MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { terms(N) -> cons(recip(sqr(N)), n__terms(s(N))) , terms(X) -> n__terms(X) , sqr(s(X)) -> s(n__add(sqr(activate(X)), dbl(activate(X)))) , sqr(0()) -> 0() , s(X) -> n__s(X) , activate(X) -> X , activate(n__terms(X)) -> terms(X) , activate(n__add(X1, X2)) -> add(X1, X2) , activate(n__s(X)) -> s(X) , activate(n__dbl(X)) -> dbl(X) , activate(n__first(X1, X2)) -> first(X1, X2) , dbl(X) -> n__dbl(X) , dbl(s(X)) -> s(n__s(n__dbl(activate(X)))) , dbl(0()) -> 0() , add(X1, X2) -> n__add(X1, X2) , add(s(X), Y) -> s(n__add(activate(X), Y)) , add(0(), X) -> X , first(X1, X2) -> n__first(X1, X2) , first(s(X), cons(Y, Z)) -> cons(Y, n__first(activate(X), activate(Z))) , first(0(), X) -> nil() } 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: { terms^#(N) -> c_1(sqr^#(N), s^#(N)) , terms^#(X) -> c_2(X) , sqr^#(s(X)) -> c_3(s^#(n__add(sqr(activate(X)), dbl(activate(X))))) , sqr^#(0()) -> c_4() , s^#(X) -> c_5(X) , activate^#(X) -> c_6(X) , activate^#(n__terms(X)) -> c_7(terms^#(X)) , activate^#(n__add(X1, X2)) -> c_8(add^#(X1, X2)) , activate^#(n__s(X)) -> c_9(s^#(X)) , activate^#(n__dbl(X)) -> c_10(dbl^#(X)) , activate^#(n__first(X1, X2)) -> c_11(first^#(X1, X2)) , add^#(X1, X2) -> c_15(X1, X2) , add^#(s(X), Y) -> c_16(s^#(n__add(activate(X), Y))) , add^#(0(), X) -> c_17(X) , dbl^#(X) -> c_12(X) , dbl^#(s(X)) -> c_13(s^#(n__s(n__dbl(activate(X))))) , dbl^#(0()) -> c_14() , first^#(X1, X2) -> c_18(X1, X2) , first^#(s(X), cons(Y, Z)) -> c_19(Y, activate^#(X), activate^#(Z)) , first^#(0(), X) -> c_20() } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { terms^#(N) -> c_1(sqr^#(N), s^#(N)) , terms^#(X) -> c_2(X) , sqr^#(s(X)) -> c_3(s^#(n__add(sqr(activate(X)), dbl(activate(X))))) , sqr^#(0()) -> c_4() , s^#(X) -> c_5(X) , activate^#(X) -> c_6(X) , activate^#(n__terms(X)) -> c_7(terms^#(X)) , activate^#(n__add(X1, X2)) -> c_8(add^#(X1, X2)) , activate^#(n__s(X)) -> c_9(s^#(X)) , activate^#(n__dbl(X)) -> c_10(dbl^#(X)) , activate^#(n__first(X1, X2)) -> c_11(first^#(X1, X2)) , add^#(X1, X2) -> c_15(X1, X2) , add^#(s(X), Y) -> c_16(s^#(n__add(activate(X), Y))) , add^#(0(), X) -> c_17(X) , dbl^#(X) -> c_12(X) , dbl^#(s(X)) -> c_13(s^#(n__s(n__dbl(activate(X))))) , dbl^#(0()) -> c_14() , first^#(X1, X2) -> c_18(X1, X2) , first^#(s(X), cons(Y, Z)) -> c_19(Y, activate^#(X), activate^#(Z)) , first^#(0(), X) -> c_20() } Strict Trs: { terms(N) -> cons(recip(sqr(N)), n__terms(s(N))) , terms(X) -> n__terms(X) , sqr(s(X)) -> s(n__add(sqr(activate(X)), dbl(activate(X)))) , sqr(0()) -> 0() , s(X) -> n__s(X) , activate(X) -> X , activate(n__terms(X)) -> terms(X) , activate(n__add(X1, X2)) -> add(X1, X2) , activate(n__s(X)) -> s(X) , activate(n__dbl(X)) -> dbl(X) , activate(n__first(X1, X2)) -> first(X1, X2) , dbl(X) -> n__dbl(X) , dbl(s(X)) -> s(n__s(n__dbl(activate(X)))) , dbl(0()) -> 