MAYBE Problem: active(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y)))) active(hd(cons(X,Y))) -> mark(X) active(tl(cons(X,Y))) -> mark(Y) active(adx(X)) -> adx(active(X)) active(incr(X)) -> incr(active(X)) active(hd(X)) -> hd(active(X)) active(tl(X)) -> tl(active(X)) adx(mark(X)) -> mark(adx(X)) incr(mark(X)) -> mark(incr(X)) hd(mark(X)) -> mark(hd(X)) tl(mark(X)) -> mark(tl(X)) proper(nats()) -> ok(nats()) proper(adx(X)) -> adx(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(s(X)) -> s(proper(X)) proper(hd(X)) -> hd(proper(X)) proper(tl(X)) -> tl(proper(X)) adx(ok(X)) -> ok(adx(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) hd(ok(X)) -> ok(hd(X)) tl(ok(X)) -> ok(tl(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Proof: DP Processor: DPs: active#(nats()) -> adx#(zeros()) active#(zeros()) -> cons#(0(),zeros()) active#(incr(cons(X,Y))) -> incr#(Y) active#(incr(cons(X,Y))) -> s#(X) active#(incr(cons(X,Y))) -> cons#(s(X),incr(Y)) active#(adx(cons(X,Y))) -> adx#(Y) active#(adx(cons(X,Y))) -> cons#(X,adx(Y)) active#(adx(cons(X,Y))) -> incr#(cons(X,adx(Y))) active#(adx(X)) -> active#(X) active#(adx(X)) -> adx#(active(X)) active#(incr(X)) -> active#(X) active#(incr(X)) -> incr#(active(X)) active#(hd(X)) -> active#(X) active#(hd(X)) -> hd#(active(X)) active#(tl(X)) -> active#(X) active#(tl(X)) -> tl#(active(X)) adx#(mark(X)) -> adx#(X) incr#(mark(X)) -> incr#(X) hd#(mark(X)) -> hd#(X) tl#(mark(X)) -> tl#(X) proper#(adx(X)) -> proper#(X) proper#(adx(X)) -> adx#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(incr(X)) -> proper#(X) proper#(incr(X)) -> incr#(proper(X)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(hd(X)) -> proper#(X) proper#(hd(X)) -> hd#(proper(X)) proper#(tl(X)) -> proper#(X) proper#(tl(X)) -> tl#(proper(X)) adx#(ok(X)) -> adx#(X) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) incr#(ok(X)) -> incr#(X) s#(ok(X)) -> s#(X) hd#(ok(X)) -> hd#(X) tl#(ok(X)) -> tl#(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(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y)))) active(hd(cons(X,Y))) -> mark(X) active(tl(cons(X,Y))) -> mark(Y) active(adx(X)) -> adx(active(X)) active(incr(X)) -> incr(active(X)) active(hd(X)) -> hd(active(X)) active(tl(X)) -> tl(active(X)) adx(mark(X)) -> mark(adx(X)) incr(mark(X)) -> mark(incr(X)) hd(mark(X)) -> mark(hd(X)) tl(mark(X)) -> mark(tl(X)) proper(nats()) -> ok(nats()) proper(adx(X)) -> adx(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(s(X)) -> s(proper(X)) proper(hd(X)) -> hd(proper(X)) proper(tl(X)) -> tl(proper(X)) adx(ok(X)) -> ok(adx(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) hd(ok(X)) -> ok(hd(X)) tl(ok(X)) -> ok(tl(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) EDG Processor: DPs: active#(nats()) -> adx#(zeros()) active#(zeros()) -> cons#(0(),zeros()) active#(incr(cons(X,Y))) -> incr#(Y) active#(incr(cons(X,Y))) -> s#(X) active#(incr(cons(X,Y))) -> cons#(s(X),incr(Y)) active#(adx(cons(X,Y))) -> adx#(Y) active#(adx(cons(X,Y))) -> cons#(X,adx(Y)) active#(adx(cons(X,Y))) -> incr#(cons(X,adx(Y))) active#(adx(X)) -> active#(X) active#(adx(X)) -> adx#(active(X)) active#(incr(X)) -> active#(X) active#(incr(X)) -> incr#(active(X)) active#(hd(X)) -> active#(X) active#(hd(X)) -> hd#(active(X)) active#(tl(X)) -> active#(X) active#(tl(X)) -> tl#(active(X)) adx#(mark(X)) -> adx#(X) incr#(mark(X)) -> incr#(X) hd#(mark(X)) -> hd#(X) tl#(mark(X)) -> tl#(X) proper#(adx(X)) -> proper#(X) proper#(adx(X)) -> adx#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(incr(X)) -> proper#(X) proper#(incr(X)) -> incr#(proper(X)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(hd(X)) -> proper#(X) proper#(hd(X)) -> hd#(proper(X)) proper#(tl(X)) -> proper#(X) proper#(tl(X)) -> tl#(proper(X)) adx#(ok(X)) -> adx#(X) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) incr#(ok(X)) -> incr#(X) s#(ok(X)) -> s#(X) hd#(ok(X)) -> hd#(X) tl#(ok(X)) -> tl#(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(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y)))) active(hd(cons(X,Y))) -> mark(X) active(tl(cons(X,Y))) -> mark(Y) active(adx(X)) -> adx(active(X)) active(incr(X)) -> incr(active(X)) active(hd(X)) -> hd(active(X)) active(tl(X)) -> tl(active(X)) adx(mark(X)) -> mark(adx(X)) incr(mark(X)) -> mark(incr(X)) hd(mark(X)) -> mark(hd(X)) tl(mark(X)) -> mark(tl(X)) proper(nats()) -> ok(nats()) proper(adx(X)) -> adx(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(s(X)) -> s(proper(X)) proper(hd(X)) -> hd(proper(X)) proper(tl(X)) -> tl(proper(X)) adx(ok(X)) -> ok(adx(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) hd(ok(X)) -> ok(hd(X)) tl(ok(X)) -> ok(tl(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#(nats()) -> adx#(zeros()) top#(ok(X)) -> active#(X) -> active#(zeros()) -> cons#(0(),zeros()) top#(ok(X)) -> active#(X) -> active#(incr(cons(X,Y))) -> incr#(Y) top#(ok(X)) -> active#(X) -> active#(incr(cons(X,Y))) -> s#(X) top#(ok(X)) -> active#(X) -> active#(incr(cons(X,Y))) -> cons#(s(X),incr(Y)) top#(ok(X)) -> active#(X) -> active#(adx(cons(X,Y))) -> adx#(Y) top#(ok(X)) -> active#(X) -> active#(adx(cons(X,Y))) -> cons#(X,adx(Y)) top#(ok(X)) -> active#(X) -> active#(adx(cons(X,Y))) -> incr#(cons(X,adx(Y))) 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#(incr(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(incr(X)) -> incr#(active(X)) top#(ok(X)) -> active#(X) -> active#(hd(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(hd(X)) -> hd#(active(X)) top#(ok(X)) -> active#(X) -> active#(tl(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(tl(X)) -> tl#(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#(adx(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(adx(X)) -> adx#(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#(incr(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(incr(X)) -> incr#(proper(X)) 