MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { a__fib(N) -> a__sel(mark(N), a__fib1(s(0()), s(0()))) , a__fib(X) -> fib(X) , a__sel(X1, X2) -> sel(X1, X2) , a__sel(s(N), cons(X, XS)) -> a__sel(mark(N), mark(XS)) , a__sel(0(), cons(X, XS)) -> mark(X) , mark(s(X)) -> s(mark(X)) , mark(0()) -> 0() , mark(cons(X1, X2)) -> cons(mark(X1), X2) , mark(fib1(X1, X2)) -> a__fib1(mark(X1), mark(X2)) , mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) , mark(fib(X)) -> a__fib(mark(X)) , mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) , a__fib1(X, Y) -> cons(mark(X), fib1(Y, add(X, Y))) , a__fib1(X1, X2) -> fib1(X1, X2) , a__add(X1, X2) -> add(X1, X2) , a__add(s(X), Y) -> s(a__add(mark(X), mark(Y))) , a__add(0(), X) -> mark(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: { a__fib^#(N) -> c_1(a__sel^#(mark(N), a__fib1(s(0()), s(0())))) , a__fib^#(X) -> c_2(X) , a__sel^#(X1, X2) -> c_3(X1, X2) , a__sel^#(s(N), cons(X, XS)) -> c_4(a__sel^#(mark(N), mark(XS))) , a__sel^#(0(), cons(X, XS)) -> c_5(mark^#(X)) , mark^#(s(X)) -> c_6(mark^#(X)) , mark^#(0()) -> c_7() , mark^#(cons(X1, X2)) -> c_8(mark^#(X1), X2) , mark^#(fib1(X1, X2)) -> c_9(a__fib1^#(mark(X1), mark(X2))) , mark^#(add(X1, X2)) -> c_10(a__add^#(mark(X1), mark(X2))) , mark^#(fib(X)) -> c_11(a__fib^#(mark(X))) , mark^#(sel(X1, X2)) -> c_12(a__sel^#(mark(X1), mark(X2))) , a__fib1^#(X, Y) -> c_13(mark^#(X), Y, X, Y) , a__fib1^#(X1, X2) -> c_14(X1, X2) , a__add^#(X1, X2) -> c_15(X1, X2) , a__add^#(s(X), Y) -> c_16(a__add^#(mark(X), mark(Y))) , a__add^#(0(), X) -> c_17(mark^#(X)) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { a__fib^#(N) -> c_1(a__sel^#(mark(N), a__fib1(s(0()), s(0())))) , a__fib^#(X) -> c_2(X) , a__sel^#(X1, X2) -> c_3(X1, X2) , a__sel^#(s(N), cons(X, XS)) -> c_4(a__sel^#(mark(N), mark(XS))) , a__sel^#(0(), cons(X, XS)) -> c_5(mark^#(X)) , mark^#(s(X)) -> c_6(mark^#(X)) , mark^#(0()) -> c_7() , mark^#(cons(X1, X2)) -> c_8(mark^#(X1), X2) , mark^#(fib1(X1, X2)) -> c_9(a__fib1^#(mark(X1), mark(X2))) , mark^#(add(X1, X2)) -> c_10(a__add^#(mark(X1), mark(X2))) , mark^#(fib(X)) -> c_11(a__fib^#(mark(X))) , mark^#(sel(X1, X2)) -> c_12(a__sel^#(mark(X1), mark(X2))) , a__fib1^#(X, Y) -> c_13(mark^#(X), Y, X, Y) , a__fib1^#(X1, X2) -> c_14(X1, X2) , a__add^#(X1, X2) -> c_15(X1, X2) , a__add^#(s(X), Y) -> c_16(a__add^#(mark(X), mark(Y))) , a__add^#(0(), X) -> c_17(mark^#(X)) } Strict Trs: { a__fib(N) -> a__sel(mark(N), a__fib1(s(0()), s(0()))) , a__fib(X) -> fib(X) , a__sel(X1, X2) -> sel(X1, X2) , a__sel(s(N), cons(X, XS)) -> a__sel(mark(N), mark(XS)) , a__sel(0(), cons(X, XS)) -> mark(X) , mark(s(X)) -> s(mark(X)) , mark(0()) -> 