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(zWquot(X1, X2)) -> a__zWquot(mark(X1), mark(X2)) , mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) , mark(minus(X1, X2)) -> a__minus(mark(X1), mark(X2)) , mark(quot(X1, X2)) -> a__quot(mark(X1), mark(X2)) , 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) , a__minus(X1, X2) -> minus(X1, X2) , a__minus(X, 0()) -> 0() , a__minus(s(X), s(Y)) -> a__minus(mark(X), mark(Y)) , a__quot(X1, X2) -> quot(X1, X2) , a__quot(s(X), s(Y)) -> s(a__quot(a__minus(mark(X), mark(Y)), s(mark(Y)))) , a__quot(0(), s(Y)) -> 0() , a__zWquot(X1, X2) -> zWquot(X1, X2) , a__zWquot(XS, nil()) -> nil() , a__zWquot(cons(X, XS), cons(Y, YS)) -> cons(a__quot(mark(X), mark(Y)), zWquot(XS, YS)) , a__zWquot(nil(), XS) -> nil() } 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^#(zWquot(X1, X2)) -> c_8(a__zWquot^#(mark(X1), mark(X2))) , mark^#(sel(X1, X2)) -> c_9(a__sel^#(mark(X1), mark(X2))) , mark^#(minus(X1, X2)) -> c_10(a__minus^#(mark(X1), mark(X2))) , mark^#(quot(X1, X2)) -> c_11(a__quot^#(mark(X1), mark(X2))) , a__zWquot^#(X1, X2) -> c_21(X1, X2) , a__zWquot^#(XS, nil()) -> c_22() , a__zWquot^#(cons(X, XS), cons(Y, YS)) -> c_23(a__quot^#(mark(X), mark(Y)), XS, YS) , a__zWquot^#(nil(), XS) -> c_24() , a__sel^#(X1, X2) -> c_12(X1, X2) , a__sel^#(s(N), cons(X, XS)) -> c_13(a__sel^#(mark(N), mark(XS))) , a__sel^#(0(), cons(X, XS)) -> c_14(mark^#(X)) , a__minus^#(X1, X2) -> c_15(X1, X2) , a__minus^#(X, 0()) -> c_16() , a__minus^#(s(X), s(Y)) -> c_17(a__minus^#(mark(X), mark(Y))) , a__quot^#(X1, X2) -> c_18(X1, X2) , a__quot^#(s(X), s(Y)) -> c_19(a__quot^#(a__minus(mark(X), mark(Y)), s(mark(Y)))) , a__quot^#(0(), s(Y)) -> c_20() } 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^#(zWquot(X1, X2)) -> c_8(a__zWquot^#(mark(X1), mark(X2))) , mark^#(sel(X1, X2)) -> c_9(a__sel^#(mark(X1), mark(X2))) , mark^#(minus(X1, X2)) -> c_10(a__minus^#(mark(X1), mark(X2))) , mark^#(quot(X1, X2)) -> c_11(a__quot^#(mark(X1), mark(X2))) , a__zWquot^#(X1, X2) -> c_21(X1, X2) , a__zWquot^#(XS, nil()) -> c_22() , a__zWquot^#(cons(X, XS), cons(Y, YS)) -> c_23(a__quot^#(mark(X), mark(Y)), XS, YS) , a__zWquot^#(nil(), XS) -> c_24() , a__sel^#(X1, X2) -> c_12(X1, X2) , a__sel^#(s(N), cons(X, XS)) -> c_13(a__sel^#(mark(N), mark(XS))) , a__sel^#(0(), cons(X, XS)) -> c_14(mark^#(X)) , a__minus^#(X1, X2) -> c_15(X1, X2) , a__minus^#(X, 0()) -> c_16() , a__minus^#(s(X), s(Y)) -> c_17(a__minus^#(mark(X), mark(Y))) , a__quot^#(X1, X2) -> c_18(X1, X2) , a__quot^#(s(X), s(Y)) -> c_19(a__quot^#(a__minus(mark(X), mark(Y)), s(mark(Y)))) , a__quot^#(0(), s(Y)) -> c_20() } 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(zWquot(X1, X2)) -> a__zWquot(mark(X1), mark(X2)) , mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) , mark(minus(X1, X2)) -> a__minus(mark(X1), mark(X2)) , mark(quot(X1, X2)) -> a__quot(mark(X1), mark(X2)) , 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) , a__minus(X1, X2) -> minus(X1, X2) , a__minus(X, 0()) -> 0() , a__minus(s(X), s(Y)) -> a__minus(mark(X), mark(Y)) , a__quot(X1, X2) -> quot(X1, X2) , a__quot(s(X), s(Y)) -> s(a__quot(a__minus(mark(X), mark(Y)), s(mark(Y)))) , a__quot(0(), s(Y)) -> 0() , a__zWquot(X1, X2) -> zWquot(X1, X2) , a__zWquot(XS, nil()) -> nil() , a__zWquot(cons(X, XS), cons(Y, YS)) -> cons(a__quot(mark(X), mark(Y)), zWquot(XS, YS)) , a__zWquot(nil(), XS) -> nil() } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {6,7,13,15,20,24} by applications of Pre({6,7,13,15,20,24}) = {1,2,3,5,8,10,11,12,14,16,18,19,21,22,23}. 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^#(zWquot(X1, X2)) -> c_8(a__zWquot^#(mark(X1), mark(X2))) , 9: mark^#(sel(X1, X2)) -> c_9(a__sel^#(mark(X1), mark(X2))) , 10: mark^#(minus(X1, X2)) -> c_10(a__minus^#(mark(X1), mark(X2))) , 11: mark^#(quot(X1, X2)) -> c_11(a__quot^#(mark(X1), mark(X2))) , 12: a__zWquot^#(X1, X2) -> c_21(X1, X2) , 13: a__zWquot^#(XS, nil()) -> c_22() , 14: a__zWquot^#(cons(X, XS), cons(Y, YS)) -> c_23(a__quot^#(mark(X), mark(Y)), XS, YS) , 15: a__zWquot^#(nil(), XS) -> c_24() , 16: a__sel^#(X1, X2) -> c_12(X1, X2) , 17: a__sel^#(s(N), cons(X, XS)) -> c_13(a__sel^#(mark(N), mark(XS))) , 18: a__sel^#(0(), cons(X, XS)) -> c_14(mark^#(X)) , 19: a__minus^#(X1, X2) -> c_15(X1, X2) , 20: a__minus^#(X, 0()) -> c_16() , 21: a__minus^#(s(X), s(Y)) -> c_17(a__minus^#(mark(X), mark(Y))) , 22: a__quot^#(X1, X2) -> c_18(X1, X2) , 23: a__quot^#(s(X), s(Y)) -> c_19(a__quot^#(a__minus(mark(X), mark(Y)), s(mark(Y)))) , 24: a__quot^#(0(), s(Y)) -> c_20() } 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^#(zWquot(X1, X2)) -> c_8(a__zWquot^#(mark(X1), mark(X2))) , mark^#(sel(X1, X2)) -> c_9(a__sel^#(mark(X1), mark(X2))) , mark^#(minus(X1, X2)) -> c_10(a__minus^#(mark(X1), mark(X2))) , mark^#(quot(X1, X2)) -> c_11(a__quot^#(mark(X1), mark(X2))) , a__zWquot^#(X1, X2) -> c_21(X1, X2) , a__zWquot^#(cons(X, XS), cons(Y, YS)) -> c_23(a__quot^#(mark(X), mark(Y)), XS, YS) , a__sel^#(X1, X2) -> c_12(X1, X2) , a__sel^#(s(N), cons(X, XS)) -> c_13(a__sel^#(mark(N), mark(XS))) , a__sel^#(0(), cons(X, XS)) -> c_14(mark^#(X)) , a__minus^#(X1, X2) -> c_15(X1, X2) , a__minus^#(s(X), s(Y)) -> c_17(a__minus^#(mark(X), mark(Y))) , a__quot^#(X1, X2) -> c_18(X1, X2) , a__quot^#(s(X), s(Y)) -> c_19(a__quot^#(a__minus(mark(X), mark(Y)), s(mark(Y)))) } 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(zWquot(X1, X2)) -> a__zWquot(mark(X1), mark(X2)) , mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) , mark(minus(X1, X2)) -> a__minus(mark(X1), mark(X2)) , mark(quot(X1, X2)) -> a__quot(mark(X1), mark(X2)) , 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) , a__minus(X1, X2) -> minus(X1, X2) , a__minus(X, 0()) -> 0() , a__minus(s(X), s(Y)) -> a__minus(mark(X), mark(Y)) , a__quot(X1, X2) -> quot(X1, X2) , a__quot(s(X), s(Y)) -> s(a__quot(a__minus(mark(X), mark(Y)), s(mark(Y)))) , a__quot(0(), s(Y)) -> 0() , a__zWquot(X1, X2) -> zWquot(X1, X2) , a__zWquot(XS, nil()) -> nil() , a__zWquot(cons(X, XS), cons(Y, YS)) -> cons(a__quot(mark(X), mark(Y)), zWquot(XS, YS)) , a__zWquot(nil(), XS) -> nil() } Weak DPs: { mark^#(0()) -> c_6() , mark^#(nil()) -> c_7() , a__zWquot^#(XS, nil()) -> c_22() , a__zWquot^#(nil(), XS) -> c_24() , a__minus^#(X, 0()) -> c_16() , a__quot^#(0(), s(Y)) -> c_20() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..