YES 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) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Proof: DP Processor: DPs: active#(nats()) -> incr#(nats()) active#(nats()) -> cons#(0(),incr(nats())) active#(nats()) -> mark#(cons(0(),incr(nats()))) active#(pairs()) -> incr#(odds()) active#(pairs()) -> cons#(0(),incr(odds())) active#(pairs()) -> mark#(cons(0(),incr(odds()))) active#(odds()) -> incr#(pairs()) active#(odds()) -> mark#(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#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) active#(head(cons(X,XS))) -> mark#(X) active#(tail(cons(X,XS))) -> mark#(XS) mark#(nats()) -> active#(nats()) mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> cons#(mark(X1),X2) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(0()) -> active#(0()) mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> incr#(mark(X)) mark#(incr(X)) -> active#(incr(mark(X))) mark#(pairs()) -> active#(pairs()) mark#(odds()) -> active#(odds()) mark#(s(X)) -> mark#(X) mark#(s(X)) -> s#(mark(X)) mark#(s(X)) -> active#(s(mark(X))) mark#(head(X)) -> mark#(X) mark#(head(X)) -> head#(mark(X)) mark#(head(X)) -> active#(head(mark(X))) mark#(tail(X)) -> mark#(X) mark#(tail(X)) -> tail#(mark(X)) mark#(tail(X)) -> active#(tail(mark(X))) cons#(mark(X1),X2) -> cons#(X1,X2) cons#(X1,mark(X2)) -> cons#(X1,X2) cons#(active(X1),X2) -> cons#(X1,X2) cons#(X1,active(X2)) -> cons#(X1,X2) incr#(mark(X)) -> incr#(X) incr#(active(X)) -> incr#(X) s#(mark(X)) -> s#(X) s#(active(X)) -> s#(X) head#(mark(X)) -> head#(X) head#(active(X)) -> head#(X) tail#(mark(X)) -> tail#(X) tail#(active(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) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) TDG Processor: DPs: active#(nats()) -> incr#(nats()) active#(nats()) -> cons#(0(),incr(nats())) active#(nats()) -> mark#(cons(0(),incr(nats()))) active#(pairs()) -> incr#(odds()) active#(pairs()) -> cons#(0(),incr(odds())) active#(pairs()) -> mark#(cons(0(),incr(odds()))) active#(odds()) -> incr#(pairs()) active#(odds()) -> mark#(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#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) active#(head(cons(X,XS))) -> mark#(X) active#(tail(cons(X,XS))) -> mark#(XS) mark#(nats()) -> active#(nats()) mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> cons#(mark(X1),X2) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(0()) -> active#(0()) mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> incr#(mark(X)) mark#(incr(X)) -> active#(incr(mark(X))) mark#(pairs()) -> active#(pairs()) mark#(odds()) -> active#(odds()) mark#(s(X)) -> mark#(X) mark#(s(X)) -> s#(mark(X)) mark#(s(X)) -> active#(s(mark(X))) mark#(head(X)) -> mark#(X) mark#(head(X)) -> head#(mark(X)) mark#(head(X)) -> active#(head(mark(X))) mark#(tail(X)) -> mark#(X) mark#(tail(X)) -> tail#(mark(X)) mark#(tail(X)) -> active#(tail(mark(X))) cons#(mark(X1),X2) -> cons#(X1,X2) cons#(X1,mark(X2)) -> cons#(X1,X2) cons#(active(X1),X2) -> cons#(X1,X2) cons#(X1,active(X2)) -> cons#(X1,X2) incr#(mark(X)) -> incr#(X) incr#(active(X)) -> incr#(X) s#(mark(X)) -> s#(X) s#(active(X)) -> s#(X) head#(mark(X)) -> head#(X) head#(active(X)) -> head#(X) tail#(mark(X)) -> tail#(X) tail#(active(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) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) graph: tail#(mark(X)) -> tail#(X) -> tail#(active(X)) -> tail#(X) tail#(mark(X)) -> tail#(X) -> tail#(mark(X)) -> tail#(X) tail#(active(X)) -> tail#(X) -> tail#(active(X)) -> tail#(X) tail#(active(X)) -> tail#(X) -> tail#(mark(X)) -> tail#(X) head#(mark(X)) -> head#(X) -> head#(active(X)) -> head#(X) head#(mark(X)) -> head#(X) -> head#(mark(X)) -> head#(X) head#(active(X)) -> head#(X) -> head#(active(X)) -> head#(X) head#(active(X)) -> head#(X) -> head#(mark(X)) -> head#(X) s#(mark(X)) -> s#(X) -> s#(active(X)) -> s#(X) s#(mark(X)) -> s#(X) -> s#(mark(X)) -> s#(X) s#(active(X)) -> s#(X) -> s#(active(X)) -> s#(X) s#(active(X)) -> s#(X) -> s#(mark(X)) -> s#(X) mark#(tail(X)) -> tail#(mark(X)) -> tail#(active(X)) -> tail#(X) mark#(tail(X)) -> tail#(mark(X)) -> tail#(mark(X)) -> tail#(X) mark#(tail(X)) -> mark#(X) -> mark#(tail(X)) -> active#(tail(mark(X))) mark#(tail(X)) -> mark#(X) -> mark#(tail(X)) -> tail#(mark(X)) mark#(tail(X)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) mark#(tail(X)) -> mark#(X) -> mark#(head(X)) -> active#(head(mark(X))) mark#(tail(X)) -> mark#(X) -> mark#(head(X)) -> head#(mark(X)) mark#(tail(X)) -> mark#(X) -> mark#(head(X)) -> mark#(X) mark#(tail(X)) -> mark#(X) -> mark#(s(X)) -> active#(s(mark(X))) mark#(tail(X)) -> mark#(X) -> mark#(s(X)) -> s#(mark(X)) mark#(tail(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(tail(X)) -> mark#(X) -> mark#(odds()) -> active#(odds()) mark#(tail(X)) -> mark#(X) -> mark#(pairs()) -> active#(pairs()) mark#(tail(X)) -> mark#(X) -> mark#(incr(X)) -> active#(incr(mark(X))) mark#(tail(X)) -> mark#(X) -> mark#(incr(X)) -> incr#(mark(X)) mark#(tail(X)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) mark#(tail(X)) -> mark#(X) -> mark#(0()) -> active#(0()) mark#(tail(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(tail(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> cons#(mark(X1),X2) mark#(tail(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(tail(X)) -> mark#(X) -> mark#(nats()) -> active#(nats()) mark#(tail(X)) -> active#(tail(mark(X))) -> active#(tail(cons(X,XS))) -> mark#(XS) mark#(tail(X)) -> active#(tail(mark(X))) -> active#(head(cons(X,XS))) -> mark#(X) mark#(tail(X)) -> active#(tail(mark(X))) -> active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) mark#(tail(X)) -> active#(tail(mark(X))) -> active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) mark#(tail(X)) -> active#(tail(mark(X))) -> active#(incr(cons(X,XS))) -> s#(X) mark#(tail(X)) -> active#(tail(mark(X))) -> active#(incr(cons(X,XS))) -> incr#(XS) mark#(tail(X)) -> active#(tail(mark(X))) -> active#(odds()) -> mark#(incr(pairs())) mark#(tail(X)) -> active#(tail(mark(X))) -> active#(odds()) -> incr#(pairs()) mark#(tail(X)) -> active#(tail(mark(X))) -> active#(pairs()) -> mark#(cons(0(),incr(odds()))) mark#(tail(X)) -> active#(tail(mark(X))) -> active#(pairs()) -> cons#(0(),incr(odds())) mark#(tail(X)) -> active#(tail(mark(X))) -> active#(pairs()) -> incr#(odds()) mark#(tail(X)) -> active#(tail(mark(X))) -> active#(nats()) -> mark#(cons(0(),incr(nats()))) mark#(tail(X)) -> active#(tail(mark(X))) -> active#(nats()) -> cons#(0(),incr(nats())) mark#(tail(X)) -> active#(tail(mark(X))) -> active#(nats()) -> incr#(nats()) mark#(head(X)) -> head#(mark(X)) -> head#(active(X)) -> head#(X) mark#(head(X)) -> head#(mark(X)) -> head#(mark(X)) -> head#(X) mark#(head(X)) -> mark#(X) -> mark#(tail(X)) -> active#(tail(mark(X))) mark#(head(X)) -> mark#(X) -> mark#(tail(X)) -> tail#(mark(X)) mark#(head(X)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) mark#(head(X)) -> mark#(X) -> mark#(head(X)) -> active#(head(mark(X))) mark#(head(X)) -> mark#(X) -> mark#(head(X)) -> head#(mark(X)) mark#(head(X)) -> mark#(X) -> mark#(head(X)) -> mark#(X) mark#(head(X)) -> mark#(X) -> mark#(s(X)) -> active#(s(mark(X))) mark#(head(X)) -> mark#(X) -> mark#(s(X)) -> s#(mark(X)) mark#(head(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(head(X)) -> mark#(X) -> mark#(odds()) -> active#(odds()) mark#(head(X)) -> mark#(X) -> mark#(pairs()) -> active#(pairs()) mark#(head(X)) -> mark#(X) -> mark#(incr(X)) -> active#(incr(mark(X))) mark#(head(X)) -> mark#(X) -> mark#(incr(X)) -> incr#(mark(X)) mark#(head(X)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) mark#(head(X)) -> mark#(X) -> mark#(0()) -> active#(0()) mark#(head(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(head(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> cons#(mark(X1),X2) mark#(head(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(head(X)) -> mark#(X) -> mark#(nats()) -> active#(nats()) mark#(head(X)) -> active#(head(mark(X))) -> active#(tail(cons(X,XS))) -> mark#(XS) mark#(head(X)) -> active#(head(mark(X))) -> active#(head(cons(X,XS))) -> mark#(X) mark#(head(X)) -> active#(head(mark(X))) -> active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) mark#(head(X)) -> active#(head(mark(X))) -> active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) mark#(head(X)) -> active#(head(mark(X))) -> active#(incr(cons(X,XS))) -> s#(X) mark#(head(X)) -> active#(head(mark(X))) -> active#(incr(cons(X,XS))) -> incr#(XS) mark#(head(X)) -> active#(head(mark(X))) -> active#(odds()) -> mark#(incr(pairs())) mark#(head(X)) -> active#(head(mark(X))) -> active#(odds()) -> incr#(pairs()) mark#(head(X)) -> active#(head(mark(X))) -> active#(pairs()) -> mark#(cons(0(),incr(odds()))) mark#(head(X)) -> active#(head(mark(X))) -> active#(pairs()) -> cons#(0(),incr(odds())) mark#(head(X)) -> active#(head(mark(X))) -> active#(pairs()) -> incr#(odds()) mark#(head(X)) -> active#(head(mark(X))) -> active#(nats()) -> mark#(cons(0(),incr(nats()))) mark#(head(X)) -> active#(head(mark(X))) -> active#(nats()) -> cons#(0(),incr(nats())) mark#(head(X)) -