MAYBE Problem: active(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) -> mark(X) active(tail(cons(X,XS))) -> mark(XS) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(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#(nats()) -> incr#(nats()) active#(nats()) -> cons#(0(),incr(nats())) active#(pairs()) -> incr#(odds()) active#(pairs()) -> cons#(0(),incr(odds())) active#(odds()) -> incr#(pairs()) active#(incr(cons(X,XS))) -> incr#(XS) active#(incr(cons(X,XS))) -> s#(X) active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) active#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(incr(X)) -> active#(X) active#(incr(X)) -> incr#(active(X)) active#(s(X)) -> active#(X) active#(s(X)) -> s#(active(X)) active#(head(X)) -> active#(X) active#(head(X)) -> head#(active(X)) active#(tail(X)) -> active#(X) active#(tail(X)) -> tail#(active(X)) cons#(mark(X1),X2) -> cons#(X1,X2) incr#(mark(X)) -> incr#(X) s#(mark(X)) -> s#(X) head#(mark(X)) -> head#(X) tail#(mark(X)) -> tail#(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#(head(X)) -> proper#(X) proper#(head(X)) -> head#(proper(X)) proper#(tail(X)) -> proper#(X) proper#(tail(X)) -> tail#(proper(X)) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) incr#(ok(X)) -> incr#(X) s#(ok(X)) -> s#(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(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) -> mark(X) active(tail(cons(X,XS))) -> mark(XS) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) TDG Processor: DPs: active#(nats()) -> incr#(nats()) active#(nats()) -> cons#(0(),incr(nats())) active#(pairs()) -> incr#(odds()) active#(pairs()) -> cons#(0(),incr(odds())) active#(odds()) -> incr#(pairs()) active#(incr(cons(X,XS))) -> incr#(XS) active#(incr(cons(X,XS))) -> s#(X) active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) active#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(incr(X)) -> active#(X) active#(incr(X)) -> incr#(active(X)) active#(s(X)) -> active#(X) active#(s(X)) -> s#(active(X)) active#(head(X)) -> active#(X) active#(head(X)) -> head#(active(X)) active#(tail(X)) -> active#(X) active#(tail(X)) -> tail#(active(X)) cons#(mark(X1),X2) -> cons#(X1,X2) incr#(mark(X)) -> incr#(X) s#(mark(X)) -> s#(X) head#(mark(X)) -> head#(X) tail#(mark(X)) -> tail#(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#(head(X)) -> proper#(X) proper#(head(X)) -> head#(proper(X)) proper#(tail(X)) -> proper#(X) proper#(tail(X)) -> tail#(proper(X)) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) incr#(ok(X)) -> incr#(X) s#(ok(X)) -> s#(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(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) -> mark(X) active(tail(cons(X,XS))) -> mark(XS) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) graph: top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> top#(active(X)) top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) -> top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> top#(active(X)) -> top#(mark(X)) -> proper#(X) top#(ok(X)) -> active#(X) -> active#(tail(X)) -> tail#(active(X)) top#(ok(X)) -> active#(X) -> active#(tail(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(head(X)) -> head#(active(X)) top#(ok(X)) -> active#(X) -> active#(head(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(s(X)) -> s#(active(X)) top#(ok(X)) -> active#(X) -> active#(s(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(incr(X)) -> incr#(active(X)) top#(ok(X)) -> active#(X) -> active#(incr(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) top#(ok(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) top#(ok(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) top#(ok(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> s#(X) top#(ok(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> incr#(XS) top#(ok(X)) -> active#(X) -> active#(odds()) -> incr#(pairs()) top#(ok(X)) -> active#(X) -> active#(pairs()) -> cons#(0(),incr(odds())) top#(ok(X)) -> active#(X) -> active#(pairs()) -> incr#(odds()) top#(ok(X)) -> active#(X) -> active#(nats()) -> cons#(0(),incr(nats())) top#(ok(X)) -> active#(X) -> active#(nats()) -> incr#(nats()) top#(mark(X)) -> top#(proper(X)) -> top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> top#(proper(X)) -> top#(ok(X)) -> active#(X) top#(mark(X)) -> top#(proper(X)) -> top#(mark(X)) -> top#(proper(X)) top#(mark(X)) -> top#(proper(X)) -> top#(mark(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(tail(X)) -> tail#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(tail(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(head(X)) -> head#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(head(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(incr(X)) -> incr#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(incr(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) top#(mark(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(tail(X)) -> proper#(X) -> proper#(tail(X)) -> tail#(proper(X)) proper#(tail(X)) -> proper#(X) -> proper#(tail(X)) -> proper#(X) proper#(tail(X)) -> proper#(X) -> proper#(head(X)) -> head#(proper(X)) proper#(tail(X)) -> proper#(X) -> proper#(head(X)) -> proper#(X) proper#(tail(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) proper#(tail(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(tail(X)) -> proper#(X) -> proper#(incr(X)) -> incr#(proper(X)) proper#(tail(X)) -> proper#(X) -> proper#(incr(X)) -> proper#(X) proper#(tail(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(tail(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(tail(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(tail(X)) -> tail#(proper(X)) -> tail#(ok(X)) -> tail#(X) proper#(tail(X)) -> tail#(proper(X)) -> tail#(mark(X)) -> tail#(X) proper#(head(X)) -> proper#(X) -> proper#(tail(X)) -> tail#(proper(X)) proper#(head(X)) -> proper#(X) -> proper#(tail(X)) -> proper#(X) proper#(head(X)) -> proper#(X) -> proper#(head(X)) -> head#(proper(X)) proper#(head(X)) -> proper#(X) -> proper#(head(X)) -> proper#(X) proper#(head(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) proper#(head(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(head(X)) -> proper#(X) -> proper#(incr(X)) -> incr#(proper(X)) proper#(head(X)) -> proper#(X) -> proper#(incr(X)) -> proper#(X) proper#(head(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(head(