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#(hd(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(hd(X)) -> hd#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(tl(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(tl(X)) -> tl#(proper(X)) proper#(tl(X)) -> proper#(X) -> proper#(adx(X)) -> proper#(X) proper#(tl(X)) -> proper#(X) -> proper#(adx(X)) -> adx#(proper(X)) proper#(tl(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(tl(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(tl(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(tl(X)) -> proper#(X) -> proper#(incr(X)) -> proper#(X) proper#(tl(X)) -> proper#(X) -> proper#(incr(X)) -> incr#(proper(X)) proper#(tl(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(tl(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) proper#(tl(X)) -> proper#(X) -> proper#(hd(X)) -> proper#(X) proper#(tl(X)) -> proper#(X) -> proper#(hd(X)) -> hd#(proper(X)) proper#(tl(X)) -> proper#(X) -> proper#(tl(X)) -> proper#(X) proper#(tl(X)) -> proper#(X) -> proper#(tl(X)) -> tl#(proper(X)) proper#(tl(X)) -> tl#(proper(X)) -> tl#(mark(X)) -> tl#(X) proper#(tl(X)) -> tl#(proper(X)) -> tl#(ok(X)) -> tl#(X) proper#(hd(X)) -> proper#(X) -> proper#(adx(X)) -> proper#(X) proper#(hd(X)) -> proper#(X) -> proper#(adx(X)) -> adx#(proper(X)) proper#(hd(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(hd(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(hd(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(hd(X)) -> proper#(X) -> proper#(incr(X)) -> proper#(X) proper#(hd(X)) -> proper#(X) -> proper#(incr(X)) -> incr#(proper(X)) proper#(hd(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(hd(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) proper#(hd(X)) -> proper#(X) -> proper#(hd(X)) -> proper#(X) proper#(hd(X)) -> proper#(X) -> proper#(hd(X)) -> hd#(proper(X)) proper#(hd(X)) -> proper#(X) -> proper#(tl(X)) -> proper#(X) proper#(hd(X)) -> proper#(X) -> proper#(tl(X)) -> tl#(proper(X)) proper#(hd(X)) -> hd#(proper(X)) -> hd#(mark(X)) -> hd#(X) proper#(hd(X)) -> hd#(proper(X)) -> hd#(ok(X)) -> hd#(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#(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#(incr(X)) -> proper#(X) proper#(s(X)) -> proper#(X) -> proper#(incr(X)) -> incr#(proper(X)) 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#(hd(X)) -> proper#(X) proper#(s(X)) -> proper#(X) -> proper#(hd(X)) -> hd#(proper(X)) proper#(s(X)) -> proper#(X) -> proper#(tl(X)) -> proper#(X) proper#(s(X)) -> proper#(X) -> proper#(tl(X)) -> tl#(proper(X)) proper#(s(X)) -> s#(proper(X)) -> s#(ok(X)) -> s#(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#(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#(incr(X)) -> proper#(X) proper#(incr(X)) -> proper#(X) -> proper#(incr(X)) -> incr#(proper(X)) 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#(hd(X)) -> proper#(X) proper#(incr(X)) -> proper#(X) -> proper#(hd(X)) -> hd#(proper(X)) proper#(incr(X)) -> proper#(X) -> proper#(tl(X)) -> proper#(X) proper#(incr(X)) -> proper#(X) -> proper#(tl(X)) -> tl#(proper(X)) proper#(incr(X)) -> incr#(proper(X)) -> incr#(mark(X)) -> incr#(X) proper#(incr(X)) -> incr#(proper(X)) -> incr#(ok(X)) -> incr#(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#(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#(incr(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(incr(X)) -> incr#(proper(X)) 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#(hd(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(hd(X)) -> hd#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(tl(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(tl(X)) -> tl#(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#(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#(incr(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(incr(X)) -> incr#(proper(X)) 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#(hd(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(hd(X)) -> hd#(proper(X)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(tl(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(tl(X)) -> tl#(proper(X)) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) 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#(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#(incr(X)) -> proper#(X) proper#(adx(X)) -> proper#(X) -> proper#(incr(X)) -> incr#(proper(X)) 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#(hd(X)) -> proper#(X) proper#(adx(X)) -> proper#(X) -> proper#(hd(X)) -> hd#(proper(X)) proper#(adx(X)) -> proper#(X) -> proper#(tl(X)) -> proper#(X) proper#(adx(X)) -> proper#(X) -> proper#(tl(X)) -> tl#(proper(X)) proper#(adx(X)) -> adx#(proper(X)) -> adx#(mark(X)) -> adx#(X) proper#(adx(X)) -> adx#(proper(X)) -> adx#(ok(X)) -> adx#(X) tl#(ok(X)) -> tl#(X) -> tl#(mark(X)) -> tl#(X) tl#(ok(X)) -> tl#(X) -> tl#(ok(X)) -> tl#(X) tl#(mark(X)) -> tl#(X) -> tl#(mark(X)) -> tl#(X) tl#(mark(X)) -> tl#(X) -> tl#(ok(X)) -> tl#(X) hd#(ok(X)) -> hd#(X) -> hd#(mark(X)) -> hd#(X) hd#(ok(X)) -> hd#(X) -> hd#(ok(X)) -> hd#(X) hd#(mark(X)) -> hd#(X) -> hd#(mark(X)) -> hd#(X) hd#(mark(X)) -> hd#(X) -> hd#(ok(X)) -> hd#(X) s#(ok(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) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) 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) active#(tl(X)) -> tl#(active(X)) -> tl#(mark(X)) -> tl#(X) active#(tl(X)) -> tl#(active(X)) -> tl#(ok(X)) -> tl#(X) active#(tl(X)) -> active#(X) -> active#(nats()) -> adx#(zeros()) active#(tl(X)) -> active#(X) -> active#(zeros()) -> cons#(0(),zeros()) active#(tl(X)) -> active#(X) -> active#(incr(cons(X,Y))) -> incr#(Y) active#(tl(X)) -> active#(X) -> active#(incr(cons(X,Y))) -> s#(X) active#(tl(X)) -> active#(X) -> active#(incr(cons(X,Y))) -> cons#(s(X),incr(Y)) active#(tl(X)) -> active#(X) -> active#(adx(cons(X,Y))) -> adx#(Y) active#(tl(X)) -> active#(X) -> active#(adx(cons(X,Y))) -> cons#(X,adx(Y)) active#(tl(X)) -> active#(X) -> active#(adx(cons(X,Y))) -> incr#(cons(X,adx(Y))) active#(tl(X)) -> active#(X) -> active#(adx(X)) -> active#(X) active#(tl(X)) -> active#(X) -> active#(adx(X)) -> adx#(active(X)) active#(tl(X)) -> active#(X) -> active#(incr(X)) -> active#(X) active#(tl(X)) -> active#(X) -> active#(incr(X)) -> incr#(active(X)) active#(tl(X)) -> active#(X) -> active#(hd(X)) -> active#(X) active#(tl(X)) -> active#(X) -> active#(hd(X)) -> hd#(active(X)) active#(tl(X)) -> active#(X) -> active#(tl(X)) -> active#(X) active#(tl(X)) -> active#(X) -> active#(tl(X)) -> tl#(active(X)) active#(hd(X)) -> hd#(active(X)) -> hd#(mark(X)) -> hd#(X) active#(hd(X)) -> hd#(active(X)) -> hd#(ok(X)) -> hd#(X) active#(hd(X)) -> active#(X) -> active#(nats()) -> adx#(zeros()) active#(hd(X)) -> active#(X) -> active#(zeros()) -> cons#(0(),zeros()) active#(hd(X)) -> active#(X) -> active#(incr(cons(X,Y))) -> incr#(Y) active#(hd(X)) -> active#(X) -> active#(incr(cons(X,Y))) -> s#(X) active#(hd(X)) -> active#(X) -> active#(incr(cons(X,Y))) -> cons#(s(X),incr(Y)) active#(hd(X)) -> active#(X) -> active#(adx(cons(X,Y))) -> adx#(Y) active#(hd(X)) -> active#(X) -> active#(adx(cons(X,Y))) -> cons#(X,adx(Y)) active#(hd(X)) -> active#(X) -> active#(adx(cons(X,Y))) -> incr#(cons(X,adx(Y))) active#(hd(X)) -> active#(X) -> active#(adx(X)) -> active#(X) active#(hd(X)) -> active#(X) -> active#(adx(X)) -> adx#(active(X)) active#(hd(X)) -> active#(X) -> active#(incr(X)) -> active#(X) active#(hd(X)) -> active#(X) -> active#(incr(X)) -> incr#(active(X)) active#(hd(X)) -> active#(X) -> active#(hd(X)) -> active#(X) active#(hd(X)) -> active#(X) -> active#(hd(X)) -> hd#(active(X)) active#(hd(X)) -> active#(X) -> active#(tl(X)) -> active#(X) active#(hd(X)) -> active#(X) -> active#(tl(X)) -> tl#(active(X)) active#(incr(cons(X,Y))) -> s#(X) -> s#(ok(X)) -> s#(X) active#(incr(cons(X,Y))) -> incr#(Y) -> incr#(mark(X)) -> incr#(X) active#(incr(cons(X,Y))) -> incr#(Y) -> 