> active#(head(mark(X))) -> active#(nats()) -> incr#(nats()) mark#(s(X)) -> s#(mark(X)) -> s#(active(X)) -> s#(X) mark#(s(X)) -> s#(mark(X)) -> s#(mark(X)) -> s#(X) mark#(s(X)) -> mark#(X) -> mark#(tail(X)) -> active#(tail(mark(X))) mark#(s(X)) -> mark#(X) -> mark#(tail(X)) -> tail#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(head(X)) -> active#(head(mark(X))) mark#(s(X)) -> mark#(X) -> mark#(head(X)) -> head#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(head(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> active#(s(mark(X))) mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> s#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(odds()) -> active#(odds()) mark#(s(X)) -> mark#(X) -> mark#(pairs()) -> active#(pairs()) mark#(s(X)) -> mark#(X) -> mark#(incr(X)) -> active#(incr(mark(X))) mark#(s(X)) -> mark#(X) -> mark#(incr(X)) -> incr#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(0()) -> active#(0()) mark#(s(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(s(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> cons#(mark(X1),X2) mark#(s(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(nats()) -> active#(nats()) mark#(s(X)) -> active#(s(mark(X))) -> active#(tail(cons(X,XS))) -> mark#(XS) mark#(s(X)) -> active#(s(mark(X))) -> active#(head(cons(X,XS))) -> mark#(X) mark#(s(X)) -> active#(s(mark(X))) -> active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) mark#(s(X)) -> active#(s(mark(X))) -> active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) mark#(s(X)) -> active#(s(mark(X))) -> active#(incr(cons(X,XS))) -> s#(X) mark#(s(X)) -> active#(s(mark(X))) -> active#(incr(cons(X,XS))) -> incr#(XS) mark#(s(X)) -> active#(s(mark(X))) -> active#(odds()) -> mark#(incr(pairs())) mark#(s(X)) -> active#(s(mark(X))) -> active#(odds()) -> incr#(pairs()) mark#(s(X)) -> active#(s(mark(X))) -> active#(pairs()) -> mark#(cons(0(),incr(odds()))) mark#(s(X)) -> active#(s(mark(X))) -> active#(pairs()) -> cons#(0(),incr(odds())) mark#(s(X)) -> active#(s(mark(X))) -> active#(pairs()) -> incr#(odds()) mark#(s(X)) -> active#(s(mark(X))) -> active#(nats()) -> mark#(cons(0(),incr(nats()))) mark#(s(X)) -> active#(s(mark(X))) -> active#(nats()) -> cons#(0(),incr(nats())) mark#(s(X)) -> active#(s(mark(X))) -> active#(nats()) -> incr#(nats()) mark#(odds()) -> active#(odds()) -> active#(tail(cons(X,XS))) -> mark#(XS) mark#(odds()) -> active#(odds()) -> active#(head(cons(X,XS))) -> mark#(X) mark#(odds()) -> active#(odds()) -> active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) mark#(odds()) -> active#(odds()) -> active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) mark#(odds()) -> active#(odds()) -> active#(incr(cons(X,XS))) -> s#(X) mark#(odds()) -> active#(odds()) -> active#(incr(cons(X,XS))) -> incr#(XS) mark#(odds()) -> active#(odds()) -> active#(odds()) -> mark#(incr(pairs())) mark#(odds()) -> active#(odds()) -> active#(odds()) -> incr#(pairs()) mark#(odds()) -> active#(odds()) -> active#(pairs()) -> mark#(cons(0(),incr(odds()))) mark#(odds()) -> active#(odds()) -> active#(pairs()) -> cons#(0(),incr(odds())) mark#(odds()) -> active#(odds()) -> active#(pairs()) -> incr#(odds()) mark#(odds()) -> active#(odds()) -> active#(nats()) -> mark#(cons(0(),incr(nats()))) mark#(odds()) -> active#(odds()) -> active#(nats()) -> cons#(0(),incr(nats())) mark#(odds()) -> active#(odds()) -> active#(nats()) -> incr#(nats()) mark#(pairs()) -> active#(pairs()) -> active#(tail(cons(X,XS))) -> mark#(XS) mark#(pairs()) -> active#(pairs()) -> active#(head(cons(X,XS))) -> mark#(X) mark#(pairs()) -> active#(pairs()) -> active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) mark#(pairs()) -> active#(pairs()) -> active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) mark#(pairs()) -> active#(pairs()) -> active#(incr(cons(X,XS))) -> s#(X) mark#(pairs()) -> active#(pairs()) -> active#(incr(cons(X,XS))) -> incr#(XS) mark#(pairs()) -> active#(pairs()) -> active#(odds()) -> mark#(incr(pairs())) mark#(pairs()) -> active#(pairs()) -> active#(odds()) -> incr#(pairs()) mark#(pairs()) -> active#(pairs()) -> active#(pairs()) -> mark#(cons(0(),incr(odds()))) mark#(pairs()) -> active#(pairs()) -> active#(pairs()) -> cons#(0(),incr(odds())) mark#(pairs()) -> active#(pairs()) -> active#(pairs()) -> incr#(odds()) mark#(pairs()) -> active#(pairs()) -> active#(nats()) -> mark#(cons(0(),incr(nats()))) mark#(pairs()) -> active#(pairs()) -> active#(nats()) -> cons#(0(),incr(nats())) mark#(pairs()) -> active#(pairs()) -> active#(nats()) -> incr#(nats()) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(tail(X)) -> active#(tail(mark(X))) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(tail(X)) -> tail#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(tail(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(head(X)) -> active#(head(mark(X))) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(head(X)) -> head#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(head(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> active#(s(mark(X))) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> s#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(odds()) -> active#(odds()) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(pairs()) -> active#(pairs()) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(incr(X)) -> active#(incr(mark(X))) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(incr(X)) -> incr#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(incr(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(0()) -> active#(0()) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> cons#(mark(X1),X2) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(nats()) -> active#(nats()) mark#(cons(X1,X2)) -> cons#(mark(X1),X2) -> cons#(X1,active(X2)) -> cons#(X1,X2) mark#(cons(X1,X2)) -> cons#(mark(X1),X2) -> cons#(active(X1),X2) -> cons#(X1,X2) mark#(cons(X1,X2)) -> cons#(mark(X1),X2) -> cons#(X1,mark(X2)) -> cons#(X1,X2) mark#(cons(X1,X2)) -> cons#(mark(X1),X2) -> cons#(mark(X1),X2) -> cons#(X1,X2) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) -> active#(tail(cons(X,XS))) -> mark#(XS) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) -> active#(head(cons(X,XS))) -> mark#(X) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) -> active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) -> active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) -> active#(incr(cons(X,XS))) -> s#(X) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) -> active#(incr(cons(X,XS))) -> incr#(XS) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) -> active#(odds()) -> mark#(incr(pairs())) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) -> active#(odds()) -> incr#(pairs()) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) -> active#(pairs()) -> mark#(cons(0(),incr(odds()))) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) -> active#(pairs()) -> cons#(0(),incr(odds())) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) -> active#(pairs()) -> incr#(odds()) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) -> active#(nats()) -> mark#(cons(0(),incr(nats()))) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) -> active#(nats()) -> cons#(0(),incr(nats())) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) -> active#(nats()) -> incr#(nats()) mark#(incr(X)) -> mark#(X) -> mark#(tail(X)) -> active#(tail(mark(X))) mark#(incr(X)) -> mark#(X) -> mark#(tail(X)) -> tail#(mark(X)) mark#(incr(X)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) mark#(incr(X)) -> mark#(X) -> mark#(head(X)) -> active#(head(mark(X))) mark#(incr(X)) -> mark#(X) -> mark#(head(X)) -> head#(mark(X)) mark#(incr(X)) -> mark#(X) -> mark#(head(X)) -> mark#(X) mark#(incr(X)) -> mark#(X) -> mark#(s(X)) -> active#(s(mark(X))) mark#(incr(X)) -> mark#(X) -> mark#(s(X)) -> s#(mark(X)) mark#(incr(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(incr(X)) -> mark#(X) -> mark#(odds()) -> active#(odds()) mark#(incr(X)) -> mark#(X) -> mark#(pairs()) -> active#(pairs()) mark#(incr(X)) -> mark#(X) -> mark#(incr(X)) -> active#(incr(mark(X))) mark#(incr(X)) -> mark#(X) -> mark#(incr(X)) -> incr#(mark(X)) mark#(incr(X)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> mark#(X) -> mark#(0()) -> active#(0()) mark#(incr(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(incr(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> cons#(mark(X1),X2) mark#(incr(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(incr(X)) -> mark#(X) -> mark#(nats()) -> active#(nats()) mark#(incr(X)) -> incr#(mark(X)) -> incr#(active(X)) -> incr#(X) mark#(incr(X)) -> incr#(mark(X)) -> incr#(mark(X)) -> incr#(X) mark#(incr(X)) -> active#(incr(mark(X))) -> active#(tail(cons(X,XS))) -> mark#(XS) mark#(incr(X)) -> active#(incr(mark(X))) -> active#(head(cons(X,XS))) -> mark#(X) mark#(incr(X)) -> active#(incr(mark(X))) -> active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) mark#(incr(X)) -> active#(incr(mark(X))) -> active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) mark#(incr(X)) -> active#(incr(mark(X))) -> active#(incr(cons(X,XS))) -> s#(X) mark#(incr(X)) -> active#(incr(mark(X))) -> active#(incr(cons(X,XS))) -> incr#(XS) mark#(incr(X)) -> active#(incr(mark(X))) -> active#(odds()) -> mark#(incr(pairs())) mark#(incr(X)) -> active#(incr(mark(X))) -> active#(odds()) -> incr#(pairs()) mark#(incr(X)) -> active#(incr(mark(X))) -> active#(pairs()) -> mark#(cons(0(),incr(odds()))) mark#(incr(X)) -> active#(incr(mark(X))) -> active#(pairs()) -> cons#(0(),incr(odds())) mark#(incr(X)) -> active#(incr(mark(X))) -> active#(pairs()) -> incr#(odds()) mark#(incr(X)) -> active#(incr(mark(X))) -> active#(nats()) -> mark#(cons(0(),incr(nats()))) mark#(incr(X)) -> active#(incr(mark(X))) -> active#(nats()) -> cons#(0(),incr(nats())) mark#(incr(X)) -> active#(incr(mark(X))) -> active#(nats()) -> incr#(nats()) mark#(0()) -> active#(0()) -> active#(tail(cons(X,XS))) -> mark#(XS) mark#(0()) -> active#(0()) -> active#(head(cons(X,XS))) -> mark#(X) mark#(0()) -> active#(0()) -> active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) mark#(0()) -> active#(0()) -> active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) mark#(0()) -> active#(0()) -> active#(incr(cons(X,XS))) -> s#(X) mark#(0()) -> active#(0()) -> active#(incr(cons(X,XS))) -> incr#(XS) mark#(0()) -> active#(0()) -> active#(odds()) -> mark#(incr(pairs())) mark#(0()) -> active#(0()) -> active#(odds()) -> incr#(pairs()) mark#(0()) -> active#(0()) -> active#(pairs()) -> mark#(cons(0(),incr(odds()))) mark#(0()) -> active#(0()) -> active#(pairs()) -> cons#(0(),incr(odds())) mark#(0()) -> active#(0()) -> active#(pairs()) -> incr#(odds()) mark#(0()) -> active#(0()) -> active#(nats()) -> mark#(cons(0(),incr(nats()))) mark#(0()) -> active#(0()) -> active#(nats()) -> cons#(0(),incr(nats())) mark#(0()) -> active#(0()) -> active#(nats()) -> incr#(nats()) mark#(nats()) -> active#(nats()) -> active#(tail(cons(X,XS))) -> mark#(XS) mark#(nats()) -> active#(nats()) -> active#(head(cons(X,XS))) -> mark#(X) mark#(nats()) -> active#(nats()) -> active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) mark#(nats()) -> active#(nats()) -> active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) mark#(nats()) -> active#(nats()) -> active#(incr(cons(X,XS))) -> s#(X) mark#(nats()) -> active#(nats()) -> active#(incr(cons(X,XS))) -> incr#(XS) mark#(nats()) -> active#(nats()) -> active#(odds()) -> mark#(incr(pairs())) mark#(nats()) -> active#(nats()) -> active#(odds()) -> incr#(pairs()) mark#(nats()) -> active#(nats()) -> active#(pairs()) -> mark#(cons(0(),incr(odds()))) mark#(nats()) -> active#(nats()) -> active#(pairs()) -> cons#(0(),incr(odds())) mark#(nats()) -> active#(nats()) -> active#(pairs()) -> incr#(odds()) mark#(nats()) -> active#(nats()) -> active#(nats()) -> mark#(cons(0(),incr(nats()))) mark#(nats()) -> active#(nats()) -> active#(nats()) -> cons#(0(),incr(nats())) mark#(nats()) -> active#(nats()) -> active#(nats()) -> incr#(nats()) cons#(mark(X1),X2) -> cons#(X1,X2) -> cons#(X1,active(X2)) -> cons#(X1,X2) cons#(mark(X1),X2) -> cons#(X1,X2) -> cons#(active(X1),X2) -> cons#(X1,X2) cons#(mark(X1),X2) -> cons#(X1,X2) -> cons#(X1,mark(X2)) -> cons#(X1,X2) cons#(mark(X1),X2) -> cons#(X1,X2) -> cons#(mark(X1),X2) -> cons#(X1,X2) cons#(active(X1),X2) -> cons#(X1,X2) -> cons#(X1,active(X2)) -> cons#(X1,X2) cons#(active(X1),X2) -> cons#(X1,X2) -> cons#(active(X1),X2) -> cons#(X1,X2) cons#(active(X1),X2) -> cons#(X1,X2) -> cons#(X1,mark(X2)) -> cons#(X1,X2) cons#(active(X1),X2) -> cons#(X1,X2) -> cons#(mark(X1),X2) -> cons#(X1,X2) cons#(X1,mark(X2)) -> cons#(X1,X2) -> cons#(X1,active(X2)) -> cons#(X1,X2) cons#(X1,mark(X2)) -> cons#(X1,X2) -> cons#(active(X1),X2) -> cons#(X1,X2) cons#(X1,mark(X2)) -> cons#(X1,X2) -> cons#(X1,mark(X2)) -> cons#(X1,X2) cons#(X1,mark(X2)) -> cons#(X1,X2) -> cons#(mark(X1),X2) -> cons#(X1,X2) cons#(X1,active(X2)) -> cons#(X1,X2) -> cons#(X1,active(X2)) -> cons#(X1,X2) cons#(X1,active(X2)) -> cons#(X1,X2) -> cons#(active(X1),X2) -> cons#(X1,X2) cons#(X1,active(X2)) -> cons#(X1,X2) -> cons#(X1,mark(X2)) -> cons#(X1,X2) cons#(X1,active(X2)) -> cons#(X1,X2) -> cons#(mark(X1),X2) -> cons#(X1,X2) incr#(mark(X)) -> incr#(X) -> incr#(active(X)) -> incr#(X) incr#(mark(X)) -> incr#(X) -> incr#(mark(X)) -> incr#(X) incr#(active(X)) -> incr#(X) -> incr#(active(X)) -> incr#(X) incr#(active(X)) -> incr#(X) -> incr#(mark(X)) -> incr#(X) active#(tail(cons(X,XS))) -> mark#(XS) -> mark#(tail(X)) -> active#(tail(mark(X))) active#(tail(cons(X,XS))) -> mark#(XS) -> mark#(tail(X)) -> tail#(mark(X)) active#(tail(cons(X,XS))) -> mark#(XS) -> mark#(tail(X)) -> mark#(X) active#(tail(cons(X,XS))) -> mark#(XS) -> mark#(head(X)) -> active#(head(mark(X))) active#(tail(cons(X,XS))) -> mark#(XS) -> mark#(head(X)) -> head#(mark(X)) active#(tail(cons(X,XS))) -> mark#(XS) -> mark#(head(X)) -> mark#(X) active#(tail(cons(X,XS))) -> mark#(XS) -> mark#(s(X)) -> active#(s(mark(X))) active#(tail(cons(X,XS))) -> mark#(XS) -> mark#(s(X)) -> s#(mark(X)) active#(tail(cons(X,XS))) -> mark#(XS) -> mark#(s(X)) -> mark#(X) active#(tail(cons(X,XS))) -> mark#(XS) -> mark#(odds()) -> active#(odds()) active#(tail(cons(X,XS))) -> mark#(XS) -> mark#(pairs()) -> active#(pairs()) active#(tail(cons(X,XS))) -> mark#(XS) -> mark#(incr(X)) -> active#(incr(mark(X))) active#(tail(cons(X,XS))) -> mark#(XS) -> mark#(incr(X)) -> incr#(mark(X)) active#(tail(cons(X,XS))) -> mark#(XS) -> mark#(incr(X)) -> mark#(X) active#(tail(cons(X,XS))) -> mark#(XS) -> mark#(0()) -> active#(0()) active#(tail(cons(X,XS))) -> mark#(XS) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(tail(cons(X,XS))) -> mark#(XS) -> mark#(cons(X1,X2)) -> cons#(mark(X1),X2) active#(tail(cons(X,XS))) -> mark#(XS) -> mark#(cons(X1,X2)) -> mark#(X1) active#(tail(cons(X,XS))) -> mark#(XS) -> mark#(nats()) -> active#(nats()) active#(head(cons(X,XS))) -> mark#(X) -> mark#(tail(X)) -> active#(tail(mark(X))) active#(head(cons(X,XS))) -> mark#(X) -> mark#(tail(X)) -> tail#(mark(X)) active#(head(cons(X,XS))) -> mark#(X) -> mark#(tail(X)) -> mark#(X) active#(head(cons(X,XS))) -> mark#(X) -> mark#(head(X)) -> active#(head(mark(X))) active#(head(cons(X,XS))) -> mark#(X) -> mark#(head(X)) -> head#(mark(X)) active#(head(cons(X,XS))) -> mark#(X) -> mark#(head(X)) -> mark#(X) active#(head(cons(X,XS))) -> mark#(X) -> mark#(s(X)) -> active#(s(mark(X))) active#(head(cons(X,XS))) -> mark#(X) -> mark#(s(X)) -> s#(mark(X)) active#(head(cons(X,XS))) -> mark#(X) -> mark#(s(X)) -> mark#(X) active#(head(cons(X,XS))) -> mark#(X) -> mark#(odds()) -> active#(odds()) active#(head(cons(X,XS))) -> mark#(X) -> mark#(pairs()) -> active#(pairs()) active#(head(cons(X,XS))) -> mark#(X) -> mark#(incr(X)) -> active#(incr(mark(X))) active#(head(cons(X,XS))) -> mark#(X) -> mark#(incr(X)) -> incr#(mark(X)) active#(head(cons(X,XS))) -> mark#(X) -> mark#(incr(X)) -> mark#(X) active#(head(cons(X,XS))) -> mark#(X) -> mark#(0()) -> active#(0()) active#(head(cons(X,XS))) -> mark#(X) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(head(cons(X,XS))) -> mark#(X) -> mark#(cons(X1,X2)) -> cons#(mark(X1),X2) active#(head(cons(X,XS))) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) active#(head(cons(X,XS))) -> mark#(X) -> mark#(nats()) -> active#(nats()) active#(odds()) -> mark#(incr(pairs())) -> mark#(tail(X)) -> active#(tail(mark(X))) active#(odds()) -> mark#(incr(pairs())) -> mark#(tail(X)) -> tail#(mark(X)) active#(odds()) -> mark#(incr(pairs())) -> mark#(tail(X)) -> mark#(X) active#(odds()) -> mark#(incr(pairs())) -> mark#(head(X)) -> active#(head(mark(X))) active#(odds()) -> mark#(incr(pairs())) -> mark#(head(X)) -> head#(mark(X)) active#(odds()) -> mark#(incr(pairs())) -> mark#(head(X)) -> mark#(X) active#(odds()) -> mark#(incr(pairs())) -> mark#(s(X)) -> active#(s(mark(X))) active#(odds()) -> mark#(incr(pairs())) -> mark#(s(X)) -> s#(mark(X)) active#(odds()) -> mark#(incr(pairs())) -> mark#(s(X)) -> mark#(X) active#(odds()) -> mark#(incr(pairs())) -> mark#(odds()) -> active#(odds()) active#(odds()) -> mark#(incr(pairs())) -> mark#(pairs()) -> active#(pairs()) active#(odds()) -> mark#(incr(pairs())) -> mark#(incr(X)) -> active#(incr(mark(X))) active#(odds()) -> mark#(incr(pairs())) -> mark#(incr(X)) -> incr#(mark(X)) active#(odds()) -> mark#(incr(pairs())) -> mark#(incr(X)) -> mark#(X) active#(odds()) -> mark#(incr(pairs())) -> mark#(0()) -> active#(0()) active#(odds()) -> mark#(incr(pairs())) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(odds()) -> mark#(incr(pairs())) -> mark#(cons(X1,X2)) -> cons#(mark(X1),X2) active#(odds()) -> mark#(incr(pairs())) -> mark#(cons(X1,X2)) -> mark#(X1) active#(odds()) -> mark#(incr(pairs())) -> mark#(nats()) -> active#(nats()) active#(odds()) -> incr#(pairs()) -> incr#(active(X)) -> incr#(X) active#(odds()) -> incr#(pairs()) -> incr#(mark(X)) -> incr#(X) active#(pairs()) -> mark#(cons(0(),incr(odds()))) -> mark#(tail(X)) -> active#(tail(mark(X))) active#(pairs()) -> mark#(cons(0(),incr(odds()))) -> mark#(tail(X)) -> tail#(mark(X)) active#(pairs()) -> mark#(cons(0(),incr(odds()))) -> mark#(tail(X)) -> mark#(X) active#(pairs()) -> mark#(cons(0(),incr(odds()))) -> mark#(head(X)) -> active#(head(mark(X))) active#(pairs()) -> mark#(cons(0(),incr(odds()))) -> mark#(head(X)) -> head#(mark(X)) active#(pairs()) -> mark#(cons(0(),incr(odds()))) -> mark#(head(X)) -> mark#(X) active#(pairs()) -> mark#(cons(0(),incr(odds()))) -> mark#(s(X)) -> active#(s(mark(X))) active#(pairs()) -> mark#(cons(0(),incr(odds()))) -> mark#(s(X)) -> s#(mark(X)) active#(pairs()) -> mark#(cons(0(),incr(odds()))) -> mark#(s(X)) -> mark#(X) active#(pairs()) -> mark#(cons(0(),incr(odds()))) -> mark#(odds()) -> active#(odds()) active#(pairs()) -> mark#(cons(0(),incr(odds()))) -> mark#(pairs()) -> active#(pairs()) active#(pairs()) -> mark#(cons(0(),incr(odds()))) -> mark#(incr(X)) -> active#(incr(mark(X))) active#(pairs()) -> mark#(cons(0(),incr(odds()))) -> mark#(incr(X)) -> incr#(mark(X)) active#(pairs()) -> mark#(cons(0(),incr(odds()))) -> mark#(incr(X)) -> mark#(X) active#(pairs()) -> mark#(cons(0(),incr(odds()))) -> mark#(0()) -> active#(0()) active#(pairs()) -> mark#(cons(0(),incr(odds()))) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(pairs()) -> mark#(cons(0(),incr(odds()))) -> mark#(cons(X1,X2)) -> cons#(mark(X1),X2) active#(pairs()) -> mark#(cons(0(),incr(odds()))) -> mark#(cons(X1,X2)) -> mark#(X1) active#(pairs()) -> mark#(cons(0(),incr(odds()))) -> mark#(nats()) -> active#(nats()) active#(pairs()) -> cons#(0(),incr(odds())) -> cons#(X1,active(X2)) -> cons#(X1,X2) active#(pairs()) -> cons#(0(),incr(odds())) -> cons#(active(X1),X2) -> cons#(X1,X2) active#(pairs()) -> cons#(0(),incr(odds())) -> cons#(X1,mark(X2)) -> cons#(X1,X2) active#(pairs()) -> cons#(0(),incr(odds())) -> cons#(mark(X1),X2) -> cons#(X1,X2) active#(pairs()) -> incr#(odds()) -> incr#(active(X)) -> incr#(X) active#(pairs()) -> incr#(odds()) -> incr#(mark(X)) -> incr#(X) active#(incr(cons(X,XS))) -> s#(X) -> s#(active(X)) -> s#(X) active#(incr(cons(X,XS))) -> s#(X) -> s#(mark(X)) -> s#(X) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(tail(X)) -> active#(tail(mark(X))) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(tail(X)) -> tail#(mark(X)) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(tail(X)) -> mark#(X) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(head(X)) -> active#(head(mark(X))) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(head(X)) -> head#(mark(X)) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(head(X)) -> mark#(X) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(s(X)) -> active#(s(mark(X))) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(s(X)) -> s#(mark(X)) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(s(X)) -> mark#(X) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(odds()) -> active#(odds()) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(pairs()) -> active#(pairs()) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(incr(X)) -> active#(incr(mark(X))) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(incr(X)) -> incr#(mark(X)) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(incr(X)) -> mark#(X) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(0()) -> active#(0()) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(cons(X1,X2)) -> cons#(mark(X1),X2) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(cons(X1,X2)) -> mark#(X1) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(nats()) -> active#(nats()) active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) -> cons#(X1,active(X2)) -> cons#(X1,X2) active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) -> cons#(active(X1),X2) -> cons#(X1,X2) active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) -> cons#(X1,mark(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#(active(X)) -> incr#(X) active#(incr(cons(X,XS))) -> incr#(XS) -> incr#(mark(X)) -> incr#(X) active#(nats()) -> mark#(cons(0(),incr(nats()))) -> mark#(tail(X)) -> active#(tail(mark(X))) active#(nats()) -> mark#(cons(0(),incr(nats()))) -> mark#(tail(X)) -> tail#(mark(X)) active#(nats()) -> mark#(cons(0(),incr(nats()))) -> mark#(tail(X)) -> mark#(X) active#(nats()) -> mark#(cons(0(),incr(nats()))) -> mark#(head(X)) -> active#(head(mark(X))) active#(nats()) -> mark#(cons(0(),incr(nats()))) -> mark#(head(X)) -> head#(mark(X)) active#(nats()) -> mark#(cons(0(),incr(nats()))) -> mark#(head(X)) -> mark#(X) active#(nats()) -> mark#(cons(0(),incr(nats()))) -> mark#(s(X)) -> active#(s(mark(X))) active#(nats()) -> mark#(cons(0(),incr(nats()))) -> mark#(s(X)) -> s#(mark(X)) active#(nats()) -> mark#(cons(0(),incr(nats()))) -> mark#(s(X)) -> mark#(X) active#(nats()) -> mark#(cons(0(),incr(nats()))) -> mark#(odds()) -> active#(odds()) active#(nats()) -> mark#(cons(0(),incr(nats()))) -> mark#(pairs()) -> active#(pairs()) active#(nats()) -> mark#(cons(0(),incr(nats()))) -> mark#(incr(X)) -> active#(incr(mark(X))) active#(nats()) -> mark#(cons(0(),incr(nats()))) -> mark#(incr(X)) -> incr#(mark(X)) active#(nats()) -> mark#(cons(0(),incr(nats()))) -> mark#(incr(X)) -> mark#(X) active#(nats()) -> mark#(cons(0(),incr(nats()))) -> mark#(0()) -> active#(0()) active#(nats()) -> mark#(cons(0(),incr(nats()))) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(nats()) -> mark#(cons(0(),incr(nats()))) -> mark#(cons(X1,X2)) -> cons#(mark(X1),X2) active#(nats()) -> mark#(cons(0(),incr(nats()))) -> mark#(cons(X1,X2)) -> mark#(X1) active#(nats()) -> mark#(cons(0(),incr(nats()))) -> mark#(nats()) -> active#(nats()) active#(nats()) -> cons#(0(),incr(nats())) -> cons#(X1,active(X2)) -> cons#(X1,X2) active#(nats()) -> cons#(0(),incr(nats())) -> cons#(active(X1),X2) -> cons#(X1,X2) active#(nats()) -> cons#(0(),incr(nats())) -> cons#(X1,mark(X2)) -> cons#(X1,X2) active#(nats()) -> cons#(0(),incr(nats())) -> cons#(mark(X1),X2) -> cons#(X1,X2) active#(nats()) -> incr#(nats()) -> incr#(active(X)) -> incr#(X) active#(nats()) -> incr#(nats()) -> incr#(mark(X)) -> incr#(X) SCC Processor: #sccs: 6 #rules: 32 #arcs: 401/2025 DPs: mark#(tail(X)) -> mark#(X) mark#(nats()) -> active#(nats()) active#(nats()) -> mark#(cons(0(),incr(nats()))) mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(pairs()) -> mark#(cons(0(),incr(odds()))) mark#(0()) -> active#(0()) active#(odds()) -> mark#(incr(pairs())) mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> active#(incr(mark(X))) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) mark#(pairs()) -> active#(pairs()) active#(head(cons(X,XS))) -> mark#(X) mark#(odds()) -> active#(odds()) active#(tail(cons(X,XS))) -> mark#(XS) mark#(s(X)) -> mark#(X) mark#(s(X)) -> active#(s(mark(X))) mark#(head(X)) -> mark#(X) mark#(head(X)) -> active#(head(mark(X))) mark#(tail(X)) -> active#(tail(mark(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) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) EDG Processor: DPs: mark#(tail(X)) -> mark#(X) mark#(nats()) -> active#(nats()) active#(nats()) -> mark#(cons(0(),incr(nats()))) mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(pairs()) -> mark#(cons(0(),incr(odds()))) mark#(0()) -> active#(0()) active#(odds()) -> mark#(incr(pairs())) mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> active#(incr(mark(X))) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) mark#(pairs()) -> active#(pairs()) active#(head(cons(X,XS))) -> mark#(X) mark#(odds()) -> active#(odds()) active#(tail(cons(X,XS))) -> mark#(XS) mark#(s(X)) -> mark#(X) mark#(s(X)) -> active#(s(mark(X))) mark#(head(X)) -> mark#(X) mark#(head(X)) -> active#(head(mark(X))) mark#(tail(X)) -> active#(tail(mark(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) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) graph: mark#(tail(X)) -> mark#(X) -> mark#(nats()) -> active#(nats()) mark#(tail(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(tail(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(tail(X)) -> mark#(X) -> mark#(0()) -> active#(0()) mark#(tail(X)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) mark#(tail(X)) -> mark#(X) -> mark#(incr(X)) -> active#(incr(mark(X))) mark#(tail(X)) -> mark#(X) -> mark#(pairs()) -> active#(pairs()) mark#(tail(X)) -> mark#(X) -> mark#(odds()) -> active#(odds()) mark#(tail(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(tail(X)) -> mark#(X) -> mark#(s(X)) -> active#(s(mark(X))) mark#(tail(X)) -> mark#(X) -> mark#(head(X)) -> mark#(X) mark#(tail(X)) -> mark#(X) -> mark#(head(X)) -> active#(head(mark(X))) mark#(tail(X)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) mark#(tail(X)) -> mark#(X) -> mark#(tail(X)) -> active#(tail(mark(X))) mark#(tail(X)) -> active#(tail(mark(X))) -> active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) mark#(tail(X)) -> active#(tail(mark(X))) -> active#(head(cons(X,XS))) -> mark#(X) mark#(tail(X)) -> active#(tail(mark(X))) -> active#(tail(cons(X,XS))) -> mark#(XS) mark#(head(X)) -> mark#(X) -> mark#(nats()) -> active#(nats()) mark#(head(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(head(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(head(X)) -> mark#(X) -> mark#(0()) -> active#(0()) mark#(head(X)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) mark#(head(X)) -> mark#(X) -> mark#(incr(X)) -> active#(incr(mark(