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)) EDG 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#(mark(X)) -> proper#(X) top#(ok(X)) -> top#(active(X)) -> top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> top#(active(X)) top#(ok(X)) -> active#(X) -> active#(incr(cons(X,L))) -> incr#(L) top#(ok(X)) -> active#(X) -> active#(incr(cons(X,L))) -> s#(X) top#(ok(X)) -> active#(X) -> active#(incr(cons(X,L))) -> cons#(s(X),incr(L)) top#(ok(X)) -> active#(X) -> active#(adx(cons(X,L))) -> 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))) -> incr#(cons(X,adx(L))) top#(ok(X)) -> active#(X) -> active#(nats()) -> adx#(zeros()) top#(ok(X)) -> active#(X) -> active#(zeros()) -> cons#(0(),zeros()) top#(ok(X)) -> active#(X) -> active#(incr(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(incr(X)) -> incr#(active(X)) top#(ok(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) top#(ok(X)) -> active#(X) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) top#(ok(X)) -> active#(X) -> active#(s(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(s(X)) -> s#(active(X)) top#(ok(X)) -> active#(X) -> active#(adx(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(adx(X)) -> adx#(active(X)) top#(ok(X)) -> active#(X) -> active#(head(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(head(X)) -> head#(active(X)) top#(ok(X)) -> active#(X) -> active#(tail(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(tail(X)) -> tail#(active(X)) top#(mark(X)) -> top#(proper(X)) -> top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) -> top#(mark(X)) -> top#(proper(X)) top#(mark(X)) -> top#(proper(X)) -> top#(ok(X)) -> active#(X) top#(mark(X)) -> top#(proper(X)) -> top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> proper#(X) -> proper#(incr(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(incr(X)) -> incr#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) top#(mark(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) top#(mark(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(adx(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(adx(X)) -> adx#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(head(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(head(X)) -> head#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(tail(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(tail(X)) -> tail#(proper(X)) proper#(tail(X)) -> proper#(X) -> proper#(incr(X)) -> proper#(X) proper#(tail(X)) -> proper#(X) -> proper#(incr(X)) -> incr#(proper(X)) proper#(tail(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(tail(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(tail(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(tail(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(tail(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) proper#(tail(X)) -> proper#(X) -> proper#(adx(X)) -> proper#(X) proper#(tail(X)) -> proper#(X) -> proper#(adx(X)) -> adx#(proper(X)) proper#(tail(X)) -> proper#(X) -> proper#(head(X)) -> proper#(X) proper#(tail(X)) -> proper#(X) -> proper#(head(X)) -> head#(proper(X)) proper#(tail(X)) -> proper#(X) -> proper#(tail(X)) -> proper#(X) proper#(tail(X)) -> proper#(X) -> proper#(tail(X)) -> tail#(proper(X)) proper#(tail(X)) -> tail#(proper(X)) -> tail#(mark(X)) -> tail#(X) proper#(tail(X)) -> tail#(proper(X)) -> tail#(ok(X)) -> tail#(X) proper#(head(X)) -> proper#(X) -> proper#(incr(X)) -> proper#(X) proper#(head(X)) -> proper#(X) -> proper#(incr(X)) -> incr#(proper(X)) proper#(head(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(head(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(head(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(head(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(head(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) proper#(head(X)) -> proper#(X) -> proper#(adx(X)) -> proper#(X) proper#(head(X)) -> proper#(X) -> proper#(adx(X)) -> adx#(proper(X)) proper#(head(X)) -> proper#(X) -> proper#(head(X)) -> proper#(X) proper#(head(X)) -> proper#(X) -> proper#(head(X)) -> head#(proper(X)) proper#(head(X)) -> proper#(X) -> proper#(tail(X)) -> proper#(X) proper#(head(X)) -> proper#(X) -> proper#(tail(X)) -> tail#(proper(X)) proper#(head(X)) -> head#(proper(X)) -> head#(mark(X)) -> head#(X) proper#(head(X)) -> head#(proper(X)) -> head#(ok(X)) -> head#(X) proper#(adx(X)) -> proper#(X) -> proper#(incr(X)) -> proper#(X) proper#(adx(X)) -> proper#(X) -> proper#(incr(X)) -> incr#(proper(X)) proper#(adx(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(adx(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(adx(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(adx(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(adx(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) proper#(adx(X)) -> proper#(X) -> proper#(adx(X)) -> proper#(X) proper#(adx(X)) -> proper#(X) -> proper#(adx(X)) -> adx#(proper(X)) proper#(adx(X)) -> proper#(X) -> proper#(head(X)) -> proper#(X) proper#(adx(X)) -> proper#(X) -> proper#(head(X)) -> head#(proper(X)) proper#(adx(X)) -> proper#(X) -> proper#(tail(X)) -> proper#(X) proper#(adx(X)) -> proper#(X) -> proper#(tail(X)) -> tail#(proper(X)) proper#(adx(X)) -> adx#(proper(X)) -> adx#(mark(X)) -> adx#(X) proper#(adx(X)) -> adx#(proper(X)) -> adx#(ok(X)) -> adx#(X) proper#(s(X)) -> proper#(X) -> proper#(incr(X)) -> proper#(X) proper#(s(X)) -> proper#(X) -> proper#(incr(X)) -> incr#(proper(X)) proper#(s(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(s(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(s(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(s(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) proper#(s(X)) -> proper#(X) -> proper#(adx(X)) -> proper#(X) proper#(s(X)) -> proper#(X) -> proper#(adx(X)) -> adx#(proper(X)) proper#(s(X)) -> proper#(X) -> proper#(head(X)) -> proper#(X) proper#(s(X)) -> proper#(X) -> proper#(head(X)) -> head#(proper(X)) proper#(s(X)) -> proper#(X) -> proper#(tail(X)) -> proper#(X) proper#(s(X)) -> proper#(X) -> proper#(tail(X)) -> tail#(proper(X)) proper#(s(X)) -> s#(proper(X)) -> s#(mark(X)) -> s#(X) proper#(s(X)) -> s#(proper(X)) -> s#(ok(X)) -> s#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(incr(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