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#(nats()) -> adx#(zeros()) active#(incr(X)) -> active#(X) -> active#(zeros()) -> cons#(0(),zeros()) active#(incr(X)) -> active#(X) -> active#(incr(cons(X,Y))) -> incr#(Y) active#(incr(X)) -> active#(X) -> active#(incr(cons(X,Y))) -> s#(X) active#(incr(X)) -> active#(X) -> active#(incr(cons(X,Y))) -> cons#(s(X),incr(Y)) active#(incr(X)) -> active#(X) -> active#(adx(cons(X,Y))) -> adx#(Y) active#(incr(X)) -> active#(X) -> active#(adx(cons(X,Y))) -> cons#(X,adx(Y)) active#(incr(X)) -> active#(X) -> active#(adx(cons(X,Y))) -> incr#(cons(X,adx(Y))) 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#(incr(X)) -> active#(X) active#(incr(X)) -> active#(X) -> active#(incr(X)) -> incr#(active(X)) active#(incr(X)) -> active#(X) -> active#(hd(X)) -> active#(X) active#(incr(X)) -> active#(X) -> active#(hd(X)) -> hd#(active(X)) active#(incr(X)) -> active#(X) -> active#(tl(X)) -> active#(X) active#(incr(X)) -> active#(X) -> active#(tl(X)) -> tl#(active(X)) active#(adx(cons(X,Y))) -> incr#(cons(X,adx(Y))) -> incr#(mark(X)) -> incr#(X) active#(adx(cons(X,Y))) -> incr#(cons(X,adx(Y))) -> incr#(ok(X)) -> incr#(X) active#(adx(cons(X,Y))) -> cons#(X,adx(Y)) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) active#(adx(cons(X,Y))) -> adx#(Y) -> adx#(mark(X)) -> adx#(X) active#(adx(cons(X,Y))) -> adx#(Y) -> adx#(ok(X)) -> adx#(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#(nats()) -> adx#(zeros()) active#(adx(X)) -> active#(X) -> active#(zeros()) -> cons#(0(),zeros()) active#(adx(X)) -> active#(X) -> active#(incr(cons(X,Y))) -> incr#(Y) active#(adx(X)) -> active#(X) -> active#(incr(cons(X,Y))) -> s#(X) active#(adx(X)) -> active#(X) -> active#(incr(cons(X,Y))) -> cons#(s(X),incr(Y)) active#(adx(X)) -> active#(X) -> active#(adx(cons(X,Y))) -> adx#(Y) active#(adx(X)) -> active#(X) -> active#(adx(cons(X,Y))) -> cons#(X,adx(Y)) active#(adx(X)) -> active#(X) -> active#(adx(cons(X,Y))) -> incr#(cons(X,adx(Y))) 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#(incr(X)) -> active#(X) active#(adx(X)) -> active#(X) -> active#(incr(X)) -> incr#(active(X)) active#(adx(X)) -> active#(X) -> active#(hd(X)) -> active#(X) active#(adx(X)) -> active#(X) -> active#(hd(X)) -> hd#(active(X)) active#(adx(X)) -> active#(X) -> active#(tl(X)) -> active#(X) active#(adx(X)) -> active#(X) -> active#(tl(X)) -> tl#(active(X)) SCC Processor: #sccs: 9 #rules: 23 #arcs: 236/1849 DPs: top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> top#(proper(X)) TRS: active(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y)))) active(hd(cons(X,Y))) -> mark(X) active(tl(cons(X,Y))) -> mark(Y) active(adx(X)) -> adx(active(X)) active(incr(X)) -> incr(active(X)) active(hd(X)) -> hd(active(X)) active(tl(X)) -> tl(active(X)) adx(mark(X)) -> mark(adx(X)) incr(mark(X)) -> mark(incr(X)) hd(mark(X)) -> mark(hd(X)) tl(mark(X)) -> mark(tl(X)) proper(nats()) -> ok(nats()) proper(adx(X)) -> adx(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(s(X)) -> s(proper(X)) proper(hd(X)) -> hd(proper(X)) proper(tl(X)) -> tl(proper(X)) adx(ok(X)) -> ok(adx(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) hd(ok(X)) -> ok(hd(X)) tl(ok(X)) -> ok(tl(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: proper#(tl(X)) -> proper#(X) proper#(hd(X)) -> proper#(X) proper#(s(X)) -> proper#(X) proper#(incr(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) proper#(adx(X)) -> proper#(X) TRS: active(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y)))) active(hd(cons(X,Y))) -> mark(X) active(tl(cons(X,Y))) -> mark(Y) active(adx(X)) -> adx(active(X)) active(incr(X)) -> incr(active(X)) active(hd(X)) -> hd(active(X)) active(tl(X)) -> tl(active(X)) adx(mark(X)) -> mark(adx(X)) incr(mark(X)) -> mark(incr(X)) hd(mark(X)) -> mark(hd(X)) tl(mark(X)) -> mark(tl(X)) proper(nats()) -> ok(nats()) proper(adx(X)) -> adx(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(s(X)) -> s(proper(X)) proper(hd(X)) -> hd(proper(X)) proper(tl(X)) -> tl(proper(X)) adx(ok(X)) -> ok(adx(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) hd(ok(X)) -> ok(hd(X)) tl(ok(X)) -> ok(tl(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) = x0, [proper](x0) = x0, [tl](x0) = x0 + 1, [hd](x0) = x0 + 1, [s](x0) = x0 + 1, [incr](x0) = x0, [cons](x0, x1) = x0 + x1 + 1, [0] = 0, [mark](x0) = x0, [adx](x0) = x0 + 1, [zeros] = 0, [active](x0) = x0 + 1, [nats] = 1 orientation: proper#(tl(X)) = X + 2 >= X + 1 = proper#(X) proper#(hd(X)) = X + 2 >= X + 1 = proper#(X) proper#(s(X)) = X + 2 >= X + 1 = proper#(X) proper#(incr(X)) = X + 1 >= 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#(adx(X)) = X + 2 >= X + 1 = proper#(X) active(nats()) = 2 >= 1 = mark(adx(zeros())) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(incr(cons(X,Y))) = X + Y + 2 >= X + Y + 2 = mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) = X + Y + 3 >= X + Y + 2 = mark(incr(cons(X,adx(Y)))) active(hd(cons(X,Y))) = X + Y + 3 >= X = mark(X) active(tl(cons(X,Y))) = X + Y + 3 >= Y = mark(Y) active(adx(X)) = X + 2 >= X + 2 = adx(active(X)) active(incr(X)) = X + 1 >= X + 1 = incr(active(X)) active(hd(X)) = X + 2 >= X + 2 = hd(active(X)) active(tl(X)) = X + 2 >= X + 2 = tl(active(X)) adx(mark(X)) = X + 1 >= X + 1 = mark(adx(X)) incr(mark(X)) = X >= X = mark(incr(X)) hd(mark(X)) = X + 1 >= X + 1 = mark(hd(X)) tl(mark(X)) = X + 1 >= X + 1 = mark(tl(X)) proper(nats()) = 1 >= 1 = ok(nats()) proper(adx(X)) = X + 1 >= X + 1 = adx(proper(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(incr(X)) = X >= X = incr(proper(X)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(hd(X)) = X + 1 >= X + 1 = hd(proper(X)) proper(tl(X)) = X + 1 >= X + 1 = tl(proper(X)) adx(ok(X)) = X + 1 >= X + 1 = ok(adx(X)) cons(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(cons(X1,X2)) incr(ok(X)) = X >= X = ok(incr(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) hd(ok(X)) = X + 1 >= X + 1 = ok(hd(X)) tl(ok(X)) = X + 1 >= X + 1 = ok(tl(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: proper#(incr(X)) -> proper#(X) TRS: