MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { terms(N) -> cons(recip(sqr(N)), n__terms(n__s(N))) , terms(X) -> n__terms(X) , sqr(X) -> n__sqr(X) , sqr(0()) -> 0() , sqr(s(X)) -> s(n__add(n__sqr(activate(X)), n__dbl(activate(X)))) , s(X) -> n__s(X) , activate(X) -> X , activate(n__terms(X)) -> terms(activate(X)) , activate(n__s(X)) -> s(X) , activate(n__add(X1, X2)) -> add(activate(X1), activate(X2)) , activate(n__sqr(X)) -> sqr(activate(X)) , activate(n__dbl(X)) -> dbl(activate(X)) , activate(n__first(X1, X2)) -> first(activate(X1), activate(X2)) , dbl(X) -> n__dbl(X) , dbl(0()) -> 0() , dbl(s(X)) -> s(n__s(n__dbl(activate(X)))) , add(X1, X2) -> n__add(X1, X2) , add(0(), X) -> X , add(s(X), Y) -> s(n__add(activate(X), Y)) , first(X1, X2) -> n__first(X1, X2) , first(0(), X) -> nil() , first(s(X), cons(Y, Z)) -> cons(Y, n__first(activate(X), activate(Z))) } 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 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) '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: { terms^#(N) -> c_1(sqr^#(N), N) , terms^#(X) -> c_2(X) , sqr^#(X) -> c_3(X) , sqr^#(0()) -> c_4() , sqr^#(s(X)) -> c_5(s^#(n__add(n__sqr(activate(X)), n__dbl(activate(X))))) , s^#(X) -> c_6(X) , activate^#(X) -> c_7(X) , activate^#(n__terms(X)) -> c_8(terms^#(activate(X))) , activate^#(n__s(X)) -> c_9(s^#(X)) , activate^#(n__add(X1, X2)) -> c_10(add^#(activate(X1), activate(X2))) , activate^#(n__sqr(X)) -> c_11(sqr^#(activate(X))) , activate^#(n__dbl(X)) -> c_12(dbl^#(activate(X))) , activate^#(n__first(X1, X2)) -> c_13(first^#(activate(X1), activate(X2))) , add^#(X1, X2) -> c_17(X1, X2) , add^#(0(), X) -> c_18(X) , add^#(s(X), Y) -> c_19(s^#(n__add(activate(X), Y))) , dbl^#(X) -> c_14(X) , dbl^#(0()) -> c_15() , dbl^#(s(X)) -> c_16(s^#(n__s(n__dbl(activate(X))))) , first^#(X1, X2) -> c_20(X1, X2) , first^#(0(), X) -> c_21() , first^#(s(X), cons(Y, Z)) -> c_22(Y, activate^#(X), activate^#(Z)) } 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), N) , terms^#(X) -> c_2(X) , sqr^#(X) -> c_3(X) , sqr^#(0()) -> c_4() , sqr^#(s(X)) -> c_5(s^#(n__add(n__sqr(activate(X)), n__dbl(activate(X))))) , s^#(X) -> c_6(X) , activate^#(X) -> c_7(X) , activate^#(n__terms(X)) -> c_8(terms^#(activate(X))) , activate^#(n__s(X)) -> c_9(s^#(X)) , activate^#(n__add(X1, X2)) -> c_10(add^#(activate(X1), activate(X2))) , activate^#(n__sqr(X)) -> c_11(sqr^#(activate(X))) , activate^#(n__dbl(X)) -> c_12(dbl^#(activate(X))) , activate^#(n__first(X1, X2)) -> c_13(first^#(activate(X1), activate(X2))) , add^#(X1, X2) -> c_17(X1, X2) , add^#(0(), X) -> c_18(X) , add^#(s(X), Y) -> c_19(s^#(n__add(activate(X), Y))) , dbl^#(X) -> c_14(X) , dbl^#(0()) -> c_15() , dbl^#(s(X)) -> c_16(s^#(n__s(n__dbl(activate(X))))) , first^#(X1, X2) -> c_20(X1, X2) , first^#(0(), X) -> c_21() , first^#(s(X), cons(Y, Z)) -> c_22(Y, activate^#(X), activate^#(Z)) } Strict Trs: { terms(N) -> cons(recip(sqr(N)), n__terms(n__s(N))) , terms(X) -> n__terms(X) , sqr(X) -> n__sqr(X) , sqr(0()) -> 0() , sqr(s(X)) -> s(n__add(n__sqr(activate(X)), n__dbl(activate(X)))) , s(X) -> n__s(X) , activate(X) -> X , activate(n__terms(X)) -> terms(activate(X)) , activate(n__s(X)) -> s(X) , activate(n__add(X1, X2)) -> add(activate(X1), activate(X2)) , activate(n__sqr(X)) -> sqr(activate(X)) , activate(n__dbl(X)) -> dbl(activate(X)) , activate(n__first(X1, X2)) -> first(activate(X1), activate(X2)) , dbl(X) -> n__dbl(X) , dbl(0()) -> 