MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { from(X) -> cons(X, n__from(n__s(X))) , from(X) -> n__from(X) , cons(X1, X2) -> n__cons(X1, X2) , 2ndspos(0(), Z) -> rnil() , 2ndspos(s(N), cons(X, n__cons(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z))) , s(X) -> n__s(X) , activate(X) -> X , activate(n__from(X)) -> from(activate(X)) , activate(n__s(X)) -> s(activate(X)) , activate(n__cons(X1, X2)) -> cons(activate(X1), X2) , 2ndsneg(0(), Z) -> rnil() , 2ndsneg(s(N), cons(X, n__cons(Y, Z))) -> rcons(negrecip(activate(Y)), 2ndspos(N, activate(Z))) , pi(X) -> 2ndspos(X, from(0())) , plus(0(), Y) -> Y , plus(s(X), Y) -> s(plus(X, Y)) , times(0(), Y) -> 0() , times(s(X), Y) -> plus(Y, times(X, Y)) , square(X) -> times(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: { from^#(X) -> c_1(cons^#(X, n__from(n__s(X)))) , from^#(X) -> c_2(X) , cons^#(X1, X2) -> c_3(X1, X2) , 2ndspos^#(0(), Z) -> c_4() , 2ndspos^#(s(N), cons(X, n__cons(Y, Z))) -> c_5(activate^#(Y), 2ndsneg^#(N, activate(Z))) , activate^#(X) -> c_7(X) , activate^#(n__from(X)) -> c_8(from^#(activate(X))) , activate^#(n__s(X)) -> c_9(s^#(activate(X))) , activate^#(n__cons(X1, X2)) -> c_10(cons^#(activate(X1), X2)) , 2ndsneg^#(0(), Z) -> c_11() , 2ndsneg^#(s(N), cons(X, n__cons(Y, Z))) -> c_12(activate^#(Y), 2ndspos^#(N, activate(Z))) , s^#(X) -> c_6(X) , pi^#(X) -> c_13(2ndspos^#(X, from(0()))) , plus^#(0(), Y) -> c_14(Y) , plus^#(s(X), Y) -> c_15(s^#(plus(X, Y))) , times^#(0(), Y) -> c_16() , times^#(s(X), Y) -> c_17(plus^#(Y, times(X, Y))) , square^#(X) -> c_18(times^#(X, X)) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { from^#(X) -> c_1(cons^#(X, n__from(n__s(X)))) , from^#(X) -> c_2(X) , cons^#(X1, X2) -> c_3(X1, X2) , 2ndspos^#(0(), Z) -> c_4() , 2ndspos^#(s(N), cons(X, n__cons(Y, Z))) -> c_5(activate^#(Y), 2ndsneg^#(N, activate(Z))) , activate^#(X) -> c_7(X) , activate^#(n__from(X)) -> c_8(from^#(activate(X))) , activate^#(n__s(X)) -> c_9(s^#(activate(X))) , activate^#(n__cons(X1, X2)) -> c_10(cons^#(activate(X1), X2)) , 2ndsneg^#(0(), Z) -> c_11() , 2ndsneg^#(s(N), cons(X, n__cons(Y, Z))) -> c_12(activate^#(Y), 2ndspos^#(N, activate(Z))) , s^#(X) -> c_6(X) , pi^#(X) -> c_13(2ndspos^#(X, from(0()))) , plus^#(0(), Y) -> c_14(Y) , plus^#(s(X), Y) -> c_15(s^#(plus(X, Y))) , times^#(0(), Y) -> c_16() , times^#(s(X), Y) -> c_17(plus^#(Y, times(X, Y))) , square^#(X) -> c_18(times^#(X, X)) } Strict Trs: { from(X) -> cons(X, n__from(n__s(X))) , from(X) -> n__from(X) , cons(X1, X2) -> n__cons(X1, X2) , 2ndspos(0(), Z) -> rnil() , 2ndspos(s(N), cons(X, n__cons(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z))) , s(X) -> n__s(X) , activate(X) -> X , activate(n__from(X)) -> from(activate(X)) , activate(n__s(X)) -> s(activate(X)) , activate(n__cons(X1, X2)) -> cons(activate(X1), X2) , 2ndsneg(0(), Z) -> rnil() , 