MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { a__filter(X1, X2, X3) -> filter(X1, X2, X3) , a__filter(cons(X, Y), 0(), M) -> cons(0(), filter(Y, M, M)) , a__filter(cons(X, Y), s(N), M) -> cons(mark(X), filter(Y, N, M)) , mark(cons(X1, X2)) -> cons(mark(X1), X2) , mark(0()) -> 0() , mark(filter(X1, X2, X3)) -> a__filter(mark(X1), mark(X2), mark(X3)) , mark(s(X)) -> s(mark(X)) , mark(sieve(X)) -> a__sieve(mark(X)) , mark(nats(X)) -> a__nats(mark(X)) , mark(zprimes()) -> a__zprimes() , a__sieve(X) -> sieve(X) , a__sieve(cons(0(), Y)) -> cons(0(), sieve(Y)) , a__sieve(cons(s(N), Y)) -> cons(s(mark(N)), sieve(filter(Y, N, N))) , a__nats(N) -> cons(mark(N), nats(s(N))) , a__nats(X) -> nats(X) , a__zprimes() -> a__sieve(a__nats(s(s(0())))) , a__zprimes() -> zprimes() } Obligation: runtime complexity Answer: MAYBE None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'WithProblem (timeout of 60 seconds)' failed due to the following reason: Computation stopped due to timeout after 60.0 seconds. 2) 'Best' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'WithProblem (timeout of 30 seconds) (timeout of 60 seconds)' failed due to the following reason: Computation stopped due to timeout after 30.0 seconds. 2) 'Best' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'Polynomial Path Order (PS) (timeout of 60 seconds)' failed due to the following reason: The processor is inapplicable, reason: Processor only applicable for innermost runtime complexity analysis 2) 'bsearch-popstar (timeout of 60 seconds)' failed due to the following reason: The processor is inapplicable, reason: Processor only applicable for innermost runtime complexity analysis 3) 'Fastest (timeout of 5 seconds) (timeout of 60 seconds)' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. 2) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. 3) 'Innermost Weak Dependency Pairs (timeout of 60 seconds)' failed due to the following reason: We add the following weak dependency pairs: Strict DPs: { a__filter^#(X1, X2, X3) -> c_1(X1, X2, X3) , a__filter^#(cons(X, Y), 0(), M) -> c_2(Y, M, M) , a__filter^#(cons(X, Y), s(N), M) -> c_3(mark^#(X), Y, N, M) , mark^#(cons(X1, X2)) -> c_4(mark^#(X1), X2) , mark^#(0()) -> c_5() , mark^#(filter(X1, X2, X3)) -> c_6(a__filter^#(mark(X1), mark(X2), mark(X3))) , mark^#(s(X)) -> c_7(mark^#(X)) , mark^#(sieve(X)) -> c_8(a__sieve^#(mark(X))) , mark^#(nats(X)) -> c_9(a__nats^#(mark(X))) , mark^#(zprimes()) -> c_10(a__zprimes^#()) , a__sieve^#(X) -> c_11(X) , a__sieve^#(cons(0(), Y)) -> c_12(Y) , a__sieve^#(cons(s(N), Y)) -> c_13(mark^#(N), Y, N, N) , a__nats^#(N) -> c_14(mark^#(N), N) , a__nats^#(X) -> c_15(X) , a__zprimes^#() -> c_16(a__sieve^#(a__nats(s(s(0()))))) , a__zprimes^#() -> c_17() } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { a__filter^#(X1, X2, X3) -> c_1(X1, X2, X3) , a__filter^#(cons(X, Y), 0(), M) -> c_2(Y, M, M) , a__filter^#(cons(X, Y), s(N), M) -> c_3(mark^#(X), Y, N, M) , mark^#(cons(X1, X2)) -> c_4(mark^#(X1), X2) , mark^#(0()) -> c_5() , mark^#(filter(X1, X2, X3)) -> c_6(a__filter^#(mark(X1), mark(X2), mark(X3))) , mark^#(s(X)) -> c_7(mark^#(X)) , mark^#(sieve(X)) -> c_8(a__sieve^#(mark(X))) , mark^#(nats(X)) -> c_9(a__nats^#(mark(X))) , mark^#(zprimes()) -> c_10(a__zprimes^#()) , a__sieve^#(X) -> c_11(X) , a__sieve^#(cons(0(), Y)) -> c_12(Y) , a__sieve^#(cons(s(N), Y)) -> c_13(mark^#(N), Y, N, N) , a__nats^#(N) -> c_14(mark^#(N), N) , a__nats^#(X) -> c_15(X) , a__zprimes^#() -> c_16(a__sieve^#(a__nats(s(s(0()))))) , a__zprimes^#() -> c_17() } Strict