(incr(X)) -> incr#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> s#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(adx(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(adx(X)) -> adx#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(head(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(head(X)) -> head#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(tail(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(tail(X)) -> tail#(proper(X)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(incr(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(incr(X)) -> incr#(proper(X)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> s#(proper(X)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(adx(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(adx(X)) -> adx#(proper(X)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(head(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(head(X)) -> head#(proper(X)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(tail(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(tail(X)) -> tail#(proper(X)) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) -> cons#(mark(X1),X2) -> cons#(X1,X2) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) proper#(incr(X)) -> proper#(X) -> proper#(incr(X)) -> proper#(X) proper#(incr(X)) -> proper#(X) -> proper#(incr(X)) -> incr#(proper(X)) proper#(incr(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(incr(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(incr(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(incr(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(incr(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) proper#(incr(X)) -> proper#(X) -> proper#(adx(X)) -> proper#(X) proper#(incr(X)) -> proper#(X) -> proper#(adx(X)) -> adx#(proper(X)) proper#(incr(X)) -> proper#(X) -> proper#(head(X)) -> proper#(X) proper#(incr(X)) -> proper#(X) -> proper#(head(X)) -> head#(proper(X)) proper#(incr(X)) -> proper#(X) -> proper#(tail(X)) -> proper#(X) proper#(incr(X)) -> proper#(X) -> proper#(tail(X)) -> tail#(proper(X)) proper#(incr(X)) -> incr#(proper(X)) -> incr#(mark(X)) -> incr#(X) proper#(incr(X)) -> incr#(proper(X)) -> incr#(ok(X)) -> incr#(X) tail#(ok(X)) -> tail#(X) -> tail#(mark(X)) -> tail#(X) tail#(ok(X)) -> tail#(X) -> tail#(ok(X)) -> tail#(X) tail#(mark(X)) -> tail#(X) -> tail#(mark(X)) -> tail#(X) tail#(mark(X)) -> tail#(X) -> tail#(ok(X)) -> tail#(X) head#(ok(X)) -> head#(X) -> head#(mark(X)) -> head#(X) head#(ok(X)) -> head#(X) -> head#(ok(X)) -> head#(X) head#(mark(X)) -> head#(X) -> head#(mark(X)) -> head#(X) head#(mark(X)) -> head#(X) -> head#(ok(X)) -> head#(X) adx#(ok(X)) -> adx#(X) -> adx#(mark(X)) -> adx#(X) adx#(ok(X)) -> adx#(X) -> adx#(ok(X)) -> adx#(X) adx#(mark(X)) -> adx#(X) -> adx#(mark(X)) -> adx#(X) adx#(mark(X)) -> adx#(X) -> adx#(ok(X)) -> adx#(X) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) -> cons#(mark(X1),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#(mark(X1),X2) -> cons#(X1,X2) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) s#(ok(X)) -> s#(X) -> s#(mark(X)) -> s#(X) s#(ok(X)) -> s#(X) -> s#(ok(X)) -> s#(X) s#(mark(X)) -> s#(X) -> s#(mark(X)) -> s#(X) s#(mark(X)) -> s#(X) -> s#(ok(X)) -> s#(X) incr#(ok(X)) -> incr#(X) -> incr#(mark(X)) -> incr#(X) incr#(ok(X)) -> incr#(X) -> incr#(ok(X)) -> incr#(X) incr#(mark(X)) -> incr#(X) -> incr#(mark(X)) -> incr#(X) incr#(mark(X)) -> incr#(X) -> incr#(ok(X)) -> incr#(X) active#(tail(X)) -> tail#(active(X)) -> tail#(mark(X)) -> tail#(X) active#(tail(X)) -> tail#(active(X)) -> tail#(ok(X)) -> tail#(X) active#(tail(X)) -> active#(X) -> active#(incr(cons(X,L))) -> incr#(L) active#(tail(X)) -> active#(X) -> active#(incr(cons(X,L))) -> s#(X) active#(tail(X)) -> active#(X) -> active#(incr(cons(X,L))) -> cons#(s(X),incr(L)) active#(tail(X)) -> active#(X) -> active#(adx(cons(X,L))) -> 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))) -> incr#(cons(X,adx(L))) active#(tail(X)) -> active#(X) -> active#(nats()) -> adx#(zeros()) active#(tail(X)) -> active#(X) -> active#(zeros()) -> cons#(0(),zeros()) active#(tail(X)) -> active#(X) -> active#(incr(X)) -> active#(X) active#(tail(X)) -> active#(X) -> active#(incr(X)) -> incr#(active(X)) active#(tail(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) active#(tail(X)) -> active#(X) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(tail(X)) -> active#(X) -> active#(s(X)) -> active#(X) active#(tail(X)) -> active#(X) -> active#(s(X)) -> s#(active(X)) active#(tail(X)) -> active#(X) -> active#(adx(X)) -> active#(X) active#(tail(X)) -> active#(X) -> active#(adx(X)) -> adx#(active(X)) active#(tail(X)) -> active#(X) -> active#(head(X)) -> active#(X) active#(tail(X)) -> active#(X) -> active#(head(X)) -> head#(active(X)) active#(tail(X)) -> active#(X) -> active#(tail(X)) -> active#(X) active#(tail(X)) -> active#(X) -> active#(tail(X)) -> tail#(active(X)) active#(head(X)) -> head#(active(X)) -> head#(mark(X)) -> head#(X) active#(head(X)) -> head#(active(X)) -> head#(ok(X)) -> head#(X) active#(head(X)) -> active#(X) -> active#(incr(cons(X,L))) -> incr#(L) active#(head(X)) -> active#(X) -> active#(incr(cons(X,L))) -> s#(X) active#(head(X)) -> active#(X) -> active#(incr(cons(X,L))) -> cons#(s(X),incr(L)) active#(head(X)) -> active#(X) -> active#(adx(cons(X,L))) -> 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))) -> incr#(cons(X,adx(L))) active#(head(X)) -> active#(X) -> active#(nats()) -> adx#(zeros()) active#(head(X)) -> active#(X) -> active#(zeros()) -> cons#(0(),zeros()) active#(head(X)) -> active#(X) -> active#(incr(X)) -> active#(X) active#(head(X)) -> active#(X) -> active#(incr(X)) -> incr#(active(X)) active#(head(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) active#(head(X)) -> active#(X) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(head(X)) -> active#(X) -> active#(s(X)) -> active#(X) active#(head(X)) -> active#(X) -> active#(s(X)) -> s#(active(X)) active#(head(X)) -> active#(X) -> active#(adx(X)) -> active#(X) active#(head(X)) -> active#(X) -> active#(adx(X)) -> adx#(active(X)) active#(head(X)) -> active#(X) -> active#(head(X)) -> active#(X) active#(head(X)) -> active#(X) -> active#(head(X)) -> head#(active(X)) active#(head(X)) -> active#(X) -> active#(tail(X)) -> active#(X) active#(head(X)) -> active#(X) -> active#(tail(X)) -> tail#(active(X)) active#(adx(cons(X,L))) -> adx#(L) -> adx#(mark(X)) -> adx#(X) active#(adx(cons(X,L))) -> adx#(L) -> adx#(ok(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))) -> incr#(cons(X,adx(L))) -> incr#(mark(X)) -> incr#(X) active#(adx(cons(X,L))) -> incr#(cons(X,adx(L))) -> incr#(ok(X)) -> incr#(X) active#(adx(X)) -> adx#(active(X)) -> adx#(mark(X)) -> adx#(X) active#(adx(X)) -> adx#(active(X)) -> adx#(ok(X)) -> adx#(X) active#(adx(X)) -> active#(X) -> active#(incr(cons(X,L))) -> incr#(L) active#(adx(X)) -> active#(X) -> active#(incr(cons(X,L))) -> s#(X) active#(adx(X)) -> active#(X) -> active#(incr(cons(X,L))) -> cons#(s(X),incr(L)) active#(adx(X)) -> active#(X) -> active#(adx(cons(X,L))) -> 