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(head(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(head(X)) -> head#(proper(X)) -> head#(ok(X)) -> head#(X) proper#(head(X)) -> head#(proper(X)) -> head#(mark(X)) -> head#(X) proper#(s(X)) -> proper#(X) -> proper#(tail(X)) -> tail#(proper(X)) proper#(s(X)) -> proper#(X) -> proper#(tail(X)) -> proper#(X) proper#(s(X)) -> proper#(X) -> proper#(head(X)) -> head#(proper(X)) proper#(s(X)) -> proper#(X) -> proper#(head(X)) -> proper#(X) proper#(s(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) proper#(s(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(s(X)) -> proper#(X) -> proper#(incr(X)) -> incr#(proper(X)) proper#(s(X)) -> proper#(X) -> proper#(incr(X)) -> proper#(X) proper#(s(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(s(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(s(X)) -> s#(proper(X)) -> s#(ok(X)) -> s#(X) proper#(s(X)) -> s#(proper(X)) -> s#(mark(X)) -> s#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(tail(X)) -> tail#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(tail(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(head(X)) -> head#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(head(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> s#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(incr(X)) -> incr#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(incr(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(tail(X)) -> tail#(proper(X)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(tail(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(head(X)) -> head#(proper(X)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(head(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> s#(proper(X)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(incr(X)) -> incr#(proper(X)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(incr(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) -> cons#(mark(X1),X2) -> cons#(X1,X2) proper#(incr(X)) -> proper#(X) -> proper#(tail(X)) -> tail#(proper(X)) proper#(incr(X)) -> proper#(X) -> proper#(tail(X)) -> proper#(X) proper#(incr(X)) -> proper#(X) -> proper#(head(X)) -> head#(proper(X)) proper#(incr(X)) -> proper#(X) -> proper#(head(X)) -> proper#(X) proper#(incr(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) proper#(incr(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(incr(X)) -> proper#(X) -> proper#(incr(X)) -> incr#(proper(X)) proper#(incr(X)) -> proper#(X) -> proper#(incr(X)) -> proper#(X) proper#(incr(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(incr(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(incr(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(incr(X)) -> incr#(proper(X)) -> incr#(ok(X)) -> incr#(X) proper#(incr(X)) -> incr#(proper(X)) -> incr#(mark(X)) -> incr#(X) tail#(ok(X)) -> tail#(X) -> tail#(ok(X)) -> tail#(X) tail#(ok(X)) -> tail#(X) -> tail#(mark(X)) -> tail#(X) tail#(mark(X)) -> tail#(X) -> tail#(ok(X)) -> tail#(X) tail#(mark(X)) -> tail#(X) -> tail#(mark(X)) -> tail#(X) head#(ok(X)) -> head#(X) -> head#(ok(X)) -> head#(X) head#(ok(X)) -> head#(X) -> head#(mark(X)) -> head#(X) head#(mark(X)) -> head#(X) -> head#(ok(X)) -> head#(X) head#(mark(X)) -> head#(X) -> head#(mark(X)) -> head#(X) s#(ok(X)) -> s#(X) -> s#(ok(X)) -> s#(X) s#(ok(X)) -> s#(X) -> s#(mark(X)) -> s#(X) s#(mark(X)) -> s#(X) -> s#(ok(X)) -> s#(X) s#(mark(X)) -> s#(X) -> s#(mark(X)) -> s#(X) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) -> cons#(mark(X1),X2) -> cons#(X1,X2) cons#(mark(X1),X2) -> cons#(X1,X2) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) cons#(mark(X1),X2) -> cons#(X1,X2) -> cons#(mark(X1),X2) -> cons#(X1,X2) incr#(ok(X)) -> incr#(X) -> incr#(ok(X)) -> incr#(X) incr#(ok(X)) -> incr#(X) -> incr#(mark(X)) -> incr#(X) incr#(mark(X)) -> incr#(X) -> incr#(ok(X)) -> incr#(X) incr#(mark(X)) -> incr#(X) -> incr#(mark(X)) -> incr#(X) active#(tail(X)) -> tail#(active(X)) -> tail#(ok(X)) -> tail#(X) active#(tail(X)) -> tail#(active(X)) -> tail#(mark(X)) -> tail#(X) active#(tail(X)) -> active#(X) -> active#(tail(X)) -> tail#(active(X)) active#(tail(X)) -> active#(X) -> active#(tail(X)) -> active#(X) active#(tail(X)) -> active#(X) -> active#(head(X)) -> head#(active(X)) active#(tail(X)) -> active#(X) -> active#(head(X)) -> active#(X) active#(tail(X)) -> active#(X) -> active#(s(X)) -> s#(active(X)) active#(tail(X)) -> active#(X) -> active#(s(X)) -> active#(X) active#(tail(X)) -> active#(X) -> active#(incr(X)) -> incr#(active(X)) active#(tail(X)) -> active#(X) -> active#(incr(X)) -> active#(X) active#(tail(X)) -> active#(X) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(tail(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) active#(tail(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) active#(tail(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> s#(X) active#(tail(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> incr#(XS) active#(tail(X)) -> active#(X) -> active#(odds()) -> incr#(pairs()) active#(tail(X)) -> active#(X) -> active#(pairs()) -> cons#(0(),incr(odds())) active#(tail(X)) -> active#(X) -> active#(pairs()) -> incr#(odds()) active#(tail(X)) -> active#(X) -> active#(nats()) -> cons#(0(),incr(nats())) active#(tail(X)) -> active#(X) -> active#(nats()) -> incr#(nats()) active#(head(X)) -> head#(active(X)) -> head#(ok(X)) -> head#(X) active#(head(X)) -> head#(active(X)) -> head#(mark(X)) -> head#(X) active#(head(X)) -> active#(X) -> active#(tail(X)) -> tail#(active(X)) active#(head(X)) -> active#(X) -> active#(tail(X)) -> active#(X) active#(head(X)) -> active#(X) -> active#(head(X)) -> head#(active(X)) active#(head(X)) -> active#(X) -> active#(head(X)) -> active#(X) active#(head(X)) -> active#(X) -> active#(s(X)) -> s#(active(X)) active#(head(X)) -> active#(X) -> active#(s(X)) -> active#(X) active#(head(X)) -> active#(X) -> active#(incr(X)) -> incr#(active(X)) active#(head(X)) -> active#(X) -> active#(incr(X)) -> active#(X) active#(head(X)) -> active#(X) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(head(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) active#(head(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) active#(head(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> s#(X) active#(head(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> incr#(XS) active#(head(X)) -> active#