2ndsneg(s(N), cons(X, n__cons(Y, Z))) -> rcons(negrecip(activate(Y)), 2ndspos(N, activate(Z))) , pi(X) -> 2ndspos(X, from(0())) , plus(0(), Y) -> Y , plus(s(X), Y) -> s(plus(X, Y)) , times(0(), Y) -> 0() , times(s(X), Y) -> plus(Y, times(X, Y)) , square(X) -> times(X, X) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {4,10,16} by applications of Pre({4,10,16}) = {2,3,5,6,11,12,13,14,18}. Here rules are labeled as follows: DPs: { 1: from^#(X) -> c_1(cons^#(X, n__from(n__s(X)))) , 2: from^#(X) -> c_2(X) , 3: cons^#(X1, X2) -> c_3(X1, X2) , 4: 2ndspos^#(0(), Z) -> c_4() , 5: 2ndspos^#(s(N), cons(X, n__cons(Y, Z))) -> c_5(activate^#(Y), 2ndsneg^#(N, activate(Z))) , 6: activate^#(X) -> c_7(X) , 7: activate^#(n__from(X)) -> c_8(from^#(activate(X))) , 8: activate^#(n__s(X)) -> c_9(s^#(activate(X))) , 9: activate^#(n__cons(X1, X2)) -> c_10(cons^#(activate(X1), X2)) , 10: 2ndsneg^#(0(), Z) -> c_11() , 11: 2ndsneg^#(s(N), cons(X, n__cons(Y, Z))) -> c_12(activate^#(Y), 2ndspos^#(N, activate(Z))) , 12: s^#(X) -> c_6(X) , 13: pi^#(X) -> c_13(2ndspos^#(X, from(0()))) , 14: plus^#(0(), Y) -> c_14(Y) , 15: plus^#(s(X), Y) -> c_15(s^#(plus(X, Y))) , 16: times^#(0(), Y) -> c_16() , 17: times^#(s(X), Y) -> c_17(plus^#(Y, times(X, Y))) , 18: square^#(X) -> c_18(times^#(X, X)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { from^#(X) -> c_1(cons^#(X, n__from(n__s(X)))) , from^#(X) -> c_2(X) , cons^#(X1, X2) -> c_3(X1, X2) , 2ndspos^#(s(N), cons(X, n__cons(Y, Z))) -> c_5(activate^#(Y), 2ndsneg^#(N, activate(Z))) , activate^#(X) -> c_7(X) , activate^#(n__from(X)) -> c_8(from^#(activate(X))) , activate^#(n__s(X)) -> c_9(s^#(activate(X))) , activate^#(n__cons(X1, X2)) -> c_10(cons^#(activate(X1), X2)) , 2ndsneg^#(s(N), cons(X, n__cons(Y, Z))) -> c_12(activate^#(Y), 2ndspos^#(N, activate(Z))) , s^#(X) -> c_6(X) , pi^#(X) -> c_13(2ndspos^#(X, from(0()))) , plus^#(0(), Y) -> c_14(Y) , plus^#(s(X), Y) -> c_15(s^#(plus(X, Y))) , times^#(s(X), Y) -> c_17(plus^#(Y, times(X, Y))) , square^#(X) -> c_18(times^#(X, X)) } Strict Trs: { from(X) -> cons(X, n__from(n__s(X))) , from(X) -> n__from(X) , cons(X1, X2) -> n__cons(X1, X2) , 2ndspos(0(), Z) -> rnil() , 2ndspos(s(N), cons(X, n__cons(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z))) , s(X) -> n__s(X) , activate(X) -> X , activate(n__from(X)) -> from(activate(X)) , activate(n__s(X)) -> s(activate(X)) , activate(n__cons(X1, X2)) -> cons(activate(X1), X2) , 2ndsneg(0(), Z) -> rnil() , 2ndsneg(s(N), cons(X, n__cons(Y, Z))) -> rcons(negrecip(activate(Y)), 2ndspos(N, activate(Z))) , pi(X) -> 2ndspos(X, from(0())) , plus(0(), Y) -> Y , plus(s(X), Y) -> s(plus(X, Y)) , times(0(), Y) -> 0() , times(s(X), Y) -> plus(Y, times(X, Y)) , square(X) -> times(X, X) } Weak DPs: { 2ndspos^#(0(), Z) -> c_4() , 2ndsneg^#(0(), Z) -> c_11() , times^#(0(), Y) -> c_16() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..