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))) -> incr#(cons(X,adx(L))) active#(adx(X)) -> active#(X) -> active#(nats()) -> adx#(zeros()) active#(adx(X)) -> active#(X) -> active#(zeros()) -> cons#(0(),zeros()) active#(adx(X)) -> active#(X) -> active#(incr(X)) -> active#(X) active#(adx(X)) -> active#(X) -> active#(incr(X)) -> incr#(active(X)) active#(adx(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) active#(adx(X)) -> active#(X) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(adx(X)) -> active#(X) -> active#(s(X)) -> active#(X) active#(adx(X)) -> active#(X) -> active#(s(X)) -> s#(active(X)) active#(adx(X)) -> active#(X) -> active#(adx(X)) -> active#(X) active#(adx(X)) -> active#(X) -> active#(adx(X)) -> adx#(active(X)) active#(adx(X)) -> active#(X) -> active#(head(X)) -> active#(X) active#(adx(X)) -> active#(X) -> active#(head(X)) -> head#(active(X)) active#(adx(X)) -> active#(X) -> active#(tail(X)) -> active#(X) active#(adx(X)) -> active#(X) -> active#(tail(X)) -> tail#(active(X)) active#(s(X)) -> s#(active(X)) -> s#(mark(X)) -> s#(X) active#(s(X)) -> s#(active(X)) -> s#(ok(X)) -> s#(X) active#(s(X)) -> active#(X) -> active#(incr(cons(X,L))) -> incr#(L) active#(s(X)) -> active#(X) -> active#(incr(cons(X,L))) -> s#(X) active#(s(X)) -> active#(X) -> active#(incr(cons(X,L))) -> cons#(s(X),incr(L)) active#(s(X)) -> active#(X) -> active#(adx(cons(X,L))) -> 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))) -> incr#(cons(X,adx(L))) active#(s(X)) -> active#(X) -> active#(nats()) -> adx#(zeros()) active#(s(X)) -> active#(X) -> active#(zeros()) -> cons#(0(),zeros()) active#(s(X)) -> active#(X) -> active#(incr(X)) -> active#(X) active#(s(X)) -> active#(X) -> active#(incr(X)) -> incr#(active(X)) active#(s(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) active#(s(X)) -> active#(X) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(s(X)) -> active#(X) -> active#(s(X)) -> active#(X) active#(s(X)) -> active#(X) -> active#(s(X)) -> s#(active(X)) active#(s(X)) -> active#(X) -> active#(adx(X)) -> active#(X) active#(s(X)) -> active#(X) -> active#(adx(X)) -> adx#(active(X)) active#(s(X)) -> active#(X) -> active#(head(X)) -> active#(X) active#(s(X)) -> active#(X) -> active#(head(X)) -> head#(active(X)) active#(s(X)) -> active#(X) -> active#(tail(X)) -> active#(X) active#(s(X)) -> active#(X) -> active#(tail(X)) -> tail#(active(X)) active#(cons(X1,X2)) -> cons#(active(X1),X2) -> cons#(mark(X1),X2) -> cons#(X1,X2) active#(cons(X1,X2)) -> cons#(active(X1),X2) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) active#(cons(X1,X2)) -> active#(X1) -> active#(incr(cons(X,L))) -> 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))) -> cons#(s(X),incr(L)) active#(cons(X1,X2)) -> active#(X1) -> active#(adx(cons(X,L))) -> 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))) -> incr#(cons(X,adx(L))) active#(cons(X1,X2)) -> active#(X1) -> active#(nats()) -> adx#(zeros()) active#(cons(X1,X2)) -> active#(X1) -> active#(zeros()) -> cons#(0(),zeros()) active#(cons(X1,X2)) -> active#(X1) -> active#(incr(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) -> active#(incr(X)) -> incr#(active(X)) active#(cons(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(cons(X1,X2)) -> active#(X1) -> active#(s(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) -> active#(s(X)) -> s#(active(X)) active#(cons(X1,X2)) -> active#(X1) -> active#(adx(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) -> active#(adx(X)) -> adx#(active(X)) active#(cons(X1,X2)) -> active#(X1) -> active#(head(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) -> active#(head(X)) -> head#(active(X)) active#(cons(X1,X2)) -> active#(X1) -> active#(tail(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) -> active#(tail(X)) -> tail#(active(X)) active#(incr(cons(X,L))) -> s#(X) -> s#(ok(X)) -> s#(X) active#(incr(cons(X,L))) -> incr#(L) -> incr#(mark(X)) -> incr#(X) active#(incr(cons(X,L))) -> incr#(L) -> incr#(ok(X)) -> incr#(X) active#(incr(X)) -> incr#(active(X)) -> incr#(mark(X)) -> incr#(X) active#(incr(X)) -> incr#(active(X)) -> incr#(ok(X)) -> incr#(X) active#(incr(X)) -> active#(X) -> active#(incr(cons(X,L))) -> incr#(L) active#(incr(X)) -> active#(X) -> active#(incr(cons(X,L))) -> s#(X) active#(incr(X)) -> active#(X) -> active#(incr(cons(X,L))) -> cons#(s(X),incr(L)) active#(incr(X)) -> active#(X) -> active#(adx(cons(X,L))) -> 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))) -> incr#(cons(X,adx(L))) active#(incr(X)) -> active#(X) -> active#(nats()) -> adx#(zeros()) active#(incr(X)) -> active#(X) -> active#(zeros()) -> cons#(0(),zeros()) active#(incr(X)) -> active#(X) -> active#(incr(X)) -> active#(X) active#(incr(X)) -> active#(X) -> active#(incr(X)) -> incr#(active(X)) active#(incr(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) active#(incr(X)) -> active#(X) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(incr(X)) -> active#(X) -> active#(s(X)) -> active#(X) active#(incr(X)) -> active#(X) -> active#(s(X)) -> s#(active(X)) active#(incr(X)) -> active#(X) -> active#(adx(X)) -> active#(X) active#(incr(X)) -> active#(X) -> active#(adx(X)) -> adx#(active(X)) active#(incr(X)) -> active#(X) -> active#(head(X)) -> active#(X) active#(incr(X)) -> active#(X) -> active#(head(X)) -> head#(active(X)) active#(incr(X)) -> active#(X) -> active#(tail(X)) -> active#(X) active#(incr(X)) -> active#(X) -> active#(tail(X)) -> tail#(active(X)) SCC