X))) mark#(head(X)) -> mark#(X) -> mark#(pairs()) -> active#(pairs()) mark#(head(X)) -> mark#(X) -> mark#(odds()) -> active#(odds()) mark#(head(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(head(X)) -> mark#(X) -> mark#(s(X)) -> active#(s(mark(X))) mark#(head(X)) -> mark#(X) -> mark#(head(X)) -> mark#(X) mark#(head(X)) -> mark#(X) -> mark#(head(X)) -> active#(head(mark(X))) mark#(head(X)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) mark#(head(X)) -> mark#(X) -> mark#(tail(X)) -> active#(tail(mark(X))) mark#(head(X)) -> active#(head(mark(X))) -> active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) mark#(head(X)) -> active#(head(mark(X))) -> active#(head(cons(X,XS))) -> mark#(X) mark#(head(X)) -> active#(head(mark(X))) -> active#(tail(cons(X,XS))) -> mark#(XS) mark#(s(X)) -> mark#(X) -> mark#(nats()) -> active#(nats()) mark#(s(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(s(X)) -> mark#(X) -> mark#(0()) -> active#(0()) mark#(s(X)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(incr(X)) -> active#(incr(mark(X))) mark#(s(X)) -> mark#(X) -> mark#(pairs()) -> active#(pairs()) mark#(s(X)) -> mark#(X) -> mark#(odds()) -> active#(odds()) mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> active#(s(mark(X))) mark#(s(X)) -> mark#(X) -> mark#(head(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(head(X)) -> active#(head(mark(X))) mark#(s(X)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(tail(X)) -> active#(tail(mark(X))) mark#(s(X)) -> active#(s(mark(X))) -> active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) mark#(s(X)) -> active#(s(mark(X))) -> active#(head(cons(X,XS))) -> mark#(X) mark#(s(X)) -> active#(s(mark(X))) -> active#(tail(cons(X,XS))) -> mark#(XS) mark#(odds()) -> active#(odds()) -> active#(odds()) -> mark#(incr(pairs())) mark#(pairs()) -> active#(pairs()) -> active#(pairs()) -> mark#(cons(0(),incr(odds()))) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(nats()) -> active#(nats()) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(0()) -> active#(0()) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(incr(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(incr(X)) -> active#(incr(mark(X))) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(pairs()) -> active#(pairs()) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(odds()) -> active#(odds()) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> active#(s(mark(X))) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(head(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(head(X)) -> active#(head(mark(X))) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(tail(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(tail(X)) -> active#(tail(mark(X))) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) -> active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) -> active#(head(cons(X,XS))) -> mark#(X) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) -> active#(tail(cons(X,XS))) -> mark#(XS) mark#(incr(X)) -> mark#(X) -> mark#(nats()) -> active#(nats()) mark#(incr(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(incr(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(incr(X)) -> mark#(X) -> mark#(0()) -> active#(0()) mark#(incr(X)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> mark#(X) -> mark#(incr(X)) -> active#(incr(mark(X))) mark#(incr(X)) -> mark#(X) -> mark#(pairs()) -> active#(pairs()) mark#(incr(X)) -> mark#(X) -> mark#(odds()) -> active#(odds()) mark#(incr(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(incr(X)) -> mark#(X) -> mark#(s(X)) -> active#(s(mark(X))) mark#(incr(X)) -> mark#(X) -> mark#(head(X)) -> mark#(X) mark#(incr(X)) -> mark#(X) -> mark#(head(X)) -> active#(head(mark(X))) mark#(incr(X)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) mark#(incr(X)) -> mark#(X) -> mark#(tail(X)) -> active#(tail(mark(X))) mark#(incr(X)) -> active#(incr(mark(X))) -> active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) mark#(incr(X)) -> active#(incr(mark(X))) -> active#(head(cons(X,XS))) -> mark#(X) mark#(incr(X)) -> active#(incr(mark(X))) -> active#(tail(cons(X,XS))) -> mark#(XS) mark#(nats()) -> active#(nats()) -> active#(nats()) -> mark#(cons(0(),incr(nats()))) active#(tail(cons(X,XS))) -> mark#(XS) -> mark#(nats()) -> active#(nats()) active#(tail(cons(X,XS))) -> mark#(XS) -> mark#(cons(X1,X2)) -> mark#(X1) active#(tail(cons(X,XS))) -> mark#(XS) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(tail(cons(X,XS))) -> mark#(XS) -> mark#(0()) -> active#(0()) active#(tail(cons(X,XS))) -> mark#(XS) -> mark#(incr(X)) -> mark#(X) active#(tail(cons(X,XS))) -> mark#(XS) -> mark#(incr(X)) -> active#(incr(mark(X))) active#(tail(cons(X,XS))) -> mark#(XS) -> mark#(pairs()) -> active#(pairs()) active#(tail(cons(X,XS))) -> mark#(XS) -> mark#(odds()) -> active#(odds()) active#(tail(cons(X,XS))) -> mark#(XS) -> mark#(s(X)) -> mark#(X) active#(tail(cons(X,XS))) -> mark#(XS) -> mark#(s(X)) -> active#(s(mark(X))) active#(tail(cons(X,XS))) -> mark#(XS) -> mark#(head(X)) -> mark#(X) active#(tail(cons(X,XS))) -> mark#(XS) -> mark#(head(X)) -> active#(head(mark(X))) active#(tail(cons(X,XS))) -> mark#(XS) -> mark#(tail(X)) -> mark#(X) active#(tail(cons(X,XS))) -> mark#(XS) -> mark#(tail(X)) -> active#(tail(mark(X))) active#(head(cons(X,XS))) -> mark#(X) -> mark#(nats()) -> active#(nats()) active#(head(cons(X,XS))) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) active#(head(cons(X,XS))) -> mark#(X) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(head(cons(X,XS))) -> mark#(X) -> mark#(0()) -> active#(0()) active#(head(cons(X,XS))) -> mark#(X) -> mark#(incr(X)) -> mark#(X) active#(head(cons(X,XS))) -> mark#(X) -> mark#(incr(X)) -> active#(incr(mark(X))) active#(head(cons(X,XS))) -> mark#(X) -> mark#(pairs()) -> active#(pairs()) active#(head(cons(X,XS))) -> mark#(X) -> mark#(odds()) -> active#(odds()) active#(head(cons(X,XS))) -> mark#(X) -> mark#(s(X)) -> mark#(X) active#(head(cons(X,XS))) -> mark#(X) -> mark#(s(X)) -> active#(s(mark(X))) active#(head(cons(X,XS))) -> mark#(X) -> mark#(head(X)) -> mark#(X) active#(head(cons(X,XS))) -> mark#(X) -> mark#(head(X)) -> active#(head(mark(X))) active#(head(cons(X,XS))) -> mark#(X) -> mark#(tail(X)) -> mark#(X) active#(head(cons(X,XS))) -> mark#(X) -> mark#(tail(X)) -> active#(tail(mark(X))) active#(odds()) -> mark#(incr(pairs())) -> mark#(incr(X)) -> mark#(X) active#(odds()) -> mark#(incr(pairs())) -> mark#(incr(X)) -> active#(incr(mark(X))) active#(pairs()) -> mark#(cons(0(),incr(odds()))) -> mark#(cons(X1,X2)) -> mark#(X1) active#(pairs()) -> mark#(cons(0(),incr(odds()))) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(cons(X1,X2)) -> mark#(X1) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(incr(X)) -> mark#(X) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(incr(X)) -> active#(incr(mark(X))) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(s(X)) -> mark#(X) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(s(X)) -> active#(s(mark(X))) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(head(X)) -> mark#(X) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(head(X)) -> active#(head(mark(X))) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(tail(X)) -> mark#(X) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) -> mark#(tail(X)) -> active#(tail(mark(X))) active#(nats()) -> mark#(cons(0(),incr(nats()))) -> mark#(cons(X1,X2)) -> mark#(X1) active#(nats()) -> mark#(cons(0(),incr(nats()))) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) SCC Processor: #sccs: 1 #rules: 19 #arcs: 132/400 DPs: mark#(tail(X)) -> mark#(X) mark#(tail(X)) -> active#(tail(mark(X))) active#(tail(cons(X,XS))) -> mark#(XS) mark#(head(X)) -> active#(head(mark(X))) active#(head(cons(X,XS))) -> mark#(X) mark#(head(X)) -> mark#(X) mark#(s(X)) -> active#(s(mark(X))) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) mark#(s(X)) -> mark#(X) mark#(odds()) -> active#(odds()) active#(odds()) -> mark#(incr(pairs())) mark#(incr(X)) -> active#(incr(mark(X))) mark#(incr(X)) -> mark#(X) mark#(pairs()) -> active#(pairs()) active#(pairs()) -> mark#(cons(0(),incr(odds()))) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(nats()) -> active#(nats()) active#(nats()) -> mark#(cons(0(),incr(nats()))) 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) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Arctic Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0 + 0, [active#](x0) = x0 + 0, [tail](x0) = 5x0 + 4, [head](x0) = x0 + 0, [s](x0) = x0, [odds] = 0, [pairs] = 0, [mark](x0) = x0, [cons](x0, x1) = x0 + x1, [incr](x0) = x0, [0] = 0, [active](x0) = x0, [nats] = 0 orientation: mark#(tail(X)) = 5X + 4 >= X + 0 = mark#(X) mark#(tail(X)) = 5X + 4 >= 5X + 4 = active#(tail(mark(X))) active#(tail(cons(X,XS))) = 5X + 5XS + 4 >= XS + 0 = mark#(XS) mark#(head(X)) = X + 0 >= X + 0 = active#(head(mark(X))) active#(head(cons(X,XS))) = X + XS + 0 >= X + 0 = mark#(X) mark#(head(X)) = X + 0 >= X + 0 = mark#(X) mark#(s(X)) = X + 0 >= X + 0 = active#(s(mark(X))) active#(incr(cons(X,XS))) = X + XS + 0 >= X + XS + 0 = mark#(cons(s(X),incr(XS))) mark#(s(X)) = X + 0 >= X + 0 = mark#(X) mark#(odds()) = 0 >= 0 = active#(odds()) active#(odds()) = 0 >= 0 = mark#(incr(pairs())) mark#(incr(X)) = X + 0 >= X + 0 = active#(incr(mark(X))) mark#(incr(X)) = X + 0 >= X + 0 = mark#(X) mark#(pairs()) = 0 >= 0 = active#(pairs()) active#(pairs()) = 0 >= 0 = mark#(cons(0(),incr(odds()))) mark#(cons(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = active#(cons(mark(X1),X2)) mark#(cons(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(nats()) = 0 >= 0 = active#(nats()) active#(nats()) = 0 >= 0 = mark#(cons(0(),incr(nats()))) 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 + XS >= X + XS = mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) = X + XS + 0 >= X = mark(X) active(tail(cons(X,XS))) = 5X + 5XS + 4 >= XS = mark(XS) mark(nats()) = 0 >= 0 = active(nats()) mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(incr(X)) = X >= X = active(incr(mark(X))) mark(pairs()) = 0 >= 0 = active(pairs()) mark(odds()) = 0 >= 0 = active(odds()) mark(s(X)) = X >= X = active(s(mark(X))) mark(head(X)) = X + 0 >= X + 0 = active(head(mark(X))) mark(tail(X)) = 5X + 4 >= 5X + 4 = active(tail(mark(X))) cons(mark(X1),X2) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(X1,mark(X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(active(X1),X2) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(X1,active(X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) incr(mark(X)) = X >= X = incr(X) incr(active(X)) = X >= X = incr(X) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) head(mark(X)) = X + 0 >= X + 0 = head(X) head(active(X)) = X + 0 >= X + 0 = head(X) tail(mark(X)) = 5X + 4 >= 5X + 4 = tail(X) tail(active(X)) = 5X + 4 >= 5X + 4 = tail(X) problem: DPs: mark#(tail(X)) -> active#(tail(mark(X))) mark#(head(X)) -> active#(head(mark(X))) active#(head(cons(X,XS))) -> mark#(X) mark#(head(X)) -> mark#(X) mark#(s(X)) -> active#(s(mark(X))) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) mark#(s(X)) -> mark#(X) mark#(odds()) -> active#(odds()) active#(odds()) -> mark#(incr(pairs())) mark#(incr(X)) -> active#(incr(mark(X))) mark#(incr(X)) -> mark#(X) mark#(pairs()) -> active#(pairs()) active#(pairs()) -> mark#(cons(0(),incr(odds()))) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(nats()) -> active#(nats()) active#(nats()) -> mark#(cons(0(),incr(nats()))) 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) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Arctic Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 2, [active#](x0) = x0 + 0, [tail](x0) = 0, [head](x0) = 2, [s](x0) = 2, [odds] = 2, [pairs] = 2, [mark](x0) = 0, [cons](x0, x1) = 2, [incr](x0) = 2, [0] = 0, [active](x0) = 0, [nats] = 2 orientation: mark#(tail(X)) = 2 >= 0 = active#(tail(mark(X))) mark#(head(X)) = 2 >= 2 = active#(head(mark(X))) active#(head(cons(X,XS))) = 2 >= 2 = mark#(X) mark#(head(X)) = 2 >= 2 = mark#(X) mark#(s(X)) = 2 >= 2 = active#(s(mark(X))) active#(incr(cons(X,XS))) = 2 >= 2 = mark#(cons(s(X),incr(XS))) mark#(s(X)) = 2 >= 2 = mark#(X) mark#(odds()) = 2 >= 2 = active#(odds()) active#(odds()) = 2 >= 2 = mark#(incr(pairs())) mark#(incr(X)) = 2 >= 2 = active#(incr(mark(X))) mark#(incr(X)) = 2 >= 2 = mark#(X) mark#(pairs()) = 2 >= 2 = active#(pairs()) active#(pairs()) = 2 >= 2 = mark#(cons(0(),incr(odds()))) mark#(cons(X1,X2)) = 2 >= 2 = active#(cons(mark(X1),X2)) mark#(cons(X1,X2)) = 2 >= 2 = mark#(X1) mark#(nats()) = 2 >= 2 = active#(nats()) active#(nats()) = 2 >= 2 = mark#(cons(0(),incr(nats()))) 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))) = 0 >= 0 = mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) = 0 >= 0 = mark(X) active(tail(cons(X,XS))) = 0 >= 0 = mark(XS) mark(nats()) = 0 >= 0 = active(nats()) mark(cons(X1,X2)) = 0 >= 0 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(incr(X)) = 0 >= 0 = active(incr(mark(X))) mark(pairs()) = 0 >= 0 = active(pairs()) mark(odds()) = 0 >= 0 = active(odds()) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(head(X)) = 0 >= 0 = active(head(mark(X))) mark(tail(X)) = 0 >= 0 = active(tail(mark(X))) cons(mark(X1),X2) = 2 >= 2 = cons(X1,X2) cons(X1,mark(X2)) = 2 >= 2 = cons(X1,X2) cons(active(X1),X2) = 2 >= 2 = cons(X1,X2) cons(X1,active(X2)) = 2 >= 2 = cons(X1,X2) incr(mark(X)) = 2 >= 2 = incr(X) incr(active(X)) = 2 >= 2 = incr(X) s(mark(X)) = 2 >= 2 = s(X) s(active(X)) = 2 >= 2 = s(X) head(mark(X)) = 2 >= 2 = head(X) head(active(X)) = 2 >= 2 = head(X) tail(mark(X)) = 0 >= 0 = tail(X) tail(active(X)) = 0 >= 0 = tail(X) problem: DPs: mark#(head(X)) -> active#(head(mark(X))) active#(head(cons(X,XS))) -> mark#(X) mark#(head(X)) -> mark#(X) mark#(s(X)) -> active#(s(mark(X))) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) mark#(s(X)) -> mark#(X) mark#(odds()) -> active#(odds()) active#(odds()) -> mark#(incr(pairs())) mark#(incr(X)) -> active#(incr(mark(X))) mark#(incr(X)) -> mark#(X) mark#(pairs()) -> active#(pairs()) active#(pairs()) -> mark#(cons(0(),incr(odds()))) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(nats()) -> active#(nats()) active#(nats()) -> mark#(cons(0(),incr(nats()))) 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) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Arctic Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0 + 0, [active#](x0) = x0 + 0, [tail](x0) = x0 + 0, [head](x0) = 1x0 + 0, [s](x0) = x0 + 0, [odds] = 0, [pairs] = 0, [mark](x0) = x0, [cons](x0, x1) = x0 + x1 + 0, [incr](x0) = x0 + 0, [0] = 0, [active](x0) = x0, [nats] = 0 orientation: mark#(head(X)) = 1X + 0 >= 1X + 0 = active#(head(mark(X))) active#(head(cons(X,XS))) = 1X + 1XS + 1 >= X + 0 = mark#(X) mark#(head(X)) = 1X + 0 >= X + 0 = mark#(X) mark#(s(X)) = X + 0 >= X + 0 = active#(s(mark(X))) active#(incr(cons(X,XS))) = X + XS + 0 >= X + XS + 0 = mark#(cons(s(X),incr(XS))) mark#(s(X)) = X + 0 >= X + 0 = mark#(X) mark#(odds()) = 0 >= 0 = active#(odds()) active#(odds()) = 0 >= 0 = mark#(incr(pairs())) mark#(incr(X)) = X + 0 >= X + 0 = active#(incr(mark(X))) mark#(incr(X)) = X + 0 >= X + 0 = mark#(X) mark#(pairs()) = 0 >= 0 = active#(pairs()) active#(pairs()) = 0 >= 0 = mark#(cons(0(),incr(odds()))) mark#(cons(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = active#(cons(mark(X1),X2)) mark#(cons(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(nats()) = 0 >= 0 = active#(nats()) active#(nats()) = 0 >= 0 = mark#(cons(0(),incr(nats()))) 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 + XS + 0 >= X + XS + 0 = mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) = 1X + 1XS + 1 >= X = mark(X) active(tail(cons(X,XS))) = X + XS + 0 >= XS = mark(XS) mark(nats()) = 0 >= 0 = active(nats()) mark(cons(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(incr(X)) = X + 0 >= X + 0 = active(incr(mark(X))) mark(pairs()) = 0 >= 0 = active(pairs()) mark(odds()) = 0 >= 0 = active(odds()) mark(s(X)) = X + 0 >= X + 0 = active(s(mark(X))) mark(head(X)) = 1X + 0 >= 1X + 0 = active(head(mark(X))) mark(tail(X)) = X + 0 >= X + 0 = active(tail(mark(X))) cons(mark(X1),X2) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(X1,X2) cons(X1,mark(X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(X1,X2) cons(active(X1),X2) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(X1,X2) cons(X1,active(X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(X1,X2) incr(mark(X)) = X + 0 >= X + 0 = incr(X) incr(active(X)) = X + 0 >= X + 0 = incr(X) s(mark(X)) = X + 0 >= X + 0 = s(X) s(active(X)) = X + 0 >= X + 0 = s(X) head(mark(X)) = 1X + 0 >= 1X + 0 = head(X) head(active(X)) = 1X + 0 >= 1X + 0 = head(X) tail(mark(X)) = X + 0 >= X + 0 = tail(X) tail(active(X)) = X + 0 >= X + 0 = tail(X) problem: DPs: mark#(head(X)) -> active#(head(mark(X))) mark#(head(X)) -> mark#(X) mark#(s(X)) -> active#(s(mark(X))) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) mark#(s(X)) -> mark#(X) mark#(odds()) -> active#(odds()) active#(odds()) -> mark#(incr(pairs())) mark#(incr(X)) -> active#(incr(mark(X))) mark#(incr(X)) -> mark#(X) mark#(pairs()) -> active#(pairs()) active#(pairs()) -> mark#(cons(0(),incr(odds()))) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(nats()) -> active#(nats()) active#(nats()) -> mark#(cons(0(),incr(nats()))) 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) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Arctic Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0 + 0, [active#](x0) = x0 + 0, [tail](x0) = x0, [head](x0) = 2x0 + 2, [s](x0) = x0 + 0, [odds] = 0, [pairs] = 0, [mark](x0) = x0 + 0, [cons](x0, x1) = x0 + x1, [incr](x0) = x0, [0] = 0, [active](x0) = x0 + 0, [nats] = 0 orientation: mark#(head(X)) = 2X + 2 >= 2X + 2 = active#(head(mark(X))) mark#(head(X)) = 2X + 2 >= X + 0 = mark#(X) mark#(s(X)) = X + 0 >= X + 0 = active#(s(mark(X))) active#(incr(cons(X,XS))) = X + XS + 0 >= X + XS + 0 = mark#(cons(s(X),incr(XS))) mark#(s(X)) = X + 0 >= X + 0 = mark#(X) mark#(odds()) = 0 >= 0 = active#(odds()) active#(odds()) = 0 >= 0 = mark#(incr(pairs())) mark#(incr(X)) = X + 0 >= X + 0 = active#(incr(mark(X))) mark#(incr(X)) = X + 0 >= X + 0 = mark#(X) mark#(pairs()) = 0 >= 0 = active#(pairs()) active#(pairs()) = 