MAYBE Problem: active(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) -> mark(X) active(tail(cons(X,XS))) -> mark(XS) 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) EDG 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#(mark(X)) -> tail#(X) tail#(mark(X)) -> tail#(X) -> tail#(active(X)) -> tail#(X) tail#(active(X)) -> tail#(X) -> tail#(mark(X)) -> tail#(X) tail#(active(X)) -> tail#(X) -> tail#(active(X)) -> tail#(X) head#(mark(X)) -> head#(X) -> head#(mark(X)) -> head#(X) head#(mark(X)) -> head#(X) -> head#(active(X)) -> head#(X) head#(active(X)) -> head#(X) -> head#(mark(X)) -> head#(X) head#(active(X)) -> head#(X) -> head#(active(X)) -> head#(X) s#(mark(X)) -> s#(X) -> s#(mark(X)) -> s#(X) s#(mark(X)) -> s#(X) -> s#(active(X)) -> s#(X) s#(active(X)) -> s#(X) -> s#(mark(X)) -> s#(X) s#(active(X)) -> s#(X) -> s#(active(X)) -> s#(X) mark#(tail(X)) -> tail#(mark(X)) -> tail#(mark(X)) -> tail#(X) mark#(tail(X)) -> tail#(mark(X)) -> tail#(active(X)) -> tail#(X) 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)) -> cons#(mark(X1),X2) 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)) -> incr#(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)) -> s#(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)) -> head#(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)) -> tail#(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))) -> 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))) -> 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)) -> 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)) -> head#(mark(X)) -> head#(mark(X)) -> head#(X) mark#(head(X)) -> head#(mark(X)) -> head#(active(X)) -> head#(X) 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)) -> cons#(mark(X1),X2) 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)) -> incr#(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)) -> s#(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)) -> head#(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)) -> tail#(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))) -> 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))) -> 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)) -> 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)) -> s#(mark(X)) -> s#(mark(X)) -> s#(X) mark#(s(X)) -> s#(mark(X)) -> s#(active(X)) -> s#(X) 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)) -> cons#(mark(X1),X2) 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)) -> incr#(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)) -> s#(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)) -> head#(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)) -> tail#(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))) -> 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))) -> 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)) -> 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()) -> incr#(pairs()) mark#(odds()) -> active#(odds()) -> active#(odds()) -> mark#(incr(pairs())) mark#(pairs()) -> active#(pairs()) -> active#(pairs()) -> incr#(odds()) mark#(pairs()) -> active#(pairs()) -> active#(pairs()) -> cons#(0(),incr(odds())) 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)) -> cons#(mark(X1),X2) 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)) -> incr#(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)) -> s#(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)) -> head#(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)) -> tail#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(tail(X)) -> active#(tail(mark(X))) mark#(cons(X1,X2)) -> cons#(mark(X1),X2) -> cons#(mark(X1),X2) -> cons#(X1,X2) mark#(cons(X1,X2)) -> cons#(mark(X1),X2) -> cons#(active(X1),X2) -> cons#(X1,X2) 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#(incr(cons(X,XS))) -> s#(X) 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))) -> 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)) -> cons#(mark(X1),X2) 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)) -> incr#(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)) -> s#(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)) -> head#(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)) -> tail#(mark(X)) mark#(incr(X)) -> mark#(X) -> mark#(tail(X)) -> active#(tail(mark(X))) mark#(incr(X)) -> incr#(mark(X)) -> incr#(mark(X)) -> incr#(X) mark#(incr(X)) -> incr#(mark(X)) -> incr#(active(X)) -> incr#(X) mark#(incr(X)) -> active#(incr(mark(X))) -> active#(incr(cons(X,XS))) -> 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))) -> 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)) -> 