Processor: #sccs: 9 #rules: 27 #arcs: 308/2401 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: proper#(tail(X)) -> proper#(X) proper#(head(X)) -> proper#(X) proper#(adx(X)) -> proper#(X) proper#(s(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) proper#(incr(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)) Matrix Interpretation Processor: dimension: 1 interpretation: [proper#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = x0, [tail](x0) = x0 + 1, [head](x0) = x0 + 1, [0] = 0, [zeros] = 1, [nats] = 1, [adx](x0) = x0 + 1, [s](x0) = x0 + 1, [cons](x0, x1) = x0 + x1 + 1, [mark](x0) = 1, [active](x0) = x0, [incr](x0) = x0 + 1, [nil] = 0 orientation: proper#(tail(X)) = X + 2 >= X + 1 = proper#(X) proper#(head(X)) = X + 2 >= X + 1 = proper#(X) proper#(adx(X)) = X + 2 >= X + 1 = proper#(X) proper#(s(X)) = X + 2 >= X + 1 = proper#(X) proper#(cons(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(cons(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) proper#(incr(X)) = X + 2 >= X + 1 = proper#(X) active(incr(nil())) = 1 >= 1 = mark(nil()) active(incr(cons(X,L))) = L + X + 2 >= 1 = mark(cons(s(X),incr(L))) active(adx(nil())) = 1 >= 1 = mark(nil()) active(adx(cons(X,L))) = L + X + 2 >= 1 = mark(incr(cons(X,adx(L)))) active(nats()) = 1 >= 1 = mark(adx(zeros())) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(head(cons(X,L))) = L + X + 2 >= 1 = mark(X) active(tail(cons(X,L))) = L + X + 2 >= 1 = mark(L) active(incr(X)) = X + 1 >= X + 1 = incr(active(X)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(adx(X)) = X + 1 >= X + 1 = adx(active(X)) active(head(X)) = X + 1 >= X + 1 = head(active(X)) active(tail(X)) = X + 1 >= X + 1 = tail(active(X)) incr(mark(X)) = 2 >= 1 = mark(incr(X)) cons(mark(X1),X2) = X2 + 2 >= 1 = mark(cons(X1,X2)) s(mark(X)) = 2 >= 1 = mark(s(X)) adx(mark(X)) = 2 >= 1 = mark(adx(X)) head(mark(X)) = 2 >= 1 = mark(head(X)) tail(mark(X)) = 2 >= 1 = mark(tail(X)) proper(incr(X)) = X + 1 >= X + 1 = incr(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) proper(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(proper(X1),proper(X2)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(adx(X)) = X + 1 >= X + 1 = adx(proper(X)) proper(nats()) = 1 >= 0 = ok(nats()) proper(zeros()) = 1 >= 0 = ok(zeros()) proper(0()) = 0 >= 0 = ok(0()) proper(head(X)) = X + 1 >= X + 1 = head(proper(X)) proper(tail(X)) = X + 1 >= X + 1 = tail(proper(X)) incr(ok(X)) = 1 >= 0 = ok(incr(X)) cons(ok(X1),ok(X2)) = 1 >= 0 = ok(cons(X1,X2)) s(ok(X)) = 1 >= 0 = ok(s(X)) adx(ok(X)) = 1 >= 0 = ok(adx(X)) head(ok(X)) = 1 >= 0 = ok(head(X)) tail(ok(X)) = 1 >= 0 = ok(tail(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) 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: active#(tail(X)) -> active#(X) active#(head(X)) -> active#(X) active#(adx(X)) -> active#(X) active#(s(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) active#(incr(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)) Matrix Interpretation Processor: dimension: 1 interpretation: [active#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [tail](x0) = x0 + 1, [head](x0) = x0 + 1, [0] = 1, [zeros] = 1, [nats] = 0, [adx](x0) = x0 + 1, [s](x0) = x0 + 1, [cons](x0, x1) = x0 + 1, [mark](x0) = 0, [active](x0) = x0 + 1, [incr](x0) = x0 + 1, [nil] = 0 orientation: active#(tail(X)) = X + 2 >= X + 1 = active#(X) active#(head(X)) = X + 2 >= X + 1 = active#(X) active#(adx(X)) = X + 2 >= X + 1 = active#(X) active#(s(X)) = X + 2 >= X + 1 = active#(X) active#(cons(X1,X2)) = X1 + 2 >= X1 + 1 = active#(X1) active#(incr(X)) = X + 2 >= X + 1 = active#(X) active(incr(nil())) = 2 >= 0 = mark(nil()) active(incr(cons(X,L))) = X + 3 >= 0 = mark(cons(s(X),incr(L))) active(adx(nil())) = 2 >= 0 = mark(nil()) active(adx(cons(X,L))) = X + 3 >= 0 = mark(incr(cons(X,adx(L)))) active(nats()) = 1 >= 0 = mark(adx(zeros())) active(zeros()) = 2 >= 0 = mark(cons(0(),zeros())) active(head(cons(X,L))) = X + 3 >= 0 = mark(X) active(tail(cons(X,L))) = X + 3 >= 0 = mark(L) active(incr(X)) = X + 2 >= X + 2 = incr(active(X)) active(cons(X1,X2)) = X1 + 2 >= X1 + 2 = cons(active(X1),X2) active(s(X)) = X + 2 >= X + 2 = s(active(X)) active(adx(X)) = X + 2 >= X + 2 = adx(active(X)) active(head(X)) = X + 2 >= X + 2 = head(active(X)) active(tail(X)) = X + 2 >= X + 2 = tail(active(X)) incr(mark(X)) = 1 >= 0 = mark(incr(X)) cons(mark(X1),X2) = 1 >= 0 = mark(cons(X1,X2)) s(mark(X)) = 1 >= 0 = mark(s(X)) adx(mark(X)) = 1 >= 0 = mark(adx(X)) head(mark(X)) = 1 >= 0 = mark(head(X)) tail(mark(X)) = 1 >= 0 = mark(tail(X)) proper(incr(X)) = X + 2 >= X + 2 = incr(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) proper(cons(X1,X2)) = X1 + 2 >= X1 + 2 = cons(proper(X1),proper(X2)) proper(s(X)) = X + 2 >= X + 2 = s(proper(X)) proper(adx(X)) = X + 2 >= X + 2 = adx(proper(X)) proper(nats()) = 1 >= 1 = ok(nats()) proper(zeros()) = 2 >= 2 = ok(zeros()) proper(0()) = 2 >= 2 = ok(0()) proper(head(X)) = X + 2 >= X + 2 = head(proper(X)) proper(tail(X)) = X + 2 >= X + 2 = tail(proper(X)) incr(ok(X)) = X + 2 >= X + 2 = ok(incr(X)) cons(ok(X1),ok(X2)) = X1 + 2 >= X1 + 2 = ok(cons(X1,X2)) s(ok(X)) = X + 2 >= X + 2 = ok(s(X)) adx(ok(X)) = X + 2 >= X + 2 = ok(adx(X)) head(ok(X)) = X + 2 >= X + 2 = ok(head(X)) tail(ok(X)) = X + 2 >= X + 2 = ok(tail(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) 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#(ok(X)) -> incr#(X) incr#(mark(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)) Matrix Interpretation Processor: dimension: 1 interpretation: [incr#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [tail](x0) = x0, [head](x0) = x0, [0] = 0, [zeros] = 0, [nats] = 1, [adx](x0) = x0, [s](x0) = x0, [cons](x0, x1) = x0 + x1, [mark](x0) = x0 + 1, [active](x0) = x0 + 1, [incr](x0) = x0, [nil] = 1 orientation: incr#(ok(X)) = X + 1 >= X + 1 = incr#(X) incr#(mark(X)) = X + 2 >= X + 1 = incr#(X) active(incr(nil())) = 2 >= 2 = mark(nil()) active(incr(cons(X,L))) = L + X + 1 >= L + X + 1 = mark(cons(s(X),incr(L))) active(adx(nil())) = 2 >= 2 = mark(nil()) active(adx(cons(X,L))) = L + X + 1 >= L + X + 1 = mark(incr(cons(X,adx(L)))) active(nats()) = 2 >= 1 = mark(adx(zeros())) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(head(cons(X,L))) = L + X + 1 >= X + 1 = mark(X) active(tail(cons(X,L))) = L + X + 1 >= L + 1 = mark(L) active(incr(X)) = X + 1 >= X + 1 = incr(active(X)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(adx(X)) = X + 1 >= X + 1 = adx(active(X)) active(head(X)) = X + 1 >= X + 1 = head(active(X)) active(tail(X)) = X + 1 >= X + 1 = tail(active(X)) incr(mark(X)) = X + 1 >= X + 1 = mark(incr(X)) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) adx(mark(X)) = X + 1 >= X + 1 = mark(adx(X)) head(mark(X)) = X + 1 >= X + 1 = mark(head(X)) tail(mark(X)) = X + 1 >= X + 1 = mark(tail(X)) proper(incr(X)) = X >= X = incr(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) proper(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(proper(X1),proper(X2)) proper(s(X)) = X >= X = s(proper(X)) proper(adx(X)) = X >= X = adx(proper(X)) proper(nats()) = 1 >= 1 = ok(nats()) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(0()) = 0 >= 0 = ok(0()) proper(head(X)) = X >= X = head(proper(X)) proper(tail(X)) = X >= X = tail(proper(X)) incr(ok(X)) = X >= X = ok(incr(X)) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) s(ok(X)) = X >= X = ok(s(X)) adx(ok(X)) = X >= X = ok(adx(X)) head(ok(X)) = X >= X = ok(head(X)) tail(ok(X)) = X >= X = ok(tail(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: 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)) Matrix Interpretation Processor: dimension: 1 interpretation: [incr#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [tail](x0) = x0, [head](x0) = x0, [0] = 1, [zeros] = 0, [nats] = 1, [adx](x0) = x0 + 1, [s](x0) = x0, [cons](x0, x1) = x1, [mark](x0) = 0, [active](x0) = x0 + 1, [incr](x0) = x0, [nil] = 0 orientation: incr#(ok(X)) = X + 2 >= X + 1 = incr#(X) active(incr(nil())) = 1 >= 0 = mark(nil()) active(incr(cons(X,L))) = L + 1 >= 0 = mark(cons(s(X),incr(L))) active(adx(nil())) = 2 >= 0 = mark(nil()) active(adx(cons(X,L))) = L + 2 >= 0 = mark(incr(cons(X,adx(L)))) active(nats()) = 2 >= 0 = mark(adx(zeros())) active(zeros()) = 1 >= 0 = mark(cons(0(),zeros())) active(head(cons(X,L))) = L + 1 >= 0 = mark(X) active(tail(cons(X,L))) = L + 1 >= 0 = mark(L) active(incr(X)) = X + 1 >= X + 1 = incr(active(X)) active(cons(X1,X2)) = X2 + 1 >= X2 = cons(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(adx(X)) = X + 2 >= X + 2 = adx(active(X)) active(head(X)) = X + 1 >= X + 1 = head(active(X)) active(tail(X)) = X + 1 >= X + 1 = tail(active(X)) incr(mark(X)) = 0 >= 0 = mark(incr(X)) cons(mark(X1),X2) = X2 >= 0 = mark(cons(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) adx(mark(X)) = 1 >= 0 = mark(adx(X)) head(mark(X)) = 0 >= 0 = mark(head(X)) tail(mark(X)) = 0 >= 0 = mark(tail(X)) proper(incr(X)) = X + 1 >= X + 1 = incr(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) proper(cons(X1,X2)) = X2 + 1 >= X2 + 1 = cons(proper(X1),proper(X2)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(adx(X)) = X + 2 >= X + 2 = adx(proper(X)) proper(nats()) = 2 >= 2 = ok(nats()) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(0()) = 2 >= 2 = ok(0()) proper(head(X)) = X + 1 >= X + 1 = head(proper(X)) proper(tail(X)) = X + 1 >= X + 1 = tail(proper(X)) incr(ok(X)) = X + 1 >= X + 1 = ok(incr(X)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) adx(ok(X)) = X + 2 >= X + 2 = ok(adx(X)) head(ok(X)) = X + 1 >= X + 1 = ok(head(X)) tail(ok(X)) = X + 1 >= X + 1 = ok(tail(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) 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#(ok(X1),ok(X2)) -> cons#(X1,X2) 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)) Matrix Interpretation Processor: dimension: 1 interpretation: [cons#](x0, x1) = x1 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [tail](x0) = x0, [head](x0) = x0, [0] = 0, [zeros] = 0, [nats] = 0, [adx](x0) = x0 + 1, [s](x0) = x0 + 1, [cons](x0, x1) = x1, [mark](x0) = 0, [active](x0) = x0, [incr](x0) = x0, [nil] = 0 orientation: cons#(ok(X1),ok(X2)) = X2 + 2 >= X2 + 1 = cons#(X1,X2) cons#(mark(X1),X2) = X2 + 1 >= X2 + 1 = cons#(X1,X2) active(incr(nil())) = 0 >= 0 = mark(nil()) active(incr(cons(X,L))) = L >= 0 = mark(cons(s(X),incr(L))) active(adx(nil())) = 1 >= 0 = mark(nil()) active(adx(cons(X,L))) = L + 1 >= 0 = mark(incr(cons(X,adx(L)))) active(nats()) = 0 >= 0 = mark(adx(zeros())) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(head(cons(X,L))) = L >= 0 = mark(X) active(tail(cons(X,L))) = L >= 0 = mark(L) active(incr(X)) = X >= X = incr(active(X)) active(cons(X1,X2)) = X2 >= X2 = cons(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(adx(X)) = X + 1 >= X + 1 = adx(active(X)) active(head(X)) = X >= X = head(active(X)) active(tail(X)) = X >= X = tail(active(X)) incr(mark(X)) = 0 >= 0 = mark(incr(X)) cons(mark(X1),X2) = X2 >= 0 = mark(cons(X1,X2)) s(mark(X)) = 1 >= 0 = mark(s(X)) adx(mark(X)) = 1 >= 0 = mark(adx(X)) head(mark(X)) = 0 >= 0 = mark(head(X)) tail(mark(X)) = 0 >= 0 = mark(tail(X)) proper(incr(X)) = X + 1 >= X + 1 = incr(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) proper(cons(X1,X2)) = X2 + 1 >= X2 + 1 = cons(proper(X1),proper(X2)) proper(s(X)) = X + 2 >= X + 2 = s(proper(X)) proper(adx(X)) = X + 2 >= X + 2 = adx(proper(X)) proper(nats()) = 1 >= 1 = ok(nats()) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(0()) = 1 >= 1 = ok(0()) proper(head(X)) = X + 1 >= X + 1 = head(proper(X)) proper(tail(X)) = X + 1 >= X + 1 = tail(proper(X)) incr(ok(X)) = X + 1 >= X + 1 = ok(incr(X)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) s(ok(X)) = X + 2 >= X + 2 = ok(s(X)) adx(ok(X)) = X + 2 >= X + 2 = ok(adx(X)) head(ok(X)) = X + 1 >= X + 1 = ok(head(X)) tail(ok(X)) = X + 1 >= X + 1 = ok(tail(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: cons#(mark(X1),X2) -> cons#(X1,X2) TRS: active(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)) Matrix Interpretation Processor: dimension: 1 interpretation: [cons#](x0, x1) = x0 + 1, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = 0, [tail](x0) = x0, [head](x0) = x0, [0] = 0, [zeros] = 0, [nats] = 1, [adx](x0) = x0, [s](x0) = x0, [cons](x0, x1) = x0 + x1, [mark](x0) = x0 + 1, [active](x0) = x0 + 1, [incr](x0) = x0, [nil] = 0 orientation: cons#(mark(X1),X2) = X1 + 2 >= X1 + 1 = cons#(X1,X2) active(incr(nil())) = 1 >= 1 = mark(nil()) active(incr(cons(X,L))) = L + X + 1 >= L + X + 1 = mark(cons(s(X),incr(L))) active(adx(nil())) = 1 >= 1 = mark(nil()) active(adx(cons(X,L))) = L + X + 1 >= L + X + 1 = mark(incr(cons(X,adx(L)))) active(nats()) = 2 >= 1 = mark(adx(zeros())) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(head(cons(X,L))) = L + X + 1 >= X + 1 = mark(X) active(tail(cons(X,L))) = L + X + 1 >= L + 1 = mark(L) active(incr(X)) = X + 1 >= X + 1 = incr(active(X)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(adx(X)) = X + 1 >= X + 1 = adx(active(X)) active(head(X)) = X + 1 >= X + 1 = head(active(X)) active(tail(X)) = X + 1 >= X + 1 = tail(active(X)) incr(mark(X)) = X + 1 >= X + 1 = mark(incr(X)) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) adx(mark(X)) = X + 1 >= X + 1 = mark(adx(X)) head(mark(X)) = X + 1 >= X + 1 = mark(head(X)) tail(mark(X)) = X + 1 >= X + 1 = mark(tail(X)) proper(incr(X)) = 0 >= 0 = incr(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(adx(X)) = 0 >= 0 = adx(proper(X)) proper(nats()) = 0 >= 0 = ok(nats()) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(0()) = 0 >= 0 = ok(0()) proper(head(X)) = 0 >= 0 = head(proper(X)) proper(tail(X)) = 0 >= 0 = tail(proper(X)) incr(ok(X)) = 0 >= 0 = ok(incr(X)) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) s(ok(X)) = 0 >= 0 = ok(s(X)) adx(ok(X)) = 0 >= 0 = ok(adx(X)) head(ok(X)) = 0 >= 0 = ok(head(X)) tail(ok(X)) = 0 >= 0 = ok(tail(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) 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#(ok(X)) -> s#(X) s#(mark(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)) Matrix Interpretation Processor: dimension: 1 interpretation: [s#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [tail](x0) = x0, [head](x0) = x0, [0] = 0, [zeros] = 0, [nats] = 1, [adx](x0) = x0, [s](x0) = x0, [cons](x0, x1) = x0 + x1, [mark](x0) = x0 + 1, [active](x0) = x0 + 1, [incr](x0) = x0, [nil] = 1 orientation: s#(ok(X)) = X + 1 >= X + 1 = s#(X) s#(mark(X)) = X + 2 >= X + 1 = s#(X) active(incr(nil())) = 2 >= 2 = mark(nil()) active(incr(cons(X,L))) = L + X + 1 >= L + X + 1 = mark(cons(s(X),incr(L))) active(adx(nil())) = 2 >= 2 = mark(nil()) active(adx(cons(X,L))) = L + X + 1 >= L + X + 1 = mark(incr(cons(X,adx(L)))) active(nats()) = 2 >= 1 = mark(adx(zeros())) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(head(cons(X,L))) = L + X + 1 >= X + 1 = mark(X) active(tail(cons(X,L))) = L + X + 1 >= L + 1 = mark(L) active(incr(X)) = X + 1 >= X + 1 = incr(active(X)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(adx(X)) = X + 1 >= X + 1 = adx(active(X)) active(head(X)) = X + 1 >= X + 1 = head(active(X)) active(tail(X)) = X + 1 >= X + 1 = tail(active(X)) incr(mark(X)) = X + 1 >= X + 1 = mark(incr(X)) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) adx(mark(X)) = X + 1 >= X + 1 = mark(adx(X)) head(mark(X)) = X + 1 >= X + 1 = mark(head(X)) tail(mark(X)) = X + 1 >= X + 1 = mark(tail(X)) proper(incr(X)) = X >= X = incr(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) proper(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(proper(X1),proper(X2)) proper(s(X)) = X >= X = s(proper(X)) proper(adx(X)) = X >= X = adx(proper(X)) proper(nats()) = 1 >= 1 = ok(nats()) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(0()) = 0 >= 0 = ok(0()) proper(head(X)) = X >= X = head(proper(X)) proper(tail(X)) = X >= X = tail(proper(X)) incr(ok(X)) = X >= X = ok(incr(X)) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) s(ok(X)) = X >= X = ok(s(X)) adx(ok(X)) = X >= X = ok(adx(X)) head(ok(X)) = X >= X = ok(head(X)) tail(ok(X)) = X >= X = ok(tail(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: 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)) Matrix Interpretation Processor: dimension: 1 interpretation: [s#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [tail](x0) = x0, [head](x0) = x0, [0] = 1, [zeros] = 0, [nats] = 1, [adx](x0) = x0 + 1, [s](x0) = x0, [cons](x0, x1) = x1, [mark](x0) = 0, [active](x0) = x0 + 1, [incr](x0) = x0, [nil] = 0 orientation: s#(ok(X)) = X + 2 >= X + 1 = s#(X) active(incr(nil())) = 1 >= 0 = mark(nil()) active(incr(cons(X,L))) = L + 1 >= 0 = mark(cons(s(X),incr(L))) active(adx(nil())) = 2 >= 0 = mark(nil()) active(adx(cons(X,L))) = L + 2 >= 0 = mark(incr(cons(X,adx(L)))) active(nats()) = 2 >= 0 = mark(adx(zeros())) active(zeros()) = 1 >= 0 = mark(cons(0(),zeros())) active(head(cons(X,L))) = L + 1 >= 0 = mark(X) active(tail(cons(X,L))) = L + 1 >= 0 = mark(L) active(incr(X)) = X + 1 >= X + 1 = incr(active(X)) active(cons(X1,X2)) = X2 + 1 >= X2 = cons(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(adx(X)) = X + 2 >= X + 2 = adx(active(X)) active(head(X)) = X + 1 >= X + 1 = head(active(X)) active(tail(X)) = X + 1 >= X + 1 = tail(active(X)) incr(mark(X)) = 0 >= 0 = mark(incr(X)) cons(mark(X1),X2) = X2 >= 0 = mark(cons(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) adx(mark(X)) = 1 >= 0 = mark(adx(X)) head(mark(X)) = 0 >= 0 = mark(head(X)) tail(mark(X)) = 0 >= 0 = mark(tail(X)) proper(incr(X)) = X + 1 >= X + 1 = incr(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) proper(cons(X1,X2)) = X2 + 1 >= X2 + 1 = cons(proper(X1),proper(X2)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(adx(X)) = X + 2 >= X + 2 = adx(proper(X)) proper(nats()) = 2 >= 2 = ok(nats()) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(0()) = 2 >= 2 = ok(0()) proper(head(X)) = X + 1 >= X + 1 = head(proper(X)) proper(tail(X)) = X + 1 >= X + 1 = tail(proper(X)) incr(ok(X)) = X + 1 >= X + 1 = ok(incr(X)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) adx(ok(X)) = X + 2 >= X + 2 = ok(adx(X)) head(ok(X)) = X + 1 >= X + 1 = ok(head(X)) tail(ok(X)) = X + 1 >= X + 1 = ok(tail(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) 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#(ok(X)) -> adx#(X) adx#(mark(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)) Matrix Interpretation Processor: dimension: 1 interpretation: [adx#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [tail](x0) = x0, [head](x0) = x0, [0] = 0, [zeros] = 0, [nats] = 1, [adx](x0) = x0, [s](x0) = x0, [cons](x0, x1) = x0 + x1, [mark](x0) = x0 + 1, [active](x0) = x0 + 1, [incr](x0) = x0, [nil] = 1 orientation: adx#(ok(X)) = X + 1 >= X + 