0() , mark(cons(X1, X2)) -> cons(mark(X1), X2) , mark(fib1(X1, X2)) -> a__fib1(mark(X1), mark(X2)) , mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) , mark(fib(X)) -> a__fib(mark(X)) , mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) , a__fib1(X, Y) -> cons(mark(X), fib1(Y, add(X, Y))) , a__fib1(X1, X2) -> fib1(X1, X2) , a__add(X1, X2) -> add(X1, X2) , a__add(s(X), Y) -> s(a__add(mark(X), mark(Y))) , a__add(0(), X) -> mark(X) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {7} by applications of Pre({7}) = {2,3,5,6,8,13,14,15,17}. Here rules are labeled as follows: DPs: { 1: a__fib^#(N) -> c_1(a__sel^#(mark(N), a__fib1(s(0()), s(0())))) , 2: a__fib^#(X) -> c_2(X) , 3: a__sel^#(X1, X2) -> c_3(X1, X2) , 4: a__sel^#(s(N), cons(X, XS)) -> c_4(a__sel^#(mark(N), mark(XS))) , 5: a__sel^#(0(), cons(X, XS)) -> c_5(mark^#(X)) , 6: mark^#(s(X)) -> c_6(mark^#(X)) , 7: mark^#(0()) -> c_7() , 8: mark^#(cons(X1, X2)) -> c_8(mark^#(X1), X2) , 9: mark^#(fib1(X1, X2)) -> c_9(a__fib1^#(mark(X1), mark(X2))) , 10: mark^#(add(X1, X2)) -> c_10(a__add^#(mark(X1), mark(X2))) , 11: mark^#(fib(X)) -> c_11(a__fib^#(mark(X))) , 12: mark^#(sel(X1, X2)) -> c_12(a__sel^#(mark(X1), mark(X2))) , 13: a__fib1^#(X, Y) -> c_13(mark^#(X), Y, X, Y) , 14: a__fib1^#(X1, X2) -> c_14(X1, X2) , 15: a__add^#(X1, X2) -> c_15(X1, X2) , 16: a__add^#(s(X), Y) -> c_16(a__add^#(mark(X), mark(Y))) , 17: a__add^#(0(), X) -> c_17(mark^#(X)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { a__fib^#(N) -> c_1(a__sel^#(mark(N), a__fib1(s(0()), s(0())))) , a__fib^#(X) -> c_2(X) , a__sel^#(X1, X2) -> c_3(X1, X2) , a__sel^#(s(N), cons(X, XS)) -> c_4(a__sel^#(mark(N), mark(XS))) , a__sel^#(0(), cons(X, XS)) -> c_5(mark^#(X)) , mark^#(s(X)) -> c_6(mark^#(X)) , mark^#(cons(X1, X2)) -> c_8(mark^#(X1), X2) , mark^#(fib1(X1, X2)) -> c_9(a__fib1^#(mark(X1), mark(X2))) , mark^#(add(X1, X2)) -> c_10(a__add^#(mark(X1), mark(X2))) , mark^#(fib(X)) -> c_11(a__fib^#(mark(X))) , mark^#(sel(X1, X2)) -> c_12(a__sel^#(mark(X1), mark(X2))) , a__fib1^#(X, Y) -> c_13(mark^#(X), Y, X, Y) , a__fib1^#(X1, X2) -> c_14(X1, X2) , a__add^#(X1, X2) -> c_15(X1, X2) , a__add^#(s(X), Y) -> c_16(a__add^#(mark(X), mark(Y))) , a__add^#(0(), X) -> c_17(mark^#(X)) } Strict Trs: { a__fib(N) -> a__sel(mark(N), a__fib1(s(0()), s(0()))) , a__fib(X) -> fib(X) , a__sel(X1, X2) -> sel(X1, X2) , a__sel(s(N), cons(X, XS)) -> a__sel(mark(N), mark(XS)) , a__sel(0(), cons(X, XS)) -> mark(X) , mark(s(X)) -> s(mark(X)) , mark(0()) -> 0() , mark(cons(X1, X2)) -> cons(mark(X1), X2) , mark(fib1(X1, X2)) -> a__fib1(mark(X1), mark(X2)) , mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) , mark(fib(X)) -> a__fib(mark(X)) , mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) , a__fib1(X, Y) -> cons(mark(X), fib1(Y, add(X, Y))) , a__fib1(X1, X2) -> fib1(X1, X2) , a__add(X1, X2) -> add(X1, X2) , a__add(s(X), Y) -> s(a__add(mark(X), mark(Y))) , a__add(0(), X) -> mark(X) } Weak DPs: { mark^#(0()) -> c_7() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..