Trs: { a__filter(X1, X2, X3) -> filter(X1, X2, X3) , a__filter(cons(X, Y), 0(), M) -> cons(0(), filter(Y, M, M)) , a__filter(cons(X, Y), s(N), M) -> cons(mark(X), filter(Y, N, M)) , mark(cons(X1, X2)) -> cons(mark(X1), X2) , mark(0()) -> 0() , mark(filter(X1, X2, X3)) -> a__filter(mark(X1), mark(X2), mark(X3)) , mark(s(X)) -> s(mark(X)) , mark(sieve(X)) -> a__sieve(mark(X)) , mark(nats(X)) -> a__nats(mark(X)) , mark(zprimes()) -> a__zprimes() , a__sieve(X) -> sieve(X) , a__sieve(cons(0(), Y)) -> cons(0(), sieve(Y)) , a__sieve(cons(s(N), Y)) -> cons(s(mark(N)), sieve(filter(Y, N, N))) , a__nats(N) -> cons(mark(N), nats(s(N))) , a__nats(X) -> nats(X) , a__zprimes() -> a__sieve(a__nats(s(s(0())))) , a__zprimes() -> zprimes() } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {5,17} by applications of Pre({5,17}) = {1,2,3,4,7,10,11,12,13,14,15}. Here rules are labeled as follows: DPs: { 1: a__filter^#(X1, X2, X3) -> c_1(X1, X2, X3) , 2: a__filter^#(cons(X, Y), 0(), M) -> c_2(Y, M, M) , 3: a__filter^#(cons(X, Y), s(N), M) -> c_3(mark^#(X), Y, N, M) , 4: mark^#(cons(X1, X2)) -> c_4(mark^#(X1), X2) , 5: mark^#(0()) -> c_5() , 6: mark^#(filter(X1, X2, X3)) -> c_6(a__filter^#(mark(X1), mark(X2), mark(X3))) , 7: mark^#(s(X)) -> c_7(mark^#(X)) , 8: mark^#(sieve(X)) -> c_8(a__sieve^#(mark(X))) , 9: mark^#(nats(X)) -> c_9(a__nats^#(mark(X))) , 10: mark^#(zprimes()) -> c_10(a__zprimes^#()) , 11: a__sieve^#(X) -> c_11(X) , 12: a__sieve^#(cons(0(), Y)) -> c_12(Y) , 13: a__sieve^#(cons(s(N), Y)) -> c_13(mark^#(N), Y, N, N) , 14: a__nats^#(N) -> c_14(mark^#(N), N) , 15: a__nats^#(X) -> c_15(X) , 16: a__zprimes^#() -> c_16(a__sieve^#(a__nats(s(s(0()))))) , 17: a__zprimes^#() -> c_17() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { a__filter^#(X1, X2, X3) -> c_1(X1, X2, X3) , a__filter^#(cons(X, Y), 0(), M) -> c_2(Y, M, M) , a__filter^#(cons(X, Y), s(N), M) -> c_3(mark^#(X), Y, N, M) , mark^#(cons(X1, X2)) -> c_4(mark^#(X1), X2) , mark^#(filter(X1, X2, X3)) -> c_6(a__filter^#(mark(X1), mark(X2), mark(X3))) , mark^#(s(X)) -> c_7(mark^#(X)) , mark^#(sieve(X)) -> c_8(a__sieve^#(mark(X))) , mark^#(nats(X)) -> c_9(a__nats^#(mark(X))) , mark^#(zprimes()) -> c_10(a__zprimes^#()) , a__sieve^#(X) -> c_11(X) , a__sieve^#(cons(0(), Y)) -> c_12(Y) , a__sieve^#(cons(s(N), Y)) -> c_13(mark^#(N), Y, N, N) , a__nats^#(N) -> c_14(mark^#(N), N) , a__nats^#(X) -> c_15(X) , a__zprimes^#() -> c_16(a__sieve^#(a__nats(s(s(0()))))) } Strict Trs: { a__filter(X1, X2, X3) -> filter(X1, X2, X3) , a__filter(cons(X, Y), 0(), M) -> cons(0(), filter(Y, M, M)) , a__filter(cons(X, Y), s(N), M) -> cons(mark(X), filter(Y, N, M)) , mark(cons(X1, X2)) -> cons(mark(X1), X2) , mark(0()) -> 0() , mark(filter(X1, X2, X3)) -> a__filter(mark(X1), mark(X2), mark(X3)) , mark(s(X)) -> s(mark(X)) , mark(sieve(X)) -> a__sieve(mark(X)) , mark(nats(X)) -> a__nats(mark(X)) , mark(zprimes()) -> a__zprimes() , a__sieve(X) -> sieve(X) , a__sieve(cons(0(), Y)) -> cons(0(), sieve(Y)) , a__sieve(cons(s(N), Y)) -> cons(s(mark(N)), sieve(filter(Y, N, N))) , a__nats(N) -> cons(mark(N), nats(s(N))) , a__nats(X) -> nats(X) , a__zprimes() -> a__sieve(a__nats(s(s(0())))) , a__zprimes() -> zprimes() } Weak DPs: { mark^#(0()) -> c_5() , a__zprimes^#() -> c_17() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..