active(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y)))) active(hd(cons(X,Y))) -> mark(X) active(tl(cons(X,Y))) -> mark(Y) active(adx(X)) -> adx(active(X)) active(incr(X)) -> incr(active(X)) active(hd(X)) -> hd(active(X)) active(tl(X)) -> tl(active(X)) adx(mark(X)) -> mark(adx(X)) incr(mark(X)) -> mark(incr(X)) hd(mark(X)) -> mark(hd(X)) tl(mark(X)) -> mark(tl(X)) proper(nats()) -> ok(nats()) proper(adx(X)) -> adx(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(s(X)) -> s(proper(X)) proper(hd(X)) -> hd(proper(X)) proper(tl(X)) -> tl(proper(X)) adx(ok(X)) -> ok(adx(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) hd(ok(X)) -> ok(hd(X)) tl(ok(X)) -> ok(tl(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 + 1, [tl](x0) = 0, [hd](x0) = x0 + 1, [s](x0) = x0, [incr](x0) = x0 + 1, [cons](x0, x1) = x0 + 1, [0] = 1, [mark](x0) = 0, [adx](x0) = 0, [zeros] = 0, [active](x0) = x0, [nats] = 1 orientation: proper#(incr(X)) = X + 2 >= X + 1 = proper#(X) active(nats()) = 1 >= 0 = mark(adx(zeros())) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(incr(cons(X,Y))) = X + 2 >= 0 = mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) = 0 >= 0 = mark(incr(cons(X,adx(Y)))) active(hd(cons(X,Y))) = X + 2 >= 0 = mark(X) active(tl(cons(X,Y))) = 0 >= 0 = mark(Y) active(adx(X)) = 0 >= 0 = adx(active(X)) active(incr(X)) = X + 1 >= X + 1 = incr(active(X)) active(hd(X)) = X + 1 >= X + 1 = hd(active(X)) active(tl(X)) = 0 >= 0 = tl(active(X)) adx(mark(X)) = 0 >= 0 = mark(adx(X)) incr(mark(X)) = 1 >= 0 = mark(incr(X)) hd(mark(X)) = 1 >= 0 = mark(hd(X)) tl(mark(X)) = 0 >= 0 = mark(tl(X)) proper(nats()) = 2 >= 0 = ok(nats()) proper(adx(X)) = 1 >= 0 = adx(proper(X)) proper(zeros()) = 1 >= 0 = ok(zeros()) proper(cons(X1,X2)) = X1 + 2 >= X1 + 2 = cons(proper(X1),proper(X2)) proper(0()) = 2 >= 0 = ok(0()) proper(incr(X)) = X + 2 >= X + 2 = incr(proper(X)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(hd(X)) = X + 2 >= X + 2 = hd(proper(X)) proper(tl(X)) = 1 >= 0 = tl(proper(X)) adx(ok(X)) = 0 >= 0 = ok(adx(X)) cons(ok(X1),ok(X2)) = 1 >= 0 = ok(cons(X1,X2)) incr(ok(X)) = 1 >= 0 = ok(incr(X)) s(ok(X)) = 0 >= 0 = ok(s(X)) hd(ok(X)) = 1 >= 0 = ok(hd(X)) tl(ok(X)) = 0 >= 0 = ok(tl(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y)))) active(hd(cons(X,Y))) -> mark(X) active(tl(cons(X,Y))) -> mark(Y) active(adx(X)) -> adx(active(X)) active(incr(X)) -> incr(active(X)) active(hd(X)) -> hd(active(X)) active(tl(X)) -> tl(active(X)) adx(mark(X)) -> mark(adx(X)) incr(mark(X)) -> mark(incr(X)) hd(mark(X)) -> mark(hd(X)) tl(mark(X)) -> mark(tl(X)) proper(nats()) -> ok(nats()) proper(adx(X)) -> adx(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(s(X)) -> s(proper(X)) proper(hd(X)) -> hd(proper(X)) proper(tl(X)) -> tl(proper(X)) adx(ok(X)) -> ok(adx(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) hd(ok(X)) -> ok(hd(X)) tl(ok(X)) -> ok(tl(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: active#(tl(X)) -> active#(X) active#(hd(X)) -> active#(X) active#(incr(X)) -> active#(X) active#(adx(X)) -> active#(X) TRS: active(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y)))) active(hd(cons(X,Y))) -> mark(X) active(tl(cons(X,Y))) -> mark(Y) active(adx(X)) -> adx(active(X)) active(incr(X)) -> incr(active(X)) active(hd(X)) -> hd(active(X)) active(tl(X)) -> tl(active(X)) adx(mark(X)) -> mark(adx(X)) incr(mark(X)) -> mark(incr(X)) hd(mark(X)) -> mark(hd(X)) tl(mark(X)) -> mark(tl(X)) proper(nats()) -> ok(nats()) proper(adx(X)) -> adx(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(s(X)) -> s(proper(X)) proper(hd(X)) -> hd(proper(X)) proper(tl(X)) -> tl(proper(X)) adx(ok(X)) -> ok(adx(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) hd(ok(X)) -> ok(hd(X)) tl(ok(X)) -> ok(tl(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, [proper](x0) = x0, [tl](x0) = x0 + 1, [hd](x0) = x0 + 1, [s](x0) = 1, [incr](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 1, [mark](x0) = x0, [adx](x0) = x0 + 1, [zeros] = 0, [active](x0) = x0 + 1, [nats] = 0 orientation: active#(tl(X)) = X + 2 >= X + 1 = active#(X) active#(hd(X)) = X + 2 >= X + 1 = active#(X) active#(incr(X)) = X + 2 >= X + 1 = active#(X) active#(adx(X)) = X + 2 >= X + 1 = active#(X) active(nats()) = 1 >= 1 = mark(adx(zeros())) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(incr(cons(X,Y))) = X + Y + 2 >= Y + 2 = mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) = X + Y + 2 >= X + Y + 2 = mark(incr(cons(X,adx(Y)))) active(hd(cons(X,Y))) = X + Y + 2 >= X = mark(X) active(tl(cons(X,Y))) = X + Y + 2 >= Y = mark(Y) active(adx(X)) = X + 2 >= X + 2 = adx(active(X)) active(incr(X)) = X + 2 >= X + 2 = incr(active(X)) active(hd(X)) = X + 2 >= X + 2 = hd(active(X)) active(tl(X)) = X + 2 >= X + 2 = tl(active(X)) adx(mark(X)) = X + 1 >= X + 1 = mark(adx(X)) incr(mark(X)) = X + 1 >= X + 1 = mark(incr(X)) hd(mark(X)) = X + 1 >= X + 1 = mark(hd(X)) tl(mark(X)) = X + 1 >= X + 1 = mark(tl(X)) proper(nats()) = 0 >= 0 = ok(nats()) proper(adx(X)) = X + 1 >= X + 1 = adx(proper(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(incr(X)) = X + 1 >= X + 1 = incr(proper(X)) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(hd(X)) = X + 1 >= X + 1 = hd(proper(X)) proper(tl(X)) = X + 1 >= X + 1 = tl(proper(X)) adx(ok(X)) = X + 1 >= X + 1 = ok(adx(X)) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) incr(ok(X)) = X + 1 >= X + 1 = ok(incr(X)) s(ok(X)) = 1 >= 1 = ok(s(X)) hd(ok(X)) = X + 1 >= X + 1 = ok(hd(X)) tl(ok(X)) = X + 1 >= X + 1 = ok(tl(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y)))) active(hd(cons(X,Y))) -> mark(X) active(tl(cons(X,Y))) -> mark(Y) active(adx(X)) -> adx(active(X)) active(incr(X)) -> incr(active(X)) active(hd(X)) -> hd(active(X)) active(tl(X)) -> tl(active(X)) adx(mark(X)) -> mark(adx(X)) incr(mark(X)) -> mark(incr(X)) hd(mark(X)) -> mark(hd(X)) tl(mark(X)) -> mark(tl(X)) proper(nats()) -> ok(nats()) proper(adx(X)) -> adx(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(s(X)) -> s(proper(X)) proper(hd(X)) -> hd(proper(X)) proper(tl(X)) -> tl(proper(X)) adx(ok(X)) -> ok(adx(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) hd(ok(X)) -> ok(hd(X)) tl(ok(X)) -> ok(tl(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: s#(ok(X)) -> s#(X) TRS: active(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y)))) active(hd(cons(X,Y))) -> mark(X) active(tl(cons(X,Y))) -> mark(Y) active(adx(X)) -> adx(active(X)) active(incr(X)) -> incr(active(X)) active(hd(X)) -> hd(active(X)) active(tl(X)) -> tl(active(X)) adx(mark(X)) -> mark(adx(X)) incr(mark(X)) -> mark(incr(X)) hd(mark(X)) -> mark(hd(X)) tl(mark(X)) -> mark(tl(X)) proper(nats()) -> ok(nats()) proper(adx(X)) -> adx(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(s(X)) -> s(proper(X)) proper(hd(X)) -> hd(proper(X)) proper(tl(X)) -> tl(proper(X)) adx(ok(X)) -> ok(adx(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) hd(ok(X)) -> ok(hd(X)) tl(ok(X)) -> ok(tl(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) = 1, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [tl](x0) = x0 + 1, [hd](x0) = x0, [s](x0) = x0 + 