(X) -> active#(odds()) -> incr#(pairs()) active#(head(X)) -> active#(X) -> active#(pairs()) -> cons#(0(),incr(odds())) active#(head(X)) -> active#(X) -> active#(pairs()) -> incr#(odds()) active#(head(X)) -> active#(X) -> active#(nats()) -> cons#(0(),incr(nats())) active#(head(X)) -> active#(X) -> active#(nats()) -> incr#(nats()) active#(s(X)) -> s#(active(X)) -> s#(ok(X)) -> s#(X) active#(s(X)) -> s#(active(X)) -> s#(mark(X)) -> s#(X) active#(s(X)) -> active#(X) -> active#(tail(X)) -> tail#(active(X)) active#(s(X)) -> active#(X) -> active#(tail(X)) -> active#(X) active#(s(X)) -> active#(X) -> active#(head(X)) -> head#(active(X)) active#(s(X)) -> active#(X) -> active#(head(X)) -> active#(X) active#(s(X)) -> active#(X) -> active#(s(X)) -> s#(active(X)) active#(s(X)) -> active#(X) -> active#(s(X)) -> active#(X) active#(s(X)) -> active#(X) -> active#(incr(X)) -> incr#(active(X)) active#(s(X)) -> active#(X) -> active#(incr(X)) -> active#(X) active#(s(X)) -> active#(X) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(s(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) active#(s(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) active#(s(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> s#(X) active#(s(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> incr#(XS) active#(s(X)) -> active#(X) -> active#(odds()) -> incr#(pairs()) active#(s(X)) -> active#(X) -> active#(pairs()) -> cons#(0(),incr(odds())) active#(s(X)) -> active#(X) -> active#(pairs()) -> incr#(odds()) active#(s(X)) -> active#(X) -> active#(nats()) -> cons#(0(),incr(nats())) active#(s(X)) -> active#(X) -> active#(nats()) -> incr#(nats()) active#(odds()) -> incr#(pairs()) -> incr#(ok(X)) -> incr#(X) active#(odds()) -> incr#(pairs()) -> incr#(mark(X)) -> incr#(X) active#(pairs()) -> cons#(0(),incr(odds())) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) active#(pairs()) -> cons#(0(),incr(odds())) -> cons#(mark(X1),X2) -> cons#(X1,X2) active#(pairs()) -> incr#(odds()) -> incr#(ok(X)) -> incr#(X) active#(pairs()) -> incr#(odds()) -> incr#(mark(X)) -> incr#(X) active#(cons(X1,X2)) -> cons#(active(X1),X2) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) active#(cons(X1,X2)) -> cons#(active(X1),X2) -> cons#(mark(X1),X2) -> cons#(X1,X2) active#(cons(X1,X2)) -> active#(X1) -> active#(tail(X)) -> tail#(active(X)) active#(cons(X1,X2)) -> active#(X1) -> active#(tail(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) -> active#(head(X)) -> head#(active(X)) active#(cons(X1,X2)) -> active#(X1) -> active#(head(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) -> active#(s(X)) -> s#(active(X)) active#(cons(X1,X2)) -> active#(X1) -> active#(s(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) -> active#(incr(X)) -> incr#(active(X)) active#(cons(X1,X2)) -> active#(X1) -> active#(incr(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(cons(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> active#(X1) -> active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) active#(cons(X1,X2)) -> active#(X1) -> active#(incr(cons(X,XS))) -> s#(X) active#(cons(X1,X2)) -> active#(X1) -> active#(incr(cons(X,XS))) -> incr#(XS) active#(cons(X1,X2)) -> active#(X1) -> active#(odds()) -> incr#(pairs()) active#(cons(X1,X2)) -> active#(X1) -> active#(pairs()) -> cons#(0(),incr(odds())) active#(cons(X1,X2)) -> active#(X1) -> active#(pairs()) -> incr#(odds()) active#(cons(X1,X2)) -> active#(X1) -> active#(nats()) -> cons#(0(),incr(nats())) active#(cons(X1,X2)) -> active#(X1) -> active#(nats()) -> incr#(nats()) active#(incr(cons(X,XS))) -> s#(X) -> s#(ok(X)) -> s#(X) active#(incr(cons(X,XS))) -> s#(X) -> s#(mark(X)) -> s#(X) active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) -> cons#(mark(X1),X2) -> cons#(X1,X2) active#(incr(cons(X,XS))) -> incr#(XS) -> incr#(ok(X)) -> incr#(X) active#(incr(cons(X,XS))) -> incr#(XS) -> incr#(mark(X)) -> incr#(X) active#(incr(X)) -> incr#(active(X)) -> incr#(ok(X)) -> incr#(X) active#(incr(X)) -> incr#(active(X)) -> incr#(mark(X)) -> incr#(X) active#(incr(X)) -> active#(X) -> active#(tail(X)) -> tail#(active(X)) active#(incr(X)) -> active#(X) -> active#(tail(X)) -> active#(X) active#(incr(X)) -> active#(X) -> active#(head(X)) -> head#(active(X)) active#(incr(X)) -> active#(X) -> active#(head(X)) -> active#(X) active#(incr(X)) -> active#(X) -> active#(s(X)) -> s#(active(X)) active#(incr(X)) -> active#(X) -> active#(s(X)) -> active#(X) active#(incr(X)) -> active#(X) -> active#(incr(X)) -> incr#(active(X)) active#(incr(X)) -> active#(X) -> active#(incr(X)) -> active#(X) active#(incr(X)) -> active#(X) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(incr(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) active#(incr(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) active#(incr(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> s#(X) active#(incr(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> incr#(XS) active#(incr(X)) -> active#(X) -> active#(odds()) -> incr#(pairs()) active#(incr(X)) -> active#(X) -> active#(pairs()) -> cons#(0(),incr(odds())) active#(incr(X)) -> active#(X) -> active#(pairs()) -> incr#(odds()) active#(incr(X)) -> active#(X) -> active#(nats()) -> cons#(0(),incr(nats())) active#(incr(X)) -> active#(X) -> active#(nats()) -> incr#(nats()) active#(nats()) -> cons#(0(),incr(nats())) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) active#(nats()) -> cons#(0(),incr(nats())) -> cons#(mark(X1),X2) -> cons#(X1,X2) active#(nats()) -> incr#(nats()) -> incr#(ok(X)) -> incr#(X) active#(nats()) -> incr#(nats()) -> incr#(mark(X)) -> incr#(X) EDG Processor: DPs: active#(nats()) -> incr#(nats()) active#(nats()) -> cons#(0(),incr(nats())) active#(pairs()) -> incr#(odds()) active#(pairs()) -> cons#(0(),incr(odds())) active#(odds()) -> incr#(pairs()) active#(incr(cons(X,XS))) -> incr#(XS) active#(incr(cons(X,XS))) -> s#(X) active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) active#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(incr(X)) -> active#(X) active#(incr(X)) -> incr#(active(X)) active#(s(X)) -> active#(X) active#(s(X)) -> s#(active(X)) active#(head(X)) -> active#(X) active#(head(X)) -> head#(active(X)) active#(tail(X)) -> active#(X) active#(tail(X)) -> tail#(active(X)) cons#(mark(X1),X2) -> cons#(X1,X2) incr#(mark(X)) -> incr#(X) s#(mark(X)) -> s#(X) head#(mark(X)) -> head#(X) tail#(mark(X)) -> tail#(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#(head(X)) -> proper#(X) proper#(head(X)) -> head#(proper(X)) proper#(tail(X)) -> proper#(X) proper#(tail(X)) -> tail#(proper(X)) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) incr#(ok(X)) -> incr#(X) s#(ok(X)) -> s#(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(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) -> mark(X) active(tail(cons(X,XS))) -> mark(XS) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(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#(nats()) -> incr#(nats()) top#(ok(X)) -> active#(X) -> active#(nats()) -> cons#(0(),incr(nats())) top#(ok(X)) -> active#(X) -> active#(pairs()) -> incr#(odds()) top#(ok(X)) -> active#(X) -> active#(pairs()) -> cons#(0(),incr(odds())) top#(ok(X)) -> active#(X) -> active#(odds()) -> incr#(pairs()) top#(ok(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> incr#(XS) top#(ok(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> s#(X) top#(ok(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) 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#(incr(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(incr(X)) -> incr#(active(X)) 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#(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#(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#(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#(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#(incr(X)) -> proper#(X) proper#(tail(X)) -> proper#(X) -> proper#(incr(X)) -> incr#(proper(X)) 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#(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#(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#(incr(X)) -> proper#(X) proper#(head(X)) -> proper#(X) -> proper#(incr(X)) -> incr#(proper(X)) 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#(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#(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#(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#(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#(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#(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#(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#(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#(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) 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) 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) 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#(nats()) -> incr#(nats()) active#(tail(X)) -> active#(X) -> active#(nats()) -> cons#(0(),incr(nats())) active#(tail(X)) -> active#(X) -> active#(pairs()) -> incr#(odds()) active#(tail(X)) -> active#(X) -> active#(pairs()) -> cons#(0(),incr(odds())) active#(tail(X)) -> active#(X) -> active#(odds()) -> incr#(pairs()) active#(tail(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> incr#(XS) active#(tail(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> s#(X) active#(tail(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) 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#(incr(X)) -> active#(X) active#(tail(X)) -> active#(X) -> active#(incr(X)) -> incr#(active(X)) 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#(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#(nats()) -> incr#(nats()) active#(head(X)) -> active#(X) -> active#(nats()) -> cons#(0(),incr(nats())) active#(head(X)) -> active#(X) -> active#(pairs()) -> incr#(odds()) active#(head(X)) -> active#(X) -> active#(pairs()) -> cons#(0(),incr(odds())) active#(head(X)) -> active#(X) -> active#(odds()) -> incr#(pairs()) active#(head(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> incr#(XS) active#(head(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> s#(X) active#(head(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) 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#(incr(X)) -> active#(X) active#(head(X)) -> active#(X) -> active#(incr(X)) -> incr#(active(X)) 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#(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#(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#(nats()) -> incr#(nats()) active#(s(X)) -> active#(X) -> active#(nats()) -> cons#(0(),incr(nats())) active#(s(X)) -> active#(X) -> active#(pairs()) -> incr#(odds()) active#(s(X)) -> active#(X) -> active#(pairs()) -> cons#(0(),incr(odds())) active#(s(X)) -> active#(X) -> active#(odds()) -> incr#(pairs()) active#(s(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> incr#(XS) active#(s(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> s#(X) active#(s(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) 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#(incr(X)) -> active#(X) active#(s(X)) -> active#(X) -> active#(incr(X)) -> incr#(active(X)) 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#(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#(nats()) -> incr#(nats()) active#(cons(X1,X2)) -> active#(X1) -> active#(nats()) -> cons#(0(),incr(nats())) active#(cons(X1,X2)) -> active#(X1) -> active#(pairs()) -> incr#(odds()) active#(cons(X1,X2)) -> active#(X1) -> active#(pairs()) -> cons#(0(),incr(odds())) active#(cons(X1,X2)) -> active#(X1) -> active#(odds()) -> incr#(pairs()) active#(cons(X1,X2)) -> active#(X1) -> active#(incr(cons(X,XS))) -> incr#(XS) active#(cons(X1,X2)) -> active#(X1) -> active#(incr(cons(X,XS))) -> s#(X) active#(cons(X1,X2)) -> active#(X1) -> active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) 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#(incr(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) -> active#(incr(X)) -> incr#(active(X)) 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#(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,XS))) -> s#(X) -> s#(ok(X)) -> s#(X) active#(incr(cons(X,XS))) -> incr#(XS) -> incr#(mark(X)) -> incr#(X) active#(incr(cons(X,XS))) -> incr#(XS) -> 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()) -> incr#(nats()) active#(incr(X)) -> active#(X) -> active#(nats()) -> cons#(0(),incr(nats())) active#(incr(X)) -> active#(X) -> active#(pairs()) -> incr#(odds()) active#(incr(X)) -> active#(X) -> active#(pairs()) -> cons#(0(),incr(odds())) active#(incr(X)) -> active#(X) -> active#(odds()) -> incr#(pairs()) active#(incr(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> incr#(XS) active#(incr(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> s#(X) active#(incr(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) 