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()) -> incr#(nats()) mark#(nats()) -> active#(nats()) -> active#(nats()) -> cons#(0(),incr(nats())) mark#(nats()) -> active#(nats()) -> active#(nats()) -> mark#(cons(0(),incr(nats()))) cons#(mark(X1),X2) -> cons#(X1,X2) -> cons#(mark(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#(active(X1),X2) -> cons#(X1,X2) cons#(mark(X1),X2) -> cons#(X1,X2) -> cons#(X1,active(X2)) -> cons#(X1,X2) cons#(active(X1),X2) -> cons#(X1,X2) -> cons#(mark(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#(active(X1),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#(mark(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#(active(X1),X2) -> cons#(X1,X2) cons#(X1,mark(X2)) -> cons#(X1,X2) -> cons#(X1,active(X2)) -> cons#(X1,X2) cons#(X1,active(X2)) -> cons#(X1,X2) -> cons#(mark(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#(active(X1),X2) -> cons#(X1,X2) cons#(X1,active(X2)) -> cons#(X1,X2) -> cons#(X1,active(X2)) -> cons#(X1,X2) incr#(mark(X)) -> incr#(X) -> incr#(mark(X)) -> incr#(X) incr#(mark(X)) -> incr#(X) -> incr#(active(X)) -> incr#(X) incr#(active(X)) -> incr#(X) -> incr#(mark(X)) -> incr#(X) incr#(active(X)) -> incr#(X) -> incr#(active(X)) -> incr#(X) 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)) -> cons#(mark(X1),X2) 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)) -> incr#(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)) -> s#(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)) -> head#(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)) -> 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#(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)) -> cons#(mark(X1),X2) 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)) -> incr#(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)) -> s#(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)) -> head#(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)) -> tail#(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)) -> incr#(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)) -> cons#(mark(X1),X2) 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)) -> 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))) -> 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)) -> 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))) -> 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)) -> 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))) -> 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)) -> 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))) -> 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)) -> 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)) -> cons#(X1,mark(X2)) -> cons#(X1,X2) active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) -> cons#(X1,active(X2)) -> cons#(X1,X2) active#(nats()) -> mark#(cons(0(),incr(nats()))) -> mark#(cons(X1,X2)) -> mark#(X1) 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)) -> active#(cons(mark(X1),X2)) SCC Processor: #sccs: 6 #rules: 31 #arcs: 239/2025 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) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0 + 1, [active#](x0) = x0 + 1, [tail](x0) = x0 + 1, [head](x0) = x0, [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)) = X + 2 >= X + 1 = mark#(X) mark#(tail(X)) = X + 2 >= X + 2 = active#(tail(mark(X))) active#(tail(cons(X,XS))) = X + XS + 2 >= XS + 1 = mark#(XS) mark#(head(X)) = X + 1 >= X + 1 = active#(head(mark(X))) active#(head(cons(X,XS))) = X + XS + 1 >= X + 1 = mark#(X) mark#(head(X)) = X + 1 >= X + 1 = mark#(X) mark#(s(X)) = X + 1 >= X + 1 = active#(s(mark(X))) active#(incr(cons(X,XS))) = X + XS + 1 >= X + XS + 1 = mark#(cons(s(X),incr(XS))) mark#(s(X)) = X + 1 >= X + 1 = mark#(X) mark#(odds()) = 1 >= 1 = active#(odds()) active#(odds()) = 1 >= 1 = mark#(incr(pairs())) mark#(incr(X)) = X + 1 >= X + 1 = active#(incr(mark(X))) mark#(incr(X)) = X + 1 >= X + 1 = mark#(X) mark#(pairs()) = 1 >= 1 = active#(pairs()) active#(pairs()) = 1 >= 1 = mark#(cons(0(),incr(odds()))) mark#(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active#(cons(mark(X1),X2)) mark#(cons(X1,X2)) = X1 + X2 + 1 >= X1 + 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))) = X + XS >= X + XS = mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) = X + XS >= X = mark(X) active(tail(cons(X,XS))) = X + XS + 1 >= 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 >= X = active(head(mark(X))) mark(tail(X)) = X + 1 >= X + 1 = 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 >= X = head(X) head(active(X)) = X >= X = head(X) tail(mark(X)) = X + 1 >= X + 1 = tail(X) tail(active(X)) = X + 1 >= X + 1 = 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) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [tail](x0) = x0, [head](x0) = x0 + 1, [s](x0) = x0, [odds] = 1, [pairs] = 1, [mark](x0) = x0, [cons](x0, x1) = x0 + x1, [incr](x0) = x0, [0] = 0, [active](x0) = x0, [nats] = 0 orientation: mark#(tail(X)) = X >= X = active#(tail(mark(X))) mark#(head(X)) = X + 1 >= X + 1 = active#(head(mark(X))) active#(head(cons(X,XS))) = X + XS + 1 >= X = mark#(X) mark#(head(X)) = X + 1 >= X = mark#(X) mark#(s(X)) = X >= X = active#(s(mark(X))) active#(incr(cons(X,XS))) = X + XS >= X + XS = mark#(cons(s(X),incr(XS))) mark#(s(X)) = X >= X = mark#(X) mark#(odds()) = 1 >= 1 = active#(odds()) active#(odds()) = 1 >= 1 = mark#(incr(pairs())) mark#(incr(X)) = X >= X = active#(incr(mark(X))) mark#(incr(X)) = X >= X = mark#(X) mark#(pairs()) = 