MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { lt(x, 0()) -> false() , lt(0(), s(x)) -> true() , lt(s(x), s(y)) -> lt(x, y) , logarithm(x) -> ifa(lt(0(), x), x) , ifa(true(), x) -> help(x, 1()) , ifa(false(), x) -> logZeroError() , help(x, y) -> ifb(lt(y, x), x, y) , ifb(true(), x, y) -> help(half(x), s(y)) , ifb(false(), x, y) -> y , half(0()) -> 0() , half(s(0())) -> 0() , half(s(s(x))) -> s(half(x)) } Obligation: runtime complexity Answer: MAYBE None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) '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: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'WithProblem' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'Fastest' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'WithProblem' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'WithProblem' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'WithProblem' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'WithProblem' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'WithProblem' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'WithProblem' failed due to the following reason: Empty strict component of the problem is NOT empty. 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. 2) 'Innermost Weak Dependency Pairs (timeout of 60 seconds)' failed due to the following reason: We add the following weak dependency pairs: Strict DPs: { lt^#(x, 0()) -> c_1() , lt^#(0(), s(x)) -> c_2() , lt^#(s(x), s(y)) -> c_3(lt^#(x, y)) , logarithm^#(x) -> c_4(ifa^#(lt(0(), x), x)) , ifa^#(true(), x) -> c_5(help^#(x, 1())) , ifa^#(false(), x) -> c_6() , help^#(x, y) -> c_7(ifb^#(lt(y, x), x, y)) , ifb^#(true(), x, y) -> c_8(help^#(half(x), s(y))) , ifb^#(false(), x, y) -> c_9(y) , half^#(0()) -> c_10() , half^#(s(0())) -> c_11() , half^#(s(s(x))) -> c_12(half^#(x)) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { lt^#(x, 0()) -> c_1() , lt^#(0(), s(x)) -> c_2() , lt^#(s(x), s(y)) -> c_3(lt^#(x, y)) , logarithm^#(x) -> c_4(ifa^#(lt(0(), x), x)) , ifa^#(true(), x) -> c_5(help^#(x, 1())) , ifa^#(false(), x) -> c_6() , help^#(x, y) -> c_7(ifb^#(lt(y, x), x, y)) , ifb^#(true(), x, y) -> c_8(help^#(half(x), s(y))) , ifb^#(false(), x, y) -> c_9(y) , half^#(0()) -> c_10() , half^#(s(0())) -> c_11() , half^#(s(s(x))) -> c_12(half^#(x)) } Strict Trs: { lt(x, 0()) -> false() , lt(0(), s(x)) -> true() , lt(s(x), s(y)) -> lt(x, y) , logarithm(x) -> ifa(lt(0(), x), x) , ifa(true(), x) -> help(x, 1()) , ifa(false(), x) -> logZeroError() , help(x, y) -> ifb(lt(y, x), x, y) , ifb(true(), x, y) -> help(half(x), s(y)) , ifb(false(), x, y) -> y , half(0()) -> 0() , half(s(0())) -> 0() , half(s(s(x))) -> s(half(x)) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {1,2,6,10,11} by applications of Pre({1,2,6,10,11}) = {3,4,9,12}. Here rules are labeled as follows: DPs: { 1: lt^#(x, 0()) -> c_1() , 2: lt^#(0(), s(x)) -> c_2() , 3: lt^#(s(x), s(y)) -> c_3(lt^#(x, y)) , 4: logarithm^#(x) -> c_4(ifa^#(lt(0(), x), x)) , 5: ifa^#(true(), x) -> c_5(help^#(x, 1())) , 6: ifa^#(false(), x) -> c_6() , 7: help^#(x, y) -> c_7(ifb^#(lt(y, x), x, y)) , 8: ifb^#(true(), x, y) -> c_8(help^#(half(x), s(y))) , 9: ifb^#(false(), x, y) -> c_9(y) , 10: half^#(0()) -> c_10() , 11: half^#(s(0())) -> c_11() , 12: half^#(s(s(x))) -> c_12(half^#(x)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { lt^#(s(x), s(y)) -> c_3(lt^#(x, y)) , logarithm^#(x) -> c_4(ifa^#(lt(0(), x), x)) , ifa^#(true(), x) -> c_5(help^#(x, 1())) , help^#(x, y) -> c_7(ifb^#(lt(y, x), x, y)) , ifb^#(true(), x, y) -> c_8(help^#(half(x), s(y))) , ifb^#(false(), x, y) -> c_9(y) , half^#(s(s(x))) -> c_12(half^#(x)) } Strict Trs: { lt(x, 0()) -> false() , lt(0(), s(x)) -> true() , lt(s(x), s(y)) -> lt(x, y) , logarithm(x) -> ifa(lt(0(), x), x) , ifa(true(), x) -> help(x, 1()) , ifa(false(), x) -> logZeroError() , help(x, y) -> ifb(lt(y, x), x, y) , ifb(true(), x, y) -> help(half(x), s(y)) , ifb(false(), x, y) -> y , half(0()) -> 0() , half(s(0())) -> 0() , half(s(s(x))) -> s(half(x)) } Weak DPs: { lt^#(x, 0()) -> c_1() , lt^#(0(), s(x)) -> c_2() , ifa^#(false(), x) -> c_6() , half^#(0()) -> c_10() , half^#(s(0())) -> c_11() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..