1 = adx#(X) adx#(mark(X)) = X + 2 >= X + 1 = adx#(X) active(incr(nil())) = 2 >= 2 = mark(nil()) active(incr(cons(X,L))) = L + X + 1 >= L + X + 1 = mark(cons(s(X),incr(L))) active(adx(nil())) = 2 >= 2 = mark(nil()) active(adx(cons(X,L))) = L + X + 1 >= L + X + 1 = mark(incr(cons(X,adx(L)))) active(nats()) = 2 >= 1 = mark(adx(zeros())) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(head(cons(X,L))) = L + X + 1 >= X + 1 = mark(X) active(tail(cons(X,L))) = L + X + 1 >= L + 1 = mark(L) active(incr(X)) = X + 1 >= X + 1 = incr(active(X)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(adx(X)) = X + 1 >= X + 1 = adx(active(X)) active(head(X)) = X + 1 >= X + 1 = head(active(X)) active(tail(X)) = X + 1 >= X + 1 = tail(active(X)) incr(mark(X)) = X + 1 >= X + 1 = mark(incr(X)) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) adx(mark(X)) = X + 1 >= X + 1 = mark(adx(X)) head(mark(X)) = X + 1 >= X + 1 = mark(head(X)) tail(mark(X)) = X + 1 >= X + 1 = mark(tail(X)) proper(incr(X)) = X >= X = incr(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) proper(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(proper(X1),proper(X2)) proper(s(X)) = X >= X = s(proper(X)) proper(adx(X)) = X >= X = adx(proper(X)) proper(nats()) = 1 >= 1 = ok(nats()) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(0()) = 0 >= 0 = ok(0()) proper(head(X)) = X >= X = head(proper(X)) proper(tail(X)) = X >= X = tail(proper(X)) incr(ok(X)) = X >= X = ok(incr(X)) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) s(ok(X)) = X >= X = ok(s(X)) adx(ok(X)) = X >= X = ok(adx(X)) head(ok(X)) = X >= X = ok(head(X)) tail(ok(X)) = X >= X = ok(tail(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: 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)) Matrix Interpretation Processor: dimension: 1 interpretation: [adx#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [tail](x0) = x0, [head](x0) = x0, [0] = 1, [zeros] = 0, [nats] = 1, [adx](x0) = x0 + 1, [s](x0) = x0, [cons](x0, x1) = x1, [mark](x0) = 0, [active](x0) = x0 + 1, [incr](x0) = x0, [nil] = 0 orientation: adx#(ok(X)) = X + 2 >= X + 1 = adx#(X) active(incr(nil())) = 1 >= 0 = mark(nil()) active(incr(cons(X,L))) = L + 1 >= 0 = mark(cons(s(X),incr(L))) active(adx(nil())) = 2 >= 0 = mark(nil()) active(adx(cons(X,L))) = L + 2 >= 0 = mark(incr(cons(X,adx(L)))) active(nats()) = 2 >= 0 = mark(adx(zeros())) active(zeros()) = 1 >= 0 = mark(cons(0(),zeros())) active(head(cons(X,L))) = L + 1 >= 0 = mark(X) active(tail(cons(X,L))) = L + 1 >= 0 = mark(L) active(incr(X)) = X + 1 >= X + 1 = incr(active(X)) active(cons(X1,X2)) = X2 + 1 >= X2 = cons(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(adx(X)) = X + 2 >= X + 2 = adx(active(X)) active(head(X)) = X + 1 >= X + 1 = head(active(X)) active(tail(X)) = X + 1 >= X + 1 = tail(active(X)) incr(mark(X)) = 0 >= 0 = mark(incr(X)) cons(mark(X1),X2) = X2 >= 0 = mark(cons(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) adx(mark(X)) = 1 >= 0 = mark(adx(X)) head(mark(X)) = 0 >= 0 = mark(head(X)) tail(mark(X)) = 0 >= 0 = mark(tail(X)) proper(incr(X)) = X + 1 >= X + 1 = incr(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) proper(cons(X1,X2)) = X2 + 1 >= X2 + 1 = cons(proper(X1),proper(X2)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(adx(X)) = X + 2 >= X + 2 = adx(proper(X)) proper(nats()) = 2 >= 2 = ok(nats()) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(0()) = 2 >= 2 = ok(0()) proper(head(X)) = X + 1 >= X + 1 = head(proper(X)) proper(tail(X)) = X + 1 >= X + 1 = tail(proper(X)) incr(ok(X)) = X + 1 >= X + 1 = ok(incr(X)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) adx(ok(X)) = X + 2 >= X + 2 = ok(adx(X)) head(ok(X)) = X + 1 >= X + 1 = ok(head(X)) tail(ok(X)) = X + 1 >= X + 1 = ok(tail(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) 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#(ok(X)) -> head#(X) head#(mark(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)) Matrix Interpretation Processor: dimension: 1 interpretation: [head#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [tail](x0) = x0, [head](x0) = x0, [0] = 0, [zeros] = 0, [nats] = 1, [adx](x0) = x0, [s](x0) = x0, [cons](x0, x1) = x0 + x1, [mark](x0) = x0 + 1, [active](x0) = x0 + 1, [incr](x0) = x0, [nil] = 1 orientation: head#(ok(X)) = X + 1 >= X + 1 = head#(X) head#(mark(X)) = X + 2 >= X + 1 = head#(X) active(incr(nil())) = 2 >= 2 = mark(nil()) active(incr(cons(X,L))) = L + X + 1 >= L + X + 1 = mark(cons(s(X),incr(L))) active(adx(nil())) = 2 >= 2 = mark(nil()) active(adx(cons(X,L))) = L + X + 1 >= L + X + 1 = mark(incr(cons(X,adx(L)))) active(nats()) = 2 >= 1 = mark(adx(zeros())) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(head(cons(X,L))) = L + X + 1 >= X + 1 = mark(X) active(tail(cons(X,L))) = L + X + 1 >= L + 1 = mark(L) active(incr(X)) = X + 1 >= X + 1 = incr(active(X)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(adx(X)) = X + 1 >= X + 1 = adx(active(X)) active(head(X)) = X + 1 >= X + 1 = head(active(X)) active(tail(X)) = X + 1 >= X + 1 = tail(active(X)) incr(mark(X)) = X + 1 >= X + 1 = mark(incr(X)) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) adx(mark(X)) = X + 1 >= X + 1 = mark(adx(X)) head(mark(X)) = X + 1 >= X + 1 = mark(head(X)) tail(mark(X)) = X + 1 >= X + 1 = mark(tail(X)) proper(incr(X)) = X >= X = incr(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) proper(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(proper(X1),proper(X2)) proper(s(X)) = X >= X = s(proper(X)) proper(adx(X)) = X >= X = adx(proper(X)) proper(nats()) = 1 >= 1 = ok(nats()) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(0()) = 0 >= 0 = ok(0()) proper(head(X)) = X >= X = head(proper(X)) proper(tail(X)) = X >= X = tail(proper(X)) incr(ok(X)) = X >= X = ok(incr(X)) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) s(ok(X)) = X >= X = ok(s(X)) adx(ok(X)) = X >= X = ok(adx(X)) head(ok(X)) = X >= X = ok(head(X)) tail(ok(X)) = X >= X = ok(tail(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: 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)) Matrix Interpretation Processor: dimension: 1 interpretation: [head#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [tail](x0) = x0, [head](x0) = x0, [0] = 1, [zeros] = 