1 >= 1 = active#(pairs()) active#(pairs()) = 1 >= 1 = mark#(cons(0(),incr(odds()))) mark#(cons(X1,X2)) = X1 + X2 >= X1 + X2 = active#(cons(mark(X1),X2)) mark#(cons(X1,X2)) = X1 + X2 >= X1 = 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()) = 1 >= 1 = mark(cons(0(),incr(odds()))) active(odds()) = 1 >= 1 = 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 + 1 >= X = mark(X) active(tail(cons(X,XS))) = X + XS >= 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()) = 1 >= 1 = active(pairs()) mark(odds()) = 1 >= 1 = active(odds()) mark(s(X)) = X >= X = active(s(mark(X))) mark(head(X)) = X + 1 >= X + 1 = active(head(mark(X))) mark(tail(X)) = X >= X = 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 + 1 >= X + 1 = head(X) head(active(X)) = X + 1 >= X + 1 = head(X) tail(mark(X)) = X >= X = tail(X) tail(active(X)) = X >= X = tail(X) problem: DPs: mark#(tail(X)) -> active#(tail(mark(X))) 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) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [tail](x0) = 1, [head](x0) = 1, [s](x0) = 1, [odds] = 1, [pairs] = 1, [mark](x0) = 1, [cons](x0, x1) = 0, [incr](x0) = 1, [0] = 0, [active](x0) = 1, [nats] = 1 orientation: mark#(tail(X)) = 1 >= 1 = active#(tail(mark(X))) mark#(head(X)) = 1 >= 1 = active#(head(mark(X))) mark#(s(X)) = 1 >= 1 = active#(s(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 >= 0 = active#(cons(mark(X1),X2)) 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()) = 1 >= 1 = mark(cons(0(),incr(nats()))) active(pairs()) = 1 >= 1 = mark(cons(0(),incr(odds()))) active(odds()) = 1 >= 1 = mark(incr(pairs())) active(incr(cons(X,XS))) = 1 >= 1 = mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) = 1 >= 1 = mark(X) active(tail(cons(X,XS))) = 1 >= 1 = mark(XS) mark(nats()) = 1 >= 1 = active(nats()) mark(cons(X1,X2)) = 1 >= 1 = active(cons(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(incr(X)) = 1 >= 1 = active(incr(mark(X))) mark(pairs()) = 1 >= 1 = active(pairs()) mark(odds()) = 1 >= 1 = active(odds()) mark(s(X)) = 1 >= 1 = active(s(mark(X))) mark(head(X)) = 1 >= 1 = active(head(mark(X))) mark(tail(X)) = 1 >= 1 = 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)) = 1 >= 1 = s(X) s(active(X)) = 1 >= 1 = s(X) head(mark(X)) = 1 >= 1 = head(X) head(active(X)) = 1 >= 1 = head(X) tail(mark(X)) = 1 >= 1 = tail(X) tail(active(X)) = 1 >= 1 = tail(X) problem: DPs: mark#(tail(X)) -> active#(tail(mark(X))) 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)) -> 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) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [tail](x0) = 1, [head](x0) = 1, [s](x0) = 0, [odds] = 1, [pairs] = 1, [mark](x0) = 0, [cons](x0, x1) = 0, [incr](x0) = 1, [0] = 0, [active](x0) = 0, [nats] = 1 orientation: mark#(tail(X)) = 1 >= 1 = active#(tail(mark(X))) mark#(head(X)) = 1 >= 1 = active#(head(mark(X))) mark#(s(X)) = 1 >= 0 = active#(s(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)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) head(mark(X)) = 1 >= 1 = head(X) head(active(X)) = 1 >= 1 = head(X) tail(mark(X)) = 1 >= 1 = tail(X) tail(active(X)) = 1 >= 1 = tail(X) problem: DPs: 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#(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) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [tail](x0) = 1, [head](x0) = 0, [s](x0) = 0, [odds] = 1, [pairs] = 1, [mark](x0) = 0, [cons](x0, x1) = 0, [incr](x0) = 1, [0] = 0, [active](x0) = 0, [nats] = 1 orientation: mark#(tail(X)) = 1 >= 1 = active#(tail(mark(X))) 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)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) head(mark(X)) = 0 >= 0 = head(X) head(active(X)) = 0 >= 0 = head(X) tail(mark(X)) = 1 >= 1 = tail(X) tail(active(X)) = 1 >= 1 = tail(X) problem: DPs: mark#(tail(X)) -> active#(tail(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) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [tail](x0) = 0, [head](x0) = 0, [s](x0) = 0, [odds] = 1, [pairs] = 1, [mark](x0) = 0, [cons](x0, x1) = 0, [incr](x0) = 1, [0] = 0, [active](x0) = 0, [nats] = 1 orientation: mark#(tail(X)) = 1 >= 0 = active#(tail(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)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = 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) Open DPs: incr#(active(X)) -> incr#(X) incr#(mark(X)) -> incr#(X) TRS: active(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) -> mark(X) active(tail(cons(X,XS))) -> mark(XS) 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) Open DPs: cons#(X1,active(X2)) -> cons#(X1,X2) cons#(active(X1),X2) -> cons#(X1,X2) cons#(X1,mark(X2)) -> cons#(X1,X2) cons#(mark(X1),X2) -> cons#(X1,X2) TRS: active(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) -> mark(X) active(tail(cons(X,XS))) -> mark(XS) 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) Open 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) Open 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) Open 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) Open