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) , fibo(0()) -> fib(0()) , fibo(s(0())) -> fib(s(0())) , fibo(s(s(x))) -> sum(fibo(s(x)), fibo(x)) , fib(0()) -> s(0()) , fib(s(0())) -> s(0()) , fib(s(s(x))) -> if(true(), 0(), s(s(x)), 0(), 0()) , sum(x, 0()) -> x , sum(x, s(y)) -> s(sum(x, y)) , if(true(), c, s(s(x)), a, b) -> if(lt(s(c), s(s(x))), s(c), s(s(x)), b, c) , if(false(), c, s(s(x)), a, b) -> sum(fibo(a), fibo(b)) } 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: { lt^#(x, 0()) -> c_1() , lt^#(0(), s(x)) -> c_2() , lt^#(s(x), s(y)) -> c_3(lt^#(x, y)) , fibo^#(0()) -> c_4(fib^#(0())) , fibo^#(s(0())) -> c_5(fib^#(s(0()))) , fibo^#(s(s(x))) -> c_6(sum^#(fibo(s(x)), fibo(x))) , fib^#(0()) -> c_7() , fib^#(s(0())) -> c_8() , fib^#(s(s(x))) -> c_9(if^#(true(), 0(), s(s(x)), 0(), 0())) , sum^#(x, 0()) -> c_10(x) , sum^#(x, s(y)) -> c_11(sum^#(x, y)) , if^#(true(), c, s(s(x)), a, b) -> c_12(if^#(lt(s(c), s(s(x))), s(c), s(s(x)), b, c)) , if^#(false(), c, s(s(x)), a, b) -> c_13(sum^#(fibo(a), fibo(b))) } 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)) , fibo^#(0()) -> c_4(fib^#(0())) , fibo^#(s(0())) -> c_5(fib^#(s(0()))) , fibo^#(s(s(x))) -> c_6(sum^#(fibo(s(x)), fibo(x))) , fib^#(0()) -> c_7() , fib^#(s(0())) -> c_8() , fib^#(s(s(x))) -> c_9(if^#(true(), 0(), s(s(x)), 0(), 0())) , sum^#(x, 0()) -> c_10(x) , sum^#(x, s(y)) -> c_11(sum^#(x, y)) , if^#(true(), c, s(s(x)), a, b) -> c_12(if^#(lt(s(c), s(s(x))), s(c), s(s(x)), b, c)) , if^#(false(), c, s(s(x)), a, b) -> c_13(sum^#(fibo(a), fibo(b))) } Strict Trs: { lt(x, 0()) -> false() , lt(0(), s(x)) -> true() , lt(s(x), s(y)) -> lt(x, y) , fibo(0()) -> fib(0()) , fibo(s(0())) -> fib(s(0())) , fibo(s(s(x))) -> sum(fibo(s(x)), fibo(x)) , fib(0()) -> s(0()) , fib(s(0())) -> s(0()) , fib(s(s(x))) -> if(true(), 0(), s(s(x)), 0(), 0()) , sum(x, 0()) -> x , sum(x, s(y)) -> s(sum(x, y)) , if(true(), c, s(s(x)), a, b) -> if(lt(s(c), s(s(x))), s(c), s(s(x)), b, c) , if(false(), c, s(s(x)), a, b) -> sum(fibo(a), fibo(b)) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {1,2,7,8} by applications of Pre({1,2,7,8}) = {3,4,5,10}. 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: fibo^#(0()) -> c_4(fib^#(0())) , 5: fibo^#(s(0())) -> c_5(fib^#(s(0()))) , 6: fibo^#(s(s(x))) -> c_6(sum^#(fibo(s(x)), fibo(x))) , 7: fib^#(0()) -> c_7() , 8: fib^#(s(0())) -> c_8() , 9: fib^#(s(s(x))) -> c_9(if^#(true(), 0(), s(s(x)), 0(), 0())) , 10: sum^#(x, 0()) -> c_10(x) , 11: sum^#(x, s(y)) -> c_11(sum^#(x, y)) , 12: if^#(true(), c, s(s(x)), a, b) -> c_12(if^#(lt(s(c), s(s(x))), s(c), s(s(x)), b, c)) , 13: if^#(false(), c, s(s(x)), a, b) -> c_13(sum^#(fibo(a), fibo(b))) } 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)) , fibo^#(0()) -> c_4(fib^#(0())) , fibo^#(s(0())) -> c_5(fib^#(s(0()))) , fibo^#(s(s(x))) -> c_6(sum^#(fibo(s(x)), fibo(x))) , fib^#(s(s(x))) -> c_9(if^#(true(), 0(), s(s(x)), 0(), 0())) , sum^#(x, 0()) -> c_10(x) , sum^#(x, s(y)) -> c_11(sum^#(x, y)) , if^#(true(), c, s(s(x)), a, b) -> c_12(if^#(lt(s(c), s(s(x))), s(c), s(s(x)), b, c)) , if^#(false(), c, s(s(x)), a, b) -> c_13(sum^#(fibo(a), fibo(b))) } Strict Trs: { lt(x, 0()) -> false() , lt(0(), s(x)) -> true() , lt(s(x), s(y)) -> lt(x, y) , fibo(0()) -> fib(0()) , fibo(s(0())) -> fib(s(0())) , fibo(s(s(x))) -> sum(fibo(s(x)), fibo(x)) , fib(0()) -> s(0()) , fib(s(0())) -> s(0()) , fib(s(s(x))) -> if(true(), 0(), s(s(x)), 0(), 0()) , sum(x, 0()) -> x , sum(x, s(y)) -> s(sum(x, y)) , if(true(), c, s(s(x)), a, b) -> if(lt(s(c), s(s(x))), s(c), s(s(x)), b, c) , if(false(), c, s(s(x)), a, b) -> sum(fibo(a), fibo(b)) } Weak DPs: { lt^#(x, 0()) -> c_1() , lt^#(0(), s(x)) -> c_2() , fib^#(0()) -> c_7() , fib^#(s(0())) -> c_8() } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {2,3} by applications of Pre({2,3}) = {6}. Here rules are labeled as follows: DPs: { 1: lt^#(s(x), s(y)) -> c_3(lt^#(x, y)) , 2: fibo^#(0()) -> c_4(fib^#(0())) , 3: fibo^#(s(0())) -> c_5(fib^#(s(0()))) , 4: fibo^#(s(s(x))) -> c_6(sum^#(fibo(s(x)), fibo(x))) , 5: fib^#(s(s(x))) -> c_9(if^#(true(), 0(), s(s(x)), 0(), 0())) , 6: sum^#(x, 0()) -> c_10(x) , 7: sum^#(x, s(y)) -> c_11(sum^#(x, y)) , 8: if^#(true(), c, s(s(x)), a, b) -> c_12(if^#(lt(s(c), s(s(x))), s(c), s(s(x)), b, c)) , 9: if^#(false(), c, s(s(x)), a, b) -> c_13(sum^#(fibo(a), fibo(b))) , 10: lt^#(x, 0()) -> c_1() , 11: lt^#(0(), s(x)) -> c_2() , 12: fib^#(0()) -> c_7() , 13: fib^#(s(0())) -> c_8() } 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)) , fibo^#(s(s(x))) -> c_6(sum^#(fibo(s(x)), fibo(x))) , fib^#(s(s(x))) -> c_9(if^#(true(), 0(), s(s(x)), 0(), 0())) , sum^#(x, 0()) -> c_10(x) , sum^#(x, s(y)) -> c_11(sum^#(x, y)) , if^#(true(), c, s(s(x)), a, b) -> c_12(if^#(lt(s(c), s(s(x))), s(c), s(s(x)), b, c)) , if^#(false(), c, s(s(x)), a, b) -> c_13(sum^#(fibo(a), fibo(b))) } Strict Trs: { lt(x, 0()) -> false() , lt(0(), s(x)) -> true() , lt(s(x), s(y)) -> lt(x, y) , fibo(0()) -> fib(0()) , fibo(s(0())) -> fib(s(0())) , fibo(s(s(x))) -> sum(fibo(s(x)), fibo(x)) , fib(0()) -> s(0()) , fib(s(0())) -> s(0()) , fib(s(s(x))) -> if(true(), 0(), s(s(x)), 0(), 0()) , sum(x, 0()) -> x , sum(x, s(y)) -> s(sum(x, y)) , if(true(), c, s(s(x)), a, b) -> if(lt(s(c), s(s(x))), s(c), s(s(x)), b, c) , if(false(), c, s(s(x)), a, b) -> sum(fibo(a), fibo(b)) } Weak DPs: { lt^#(x, 0()) -> c_1() , lt^#(0(), s(x)) -> c_2() , fibo^#(0()) -> c_4(fib^#(0())) , fibo^#(s(0())) -> c_5(fib^#(s(0()))) , fib^#(0()) -> c_7() , fib^#(s(0())) -> c_8() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..