0() , add(X1, X2) -> n__add(X1, X2) , add(s(X), Y) -> s(n__add(activate(X), Y)) , add(0(), X) -> X , first(X1, X2) -> n__first(X1, X2) , first(s(X), cons(Y, Z)) -> cons(Y, n__first(activate(X), activate(Z))) , first(0(), X) -> nil() } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {4,17,20} by applications of Pre({4,17,20}) = {1,2,5,6,10,11,12,14,15,18,19}. Here rules are labeled as follows: DPs: { 1: terms^#(N) -> c_1(sqr^#(N), s^#(N)) , 2: terms^#(X) -> c_2(X) , 3: sqr^#(s(X)) -> c_3(s^#(n__add(sqr(activate(X)), dbl(activate(X))))) , 4: sqr^#(0()) -> c_4() , 5: s^#(X) -> c_5(X) , 6: activate^#(X) -> c_6(X) , 7: activate^#(n__terms(X)) -> c_7(terms^#(X)) , 8: activate^#(n__add(X1, X2)) -> c_8(add^#(X1, X2)) , 9: activate^#(n__s(X)) -> c_9(s^#(X)) , 10: activate^#(n__dbl(X)) -> c_10(dbl^#(X)) , 11: activate^#(n__first(X1, X2)) -> c_11(first^#(X1, X2)) , 12: add^#(X1, X2) -> c_15(X1, X2) , 13: add^#(s(X), Y) -> c_16(s^#(n__add(activate(X), Y))) , 14: add^#(0(), X) -> c_17(X) , 15: dbl^#(X) -> c_12(X) , 16: dbl^#(s(X)) -> c_13(s^#(n__s(n__dbl(activate(X))))) , 17: dbl^#(0()) -> c_14() , 18: first^#(X1, X2) -> c_18(X1, X2) , 19: first^#(s(X), cons(Y, Z)) -> c_19(Y, activate^#(X), activate^#(Z)) , 20: first^#(0(), X) -> c_20() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { terms^#(N) -> c_1(sqr^#(N), s^#(N)) , terms^#(X) -> c_2(X) , sqr^#(s(X)) -> c_3(s^#(n__add(sqr(activate(X)), dbl(activate(X))))) , s^#(X) -> c_5(X) , activate^#(X) -> c_6(X) , activate^#(n__terms(X)) -> c_7(terms^#(X)) , activate^#(n__add(X1, X2)) -> c_8(add^#(X1, X2)) , activate^#(n__s(X)) -> c_9(s^#(X)) , activate^#(n__dbl(X)) -> c_10(dbl^#(X)) , activate^#(n__first(X1, X2)) -> c_11(first^#(X1, X2)) , add^#(X1, X2) -> c_15(X1, X2) , add^#(s(X), Y) -> c_16(s^#(n__add(activate(X), Y))) , add^#(0(), X) -> c_17(X) , dbl^#(X) -> c_12(X) , dbl^#(s(X)) -> c_13(s^#(n__s(n__dbl(activate(X))))) , first^#(X1, X2) -> c_18(X1, X2) , first^#(s(X), cons(Y, Z)) -> c_19(Y, activate^#(X), activate^#(Z)) } Strict Trs: { terms(N) -> cons(recip(sqr(N)), n__terms(s(N))) , terms(X) -> n__terms(X) , sqr(s(X)) -> s(n__add(sqr(activate(X)), dbl(activate(X)))) , sqr(0()) -> 0() , s(X) -> n__s(X) , activate(X) -> X , activate(n__terms(X)) -> terms(X) , activate(n__add(X1, X2)) -> add(X1, X2) , activate(n__s(X)) -> s(X) , activate(n__dbl(X)) -> dbl(X) , activate(n__first(X1, X2)) -> first(X1, X2) , dbl(X) -> n__dbl(X) , dbl(s(X)) -> s(n__s(n__dbl(activate(X)))) , dbl(0()) -> 0() , add(X1, X2) -> n__add(X1, X2) , add(s(X), Y) -> s(n__add(activate(X), Y)) , add(0(), X) -> X , first(X1, X2) -> n__first(X1, X2) , first(s(X), cons(Y, Z)) -> cons(Y, n__first(activate(X), activate(Z))) , first(0(), X) -> nil() } Weak DPs: { sqr^#(0()) -> c_4() , dbl^#(0()) -> c_14() , first^#(0(), X) -> c_20() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..