MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { t(N) -> cs(r(q(N)), nt(ns(N))) , t(X) -> nt(X) , q(0()) -> 0() , q(s(X)) -> s(p(q(X), d(X))) , s(X) -> ns(X) , p(X, 0()) -> X , p(0(), X) -> X , p(s(X), s(Y)) -> s(s(p(X, Y))) , d(0()) -> 0() , d(s(X)) -> s(s(d(X))) , f(X1, X2) -> nf(X1, X2) , f(0(), X) -> nil() , f(s(X), cs(Y, Z)) -> cs(Y, nf(X, a(Z))) , a(X) -> X , a(nt(X)) -> t(a(X)) , a(ns(X)) -> s(a(X)) , a(nf(X1, X2)) -> f(a(X1), a(X2)) } 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) '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 minimal-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. 2) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. 3) '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) 'Innermost Weak Dependency Pairs (timeout of 60 seconds)' failed due to the following reason: We add the following weak dependency pairs: Strict DPs: { t^#(N) -> c_1(q^#(N), N) , t^#(X) -> c_2(X) , q^#(0()) -> c_3() , q^#(s(X)) -> c_4(s^#(p(q(X), d(X)))) , s^#(X) -> c_5(X) , p^#(X, 0()) -> c_6(X) , p^#(0(), X) -> c_7(X) , p^#(s(X), s(Y)) -> c_8(s^#(s(p(X, Y)))) , d^#(0()) -> c_9() , d^#(s(X)) -> c_10(s^#(s(d(X)))) , f^#(X1, X2) -> c_11(X1, X2) , f^#(0(), X) -> c_12() , f^#(s(X), cs(Y, Z)) -> c_13(Y, X, a^#(Z)) , a^#(X) -> c_14(X) , a^#(nt(X)) -> c_15(t^#(a(X))) , a^#(ns(X)) -> c_16(s^#(a(X))) , a^#(nf(X1, X2)) -> c_17(f^#(a(X1), a(X2))) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { t^#(N) -> c_1(q^#(N), N) , t^#(X) -> c_2(X) , q^#(0()) -> c_3() , q^#(s(X)) -> c_4(s^#(p(q(X), d(X)))) , s^#(X) -> c_5(X) , p^#(X, 0()) -> c_6(X) , p^#(0(), X) -> c_7(X) , p^#(s(X), s(Y)) -> c_8(s^#(s(p(X, Y)))) , d^#(0()) -> c_9() , d^#(s(X)) -> c_10(s^#(s(d(X)))) , f^#(X1, X2) -> c_11(X1, X2) , f^#(0(), X) -> c_12() , f^#(s(X), cs(Y, Z)) -> c_13(Y, X, a^#(Z)) , a^#(X) -> c_14(X) , a^#(nt(X)) -> c_15(t^#(a(X))) , a^#(ns(X)) -> c_16(s^#(a(X))) , a^#(nf(X1, X2)) -> c_17(f^#(a(X1), a(X2))) } Strict Trs: { t(N) -> cs(r(q(N)), nt(ns(N))) , t(X) -> nt(X) , q(0()) -> 0() , q(s(X)) -> s(p(q(X), d(X))) , s(X) -> ns(X) , p(X, 0()) -> X , p(0(), X) -> X , p(s(X), s(Y)) -> s(s(p(X, Y))) , d(0()) -> 0() , d(s(X)) -> s(s(d(X))) , f(X1, X2) -> nf(X1, X2) , f(0(), X) -> nil() , f(s(X), cs(Y, Z)) -> cs(Y, nf(X, a(Z))) , a(X) -> X , a(nt(X)) -> t(a(X)) , a(ns(X)) -> s(a(X)) , a(nf(X1, X2)) -> f(a(X1), a(X2)) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {3,9,12} by applications of Pre({3,9,12}) = {1,2,5,6,7,11,13,14,17}. Here rules are labeled as follows: DPs: { 1: t^#(N) -> c_1(q^#(N), N) , 2: t^#(X) -> c_2(X) , 3: q^#(0()) -> c_3() , 4: q^#(s(X)) -> c_4(s^#(p(q(X), d(X)))) , 5: s^#(X) -> c_5(X) , 6: p^#(X, 0()) -> c_6(X) , 7: p^#(0(), X) -> c_7(X) , 8: p^#(s(X), s(Y)) -> c_8(s^#(s(p(X, Y)))) , 9: d^#(0()) -> c_9() , 10: d^#(s(X)) -> c_10(s^#(s(d(X)))) , 11: f^#(X1, X2) -> c_11(X1, X2) , 12: f^#(0(), X) -> c_12() , 13: f^#(s(X), cs(Y, Z)) -> c_13(Y, X, a^#(Z)) , 14: a^#(X) -> c_14(X) , 15: a^#(nt(X)) -> c_15(t^#(a(X))) , 16: a^#(ns(X)) -> c_16(s^#(a(X))) , 17: a^#(nf(X1, X2)) -> c_17(f^#(a(X1), a(X2))) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { t^#(N) -> c_1(q^#(N), N) , t^#(X) -> c_2(X) , q^#(s(X)) -> c_4(s^#(p(q(X), d(X)))) , s^#(X) -> c_5(X) , p^#(X, 0()) -> c_6(X) , p^#(0(), X) -> c_7(X) , p^#(s(X), s(Y)) -> c_8(s^#(s(p(X, Y)))) , d^#(s(X)) -> c_10(s^#(s(d(X)))) , f^#(X1, X2) -> c_11(X1, X2) , f^#(s(X), cs(Y, Z)) -> c_13(Y, X, a^#(Z)) , a^#(X) -> c_14(X) , a^#(nt(X)) -> c_15(t^#(a(X))) , a^#(ns(X)) -> c_16(s^#(a(X))) , a^#(nf(X1, X2)) -> c_17(f^#(a(X1), a(X2))) } Strict Trs: { t(N) -> cs(r(q(N)), nt(ns(N))) , t(X) -> nt(X) , q(0()) -> 0() , q(s(X)) -> s(p(q(X), d(X))) , s(X) -> ns(X) , p(X, 0()) -> X , p(0(), X) -> X , p(s(X), s(Y)) -> s(s(p(X, Y))) , d(0()) -> 0() , d(s(X)) -> s(s(d(X))) , f(X1, X2) -> nf(X1, X2) , f(0(), X) -> nil() , f(s(X), cs(Y, Z)) -> cs(Y, nf(X, a(Z))) , a(X) -> X , a(nt(X)) -> t(a(X)) , a(ns(X)) -> s(a(X)) , a(nf(X1, X2)) -> f(a(X1), a(X2)) } Weak DPs: { q^#(0()) -> c_3() , d^#(0()) -> c_9() , f^#(0(), X) -> c_12() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..