0, [nats] = 1, [adx](x0) = x0 + 1, [s](x0) = x0, [cons](x0, x1) = x1, [mark](x0) = 0, [active](x0) = x0 + 1, [incr](x0) = x0, [nil] = 0 orientation: head#(ok(X)) = X + 2 >= X + 1 = head#(X) active(incr(nil())) = 1 >= 0 = mark(nil()) active(incr(cons(X,L))) = L + 1 >= 0 = mark(cons(s(X),incr(L))) active(adx(nil())) = 2 >= 0 = mark(nil()) active(adx(cons(X,L))) = L + 2 >= 0 = mark(incr(cons(X,adx(L)))) active(nats()) = 2 >= 0 = mark(adx(zeros())) active(zeros()) = 1 >= 0 = mark(cons(0(),zeros())) active(head(cons(X,L))) = L + 1 >= 0 = mark(X) active(tail(cons(X,L))) = L + 1 >= 0 = mark(L) active(incr(X)) = X + 1 >= X + 1 = incr(active(X)) active(cons(X1,X2)) = X2 + 1 >= X2 = cons(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(adx(X)) = X + 2 >= X + 2 = adx(active(X)) active(head(X)) = X + 1 >= X + 1 = head(active(X)) active(tail(X)) = X + 1 >= X + 1 = tail(active(X)) incr(mark(X)) = 0 >= 0 = mark(incr(X)) cons(mark(X1),X2) = X2 >= 0 = mark(cons(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) adx(mark(X)) = 1 >= 0 = mark(adx(X)) head(mark(X)) = 0 >= 0 = mark(head(X)) tail(mark(X)) = 0 >= 0 = mark(tail(X)) proper(incr(X)) = X + 1 >= X + 1 = incr(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) proper(cons(X1,X2)) = X2 + 1 >= X2 + 1 = cons(proper(X1),proper(X2)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(adx(X)) = X + 2 >= X + 2 = adx(proper(X)) proper(nats()) = 2 >= 2 = ok(nats()) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(0()) = 2 >= 2 = ok(0()) proper(head(X)) = X + 1 >= X + 1 = head(proper(X)) proper(tail(X)) = X + 1 >= X + 1 = tail(proper(X)) incr(ok(X)) = X + 1 >= X + 1 = ok(incr(X)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) adx(ok(X)) = X + 2 >= X + 2 = ok(adx(X)) head(ok(X)) = X + 1 >= X + 1 = ok(head(X)) tail(ok(X)) = X + 1 >= X + 1 = ok(tail(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) 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#(ok(X)) -> tail#(X) tail#(mark(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)) Matrix Interpretation Processor: dimension: 1 interpretation: [tail#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [tail](x0) = x0, [head](x0) = x0, [0] = 0, [zeros] = 0, [nats] = 1, [adx](x0) = x0, [s](x0) = x0, [cons](x0, x1) = x0 + x1, [mark](x0) = x0 + 1, [active](x0) = x0 + 1, [incr](x0) = x0, [nil] = 1 orientation: tail#(ok(X)) = X + 1 >= X + 1 = tail#(X) tail#(mark(X)) = X + 2 >= X + 1 = tail#(X) active(incr(nil())) = 2 >= 2 = mark(nil()) active(incr(cons(X,L))) = L + X + 1 >= L + X + 1 = mark(cons(s(X),incr(L))) active(adx(nil())) = 2 >= 2 = mark(nil()) active(adx(cons(X,L))) = L + X + 1 >= L + X + 1 = mark(incr(cons(X,adx(L)))) active(nats()) = 2 >= 1 = mark(adx(zeros())) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(head(cons(X,L))) = L + X + 1 >= X + 1 = mark(X) active(tail(cons(X,L))) = L + X + 1 >= L + 1 = mark(L) active(incr(X)) = X + 1 >= X + 1 = incr(active(X)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(adx(X)) = X + 1 >= X + 1 = adx(active(X)) active(head(X)) = X + 1 >= X + 1 = head(active(X)) active(tail(X)) = X + 1 >= X + 1 = tail(active(X)) incr(mark(X)) = X + 1 >= X + 1 = mark(incr(X)) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) adx(mark(X)) = X + 1 >= X + 1 = mark(adx(X)) head(mark(X)) = X + 1 >= X + 1 = mark(head(X)) tail(mark(X)) = X + 1 >= X + 1 = mark(tail(X)) proper(incr(X)) = X >= X = incr(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) proper(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(proper(X1),proper(X2)) proper(s(X)) = X >= X = s(proper(X)) proper(adx(X)) = X >= X = adx(proper(X)) proper(nats()) = 1 >= 1 = ok(nats()) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(0()) = 0 >= 0 = ok(0()) proper(head(X)) = X >= X = head(proper(X)) proper(tail(X)) = X >= X = tail(proper(X)) incr(ok(X)) = X >= X = ok(incr(X)) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) s(ok(X)) = X >= X = ok(s(X)) adx(ok(X)) = X >= X = ok(adx(X)) head(ok(X)) = X >= X = ok(head(X)) tail(ok(X)) = X >= X = ok(tail(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: 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)) Matrix Interpretation Processor: dimension: 1 interpretation: [tail#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [tail](x0) = x0, [head](x0) = x0, [0] = 1, [zeros] = 0, [nats] = 1, [adx](x0) = x0 + 1, [s](x0) = x0, [cons](x0, x1) = x1, [mark](x0) = 0, [active](x0) = x0 + 1, [incr](x0) = x0, [nil] = 0 orientation: tail#(ok(X)) = X + 2 >= X + 1 = tail#(X) active(incr(nil())) = 1 >= 0 = mark(nil()) active(incr(cons(X,L))) = L + 1 >= 0 = mark(cons(s(X),incr(L))) active(adx(nil())) = 2 >= 0 = mark(nil()) active(adx(cons(X,L))) = L + 2 >= 0 = mark(incr(cons(X,adx(L)))) active(nats()) = 2 >= 0 = mark(adx(zeros())) active(zeros()) = 1 >= 0 = mark(cons(0(),zeros())) active(head(cons(X,L))) = L + 1 >= 0 = mark(X) active(tail(cons(X,L))) = L + 1 >= 0 = mark(L) active(incr(X)) = X + 1 >= X + 1 = incr(active(X)) active(cons(X1,X2)) = X2 + 1 >= X2 = cons(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(adx(X)) = X + 2 >= X + 2 = adx(active(X)) active(head(X)) = X + 1 >= X + 1 = head(active(X)) active(tail(X)) = X + 1 >= X + 1 = tail(active(X)) incr(mark(X)) = 0 >= 0 = mark(incr(X)) cons(mark(X1),X2) = X2 >= 0 = mark(cons(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) adx(mark(X)) = 1 >= 0 = mark(adx(X)) head(mark(X)) = 0 >= 0 = mark(head(X)) tail(mark(X)) = 0 >= 0 = mark(tail(X)) proper(incr(X)) = X + 1 >= X + 1 = incr(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) proper(cons(X1,X2)) = X2 + 1 >= X2 + 1 = cons(proper(X1),proper(X2)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(adx(X)) = X + 2 >= X + 2 = adx(proper(X)) proper(nats()) = 2 >= 2 = ok(nats()) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(0()) = 2 >= 2 = ok(0()) proper(head(X)) = X + 1 >= X + 1 = head(proper(X)) proper(tail(X)) = X + 1 >= X + 1 = tail(proper(X)) incr(ok(X)) = X + 1 >= X + 1 = ok(incr(X)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) adx(ok(X)) = X + 2 >= X + 2 = ok(adx(X)) head(ok(X)) = X + 1 >= X + 1 = ok(head(X)) tail(ok(X)) = X + 1 >= X + 1 = ok(tail(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) 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