0() , dbl(s(X)) -> s(n__s(n__dbl(activate(X)))) , add(X1, X2) -> n__add(X1, X2) , add(0(), X) -> X , add(s(X), Y) -> s(n__add(activate(X), Y)) , first(X1, X2) -> n__first(X1, X2) , first(0(), X) -> nil() , first(s(X), cons(Y, Z)) -> cons(Y, n__first(activate(X), activate(Z))) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {4,18,21} by applications of Pre({4,18,21}) = {1,2,3,6,7,11,12,13,14,15,17,20,22}. Here rules are labeled as follows: DPs: { 1: terms^#(N) -> c_1(sqr^#(N), N) , 2: terms^#(X) -> c_2(X) , 3: sqr^#(X) -> c_3(X) , 4: sqr^#(0()) -> c_4() , 5: sqr^#(s(X)) -> c_5(s^#(n__add(n__sqr(activate(X)), n__dbl(activate(X))))) , 6: s^#(X) -> c_6(X) , 7: activate^#(X) -> c_7(X) , 8: activate^#(n__terms(X)) -> c_8(terms^#(activate(X))) , 9: activate^#(n__s(X)) -> c_9(s^#(X)) , 10: activate^#(n__add(X1, X2)) -> c_10(add^#(activate(X1), activate(X2))) , 11: activate^#(n__sqr(X)) -> c_11(sqr^#(activate(X))) , 12: activate^#(n__dbl(X)) -> c_12(dbl^#(activate(X))) , 13: activate^#(n__first(X1, X2)) -> c_13(first^#(activate(X1), activate(X2))) , 14: add^#(X1, X2) -> c_17(X1, X2) , 15: add^#(0(), X) -> c_18(X) , 16: add^#(s(X), Y) -> c_19(s^#(n__add(activate(X), Y))) , 17: dbl^#(X) -> c_14(X) , 18: dbl^#(0()) -> c_15() , 19: dbl^#(s(X)) -> c_16(s^#(n__s(n__dbl(activate(X))))) , 20: first^#(X1, X2) -> c_20(X1, X2) , 21: first^#(0(), X) -> c_21() , 22: first^#(s(X), cons(Y, Z)) -> c_22(Y, activate^#(X), activate^#(Z)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { terms^#(N) -> c_1(sqr^#(N), N) , terms^#(X) -> c_2(X) , sqr^#(X) -> c_3(X) , sqr^#(s(X)) -> c_5(s^#(n__add(n__sqr(activate(X)), n__dbl(activate(X))))) , s^#(X) -> c_6(X) , activate^#(X) -> c_7(X) , activate^#(n__terms(X)) -> c_8(terms^#(activate(X))) , activate^#(n__s(X)) -> c_9(s^#(X)) , activate^#(n__add(X1, X2)) -> c_10(add^#(activate(X1), activate(X2))) , activate^#(n__sqr(X)) -> c_11(sqr^#(activate(X))) , activate^#(n__dbl(X)) -> c_12(dbl^#(activate(X))) , activate^#(n__first(X1, X2)) -> c_13(first^#(activate(X1), activate(X2))) , add^#(X1, X2) -> c_17(X1, X2) , add^#(0(), X) -> c_18(X) , add^#(s(X), Y) -> c_19(s^#(n__add(activate(X), Y))) , dbl^#(X) -> c_14(X) , dbl^#(s(X)) -> c_16(s^#(n__s(n__dbl(activate(X))))) , first^#(X1, X2) -> c_20(X1, X2) , first^#(s(X), cons(Y, Z)) -> c_22(Y, activate^#(X), activate^#(Z)) } Strict Trs: { terms(N) -> cons(recip(sqr(N)), n__terms(n__s(N))) , terms(X) -> n__terms(X) , sqr(X) -> n__sqr(X) , sqr(0()) -> 0() , sqr(s(X)) -> s(n__add(n__sqr(activate(X)), n__dbl(activate(X)))) , s(X) -> n__s(X) , activate(X) -> X , activate(n__terms(X)) -> terms(activate(X)) , activate(n__s(X)) -> s(X) , activate(n__add(X1, X2)) -> add(activate(X1), activate(X2)) , activate(n__sqr(X)) -> sqr(activate(X)) , activate(n__dbl(X)) -> dbl(activate(X)) , activate(n__first(X1, X2)) -> first(activate(X1), activate(X2)) , dbl(X) -> n__dbl(X) , dbl(0()) -> 0() , dbl(s(X)) -> s(n__s(n__dbl(activate(X)))) , add(X1, X2) -> n__add(X1, X2) , add(0(), X) -> X , add(s(X), Y) -> s(n__add(activate(X), Y)) , first(X1, X2) -> n__first(X1, X2) , first(0(), X) -> nil() , first(s(X), cons(Y, Z)) -> cons(Y, n__first(activate(X), activate(Z))) } Weak DPs: { sqr^#(0()) -> c_4() , dbl^#(0()) -> c_15() , first^#(0(), X) -> c_21() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..