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