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#(incr(X)) -> active#(X) active#(incr(X)) -> active#(X) -> active#(incr(X)) -> incr#(active(X)) 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#(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: 8 #rules: 23 #arcs: 236/1849 DPs: top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> top#(proper(X)) TRS: active(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) -> mark(X) active(tail(cons(X,XS))) -> mark(XS) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(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#(s(X)) -> proper#(X) proper#(incr(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) TRS: active(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) -> mark(X) active(tail(cons(X,XS))) -> mark(XS) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(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) = x0, [proper](x0) = x0, [tail](x0) = x0 + 1, [head](x0) = x0 + 1, [s](x0) = x0 + 1, [odds] = 1, [pairs] = 1, [mark](x0) = 0, [cons](x0, x1) = x0 + x1, [incr](x0) = x0 + 1, [0] = 0, [active](x0) = x0, [nats] = 1 orientation: proper#(tail(X)) = X + 2 >= X + 1 = proper#(X) proper#(head(X)) = X + 2 >= X + 1 = proper#(X) proper#(s(X)) = X + 2 >= X + 1 = proper#(X) proper#(incr(X)) = X + 2 >= X + 1 = proper#(X) proper#(cons(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = proper#(X1) proper#(cons(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = proper#(X2) active(nats()) = 1 >= 0 = mark(cons(0(),incr(nats()))) active(pairs()) = 1 >= 0 = mark(cons(0(),incr(odds()))) active(odds()) = 1 >= 0 = mark(incr(pairs())) active(incr(cons(X,XS))) = X + XS + 1 >= 0 = mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) = X + XS + 1 >= 0 = mark(X) active(tail(cons(X,XS))) = X + XS + 1 >= 0 = mark(XS) active(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(active(X1),X2) active(incr(X)) = X + 1 >= X + 1 = incr(active(X)) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(head(X)) = X + 1 >= X + 1 = head(active(X)) active(tail(X)) = X + 1 >= X + 1 = tail(active(X)) cons(mark(X1),X2) = X2 >= 0 = mark(cons(X1,X2)) incr(mark(X)) = 1 >= 0 = mark(incr(X)) s(mark(X)) = 1 >= 0 = mark(s(X)) head(mark(X)) = 1 >= 0 = mark(head(X)) tail(mark(X)) = 1 >= 0 = mark(tail(X)) proper(nats()) = 1 >= 1 = ok(nats()) proper(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(incr(X)) = X + 1 >= X + 1 = incr(proper(X)) proper(pairs()) = 1 >= 1 = ok(pairs()) proper(odds()) = 1 >= 1 = ok(odds()) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(head(X)) = X + 1 >= X + 1 = head(proper(X)) proper(tail(X)) = X + 1 >= X + 1 = tail(proper(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)) = X + 1 >= X + 1 = ok(s(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: proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) TRS: active(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) -> mark(X) active(tail(cons(X,XS))) -> mark(XS) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(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) = x0, [proper](x0) = x0, [tail](x0) = 0, [head](x0) = 0, [s](x0) = 0, [odds] = 1, [pairs] = 1, [mark](x0) = 0, [cons](x0, x1) = x0 + x1 + 1, [incr](x0) = 1, [0] = 0, [active](x0) = x0 + 1, [nats] = 1 orientation: proper#(cons(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(cons(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) active(nats()) = 2 >= 0 = mark(cons(0(),incr(nats()))) active(pairs()) = 2 >= 0 = mark(cons(0(),incr(odds()))) active(odds()) = 2 >= 0 = mark(incr(pairs())) active(incr(cons(X,XS))) = 2 >= 0 = mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) = 1 >= 0 = mark(X) active(tail(cons(X,XS))) = 1 >= 0 = mark(XS) active(cons(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = cons(active(X1),X2) active(incr(X)) = 2 >= 1 = incr(active(X)) active(s(X)) = 1 >= 0 = s(active(X)) active(head(X)) = 1 >= 0 = head(active(X)) active(tail(X)) = 1 >= 0 = tail(active(X)) cons(mark(X1),X2) = X2 + 1 >= 0 = mark(cons(X1,X2)) incr(mark(X)) = 1 >= 0 = mark(incr(X)) s(mark(X)) = 0 >= 0 = mark(s(X)) head(mark(X)) = 0 >= 0 = mark(head(X)) tail(mark(X)) = 0 >= 0 = mark(tail(X)) proper(nats()) = 1 >= 1 = ok(nats()) proper(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(incr(X)) = 1 >= 1 = incr(proper(X)) proper(pairs()) = 1 >= 1 = ok(pairs()) proper(odds()) = 1 >= 1 = ok(odds()) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(head(X)) = 0 >= 0 = head(proper(X)) proper(tail(X)) = 0 >= 0 = tail(proper(X)) cons(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(cons(X1,X2)) incr(ok(X)) = 1 >= 1 = ok(incr(X)) s(ok(X)) = 0 >= 0 = ok(s(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(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) -> mark(X) active(tail(cons(X,XS))) -> mark(XS) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(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#(s(X)) -> active#(X) active#(incr(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) TRS: active(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) -> mark(X) active(tail(cons(X,XS))) -> mark(XS) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(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, [proper](x0) = x0, [tail](x0) = x0 + 1, [head](x0) = x0 + 1, [s](x0) = x0 + 1, [odds] = 0, [pairs] = 0, [mark](x0) = 0, [cons](x0, x1) = x0 + 1, [incr](x0) = x0 + 1, [0] = 1, [active](x0) = x0 + 1, [nats] = 0 orientation: active#(tail(X)) = X + 2 >= X + 1 = active#(X) active#(head(X)) = X + 2 >= X + 1 = active#(X) active#(s(X)) = X + 2 >= X + 1 = active#(X) active#(incr(X)) = X + 2 >= X + 1 = active#(X) active#(cons(X1,X2)) = X1 + 2 >= X1 + 1 = active#(X1) active(nats()) = 1 >= 0 = mark(cons(0(),incr(nats()))) active(pairs()) = 1 >= 0 = mark(cons(0(),incr(odds()))) active(odds()) = 1 >= 0 = mark(incr(pairs())) active(incr(cons(X,XS))) = X + 3 >= 0 = mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) = X + 3 >= 0 = mark(X) active(tail(cons(X,XS))) = X + 3 >= 0 = mark(XS) active(cons(X1,X2)) = X1 + 2 >= X1 + 2 = cons(active(X1),X2) active(incr(X)) = X + 2 >= X + 2 = incr(active(X)) active(s(X)) = X + 2 >= X + 2 = s(active(X)) active(head(X)) = X + 2 >= X + 2 = head(active(X)) active(tail(X)) = X + 2 >= X + 2 = tail(active(X)) cons(mark(X1),X2) = 1 >= 0 = mark(cons(X1,X2)) incr(mark(X)) = 1 >= 0 = mark(incr(X)) s(mark(X)) = 1 >= 0 = mark(s(X)) head(mark(X)) = 1 >= 0 = mark(head(X)) tail(mark(X)) = 1 >= 0 = mark(tail(X)) proper(nats()) = 0 >= 0 = ok(nats()) proper(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(incr(X)) = X + 1 >= X + 1 = incr(proper(X)) proper(pairs()) = 0 >= 0 = ok(pairs()) proper(odds()) = 0 >= 0 = ok(odds()) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(head(X)) = X + 1 >= X + 1 = head(proper(X)) proper(tail(X)) = X + 1 >= X + 1 = tail(proper(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)) 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(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) -> mark(X) active(tail(cons(X,XS))) -> mark(XS) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(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(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) -> mark(X) active(tail(cons(X,XS))) -> mark(XS) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(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, [s](x0) = x0, [odds] = 0, [pairs] = 0, [mark](x0) = 0, [cons](x0, x1) = x0 + 1, [incr](x0) = x0, [0] = 0, [active](x0) = x0, [nats] = 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(nats()) = 0 >= 0 = mark(cons(0(),incr(nats()))) active(pairs()) = 0 >= 0 = mark(cons(0(),incr(odds()))) active(odds()) = 0 >= 0 = mark(incr(pairs())) active(incr(cons(X,XS))) = X + 1 >= 0 = mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) = X + 1 >= 0 = mark(X) active(tail(cons(X,XS))) = X + 1 >= 0 = mark(XS) active(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(active(X1),X2) active(incr(X)) = X >= X = incr(active(X)) active(s(X)) = X >= X = s(active(X)) active(head(X)) = X >= X = head(active(X)) active(tail(X)) = X >= X = tail(active(X)) cons(mark(X1),X2) = 1 >= 0 = mark(cons(X1,X2)) incr(mark(X)) = 0 >= 0 = mark(incr(X)) s(mark(X)) = 0 >= 0 = mark(s(X)) head(mark(X)) = 0 >= 0 = mark(head(X)) tail(mark(X)) = 0 >= 0 = mark(tail(X)) proper(nats()) = 1 >= 1 = ok(nats()) proper(cons(X1,X2)) = X1 + 2 >= X1 + 2 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(incr(X)) = X + 1 >= X + 1 = incr(proper(X)) proper(pairs()) = 1 >= 1 = ok(pairs()) proper(odds()) = 1 >= 1 = ok(odds()) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(head(X)) = X + 1 >= X + 1 = head(proper(X)) proper(tail(X)) = X + 1 >= X + 1 = tail(proper(X)) cons(ok(X1),ok(X2)) = X1 + 2 >= X1 + 2 = ok(cons(X1,X2)) incr(ok(X)) = X + 1 >= X + 1 = ok(incr(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(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(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) -> mark(X) active(tail(cons(X,XS))) -> mark(XS) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(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, [s](x0) = x0, [odds] = 0, [pairs] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [incr](x0) = x0, [0] = 0, [active](x0) = x0 + 1, [nats] = 0 orientation: cons#(mark(X1),X2) = X1 + 2 >= X1 + 1 = cons#(X1,X2) active(nats()) = 1 >= 1 = mark(cons(0(),incr(nats()))) active(pairs()) = 1 >= 1 = mark(cons(0(),incr(odds()))) active(odds()) = 1 >= 1 = mark(incr(pairs())) active(incr(cons(X,XS))) = X + XS + 1 >= X + XS + 1 = mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) = X + XS + 1 >= X + 1 = mark(X) active(tail(cons(X,XS))) = X + XS + 1 >= XS + 1 = mark(XS) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(incr(X)) = X + 1 >= X + 1 = incr(active(X)) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(head(X)) = X + 1 >= X + 1 = head(active(X)) active(tail(X)) = X + 1 >= X + 1 = tail(active(X)) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) incr(mark(X)) = X + 1 >= X + 1 = mark(incr(X)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) head(mark(X)) = X + 1 >= X + 1 = mark(head(X)) tail(mark(X)) = X + 1 >= X + 1 = mark(tail(X)) proper(nats()) = 0 >= 0 = ok(nats()) 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(pairs()) = 0 >= 0 = ok(pairs()) proper(odds()) = 0 >= 0 = ok(odds()) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(head(X)) = 0 >= 0 = head(proper(X)) proper(tail(X)) = 0 >= 0 = tail(proper(X)) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) incr(ok(X)) = 0 >= 0 = ok(incr(X)) s(ok(X)) = 0 >= 0 = ok(s(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(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) -> mark(X) active(tail(cons(X,XS))) -> mark(XS) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(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(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) -> mark(X) active(tail(cons(X,XS))) -> mark(XS) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(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, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = 0, [tail](x0) = x0, [head](x0) = x0, [s](x0) = x0, [odds] = 0, [pairs] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [incr](x0) = x0, [0] = 0, [active](x0) = x0 + 1, [nats] = 0 orientation: incr#(ok(X)) = X >= X = incr#(X) incr#(mark(X)) = X + 1 >= X = incr#(X) active(nats()) = 1 >= 1 = mark(cons(0(),incr(nats()))) active(pairs()) = 1 >= 1 = mark(cons(0(),incr(odds()))) active(odds()) = 1 >= 1 = mark(incr(pairs())) active(incr(cons(X,XS))) = X + XS + 1 >= X + XS + 1 = mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) = X + XS + 1 >= X + 1 = mark(X) active(tail(cons(X,XS))) = X + XS + 1 >= XS + 1 = mark(XS) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(incr(X)) = X + 1 >= X + 1 = incr(active(X)) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(head(X)) = X + 1 >= X + 1 = head(active(X)) active(tail(X)) = X + 1 >= X + 1 = tail(active(X)) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) incr(mark(X)) = X + 1 >= X + 1 = mark(incr(X)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) head(mark(X)) = X + 1 >= X + 1 = mark(head(X)) tail(mark(X)) = X + 1 >= X + 1 = mark(tail(X)) proper(nats()) = 0 >= 0 = ok(nats()) 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(pairs()) = 0 >= 0 = ok(pairs()) proper(odds()) = 0 >= 0 = ok(odds()) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(head(X)) = 0 >= 0 = head(proper(X)) proper(tail(X)) = 0 >= 0 = tail(proper(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)) 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(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) -> mark(X) active(tail(cons(X,XS))) -> mark(XS) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(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, [s](x0) = x0, [odds] = 1, [pairs] = 0, [mark](x0) = 0, [cons](x0, x1) = x1, [incr](x0) = x0, [0] = 0, [active](x0) = x0 + 1, [nats] = 1 orientation: incr#(ok(X)) = X + 2 >= X + 1 = incr#(X) active(nats()) = 2 >= 0 = mark(cons(0(),incr(nats()))) active(pairs()) = 1 >= 0 = mark(cons(0(),incr(odds()))) active(odds()) = 2 >= 0 = mark(incr(pairs())) active(incr(cons(X,XS))) = XS + 1 >= 0 = mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) = XS + 1 >= 0 = mark(X) active(tail(cons(X,XS))) = XS + 1 >= 0 = mark(XS) active(cons(X1,X2)) = X2 + 1 >= X2 = cons(active(X1),X2) active(incr(X)) = X + 1 >= X + 1 = incr(active(X)) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(head(X)) = X + 1 >= X + 1 = head(active(X)) active(tail(X)) = X + 1 >= X + 1 = tail(active(X)) cons(mark(X1),X2) = X2 >= 0 = mark(cons(X1,X2)) incr(mark(X)) = 0 >= 0 = mark(incr(X)) s(mark(X)) = 0 >= 0 = mark(s(X)) head(mark(X)) = 0 >= 0 = mark(head(X)) tail(mark(X)) = 0 >= 0 = mark(tail(X)) proper(nats()) = 2 >= 2 = ok(nats()) 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(pairs()) = 1 >= 1 = ok(pairs()) proper(odds()) = 2 >= 2 = ok(odds()) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(head(X)) = X + 1 >= X + 1 = head(proper(X)) proper(tail(X)) = X + 1 >= X + 1 = tail(proper(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 + 1 >= X + 1 = ok(s(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(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) -> mark(X) active(tail(cons(X,XS))) -> mark(XS) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(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(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) -> mark(X) active(tail(cons(X,XS))) -> mark(XS) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(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, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = 0, [tail](x0) = x0, [head](x0) = x0, [s](x0) = x0, [odds] = 0, [pairs] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [incr](x0) = x0, [0] = 0, [active](x0) = x0 + 1, [nats] = 0 orientation: s#(ok(X)) = X >= X = s#(X) s#(mark(X)) = X + 1 >= X = s#(X) active(nats()) = 1 >= 1 = mark(cons(0(),incr(nats()))) active(pairs()) = 1 >= 1 = mark(cons(0(),incr(odds()))) active(odds()) = 1 >= 1 = mark(incr(pairs())) active(incr(cons(X,XS))) = X + XS + 1 >= X + XS + 1 = mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) = X + XS + 1 >= X + 1 = mark(X) active(tail(cons(X,XS))) = X + XS + 1 >= XS + 1 = mark(XS) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(incr(X)) = X + 1 >= X + 1 = incr(active(X)) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(head(X)) = X + 1 >= X + 1 = head(active(X)) active(tail(X)) = X + 1 >= X + 1 = tail(active(X)) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) incr(mark(X)) = X + 1 >= X + 1 = mark(incr(X)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) head(mark(X)) = X + 1 >= X + 1 = mark(head(X)) tail(mark(X)) = X + 1 >= X + 1 = mark(tail(X)) proper(nats()) = 0 >= 0 = ok(nats()) 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(pairs()) = 0 >= 0 = ok(pairs()) proper(odds()) = 0 >= 0 = ok(odds()) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(head(X)) = 0 >= 0 = head(proper(X)) proper(tail(X)) = 0 >= 0 = tail(proper(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)) 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(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) -> mark(X) active(tail(cons(X,XS))) -> mark(XS) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(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, [s](x0) = x0, [odds] = 1, [pairs] = 0, [mark](x0) = 0, [cons](x0, x1) = x1, [incr](x0) = x0, [0] = 0, [active](x0) = x0 + 1, [nats] = 1 orientation: s#(ok(X)) = X + 2 >= X + 1 = s#(X) active(nats()) = 2 >= 0 = mark(cons(0(),incr(nats()))) active(pairs()) = 1 >= 0 = mark(cons(0(),incr(odds()))) active(odds()) = 2 >= 0 = mark(incr(pairs())) active(incr(cons(X,XS))) = XS + 1 >= 0 = mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) = XS + 1 >= 0 = mark(X) active(tail(cons(X,XS))) = XS + 1 >= 0 = mark(XS) active(cons(X1,X2)) = X2 + 1 >= X2 = cons(active(X1),X2) active(incr(X)) = X + 1 >= X + 1 = incr(active(X)) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(head(X)) = X + 1 >= X + 1 = head(active(X)) active(tail(X)) = X + 1 >= X + 1 = tail(active(X)) cons(mark(X1),X2) = X2 >= 0 = mark(cons(X1,X2)) incr(mark(X)) = 0 >= 0 = mark(incr(X)) s(mark(X)) = 0 >= 0 = mark(s(X)) head(mark(X)) = 0 >= 0 = mark(head(X)) tail(mark(X)) = 0 >= 0 = mark(tail(X)) proper(nats()) = 2 >= 2 = ok(nats()) 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(pairs()) = 1 >= 1 = ok(pairs()) proper(odds()) = 2 >= 2 = ok(odds()) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(head(X)) = X + 1 >= X + 1 = head(proper(X)) proper(tail(X)) = X + 1 >= X + 1 = tail(proper(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 + 1 >= X + 1 = ok(s(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(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) -> mark(X) active(tail(cons(X,XS))) -> mark(XS) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(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(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) -> mark(X) active(tail(cons(X,XS))) -> mark(XS) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(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, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = 0, [tail](x0) = x0, [head](x0) = x0, [s](x0) = x0, [odds] = 0, [pairs] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [incr](x0) = x0, [0] = 0, [active](x0) = x0 + 1, [nats] = 0 orientation: head#(ok(X)) = X >= X = head#(X) head#(mark(X)) = X + 1 >= X = head#(X) active(nats()) = 1 >= 1 = mark(cons(0(),incr(nats()))) active(pairs()) = 1 >= 1 = mark(cons(0(),incr(odds()))) active(odds()) = 1 >= 1 = mark(incr(pairs())) active(incr(cons(X,XS))) = X + XS + 1 >= X + XS + 1 = mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) = X + XS + 1 >= X + 1 = mark(X) active(tail(cons(X,XS))) = X + XS + 1 >= XS + 1 = mark(XS) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(incr(X)) = X + 1 >= X + 1 = incr(active(X)) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(head(X)) = X + 1 >= X + 1 = head(active(X)) active(tail(X)) = X + 1 >= X + 1 = tail(active(X)) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) incr(mark(X)) = X + 1 >= X + 1 = mark(incr(X)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) head(mark(X)) = X + 1 >= X + 1 = mark(head(X)) tail(mark(X)) = X + 1 >= X + 1 = mark(tail(X)) proper(nats()) = 0 >= 0 = ok(nats()) 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(pairs()) = 0 >= 0 = ok(pairs()) proper(odds()) = 0 >= 0 = ok(odds()) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(head(X)) = 0 >= 0 = head(proper(X)) proper(tail(X)) = 0 >= 0 = tail(proper(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)) 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(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) -> mark(X) active(tail(cons(X,XS))) -> mark(XS) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(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, [s](x0) = x0, [odds] = 1, [pairs] = 0, [mark](x0) = 0, [cons](x0, x1) = x1, [incr](x0) = x0, [0] = 0, [active](x0) = x0 + 1, [nats] = 1 orientation: head#(ok(X)) = X + 2 >= X + 1 = head#(X) active(nats()) = 2 >= 0 = mark(cons(0(),incr(nats()))) active(pairs()) = 1 >= 0 = mark(cons(0(),incr(odds()))) active(odds()) = 2 >= 0 = mark(incr(pairs())) active(incr(cons(X,XS))) = XS + 1 >= 0 = mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) = XS + 1 >= 0 = mark(X) active(tail(cons(X,XS))) = XS + 1 >= 0 = mark(XS) active(cons(X1,X2)) = X2 + 1 >= X2 = cons(active(X1),X2) active(incr(X)) = X + 1 >= X + 1 = incr(active(X)) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(head(X)) = X + 1 >= X + 1 = head(active(X)) active(tail(X)) = X + 1 >= X + 1 = tail(active(X)) cons(mark(X1),X2) = X2 >= 0 = mark(cons(X1,X2)) incr(mark(X)) = 0 >= 0 = mark(incr(X)) s(mark(X)) = 0 >= 0 = mark(s(X)) head(mark(X)) = 0 >= 0 = mark(head(X)) tail(mark(X)) = 0 >= 0 = mark(tail(X)) proper(nats()) = 2 >= 2 = ok(nats()) 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(pairs()) = 1 >= 1 = ok(pairs()) proper(odds()) = 2 >= 2 = ok(odds()) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(head(X)) = X + 1 >= X + 1 = head(proper(X)) proper(tail(X)) = X + 1 >= X + 1 = tail(proper(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 + 1 >= X + 1 = ok(s(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(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) -> mark(X) active(tail(cons(X,XS))) -> mark(XS) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(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(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) -> mark(X) active(tail(cons(X,XS))) -> mark(XS) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(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, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = 0, [tail](x0) = x0, [head](x0) = x0, [s](x0) = x0, [odds] = 0, [pairs] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [incr](x0) = x0, [0] = 0, [active](x0) = x0 + 1, [nats] = 0 orientation: tail#(ok(X)) = X >= X = tail#(X) tail#(mark(X)) = X + 1 >= X = tail#(X) active(nats()) = 1 >= 1 = mark(cons(0(),incr(nats()))) active(pairs()) = 1 >= 1 = mark(cons(0(),incr(odds()))) active(odds()) = 1 >= 1 = mark(incr(pairs())) active(incr(cons(X,XS))) = X + XS + 1 >= X + XS + 1 = mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) = X + XS + 1 >= X + 1 = mark(X) active(tail(cons(X,XS))) = X + XS + 1 >= XS + 1 = mark(XS) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(incr(X)) = X + 1 >= X + 1 = incr(active(X)) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(head(X)) = X + 1 >= X + 1 = head(active(X)) active(tail(X)) = X + 1 >= X + 1 = tail(active(X)) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) incr(mark(X)) = X + 1 >= X + 1 = mark(incr(X)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) head(mark(X)) = X + 1 >= X + 1 = mark(head(X)) tail(mark(X)) = X + 1 >= X + 1 = mark(tail(X)) proper(nats()) = 0 >= 0 = ok(nats()) 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(pairs()) = 0 >= 0 = ok(pairs()) proper(odds()) = 0 >= 0 = ok(odds()) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(head(X)) = 0 >= 0 = head(proper(X)) proper(tail(X)) = 0 >= 0 = tail(proper(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)) 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(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) -> mark(X) active(tail(cons(X,XS))) -> mark(XS) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(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, [s](x0) = x0, [odds] = 1, [pairs] = 0, [mark](x0) = 0, [cons](x0, x1) = x1, [incr](x0) = x0, [0] = 0, [active](x0) = x0 + 1, [nats] = 1 orientation: tail#(ok(X)) = X + 2 >= X + 1 = tail#(X) active(nats()) = 2 >= 0 = mark(cons(0(),incr(nats()))) active(pairs()) = 1 >= 0 = mark(cons(0(),incr(odds()))) active(odds()) = 2 >= 0 = mark(incr(pairs())) active(incr(cons(X,XS))) = XS + 1 >= 0 = mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) = XS + 1 >= 0 = mark(X) active(tail(cons(X,XS))) = XS + 1 >= 0 = mark(XS) active(cons(X1,X2)) = X2 + 1 >= X2 = cons(active(X1),X2) active(incr(X)) = X + 1 >= X + 1 = incr(active(X)) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(head(X)) = X + 1 >= X + 1 = head(active(X)) active(tail(X)) = X + 1 >= X + 1 = tail(active(X)) cons(mark(X1),X2) = X2 >= 0 = mark(cons(X1,X2)) incr(mark(X)) = 0 >= 0 = mark(incr(X)) s(mark(X)) = 0 >= 0 = mark(s(X)) head(mark(X)) = 0 >= 0 = mark(head(X)) tail(mark(X)) = 0 >= 0 = mark(tail(X)) proper(nats()) = 2 >= 2 = ok(nats()) 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(pairs()) = 1 >= 1 = ok(pairs()) proper(odds()) = 2 >= 2 = ok(odds()) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(head(X)) = X + 1 >= X + 1 = head(proper(X)) proper(tail(X)) = X + 1 >= X + 1 = tail(proper(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 + 1 >= X + 1 = ok(s(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(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) -> mark(X) active(tail(cons(X,XS))) -> mark(XS) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(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