0 >= 0 = mark#(cons(0(),incr(odds()))) mark#(cons(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = active#(cons(mark(X1),X2)) mark#(cons(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(nats()) = 0 >= 0 = active#(nats()) active#(nats()) = 0 >= 0 = mark#(cons(0(),incr(nats()))) 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 + XS + 0 >= X + XS + 0 = mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) = 2X + 2XS + 2 >= X + 0 = mark(X) active(tail(cons(X,XS))) = X + XS + 0 >= XS + 0 = mark(XS) mark(nats()) = 0 >= 0 = active(nats()) mark(cons(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(incr(X)) = X + 0 >= X + 0 = active(incr(mark(X))) mark(pairs()) = 0 >= 0 = active(pairs()) mark(odds()) = 0 >= 0 = active(odds()) mark(s(X)) = X + 0 >= X + 0 = active(s(mark(X))) mark(head(X)) = 2X + 2 >= 2X + 2 = active(head(mark(X))) mark(tail(X)) = X + 0 >= X + 0 = active(tail(mark(X))) cons(mark(X1),X2) = X1 + X2 + 0 >= X1 + X2 = cons(X1,X2) cons(X1,mark(X2)) = X1 + X2 + 0 >= X1 + X2 = cons(X1,X2) cons(active(X1),X2) = X1 + X2 + 0 >= X1 + X2 = cons(X1,X2) cons(X1,active(X2)) = X1 + X2 + 0 >= X1 + X2 = cons(X1,X2) incr(mark(X)) = X + 0 >= X = incr(X) incr(active(X)) = X + 0 >= X = incr(X) s(mark(X)) = X + 0 >= X + 0 = s(X) s(active(X)) = X + 0 >= X + 0 = s(X) head(mark(X)) = 2X + 2 >= 2X + 2 = head(X) head(active(X)) = 2X + 2 >= 2X + 2 = head(X) tail(mark(X)) = X + 0 >= X = tail(X) tail(active(X)) = X + 0 >= X = tail(X) problem: DPs: mark#(head(X)) -> active#(head(mark(X))) mark#(s(X)) -> active#(s(mark(X))) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) mark#(s(X)) -> mark#(X) mark#(odds()) -> active#(odds()) active#(odds()) -> mark#(incr(pairs())) mark#(incr(X)) -> active#(incr(mark(X))) mark#(incr(X)) -> mark#(X) mark#(pairs()) -> active#(pairs()) active#(pairs()) -> mark#(cons(0(),incr(odds()))) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(nats()) -> active#(nats()) active#(nats()) -> mark#(cons(0(),incr(nats()))) 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) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Arctic Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 6, [active#](x0) = x0 + 0, [tail](x0) = 1, [head](x0) = 6, [s](x0) = 0, [odds] = 6, [pairs] = 6, [mark](x0) = 0, [cons](x0, x1) = 5, [incr](x0) = 6, [0] = 1, [active](x0) = 0, [nats] = 6 orientation: mark#(head(X)) = 6 >= 6 = active#(head(mark(X))) mark#(s(X)) = 6 >= 0 = active#(s(mark(X))) active#(incr(cons(X,XS))) = 6 >= 6 = mark#(cons(s(X),incr(XS))) mark#(s(X)) = 6 >= 6 = mark#(X) mark#(odds()) = 6 >= 6 = active#(odds()) active#(odds()) = 6 >= 6 = mark#(incr(pairs())) mark#(incr(X)) = 6 >= 6 = active#(incr(mark(X))) mark#(incr(X)) = 6 >= 6 = mark#(X) mark#(pairs()) = 6 >= 6 = active#(pairs()) active#(pairs()) = 6 >= 6 = mark#(cons(0(),incr(odds()))) mark#(cons(X1,X2)) = 6 >= 5 = active#(cons(mark(X1),X2)) mark#(cons(X1,X2)) = 6 >= 6 = mark#(X1) mark#(nats()) = 6 >= 6 = active#(nats()) active#(nats()) = 6 >= 6 = mark#(cons(0(),incr(nats()))) 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))) = 0 >= 0 = mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) = 0 >= 0 = mark(X) active(tail(cons(X,XS))) = 0 >= 0 = mark(XS) mark(nats()) = 0 >= 0 = active(nats()) mark(cons(X1,X2)) = 0 >= 0 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(incr(X)) = 0 >= 0 = active(incr(mark(X))) mark(pairs()) = 0 >= 0 = active(pairs()) mark(odds()) = 0 >= 0 = active(odds()) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(head(X)) = 0 >= 0 = active(head(mark(X))) mark(tail(X)) = 0 >= 0 = active(tail(mark(X))) cons(mark(X1),X2) = 5 >= 5 = cons(X1,X2) cons(X1,mark(X2)) = 5 >= 5 = cons(X1,X2) cons(active(X1),X2) = 5 >= 5 = cons(X1,X2) cons(X1,active(X2)) = 5 >= 5 = cons(X1,X2) incr(mark(X)) = 6 >= 6 = incr(X) incr(active(X)) = 6 >= 6 = incr(X) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) head(mark(X)) = 6 >= 6 = head(X) head(active(X)) = 6 >= 6 = head(X) tail(mark(X)) = 1 >= 1 = tail(X) tail(active(X)) = 1 >= 1 = tail(X) problem: DPs: mark#(head(X)) -> active#(head(mark(X))) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) mark#(s(X)) -> mark#(X) mark#(odds()) -> active#(odds()) active#(odds()) -> mark#(incr(pairs())) mark#(incr(X)) -> active#(incr(mark(X))) mark#(incr(X)) -> mark#(X) mark#(pairs()) -> active#(pairs()) active#(pairs()) -> mark#(cons(0(),incr(odds()))) mark#(cons(X1,X2)) -> mark#(X1) mark#(nats()) -> active#(nats()) active#(nats()) -> mark#(cons(0(),incr(nats()))) 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) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Arctic Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0 + 0, [tail](x0) = 0, [head](x0) = 0, [s](x0) = 4, [odds] = 1, [pairs] = 1, [mark](x0) = 0, [cons](x0, x1) = 0, [incr](x0) = 1, [0] = 7, [active](x0) = 0, [nats] = 1 orientation: mark#(head(X)) = 1 >= 0 = active#(head(mark(X))) active#(incr(cons(X,XS))) = 1 >= 1 = mark#(cons(s(X),incr(XS))) mark#(s(X)) = 1 >= 1 = mark#(X) mark#(odds()) = 1 >= 1 = active#(odds()) active#(odds()) = 1 >= 1 = mark#(incr(pairs())) mark#(incr(X)) = 1 >= 1 = active#(incr(mark(X))) mark#(incr(X)) = 1 >= 1 = mark#(X) mark#(pairs()) = 1 >= 1 = active#(pairs()) active#(pairs()) = 1 >= 1 = mark#(cons(0(),incr(odds()))) mark#(cons(X1,X2)) = 1 >= 1 = mark#(X1) mark#(nats()) = 1 >= 1 = active#(nats()) active#(nats()) = 1 >= 1 = mark#(cons(0(),incr(nats()))) 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))) = 0 >= 0 = mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) = 0 >= 0 = mark(X) active(tail(cons(X,XS))) = 0 >= 0 = mark(XS) mark(nats()) = 0 >= 0 = active(nats()) mark(cons(X1,X2)) = 0 >= 0 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(incr(X)) = 0 >= 0 = active(incr(mark(X))) mark(pairs()) = 0 >= 0 = active(pairs()) mark(odds()) = 0 >= 0 = active(odds()) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(head(X)) = 0 >= 0 = active(head(mark(X))) mark(tail(X)) = 0 >= 0 = active(tail(mark(X))) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) incr(mark(X)) = 1 >= 1 = incr(X) incr(active(X)) = 1 >= 1 = incr(X) s(mark(X)) = 4 >= 4 = s(X) s(active(X)) = 4 >= 4 = s(X) head(mark(X)) = 0 >= 0 = head(X) head(active(X)) = 0 >= 0 = head(X) tail(mark(X)) = 0 >= 0 = tail(X) tail(active(X)) = 0 >= 0 = tail(X) problem: DPs: active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) mark#(s(X)) -> mark#(X) mark#(odds()) -> active#(odds()) active#(odds()) -> mark#(incr(pairs())) mark#(incr(X)) -> active#(incr(mark(X))) mark#(incr(X)) -> mark#(X) mark#(pairs()) -> active#(pairs()) active#(pairs()) -> mark#(cons(0(),incr(odds()))) mark#(cons(X1,X2)) -> mark#(X1) mark#(nats()) -> active#(nats()) active#(nats()) -> mark#(cons(0(),incr(nats()))) 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) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Arctic Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0 + 0, [active#](x0) = x0 + 0, [tail](x0) = x0 + 0, [head](x0) = 2x0 + 6, [s](x0) = x0 + 0, [odds] = 4, [pairs] = 4, [mark](x0) = x0 + 0, [cons](x0, x1) = 1x0 + x1 + 2, [incr](x0) = x0 + 0, [0] = 1, [active](x0) = x0 + 0, [nats] = 2 orientation: active#(incr(cons(X,XS))) = 1X + XS + 2 >= 1X + XS + 2 = mark#(cons(s(X),incr(XS))) mark#(s(X)) = X + 0 >= X + 0 = mark#(X) mark#(odds()) = 4 >= 4 = active#(odds()) active#(odds()) = 4 >= 4 = mark#(incr(pairs())) mark#(incr(X)) = X + 0 >= X + 0 = active#(incr(mark(X))) mark#(incr(X)) = X + 0 >= X + 0 = mark#(X) mark#(pairs()) = 4 >= 4 = active#(pairs()) active#(pairs()) = 4 >= 4 = mark#(cons(0(),incr(odds()))) mark#(cons(X1,X2)) = 1X1 + X2 + 2 >= X1 + 0 = mark#(X1) mark#(nats()) = 2 >= 2 = active#(nats()) active#(nats()) = 2 >= 2 = mark#(cons(0(),incr(nats()))) active(nats()) = 2 >= 2 = mark(cons(0(),incr(nats()))) active(pairs()) = 4 >= 4 = mark(cons(0(),incr(odds()))) active(odds()) = 4 >= 4 = mark(incr(pairs())) active(incr(cons(X,XS))) = 1X + XS + 2 >= 1X + XS + 2 = mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) = 3X + 2XS + 6 >= X + 0 = mark(X) active(tail(cons(X,XS))) = 1X + XS + 2 >= XS + 0 = mark(XS) mark(nats()) = 2 >= 2 = active(nats()) mark(cons(X1,X2)) = 1X1 + X2 + 2 >= 1X1 + X2 + 2 = active(cons(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(incr(X)) = X + 0 >= X + 0 = active(incr(mark(X))) mark(pairs()) = 4 >= 4 = active(pairs()) mark(odds()) = 4 >= 4 = active(odds()) mark(s(X)) = X + 0 >= X + 0 = active(s(mark(X))) mark(head(X)) = 2X + 6 >= 2X + 6 = active(head(mark(X))) mark(tail(X)) = X + 0 >= X + 0 = active(tail(mark(X))) cons(mark(X1),X2) = 1X1 + X2 + 2 >= 1X1 + X2 + 2 = cons(X1,X2) cons(X1,mark(X2)) = 1X1 + X2 + 2 >= 1X1 + X2 + 2 = cons(X1,X2) cons(active(X1),X2) = 1X1 + X2 + 2 >= 1X1 + X2 + 2 = cons(X1,X2) cons(X1,active(X2)) = 1X1 + X2 + 2 >= 1X1 + X2 + 2 = cons(X1,X2) incr(mark(X)) = X + 0 >= X + 0 = incr(X) incr(active(X)) = X + 0 >= X + 0 = incr(X) s(mark(X)) = X + 0 >= X + 0 = s(X) s(active(X)) = X + 0 >= X + 0 = s(X) head(mark(X)) = 2X + 6 >= 2X + 6 = head(X) head(active(X)) = 2X + 6 >= 2X + 6 = head(X) tail(mark(X)) = X + 0 >= X + 0 = tail(X) tail(active(X)) = X + 0 >= X + 0 = tail(X) problem: DPs: active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) mark#(s(X)) -> mark#(X) mark#(odds()) -> active#(odds()) active#(odds()) -> mark#(incr(pairs())) mark#(incr(X)) -> active#(incr(mark(X))) mark#(incr(X)) -> mark#(X) mark#(pairs()) -> active#(pairs()) active#(pairs()) -> mark#(cons(0(),incr(odds()))) mark#(nats()) -> active#(nats()) active#(nats()) -> mark#(cons(0(),incr(nats()))) 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) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) SCC Processor: #sccs: 1 #rules: 6 #arcs: 125/100 DPs: mark#(s(X)) -> mark#(X) mark#(odds()) -> active#(odds()) active#(odds()) -> mark#(incr(pairs())) mark#(incr(X)) -> active#(incr(mark(X))) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) mark#(incr(X)) -> mark#(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) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Arctic Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 2x0 + 3, [active#](x0) = 2x0 + 0, [tail](x0) = 7x0 + 2, [head](x0) = 7x0 + 0, [s](x0) = 1x0 + 4, [odds] = 2, [pairs] = 0, [mark](x0) = x0, [cons](x0, x1) = -7x0 + -3x1 + 0, [incr](x0) = 1x0 + 0, [0] = 0, [active](x0) = x0 + 0, [nats] = 3 orientation: mark#(s(X)) = 3X + 6 >= 2X + 3 = mark#(X) mark#(odds()) = 4 >= 4 = active#(odds()) active#(odds()) = 4 >= 3 = mark#(incr(pairs())) mark#(incr(X)) = 3X + 3 >= 3X + 2 = active#(incr(mark(X))) active#(incr(cons(X,XS))) = -4X + XS + 3 >= -4X + XS + 3 = mark#(cons(s(X),incr(XS))) mark#(incr(X)) = 3X + 3 >= 2X + 3 = mark#(X) active(nats()) = 3 >= 1 = mark(cons(0(),incr(nats()))) active(pairs()) = 0 >= 0 = mark(cons(0(),incr(odds()))) active(odds()) = 2 >= 1 = mark(incr(pairs())) active(incr(cons(X,XS))) = -6X + -2XS + 1 >= -6X + -2XS + 0 = mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) = X + 4XS + 7 >= X = mark(X) active(tail(cons(X,XS))) = X + 4XS + 7 >= XS = mark(XS) mark(nats()) = 3 >= 3 = active(nats()) mark(cons(X1,X2)) = -7X1 + -3X2 + 0 >= -7X1 + -3X2 + 0 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(incr(X)) = 1X + 0 >= 1X + 0 = active(incr(mark(X))) mark(pairs()) = 0 >= 0 = active(pairs()) mark(odds()) = 2 >= 2 = active(odds()) mark(s(X)) = 1X + 4 >= 1X + 4 = active(s(mark(X))) mark(head(X)) = 7X + 0 >= 7X + 0 = active(head(mark(X))) mark(tail(X)) = 7X + 2 >= 7X + 2 = active(tail(mark(X))) cons(mark(X1),X2) = -7X1 + -3X2 + 0 >= -7X1 + -3X2 + 0 = cons(X1,X2) cons(X1,mark(X2)) = -7X1 + -3X2 + 0 >= -7X1 + -3X2 + 0 = cons(X1,X2) cons(active(X1),X2) = -7X1 + -3X2 + 0 >= -7X1 + -3X2 + 0 = cons(X1,X2) cons(X1,active(X2)) = -7X1 + -3X2 + 0 >= -7X1 + -3X2 + 0 = cons(X1,X2) incr(mark(X)) = 1X + 0 >= 1X + 0 = incr(X) incr(active(X)) = 1X + 1 >= 1X + 0 = incr(X) s(mark(X)) = 1X + 4 >= 1X + 4 = s(X) s(active(X)) = 1X + 4 >= 1X + 4 = s(X) head(mark(X)) = 7X + 0 >= 7X + 0 = head(X) head(active(X)) = 7X + 7 >= 7X + 0 = head(X) tail(mark(X)) = 7X + 2 >= 7X + 2 = tail(X) tail(active(X)) = 7X + 7 >= 7X + 2 = tail(X) problem: DPs: mark#(odds()) -> active#(odds()) mark#(incr(X)) -> active#(incr(mark(X))) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) mark#(incr(X)) -> mark#(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) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) SCC Processor: #sccs: 1 #rules: 3 #arcs: 15/16 DPs: mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> active#(incr(mark(X))) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) 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) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Arctic Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0 + 0, [active#](x0) = x0 + 0, [tail](x0) = 7x0 + 7, [head](x0) = x0 + -8, [s](x0) = 0, [odds] = 5, [pairs] = 0, [mark](x0) = x0 + 0, [cons](x0, x1) = x0 + -7x1 + 0, [incr](x0) = 2x0 + 4, [0] = 0, [active](x0) = x0 + 0, [nats] = 1 orientation: mark#(incr(X)) = 2X + 4 >= X + 0 = mark#(X) mark#(incr(X)) = 2X + 4 >= 2X + 4 = active#(incr(mark(X))) active#(incr(cons(X,XS))) = 2X + -5XS + 4 >= -5XS + 0 = mark#(cons(s(X),incr(XS))) active(nats()) = 1 >= 0 = mark(cons(0(),incr(nats()))) active(pairs()) = 0 >= 0 = mark(cons(0(),incr(odds()))) active(odds()) = 5 >= 4 = mark(incr(pairs())) active(incr(cons(X,XS))) = 2X + -5XS + 4 >= -5XS + 0 = mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) = X + -7XS + 0 >= X + 0 = mark(X) active(tail(cons(X,XS))) = 7X + XS + 7 >= XS + 0 = mark(XS) mark(nats()) = 1 >= 1 = active(nats()) mark(cons(X1,X2)) = X1 + -7X2 + 0 >= X1 + -7X2 + 0 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(incr(X)) = 2X + 4 >= 2X + 4 = active(incr(mark(X))) mark(pairs()) = 0 >= 0 = active(pairs()) mark(odds()) = 5 >= 5 = active(odds()) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(head(X)) = X + 0 >= X + 0 = active(head(mark(X))) mark(tail(X)) = 7X + 7 >= 7X + 7 = active(tail(mark(X))) cons(mark(X1),X2) = X1 + -7X2 + 0 >= X1 + -7X2 + 0 = cons(X1,X2) cons(X1,mark(X2)) = X1 + -7X2 + 0 >= X1 + -7X2 + 0 = cons(X1,X2) cons(active(X1),X2) = X1 + -7X2 + 0 >= X1 + -7X2 + 0 = cons(X1,X2) cons(X1,active(X2)) = X1 + -7X2 + 0 >= X1 + -7X2 + 0 = cons(X1,X2) incr(mark(X)) = 2X + 4 >= 2X + 4 = incr(X) incr(active(X)) = 2X + 4 >= 2X + 4 = incr(X) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) head(mark(X)) = X + 0 >= X + -8 = head(X) head(active(X)) = X + 0 >= X + -8 = head(X) tail(mark(X)) = 7X + 7 >= 7X + 7 = tail(X) tail(active(X)) = 7X + 7 >= 7X + 7 = tail(X) problem: DPs: mark#(incr(X)) -> active#(incr(mark(X))) active#(incr(cons(X,XS))) -> mark#(cons(s(X),incr(XS))) 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) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Arctic Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0 + 0, [active#](x0) = x0 + 0, [tail](x0) = 1, [head](x0) = 3, [s](x0) = 4, [odds] = 0, [pairs] = 4, [mark](x0) = 0, [cons](x0, x1) = 2, [incr](x0) = 4, [0] = 0, [active](x0) = 0, [nats] = 4 orientation: mark#(incr(X)) = 4 >= 4 = active#(incr(mark(X))) active#(incr(cons(X,XS))) = 4 >= 2 = mark#(cons(s(X),incr(XS))) 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))) = 0 >= 0 = mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) = 0 >= 0 = mark(X) active(tail(cons(X,XS))) = 0 >= 0 = mark(XS) mark(nats()) = 0 >= 0 = active(nats()) mark(cons(X1,X2)) = 0 >= 0 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(incr(X)) = 0 >= 0 = active(incr(mark(X))) mark(pairs()) = 0 >= 0 = active(pairs()) mark(odds()) = 0 >= 0 = active(odds()) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(head(X)) = 0 >= 0 = active(head(mark(X))) mark(tail(X)) = 0 >= 0 = active(tail(mark(X))) cons(mark(X1),X2) = 2 >= 2 = cons(X1,X2) cons(X1,mark(X2)) = 2 >= 2 = cons(X1,X2) cons(active(X1),X2) = 2 >= 2 = cons(X1,X2) cons(X1,active(X2)) = 2 >= 2 = cons(X1,X2) incr(mark(X)) = 4 >= 4 = incr(X) incr(active(X)) = 4 >= 4 = incr(X) s(mark(X)) = 4 >= 4 = s(X) s(active(X)) = 4 >= 4 = s(X) head(mark(X)) = 3 >= 3 = head(X) head(active(X)) = 3 >= 3 = head(X) tail(mark(X)) = 1 >= 1 = tail(X) tail(active(X)) = 1 >= 1 = tail(X) problem: DPs: mark#(incr(X)) -> active#(incr(mark(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) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) SCC Processor: #sccs: 0 #rules: 0 #arcs: 5/1 DPs: cons#(mark(X1),X2) -> cons#(X1,X2) cons#(X1,mark(X2)) -> cons#(X1,X2) cons#(active(X1),X2) -> cons#(X1,X2) cons#(X1,active(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) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Subterm Criterion Processor: simple projection: pi(cons#) = 1 problem: DPs: cons#(mark(X1),X2) -> cons#(X1,X2) cons#(active(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) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Subterm Criterion Processor: simple projection: pi(cons#) = 0 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) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Qed DPs: incr#(mark(X)) -> incr#(X) incr#(active(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) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Subterm Criterion Processor: simple projection: pi(incr#) = 0 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) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Qed DPs: s#(mark(X)) -> s#(X) s#(active(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) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Subterm Criterion Processor: simple projection: pi(s#) = 0 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) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Qed DPs: head#(mark(X)) -> head#(X) head#(active(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) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Subterm Criterion Processor: simple projection: pi(head#) = 0 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) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Qed DPs: tail#(mark(X)) -> tail#(X) tail#(active(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) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Subterm Criterion Processor: simple projection: pi(tail#) = 0 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) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Qed