MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { and(true(), X) -> activate(X) , and(false(), Y) -> false() , activate(X) -> X , activate(n__add(X1, X2)) -> add(X1, X2) , activate(n__first(X1, X2)) -> first(X1, X2) , activate(n__from(X)) -> from(X) , activate(n__s(X)) -> s(X) , if(true(), X, Y) -> activate(X) , if(false(), X, Y) -> activate(Y) , add(X1, X2) -> n__add(X1, X2) , add(0(), X) -> activate(X) , add(s(X), Y) -> s(n__add(activate(X), activate(Y))) , s(X) -> n__s(X) , first(X1, X2) -> n__first(X1, X2) , first(0(), X) -> nil() , first(s(X), cons(Y, Z)) -> cons(activate(Y), n__first(activate(X), activate(Z))) , from(X) -> cons(activate(X), n__from(n__s(activate(X)))) , from(X) -> n__from(X) } 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) '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 2) '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 3) 'Innermost Weak Dependency Pairs (timeout of 60 seconds)' failed due to the following reason: We add the following weak dependency pairs: Strict DPs: { and^#(true(), X) -> c_1(activate^#(X)) , and^#(false(), Y) -> c_2() , activate^#(X) -> c_3(X) , activate^#(n__add(X1, X2)) -> c_4(add^#(X1, X2)) , activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2)) , activate^#(n__from(X)) -> c_6(from^#(X)) , activate^#(n__s(X)) -> c_7(s^#(X)) , add^#(X1, X2) -> c_10(X1, X2) , add^#(0(), X) -> c_11(activate^#(X)) , add^#(s(X), Y) -> c_12(s^#(n__add(activate(X), activate(Y)))) , first^#(X1, X2) -> c_14(X1, X2) , first^#(0(), X) -> c_15() , first^#(s(X), cons(Y, Z)) -> c_16(activate^#(Y), activate^#(X), activate^#(Z)) , from^#(X) -> c_17(activate^#(X), activate^#(X)) , from^#(X) -> c_18(X) , s^#(X) -> c_13(X) , if^#(true(), X, Y) -> c_8(activate^#(X)) , if^#(false(), X, Y) -> c_9(activate^#(Y)) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { and^#(true(), X) -> c_1(activate^#(X)) , and^#(false(), Y) -> c_2() , activate^#(X) -> c_3(X) , activate^#(n__add(X1, X2)) -> c_4(add^#(X1, X2)) , activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2)) , activate^#(n__from(X)) -> c_6(from^#(X)) , activate^#(n__s(X)) -> c_7(s^#(X)) , add^#(X1, X2) -> c_10(X1, X2) , add^#(0(), X) -> c_11(activate^#(X)) , add^#(s(X), Y) -> c_12(s^#(n__add(activate(X), activate(Y)))) , first^#(X1, X2) -> c_14(X1, X2) , first^#(0(), X) -> c_15() , first^#(s(X), cons(Y, Z)) -> c_16(activate^#(Y), activate^#(X), activate^#(Z)) , from^#(X) -> c_17(activate^#(X), activate^#(X)) , from^#(X) -> c_18(X) , s^#(X) -> c_13(X) , if^#(true(), X, Y) -> c_8(activate^#(X)) , if^#(false(), X, Y) -> c_9(activate^#(Y)) } Strict Trs: { and(true(), X) -> activate(X) , and(false(), Y) -> false() , activate(X) -> X , activate(n__add(X1, X2)) -> add(X1, X2) , activate(n__first(X1, X2)) -> first(X1, X2) , activate(n__from(X)) -> from(X) , activate(n__s(X)) -> s(X) , if(true(), X, Y) -> activate(X) , if(false(), X, Y) -> activate(Y) , add(X1, X2) -> n__add(X1, X2) , add(0(), X) -> activate(X) , add(s(X), Y) -> s(n__add(activate(X), activate(Y))) , s(X) -> n__s(X) , first(X1, X2) -> n__first(X1, X2) , first(0(), X) -> nil() , first(s(X), cons(Y, Z)) -> cons(activate(Y), n__first(activate(X), activate(Z))) , from(X) -> cons(activate(X), n__from(n__s(activate(X)))) , from(X) -> n__from(X) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {2,12} by applications of Pre({2,12}) = {3,5,8,11,15,16}. Here rules are labeled as follows: DPs: { 1: and^#(true(), X) -> c_1(activate^#(X)) , 2: and^#(false(), Y) -> c_2() , 3: activate^#(X) -> c_3(X) , 4: activate^#(n__add(X1, X2)) -> c_4(add^#(X1, X2)) , 5: activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2)) , 6: activate^#(n__from(X)) -> c_6(from^#(X)) , 7: activate^#(n__s(X)) -> c_7(s^#(X)) , 8: add^#(X1, X2) -> c_10(X1, X2) , 9: add^#(0(), X) -> c_11(activate^#(X)) , 10: add^#(s(X), Y) -> c_12(s^#(n__add(activate(X), activate(Y)))) , 11: first^#(X1, X2) -> c_14(X1, X2) , 12: first^#(0(), X) -> c_15() , 13: first^#(s(X), cons(Y, Z)) -> c_16(activate^#(Y), activate^#(X), activate^#(Z)) , 14: from^#(X) -> c_17(activate^#(X), activate^#(X)) , 15: from^#(X) -> c_18(X) , 16: s^#(X) -> c_13(X) , 17: if^#(true(), X, Y) -> c_8(activate^#(X)) , 18: if^#(false(), X, Y) -> c_9(activate^#(Y)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { and^#(true(), X) -> c_1(activate^#(X)) , activate^#(X) -> c_3(X) , activate^#(n__add(X1, X2)) -> c_4(add^#(X1, X2)) , activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2)) , activate^#(n__from(X)) -> c_6(from^#(X)) , activate^#(n__s(X)) -> c_7(s^#(X)) , add^#(X1, X2) -> c_10(X1, X2) , add^#(0(), X) -> c_11(activate^#(X)) , add^#(s(X), Y) -> c_12(s^#(n__add(activate(X), activate(Y)))) , first^#(X1, X2) -> c_14(X1, X2) , first^#(s(X), cons(Y, Z)) -> c_16(activate^#(Y), activate^#(X), activate^#(Z)) , from^#(X) -> c_17(activate^#(X), activate^#(X)) , from^#(X) -> c_18(X) , s^#(X) -> c_13(X) , if^#(true(), X, Y) -> c_8(activate^#(X)) , if^#(false(), X, Y) -> c_9(activate^#(Y)) } Strict Trs: { and(true(), X) -> activate(X) , and(false(), Y) -> false() , activate(X) -> X , activate(n__add(X1, X2)) -> add(X1, X2) , activate(n__first(X1, X2)) -> first(X1, X2) , activate(n__from(X)) -> from(X) , activate(n__s(X)) -> s(X) , if(true(), X, Y) -> activate(X) , if(false(), X, Y) -> activate(Y) , add(X1, X2) -> n__add(X1, X2) , add(0(), X) -> activate(X) , add(s(X), Y) -> s(n__add(activate(X), activate(Y))) , s(X) -> n__s(X) , first(X1, X2) -> n__first(X1, X2) , first(0(), X) -> nil() , first(s(X), cons(Y, Z)) -> cons(activate(Y), n__first(activate(X), activate(Z))) , from(X) -> cons(activate(X), n__from(n__s(activate(X)))) , from(X) -> n__from(X) } Weak DPs: { and^#(false(), Y) -> c_2() , first^#(0(), X) -> c_15() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..