1, [incr](x0) = x0, [cons](x0, x1) = x1, [0] = 0, [mark](x0) = 0, [adx](x0) = x0, [zeros] = 0, [active](x0) = x0, [nats] = 0 orientation: s#(ok(X)) = X + 2 >= X + 1 = s#(X) active(nats()) = 0 >= 0 = mark(adx(zeros())) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(incr(cons(X,Y))) = Y >= 0 = mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) = Y >= 0 = mark(incr(cons(X,adx(Y)))) active(hd(cons(X,Y))) = Y >= 0 = mark(X) active(tl(cons(X,Y))) = Y + 1 >= 0 = mark(Y) active(adx(X)) = X >= X = adx(active(X)) active(incr(X)) = X >= X = incr(active(X)) active(hd(X)) = X >= X = hd(active(X)) active(tl(X)) = X + 1 >= X + 1 = tl(active(X)) adx(mark(X)) = 0 >= 0 = mark(adx(X)) incr(mark(X)) = 0 >= 0 = mark(incr(X)) hd(mark(X)) = 0 >= 0 = mark(hd(X)) tl(mark(X)) = 1 >= 0 = mark(tl(X)) proper(nats()) = 1 >= 1 = ok(nats()) proper(adx(X)) = X + 1 >= X + 1 = adx(proper(X)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = X2 + 1 >= X2 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(incr(X)) = X + 1 >= X + 1 = incr(proper(X)) proper(s(X)) = X + 2 >= X + 2 = s(proper(X)) proper(hd(X)) = X + 1 >= X + 1 = hd(proper(X)) proper(tl(X)) = X + 2 >= X + 2 = tl(proper(X)) adx(ok(X)) = X + 1 >= X + 1 = ok(adx(X)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) incr(ok(X)) = X + 1 >= X + 1 = ok(incr(X)) s(ok(X)) = X + 2 >= X + 2 = ok(s(X)) hd(ok(X)) = X + 1 >= X + 1 = ok(hd(X)) tl(ok(X)) = X + 2 >= X + 2 = ok(tl(X)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: TRS: active(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y)))) active(hd(cons(X,Y))) -> mark(X) active(tl(cons(X,Y))) -> mark(Y) active(adx(X)) -> adx(active(X)) active(incr(X)) -> incr(active(X)) active(hd(X)) -> hd(active(X)) active(tl(X)) -> tl(active(X)) adx(mark(X)) -> mark(adx(X)) incr(mark(X)) -> mark(incr(X)) hd(mark(X)) -> mark(hd(X)) tl(mark(X)) -> mark(tl(X)) proper(nats()) -> ok(nats()) proper(adx(X)) -> adx(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(s(X)) -> s(proper(X)) proper(hd(X)) -> hd(proper(X)) proper(tl(X)) -> tl(proper(X)) adx(ok(X)) -> ok(adx(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) hd(ok(X)) -> ok(hd(X)) tl(ok(X)) -> ok(tl(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: cons#(ok(X1),ok(X2)) -> cons#(X1,X2) TRS: active(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y)))) active(hd(cons(X,Y))) -> mark(X) active(tl(cons(X,Y))) -> mark(Y) active(adx(X)) -> adx(active(X)) active(incr(X)) -> incr(active(X)) active(hd(X)) -> hd(active(X)) active(tl(X)) -> tl(active(X)) adx(mark(X)) -> mark(adx(X)) incr(mark(X)) -> mark(incr(X)) hd(mark(X)) -> mark(hd(X)) tl(mark(X)) -> mark(tl(X)) proper(nats()) -> ok(nats()) proper(adx(X)) -> adx(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(s(X)) -> s(proper(X)) proper(hd(X)) -> hd(proper(X)) proper(tl(X)) -> tl(proper(X)) adx(ok(X)) -> ok(adx(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) hd(ok(X)) -> ok(hd(X)) tl(ok(X)) -> ok(tl(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) = 1, [tl](x0) = x0, [hd](x0) = x0, [s](x0) = x0, [incr](x0) = x0, [cons](x0, x1) = x0, [0] = 0, [mark](x0) = 0, [adx](x0) = x0, [zeros] = 0, [active](x0) = 0, [nats] = 0 orientation: cons#(ok(X1),ok(X2)) = X2 + 2 >= X2 + 1 = cons#(X1,X2) active(nats()) = 0 >= 0 = mark(adx(zeros())) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(incr(cons(X,Y))) = 0 >= 0 = mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) = 0 >= 0 = mark(incr(cons(X,adx(Y)))) active(hd(cons(X,Y))) = 0 >= 0 = mark(X) active(tl(cons(X,Y))) = 0 >= 0 = mark(Y) active(adx(X)) = 0 >= 0 = adx(active(X)) active(incr(X)) = 0 >= 0 = incr(active(X)) active(hd(X)) = 0 >= 0 = hd(active(X)) active(tl(X)) = 0 >= 0 = tl(active(X)) adx(mark(X)) = 0 >= 0 = mark(adx(X)) incr(mark(X)) = 0 >= 0 = mark(incr(X)) hd(mark(X)) = 0 >= 0 = mark(hd(X)) tl(mark(X)) = 0 >= 0 = mark(tl(X)) proper(nats()) = 1 >= 1 = ok(nats()) proper(adx(X)) = 1 >= 1 = adx(proper(X)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(incr(X)) = 1 >= 1 = incr(proper(X)) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(hd(X)) = 1 >= 1 = hd(proper(X)) proper(tl(X)) = 1 >= 1 = tl(proper(X)) adx(ok(X)) = X + 1 >= X + 1 = ok(adx(X)) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) incr(ok(X)) = X + 1 >= X + 1 = ok(incr(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) hd(ok(X)) = X + 1 >= X + 1 = ok(hd(X)) tl(ok(X)) = X + 1 >= X + 1 = ok(tl(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y)))) active(hd(cons(X,Y))) -> mark(X) active(tl(cons(X,Y))) -> mark(Y) active(adx(X)) -> adx(active(X)) active(incr(X)) -> incr(active(X)) active(hd(X)) -> hd(active(X)) active(tl(X)) -> tl(active(X)) adx(mark(X)) -> mark(adx(X)) incr(mark(X)) -> mark(incr(X)) hd(mark(X)) -> mark(hd(X)) tl(mark(X)) -> mark(tl(X)) proper(nats()) -> ok(nats()) proper(adx(X)) -> adx(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(s(X)) -> s(proper(X)) proper(hd(X)) -> hd(proper(X)) proper(tl(X)) -> tl(proper(X)) adx(ok(X)) -> ok(adx(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) hd(ok(X)) -> ok(hd(X)) tl(ok(X)) -> ok(tl(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(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y)))) active(hd(cons(X,Y))) -> mark(X) active(tl(cons(X,Y))) -> mark(Y) active(adx(X)) -> adx(active(X)) active(incr(X)) -> incr(active(X)) active(hd(X)) -> hd(active(X)) active(tl(X)) -> tl(active(X)) adx(mark(X)) -> mark(adx(X)) incr(mark(X)) -> mark(incr(X)) hd(mark(X)) -> mark(hd(X)) tl(mark(X)) -> mark(tl(X)) proper(nats()) -> ok(nats()) proper(adx(X)) -> adx(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(s(X)) -> s(proper(X)) proper(hd(X)) -> hd(proper(X)) proper(tl(X)) -> tl(proper(X)) adx(ok(X)) -> ok(adx(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) hd(ok(X)) -> ok(hd(X)) tl(ok(X)) -> ok(tl(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) = 0, [tl](x0) = x0, [hd](x0) = x0, [s](x0) = x0, [incr](x0) = x0, [cons](x0, x1) = x0 + x1, [0] = 0, [mark](x0) = x0 + 1, [adx](x0) = x0, [zeros] = 0, [active](x0) = x0 + 1, [nats] = 0 orientation: adx#(ok(X)) = X + 1 >= X + 1 = adx#(X) adx#(mark(X)) = X + 2 >= X + 1 = adx#(X) active(nats()) = 1 >= 1 = mark(adx(zeros())) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(incr(cons(X,Y))) = X + Y + 1 >= X + Y + 1 = mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) = X + Y + 1 >= X + Y + 1 = mark(incr(cons(X,adx(Y)))) active(hd(cons(X,Y))) = X + Y + 1 >= X + 1 = mark(X) active(tl(cons(X,Y))) = X + Y + 1 >= Y + 1 = mark(Y) active(adx(X)) = X + 1 >= X + 1 = adx(active(X)) active(incr(X)) = X + 1 >= X + 1 = incr(active(X)) active(hd(X)) = X + 1 >= X + 1 = hd(active(X)) active(tl(X)) = X + 1 >= X + 1 = tl(active(X)) adx(mark(X)) = X + 1 >= X + 1 = mark(adx(X)) incr(mark(X)) = X + 1 >= X + 1 = mark(incr(X)) hd(mark(X)) = X + 1 >= X + 1 = mark(hd(X)) tl(mark(X)) = X + 1 >= X + 1 = mark(tl(X)) proper(nats()) = 0 >= 0 = ok(nats()) proper(adx(X)) = 0 >= 0 = adx(proper(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(incr(X)) = 0 >= 0 = incr(proper(X)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(hd(X)) = 0 >= 0 = hd(proper(X)) proper(tl(X)) = 0 >= 0 = tl(proper(X)) adx(ok(X)) = X >= X = ok(adx(X)) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) incr(ok(X)) = X >= X = ok(incr(X)) s(ok(X)) = X >= X = ok(s(X)) hd(ok(X)) = X >= X = ok(hd(X)) tl(ok(X)) = X >= X = ok(tl(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(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y)))) active(hd(cons(X,Y))) -> mark(X) active(tl(cons(X,Y))) -> mark(Y) active(adx(X)) -> adx(active(X)) active(incr(X)) -> incr(active(X)) active(hd(X)) -> hd(active(X)) active(tl(X)) -> tl(active(X)) adx(mark(X)) -> mark(adx(X)) incr(mark(X)) -> mark(incr(X)) hd(mark(X)) -> mark(hd(X)) tl(mark(X)) -> mark(tl(X)) proper(nats()) -> ok(nats()) proper(adx(X)) -> adx(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(s(X)) -> s(proper(X)) proper(hd(X)) -> hd(proper(X)) proper(tl(X)) -> tl(proper(X)) adx(ok(X)) -> ok(adx(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) hd(ok(X)) -> ok(hd(X)) tl(ok(X)) -> ok(tl(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) = 1, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [tl](x0) = x0 + 1, [hd](x0) = x0, [s](x0) = x0 + 1, [incr](x0) = x0, [cons](x0, x1) = x1, [0] = 0, [mark](x0) = 0, [adx](x0) = x0, [zeros] = 0, [active](x0) = x0, [nats] = 0 orientation: adx#(ok(X)) = X + 2 >= X + 1 = adx#(X) active(nats()) = 0 >= 0 = mark(adx(zeros())) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(incr(cons(X,Y))) = Y >= 0 = mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) = Y >= 0 = mark(incr(cons(X,adx(Y)))) active(hd(cons(X,Y))) = Y >= 0 = mark(X) active(tl(cons(X,Y))) = Y + 1 >= 0 = mark(Y) active(adx(X)) = X >= X = adx(active(X)) active(incr(X)) = X >= X = incr(active(X)) active(hd(X)) = X >= X = hd(active(X)) active(tl(X)) = X + 1 >= X + 1 = tl(active(X)) adx(mark(X)) = 0 >= 0 = mark(adx(X)) incr(mark(X)) = 0 >= 0 = mark(incr(X)) hd(mark(X)) = 0 >= 0 = mark(hd(X)) tl(mark(X)) = 1 >= 0 = mark(tl(X)) proper(nats()) = 1 >= 1 = ok(nats()) proper(adx(X)) = X + 1 >= X + 1 = adx(proper(X)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = X2 + 1 >= X2 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(incr(X)) = X + 1 >= X + 1 = incr(proper(X)) proper(s(X)) = X + 2 >= X + 2 = s(proper(X)) proper(hd(X)) = X + 1 >= X + 1 = hd(proper(X)) proper(tl(X)) = X + 2 >= X + 2 = tl(proper(X)) adx(ok(X)) = X + 1 >= X + 1 = ok(adx(X)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) incr(ok(X)) = X + 1 >= X + 1 = ok(incr(X)) s(ok(X)) = X + 2 >= X + 2 = ok(s(X)) hd(ok(X)) = X + 1 >= X + 1 = ok(hd(X)) tl(ok(X)) = X + 2 >= X + 2 = ok(tl(X)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: TRS: active(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y)))) active(hd(cons(X,Y))) -> mark(X) active(tl(cons(X,Y))) -> mark(Y) active(adx(X)) -> adx(active(X)) active(incr(X)) -> incr(active(X)) active(hd(X)) -> hd(active(X)) active(tl(X)) -> tl(active(X)) adx(mark(X)) -> mark(adx(X)) incr(mark(X)) -> mark(incr(X)) hd(mark(X)) -> mark(hd(X)) tl(mark(X)) -> mark(tl(X)) proper(nats()) -> ok(nats()) proper(adx(X)) -> adx(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(s(X)) -> s(proper(X)) proper(hd(X)) -> hd(proper(X)) proper(tl(X)) -> tl(proper(X)) adx(ok(X)) -> ok(adx(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) hd(ok(X)) -> ok(hd(X)) tl(ok(X)) -> ok(tl(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(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y)))) active(hd(cons(X,Y))) -> mark(X) active(tl(cons(X,Y))) -> mark(Y) active(adx(X)) -> adx(active(X)) active(incr(X)) -> incr(active(X)) active(hd(X)) -> hd(active(X)) active(tl(X)) -> tl(active(X)) adx(mark(X)) -> mark(adx(X)) incr(mark(X)) -> mark(incr(X)) hd(mark(X)) -> mark(hd(X)) tl(mark(X)) -> mark(tl(X)) proper(nats()) -> ok(nats()) proper(adx(X)) -> adx(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(s(X)) -> s(proper(X)) proper(hd(X)) -> hd(proper(X)) proper(tl(X)) -> tl(proper(X)) adx(ok(X)) -> ok(adx(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) hd(ok(X)) -> ok(hd(X)) tl(ok(X)) -> ok(tl(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) = 0, [tl](x0) = x0, [hd](x0) = x0, [s](x0) = x0, [incr](x0) = x0, [cons](x0, x1) = x0 + x1, [0] = 0, [mark](x0) = x0 + 1, [adx](x0) = x0, [zeros] = 0, [active](x0) = x0 + 1, [nats] = 0 orientation: incr#(ok(X)) = X + 1 >= X + 1 = incr#(X) incr#(mark(X)) = X + 2 >= X + 1 = incr#(X) active(nats()) = 1 >= 1 = mark(adx(zeros())) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(incr(cons(X,Y))) = X + Y + 1 >= X + Y + 1 = mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) = X + Y + 1 >= X + Y + 1 = mark(incr(cons(X,adx(Y)))) active(hd(cons(X,Y))) = X + Y + 1 >= X + 1 = mark(X) active(tl(cons(X,Y))) = X + Y + 1 >= Y + 1 = mark(Y) active(adx(X)) = X + 1 >= X + 1 = adx(active(X)) active(incr(X)) = X + 1 >= X + 1 = incr(active(X)) active(hd(X)) = X + 1 >= X + 1 = hd(active(X)) active(tl(X)) = X + 1 >= X + 1 = tl(active(X)) adx(mark(X)) = X + 1 >= X + 1 = mark(adx(X)) incr(mark(X)) = X + 1 >= X + 1 = mark(incr(X)) hd(mark(X)) = X + 1 >= X + 1 = mark(hd(X)) tl(mark(X)) = X + 1 >= X + 1 = mark(tl(X)) proper(nats()) = 0 >= 0 = ok(nats()) proper(adx(X)) = 0 >= 0 = adx(proper(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(incr(X)) = 0 >= 0 = incr(proper(X)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(hd(X)) = 0 >= 0 = hd(proper(X)) proper(tl(X)) = 0 >= 0 = tl(proper(X)) adx(ok(X)) = X >= X = ok(adx(X)) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) incr(ok(X)) = X >= X = ok(incr(X)) s(ok(X)) = X >= X = ok(s(X)) hd(ok(X)) = X >= X = ok(hd(X)) tl(ok(X)) = X >= X = ok(tl(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(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y)))) active(hd(cons(X,Y))) -> mark(X) active(tl(cons(X,Y))) -> mark(Y) active(adx(X)) -> adx(active(X)) active(incr(X)) -> incr(active(X)) active(hd(X)) -> hd(active(X)) active(tl(X)) -> tl(active(X)) adx(mark(X)) -> mark(adx(X)) incr(mark(X)) -> mark(incr(X)) hd(mark(X)) -> mark(hd(X)) tl(mark(X)) -> mark(tl(X)) proper(nats()) -> ok(nats()) proper(adx(X)) -> adx(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(s(X)) -> s(proper(X)) proper(hd(X)) -> hd(proper(X)) proper(tl(X)) -> tl(proper(X)) adx(ok(X)) -> ok(adx(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) hd(ok(X)) -> ok(hd(X)) tl(ok(X)) -> ok(tl(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) = 1, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [tl](x0) = x0 + 1, [hd](x0) = x0, [s](x0) = x0 + 1, [incr](x0) = x0, [cons](x0, x1) = x1, [0] = 0, [mark](x0) = 0, [adx](x0) = x0, [zeros] = 0, [active](x0) = x0, [nats] = 0 orientation: incr#(ok(X)) = X + 2 >= X + 1 = incr#(X) active(nats()) = 0 >= 0 = mark(adx(zeros())) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(incr(cons(X,Y))) = Y >= 0 = mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) = Y >= 0 = mark(incr(cons(X,adx(Y)))) active(hd(cons(X,Y))) = Y >= 0 = mark(X) active(tl(cons(X,Y))) = Y + 1 >= 0 = mark(Y) active(adx(X)) = X >= X = adx(active(X)) active(incr(X)) = X >= X = incr(active(X)) active(hd(X)) = X >= X = hd(active(X)) active(tl(X)) = X + 1 >= X + 1 = tl(active(X)) adx(mark(X)) = 0 >= 0 = mark(adx(X)) incr(mark(X)) = 0 >= 0 = mark(incr(X)) hd(mark(X)) = 0 >= 0 = mark(hd(X)) tl(mark(X)) = 1 >= 0 = mark(tl(X)) proper(nats()) = 1 >= 1 = ok(nats()) proper(adx(X)) = X + 1 >= X + 1 = adx(proper(X)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = X2 + 1 >= X2 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(incr(X)) = X + 1 >= X + 1 = incr(proper(X)) proper(s(X)) = X + 2 >= X + 2 = s(proper(X)) proper(hd(X)) = X + 1 >= X + 1 = hd(proper(X)) proper(tl(X)) = X + 2 >= X + 2 = tl(proper(X)) adx(ok(X)) = X + 1 >= X + 1 = ok(adx(X)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) incr(ok(X)) = X + 1 >= X + 1 = ok(incr(X)) s(ok(X)) = X + 2 >= X + 2 = ok(s(X)) hd(ok(X)) = X + 1 >= X + 1 = ok(hd(X)) tl(ok(X)) = X + 2 >= X + 2 = ok(tl(X)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: TRS: active(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y)))) active(hd(cons(X,Y))) -> mark(X) active(tl(cons(X,Y))) -> mark(Y) active(adx(X)) -> adx(active(X)) active(incr(X)) -> incr(active(X)) active(hd(X)) -> hd(active(X)) active(tl(X)) -> tl(active(X)) adx(mark(X)) -> mark(adx(X)) incr(mark(X)) -> mark(incr(X)) hd(mark(X)) -> mark(hd(X)) tl(mark(X)) -> mark(tl(X)) proper(nats()) -> ok(nats()) proper(adx(X)) -> adx(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(s(X)) -> s(proper(X)) proper(hd(X)) -> hd(proper(X)) proper(tl(X)) -> tl(proper(X)) adx(ok(X)) -> ok(adx(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) hd(ok(X)) -> ok(hd(X)) tl(ok(X)) -> ok(tl(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: hd#(ok(X)) -> hd#(X) hd#(mark(X)) -> hd#(X) TRS: active(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y)))) active(hd(cons(X,Y))) -> mark(X) active(tl(cons(X,Y))) -> mark(Y) active(adx(X)) -> adx(active(X)) active(incr(X)) -> incr(active(X)) active(hd(X)) -> hd(active(X)) active(tl(X)) -> tl(active(X)) adx(mark(X)) -> mark(adx(X)) incr(mark(X)) -> mark(incr(X)) hd(mark(X)) -> mark(hd(X)) tl(mark(X)) -> mark(tl(X)) proper(nats()) -> ok(nats()) proper(adx(X)) -> adx(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(s(X)) -> s(proper(X)) proper(hd(X)) -> hd(proper(X)) proper(tl(X)) -> tl(proper(X)) adx(ok(X)) -> ok(adx(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) hd(ok(X)) -> ok(hd(X)) tl(ok(X)) -> ok(tl(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [hd#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = 0, [tl](x0) = x0, [hd](x0) = x0, [s](x0) = x0, [incr](x0) = x0, [cons](x0, x1) = x0 + x1, [0] = 0, [mark](x0) = x0 + 1, [adx](x0) = x0, [zeros] = 0, [active](x0) = x0 + 1, [nats] = 0 orientation: hd#(ok(X)) = X + 1 >= X + 1 = hd#(X) hd#(mark(X)) = X + 2 >= X + 1 = hd#(X) active(nats()) = 1 >= 1 = mark(adx(zeros())) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(incr(cons(X,Y))) = X + Y + 1 >= X + Y + 1 = mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) = X + Y + 1 >= X + Y + 1 = mark(incr(cons(X,adx(Y)))) active(hd(cons(X,Y))) = X + Y + 1 >= X + 1 = mark(X) active(tl(cons(X,Y))) = X + Y + 1 >= Y + 1 = mark(Y) active(adx(X)) = X + 1 >= X + 1 = adx(active(X)) active(incr(X)) = X + 1 >= X + 1 = incr(active(X)) active(hd(X)) = X + 1 >= X + 1 = hd(active(X)) active(tl(X)) = X + 1 >= X + 1 = tl(active(X)) adx(mark(X)) = X + 1 >= X + 1 = mark(adx(X)) incr(mark(X)) = X + 1 >= X + 1 = mark(incr(X)) hd(mark(X)) = X + 1 >= X + 1 = mark(hd(X)) tl(mark(X)) = X + 1 >= X + 1 = mark(tl(X)) proper(nats()) = 0 >= 0 = ok(nats()) proper(adx(X)) = 0 >= 0 = adx(proper(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(incr(X)) = 0 >= 0 = incr(proper(X)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(hd(X)) = 0 >= 0 = hd(proper(X)) proper(tl(X)) = 0 >= 0 = tl(proper(X)) adx(ok(X)) = X >= X = ok(adx(X)) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) incr(ok(X)) = X >= X = ok(incr(X)) s(ok(X)) = X >= X = ok(s(X)) hd(ok(X)) = X >= X = ok(hd(X)) tl(ok(X)) = X >= X = ok(tl(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: hd#(ok(X)) -> hd#(X) TRS: active(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y)))) active(hd(cons(X,Y))) -> mark(X) active(tl(cons(X,Y))) -> mark(Y) active(adx(X)) -> adx(active(X)) active(incr(X)) -> incr(active(X)) active(hd(X)) -> hd(active(X)) active(tl(X)) -> tl(active(X)) adx(mark(X)) -> mark(adx(X)) incr(mark(X)) -> mark(incr(X)) hd(mark(X)) -> mark(hd(X)) tl(mark(X)) -> mark(tl(X)) proper(nats()) -> ok(nats()) proper(adx(X)) -> adx(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(s(X)) -> s(proper(X)) proper(hd(X)) -> hd(proper(X)) proper(tl(X)) -> tl(proper(X)) adx(ok(X)) -> ok(adx(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) hd(ok(X)) -> ok(hd(X)) tl(ok(X)) -> ok(tl(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [hd#](x0) = x0 + 1, [top](x0) = 1, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [tl](x0) = x0 + 1, [hd](x0) = x0, [s](x0) = x0 + 1, [incr](x0) = x0, [cons](x0, x1) = x1, [0] = 0, [mark](x0) = 0, [adx](x0) = x0, [zeros] = 0, [active](x0) = x0, [nats] = 0 orientation: hd#(ok(X)) = X + 2 >= X + 1 = hd#(X) active(nats()) = 0 >= 0 = mark(adx(zeros())) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(incr(cons(X,Y))) = Y >= 0 = mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) = Y >= 0 = mark(incr(cons(X,adx(Y)))) active(hd(cons(X,Y))) = Y >= 0 = mark(X) active(tl(cons(X,Y))) = Y + 1 >= 0 = mark(Y) active(adx(X)) = X >= X = adx(active(X)) active(incr(X)) = X >= X = incr(active(X)) active(hd(X)) = X >= X = hd(active(X)) active(tl(X)) = X + 1 >= X + 1 = tl(active(X)) adx(mark(X)) = 0 >= 0 = mark(adx(X)) incr(mark(X)) = 0 >= 0 = mark(incr(X)) hd(mark(X)) = 0 >= 0 = mark(hd(X)) tl(mark(X)) = 1 >= 0 = mark(tl(X)) proper(nats()) = 1 >= 1 = ok(nats()) proper(adx(X)) = X + 1 >= X + 1 = adx(proper(X)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = X2 + 1 >= X2 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(incr(X)) = X + 1 >= X + 1 = incr(proper(X)) proper(s(X)) = X + 2 >= X + 2 = s(proper(X)) proper(hd(X)) = X + 1 >= X + 1 = hd(proper(X)) proper(tl(X)) = X + 2 >= X + 2 = tl(proper(X)) adx(ok(X)) = X + 1 >= X + 1 = ok(adx(X)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) incr(ok(X)) = X + 1 >= X + 1 = ok(incr(X)) s(ok(X)) = X + 2 >= X + 2 = ok(s(X)) hd(ok(X)) = X + 1 >= X + 1 = ok(hd(X)) tl(ok(X)) = X + 2 >= X + 2 = ok(tl(X)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: TRS: active(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y)))) active(hd(cons(X,Y))) -> mark(X) active(tl(cons(X,Y))) -> mark(Y) active(adx(X)) -> adx(active(X)) active(incr(X)) -> incr(active(X)) active(hd(X)) -> hd(active(X)) active(tl(X)) -> tl(active(X)) adx(mark(X)) -> mark(adx(X)) incr(mark(X)) -> mark(incr(X)) hd(mark(X)) -> mark(hd(X)) tl(mark(X)) -> mark(tl(X)) proper(nats()) -> ok(nats()) proper(adx(X)) -> adx(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(s(X)) -> s(proper(X)) proper(hd(X)) -> hd(proper(X)) proper(tl(X)) -> tl(proper(X)) adx(ok(X)) -> ok(adx(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) hd(ok(X)) -> ok(hd(X)) tl(ok(X)) -> ok(tl(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: tl#(ok(X)) -> tl#(X) tl#(mark(X)) -> tl#(X) TRS: active(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y)))) active(hd(cons(X,Y))) -> mark(X) active(tl(cons(X,Y))) -> mark(Y) active(adx(X)) -> adx(active(X)) active(incr(X)) -> incr(active(X)) active(hd(X)) -> hd(active(X)) active(tl(X)) -> tl(active(X)) adx(mark(X)) -> mark(adx(X)) incr(mark(X)) -> mark(incr(X)) hd(mark(X)) -> mark(hd(X)) tl(mark(X)) -> mark(tl(X)) proper(nats()) -> ok(nats()) proper(adx(X)) -> adx(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(s(X)) -> s(proper(X)) proper(hd(X)) -> hd(proper(X)) proper(tl(X)) -> tl(proper(X)) adx(ok(X)) -> ok(adx(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) hd(ok(X)) -> ok(hd(X)) tl(ok(X)) -> ok(tl(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [tl#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = 0, [tl](x0) = x0, [hd](x0) = x0, [s](x0) = x0, [incr](x0) = x0, [cons](x0, x1) = x0 + x1, [0] = 0, [mark](x0) = x0 + 1, [adx](x0) = x0, [zeros] = 0, [active](x0) = x0 + 1, [nats] = 0 orientation: tl#(ok(X)) = X + 1 >= X + 1 = tl#(X) tl#(mark(X)) = X + 2 >= X + 1 = tl#(X) active(nats()) = 1 >= 1 = mark(adx(zeros())) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(incr(cons(X,Y))) = X + Y + 1 >= X + Y + 1 = mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) = X + Y + 1 >= X + Y + 1 = mark(incr(cons(X,adx(Y)))) active(hd(cons(X,Y))) = X + Y + 1 >= X + 1 = mark(X) active(tl(cons(X,Y))) = X + Y + 1 >= Y + 1 = mark(Y) active(adx(X)) = X + 1 >= X + 1 = adx(active(X)) active(incr(X)) = X + 1 >= X + 1 = incr(active(X)) active(hd(X)) = X + 1 >= X + 1 = hd(active(X)) active(tl(X)) = X + 1 >= X + 1 = tl(active(X)) adx(mark(X)) = X + 1 >= X + 1 = mark(adx(X)) incr(mark(X)) = X + 1 >= X + 1 = mark(incr(X)) hd(mark(X)) = X + 1 >= X + 1 = mark(hd(X)) tl(mark(X)) = X + 1 >= X + 1 = mark(tl(X)) proper(nats()) = 0 >= 0 = ok(nats()) proper(adx(X)) = 0 >= 0 = adx(proper(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(incr(X)) = 0 >= 0 = incr(proper(X)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(hd(X)) = 0 >= 0 = hd(proper(X)) proper(tl(X)) = 0 >= 0 = tl(proper(X)) adx(ok(X)) = X >= X = ok(adx(X)) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) incr(ok(X)) = X >= X = ok(incr(X)) s(ok(X)) = X >= X = ok(s(X)) hd(ok(X)) = X >= X = ok(hd(X)) tl(ok(X)) = X >= X = ok(tl(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: tl#(ok(X)) -> tl#(X) TRS: active(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y)))) active(hd(cons(X,Y))) -> mark(X) active(tl(cons(X,Y))) -> mark(Y) active(adx(X)) -> adx(active(X)) active(incr(X)) -> incr(active(X)) active(hd(X)) -> hd(active(X)) active(tl(X)) -> tl(active(X)) adx(mark(X)) -> mark(adx(X)) incr(mark(X)) -> mark(incr(X)) hd(mark(X)) -> mark(hd(X)) tl(mark(X)) -> mark(tl(X)) proper(nats()) -> ok(nats()) proper(adx(X)) -> adx(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(s(X)) -> s(proper(X)) proper(hd(X)) -> hd(proper(X)) proper(tl(X)) -> tl(proper(X)) adx(ok(X)) -> ok(adx(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) hd(ok(X)) -> ok(hd(X)) tl(ok(X)) -> ok(tl(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [tl#](x0) = x0 + 1, [top](x0) = 1, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [tl](x0) = x0 + 1, [hd](x0) = x0, [s](x0) = x0 + 1, [incr](x0) = x0, [cons](x0, x1) = x1, [0] = 0, [mark](x0) = 0, [adx](x0) = x0, [zeros] = 0, [active](x0) = x0, [nats] = 0 orientation: tl#(ok(X)) = X + 2 >= X + 1 = tl#(X) active(nats()) = 0 >= 0 = mark(adx(zeros())) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(incr(cons(X,Y))) = Y >= 0 = mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) = Y >= 0 = mark(incr(cons(X,adx(Y)))) active(hd(cons(X,Y))) = Y >= 0 = mark(X) active(tl(cons(X,Y))) = Y + 1 >= 0 = mark(Y) active(adx(X)) = X >= X = adx(active(X)) active(incr(X)) = X >= X = incr(active(X)) active(hd(X)) = X >= X = hd(active(X)) active(tl(X)) = X + 1 >= X + 1 = tl(active(X)) adx(mark(X)) = 0 >= 0 = mark(adx(X)) incr(mark(X)) = 0 >= 0 = mark(incr(X)) hd(mark(X)) = 0 >= 0 = mark(hd(X)) tl(mark(X)) = 1 >= 0 = mark(tl(X)) proper(nats()) = 1 >= 1 = ok(nats()) proper(adx(X)) = X + 1 >= X + 1 = adx(proper(X)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = X2 + 1 >= X2 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(incr(X)) = X + 1 >= X + 1 = incr(proper(X)) proper(s(X)) = X + 2 >= X + 2 = s(proper(X)) proper(hd(X)) = X + 1 >= X + 1 = hd(proper(X)) proper(tl(X)) = X + 2 >= X + 2 = tl(proper(X)) adx(ok(X)) = X + 1 >= X + 1 = ok(adx(X)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) incr(ok(X)) = X + 1 >= X + 1 = ok(incr(X)) s(ok(X)) = X + 2 >= X + 2 = ok(s(X)) hd(ok(X)) = X + 1 >= X + 1 = ok(hd(X)) tl(ok(X)) = X + 2 >= X + 2 = ok(tl(X)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: TRS: active(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y)))) active(hd(cons(X,Y))) -> mark(X) active(tl(cons(X,Y))) -> mark(Y) active(adx(X)) -> adx(active(X)) active(incr(X)) -> incr(active(X)) active(hd(X)) -> hd(active(X)) active(tl(X)) -> tl(active(X)) adx(mark(X)) -> mark(adx(X)) incr(mark(X)) -> mark(incr(X)) hd(mark(X)) -> mark(hd(X)) tl(mark(X)) -> mark(tl(X)) proper(nats()) -> ok(nats()) proper(adx(X)) -> adx(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(s(X)) -> s(proper(X)) proper(hd(X)) -> hd(proper(X)) proper(tl(X)) -> tl(proper(X)) adx(ok(X)) -> ok(adx(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) hd(ok(X)) -> ok(hd(X)) tl(ok(X)) -> ok(tl(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed