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(add(sqr(X), dbl(X))) , sqr(0()) -> 0() , add(s(X), Y) -> s(add(X, Y)) , add(0(), X) -> X , dbl(s(X)) -> s(s(dbl(X))) , dbl(0()) -> 0() , first(X1, X2) -> n__first(X1, X2) , first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z))) , first(0(), X) -> nil() , activate(X) -> X , activate(n__terms(X)) -> terms(X) , activate(n__first(X1, X2)) -> first(X1, X2) , half(s(s(X))) -> s(half(X)) , half(s(0())) -> 0() , half(0()) -> 0() , half(dbl(X)) -> 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) '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) '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) '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^#(s(X)) -> c_3(add^#(sqr(X), dbl(X))) , sqr^#(0()) -> c_4() , add^#(s(X), Y) -> c_5(add^#(X, Y)) , add^#(0(), X) -> c_6(X) , dbl^#(s(X)) -> c_7(dbl^#(X)) , dbl^#(0()) -> c_8() , first^#(X1, X2) -> c_9(X1, X2) , first^#(s(X), cons(Y, Z)) -> c_10(Y, X, activate^#(Z)) , first^#(0(), X) -> c_11() , activate^#(X) -> c_12(X) , activate^#(n__terms(X)) -> c_13(terms^#(X)) , activate^#(n__first(X1, X2)) -> c_14(first^#(X1, X2)) , half^#(s(s(X))) -> c_15(half^#(X)) , half^#(s(0())) -> c_16() , half^#(0()) -> c_17() , half^#(dbl(X)) -> c_18(X) } 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^#(s(X)) -> c_3(add^#(sqr(X), dbl(X))) , sqr^#(0()) -> c_4() , add^#(s(X), Y) -> c_5(add^#(X, Y)) , add^#(0(), X) -> c_6(X) , dbl^#(s(X)) -> c_7(dbl^#(X)) , dbl^#(0()) -> c_8() , first^#(X1, X2) -> c_9(X1, X2) , first^#(s(X), cons(Y, Z)) -> c_10(Y, X, activate^#(Z)) , first^#(0(), X) -> c_11() , activate^#(X) -> c_12(X) , activate^#(n__terms(X)) -> c_13(terms^#(X)) , activate^#(n__first(X1, X2)) -> c_14(first^#(X1, X2)) , half^#(s(s(X))) -> c_15(half^#(X)) , half^#(s(0())) -> c_16() , half^#(0()) -> c_17() , half^#(dbl(X)) -> c_18(X) } Strict Trs: { terms(N) -> cons(recip(sqr(N)), n__terms(s(N))) , terms(X) -> n__terms(X) , sqr(s(X)) -> s(add(sqr(X), dbl(X))) , sqr(0()) -> 0() , add(s(X), Y) -> s(add(X, Y)) , add(0(), X) -> X , dbl(s(X)) -> s(s(dbl(X))) , dbl(0()) -> 0() , first(X1, X2) -> n__first(X1, X2) , first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z))) , first(0(), X) -> nil() , activate(X) -> X , activate(n__terms(X)) -> terms(X) , activate(n__first(X1, X2)) -> first(X1, X2) , half(s(s(X))) -> s(half(X)) , half(s(0())) -> 0() , half(0()) -> 0() , half(dbl(X)) -> X } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {4,8,11,16,17} by applications of Pre({4,8,11,16,17}) = {1,2,6,7,9,10,12,14,15,18}. Here rules are labeled as follows: DPs: { 1: terms^#(N) -> c_1(sqr^#(N), N) , 2: terms^#(X) -> c_2(X) , 3: sqr^#(s(X)) -> c_3(add^#(sqr(X), dbl(X))) , 4: sqr^#(0()) -> c_4() , 5: add^#(s(X), Y) -> c_5(add^#(X, Y)) , 6: add^#(0(), X) -> c_6(X) , 7: dbl^#(s(X)) -> c_7(dbl^#(X)) , 8: dbl^#(0()) -> c_8() , 9: first^#(X1, X2) -> c_9(X1, X2) , 10: first^#(s(X), cons(Y, Z)) -> c_10(Y, X, activate^#(Z)) , 11: first^#(0(), X) -> c_11() , 12: activate^#(X) -> c_12(X) , 13: activate^#(n__terms(X)) -> c_13(terms^#(X)) , 14: activate^#(n__first(X1, X2)) -> c_14(first^#(X1, X2)) , 15: half^#(s(s(X))) -> c_15(half^#(X)) , 16: half^#(s(0())) -> c_16() , 17: half^#(0()) -> c_17() , 18: half^#(dbl(X)) -> c_18(X) } 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_3(add^#(sqr(X), dbl(X))) , add^#(s(X), Y) -> c_5(add^#(X, Y)) , add^#(0(), X) -> c_6(X) , dbl^#(s(X)) -> c_7(dbl^#(X)) , first^#(X1, X2) -> c_9(X1, X2) , first^#(s(X), cons(Y, Z)) -> c_10(Y, X, activate^#(Z)) , activate^#(X) -> c_12(X) , activate^#(n__terms(X)) -> c_13(terms^#(X)) , activate^#(n__first(X1, X2)) -> c_14(first^#(X1, X2)) , half^#(s(s(X))) -> c_15(half^#(X)) , half^#(dbl(X)) -> c_18(X) } Strict Trs: { terms(N) -> cons(recip(sqr(N)), n__terms(s(N))) , terms(X) -> n__terms(X) , sqr(s(X)) -> s(add(sqr(X), dbl(X))) , sqr(0()) -> 0() , add(s(X), Y) -> s(add(X, Y)) , add(0(), X) -> X , dbl(s(X)) -> s(s(dbl(X))) , dbl(0()) -> 0() , first(X1, X2) -> n__first(X1, X2) , first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z))) , first(0(), X) -> nil() , activate(X) -> X , activate(n__terms(X)) -> terms(X) , activate(n__first(X1, X2)) -> first(X1, X2) , half(s(s(X))) -> s(half(X)) , half(s(0())) -> 0() , half(0()) -> 0() , half(dbl(X)) -> X } Weak DPs: { sqr^#(0()) -> c_4() , dbl^#(0()) -> c_8() , first^#(0(), X) -> c_11() , half^#(s(0())) -> c_16() , half^#(0()) -> c_17() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..