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) adx(mark(X)) -> mark(adx(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(incr(X)) -> incr(proper(X)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(adx(X)) -> adx(proper(X)) proper(nats()) -> ok(nats()) proper(zeros()) -> ok(zeros()) proper(0()) -> ok(0()) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) incr(ok(X)) -> ok(incr(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) adx(ok(X)) -> ok(adx(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: tail#(mark(X)) -> tail#(X) tail#(ok(X)) -> tail#(X) TRS: active(incr(nil())) -> mark(nil()) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(adx(nil())) -> mark(nil()) active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(head(cons(X,L))) -> mark(X) active(tail(cons(X,L))) -> mark(L) active(incr(X)) -> incr(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(adx(X)) -> adx(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) incr(mark(X)) -> mark(incr(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) adx(mark(X)) -> mark(adx(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(incr(X)) -> incr(proper(X)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(adx(X)) -> adx(proper(X)) proper(nats()) -> ok(nats()) proper(zeros()) -> ok(zeros()) proper(0()) -> ok(0()) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) incr(ok(X)) -> ok(incr(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) adx(ok(X)) -> ok(adx(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(tail#) = 0 problem: DPs: TRS: active(incr(nil())) -> mark(nil()) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(adx(nil())) -> mark(nil()) active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(head(cons(X,L))) -> mark(X) active(tail(cons(X,L))) -> mark(L) active(incr(X)) -> incr(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(adx(X)) -> adx(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) incr(mark(X)) -> mark(incr(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) adx(mark(X)) -> mark(adx(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(incr(X)) -> incr(proper(X)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(adx(X)) -> adx(proper(X)) proper(nats()) -> ok(nats()) proper(zeros()) -> ok(zeros()) proper(0()) -> ok(0()) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) incr(ok(X)) -> ok(incr(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) adx(ok(X)) -> ok(adx(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: head#(mark(X)) -> head#(X) head#(ok(X)) -> head#(X) TRS: active(incr(nil())) -> mark(nil()) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(adx(nil())) -> mark(nil()) active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(head(cons(X,L))) -> mark(X) active(tail(cons(X,L))) -> mark(L) active(incr(X)) -> incr(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(adx(X)) -> adx(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) incr(mark(X)) -> mark(incr(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) adx(mark(X)) -> mark(adx(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(incr(X)) -> incr(proper(X)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(adx(X)) -> adx(proper(X)) proper(nats()) -> ok(nats()) proper(zeros()) -> ok(zeros()) proper(0()) -> ok(0()) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) incr(ok(X)) -> ok(incr(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) adx(ok(X)) -> ok(adx(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(head#) = 0 problem: DPs: TRS: active(incr(nil())) -> mark(nil()) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(adx(nil())) -> mark(nil()) active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(head(cons(X,L))) -> mark(X) active(tail(cons(X,L))) -> mark(L) active(incr(X)) -> incr(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(adx(X)) -> adx(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) incr(mark(X)) -> mark(incr(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) adx(mark(X)) -> mark(adx(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(incr(X)) -> incr(proper(X)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(adx(X)) -> adx(proper(X)) proper(nats()) -> ok(nats()) proper(zeros()) -> ok(zeros()) proper(0()) -> ok(0()) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) incr(ok(X)) -> ok(incr(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) adx(ok(X)) -> ok(adx(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: adx#(mark(X)) -> adx#(X) adx#(ok(X)) -> adx#(X) TRS: active(incr(nil())) -> mark(nil()) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(adx(nil())) -> mark(nil()) active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(head(cons(X,L))) -> mark(X) active(tail(cons(X,L))) -> mark(L) active(incr(X)) -> incr(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(adx(X)) -> adx(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) incr(mark(X)) -> mark(incr(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) adx(mark(X)) -> mark(adx(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(incr(X)) -> incr(proper(X)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(adx(X)) -> adx(proper(X)) proper(nats()) -> ok(nats()) proper(zeros()) -> ok(zeros()) proper(0()) -> ok(0()) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) incr(ok(X)) -> ok(incr(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) adx(ok(X)) -> ok(adx(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(adx#) = 0 problem: DPs: TRS: active(incr(nil())) -> mark(nil()) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(adx(nil())) -> mark(nil()) active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(head(cons(X,L))) -> mark(X) active(tail(cons(X,L))) -> mark(L) active(incr(X)) -> incr(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(adx(X)) -> adx(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) incr(mark(X)) -> mark(incr(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) adx(mark(X)) -> mark(adx(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(incr(X)) -> incr(proper(X)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(adx(X)) -> adx(proper(X)) proper(nats()) -> ok(nats()) proper(zeros()) -> ok(zeros()) proper(0()) -> ok(0()) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) incr(ok(X)) -> ok(incr(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) adx(ok(X)) -> ok(adx(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(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(incr(nil())) -> mark(nil()) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(adx(nil())) -> mark(nil()) active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(head(cons(X,L))) -> mark(X) active(tail(cons(X,L))) -> mark(L) active(incr(X)) -> incr(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(adx(X)) -> adx(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) incr(mark(X)) -> mark(incr(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) adx(mark(X)) -> mark(adx(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(incr(X)) -> incr(proper(X)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(adx(X)) -> adx(proper(X)) proper(nats()) -> ok(nats()) proper(zeros()) -> ok(zeros()) proper(0()) -> ok(0()) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) incr(ok(X)) -> ok(incr(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) adx(ok(X)) -> ok(adx(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(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(incr(nil())) -> mark(nil()) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(adx(nil())) -> mark(nil()) active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(head(cons(X,L))) -> mark(X) active(tail(cons(X,L))) -> mark(L) active(incr(X)) -> incr(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(adx(X)) -> adx(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) incr(mark(X)) -> mark(incr(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) adx(mark(X)) -> mark(adx(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(incr(X)) -> incr(proper(X)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(adx(X)) -> adx(proper(X)) proper(nats()) -> ok(nats()) proper(zeros()) -> ok(zeros()) proper(0()) -> ok(0()) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) incr(ok(X)) -> ok(incr(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) adx(ok(X)) -> ok(adx(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(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(incr(nil())) -> mark(nil()) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(adx(nil())) -> mark(nil()) active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(head(cons(X,L))) -> mark(X) active(tail(cons(X,L))) -> mark(L) active(incr(X)) -> incr(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(adx(X)) -> adx(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) incr(mark(X)) -> mark(incr(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) adx(mark(X)) -> mark(adx(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(incr(X)) -> incr(proper(X)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(adx(X)) -> adx(proper(X)) proper(nats()) -> ok(nats()) proper(zeros()) -> ok(zeros()) proper(0()) -> ok(0()) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) incr(ok(X)) -> ok(incr(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) adx(ok(X)) -> ok(adx(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(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(incr(nil())) -> mark(nil()) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(adx(nil())) -> mark(nil()) active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(head(cons(X,L))) -> mark(X) active(tail(cons(X,L))) -> mark(L) active(incr(X)) -> incr(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(adx(X)) -> adx(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) incr(mark(X)) -> mark(incr(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) adx(mark(X)) -> mark(adx(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(incr(X)) -> incr(proper(X)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(adx(X)) -> adx(proper(X)) proper(nats()) -> ok(nats()) proper(zeros()) -> ok(zeros()) proper(0()) -> ok(0()) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) incr(ok(X)) -> ok(incr(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) adx(ok(X)) -> ok(adx(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(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(incr(nil())) -> mark(nil()) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(adx(nil())) -> mark(nil()) active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(head(cons(X,L))) -> mark(X) active(tail(cons(X,L))) -> mark(L) active(incr(X)) -> incr(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(adx(X)) -> adx(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) incr(mark(X)) -> mark(incr(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) adx(mark(X)) -> mark(adx(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(incr(X)) -> incr(proper(X)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(adx(X)) -> adx(proper(X)) proper(nats()) -> ok(nats()) proper(zeros()) -> ok(zeros()) proper(0()) -> ok(0()) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) incr(ok(X)) -> ok(incr(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) adx(ok(X)) -> ok(adx(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: incr#(mark(X)) -> incr#(X) incr#(ok(X)) -> incr#(X) TRS: active(incr(nil())) -> mark(nil()) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(adx(nil())) -> mark(nil()) active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(head(cons(X,L))) -> mark(X) active(tail(cons(X,L))) -> mark(L) active(incr(X)) -> incr(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(adx(X)) -> adx(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) incr(mark(X)) -> mark(incr(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) adx(mark(X)) -> mark(adx(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(incr(X)) -> incr(proper(X)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(adx(X)) -> adx(proper(X)) proper(nats()) -> ok(nats()) proper(zeros()) -> ok(zeros()) proper(0()) -> ok(0()) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) incr(ok(X)) -> ok(incr(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) adx(ok(X)) -> ok(adx(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(incr#) = 0 problem: DPs: TRS: active(incr(nil())) -> mark(nil()) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(adx(nil())) -> mark(nil()) active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(head(cons(X,L))) -> mark(X) active(tail(cons(X,L))) -> mark(L) active(incr(X)) -> incr(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(adx(X)) -> adx(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) incr(mark(X)) -> mark(incr(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) adx(mark(X)) -> mark(adx(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(incr(X)) -> incr(proper(X)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(adx(X)) -> adx(proper(X)) proper(nats()) -> ok(nats()) proper(zeros()) -> ok(zeros()) proper(0()) -> ok(0()) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) incr(ok(X)) -> ok(incr(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) adx(ok(X)) -> ok(adx(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed