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