MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { a__natsFrom(N) -> cons(mark(N), natsFrom(s(N))) , a__natsFrom(X) -> natsFrom(X) , mark(cons(X1, X2)) -> cons(mark(X1), X2) , mark(natsFrom(X)) -> a__natsFrom(mark(X)) , mark(s(X)) -> s(mark(X)) , mark(pair(X1, X2)) -> pair(mark(X1), mark(X2)) , mark(0()) -> 0() , mark(nil()) -> nil() , mark(fst(X)) -> a__fst(mark(X)) , mark(snd(X)) -> a__snd(mark(X)) , mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2)) , mark(u(X1, X2, X3, X4)) -> a__u(mark(X1), X2, X3, X4) , mark(head(X)) -> a__head(mark(X)) , mark(tail(X)) -> a__tail(mark(X)) , mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) , mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2)) , mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) , a__fst(X) -> fst(X) , a__fst(pair(XS, YS)) -> mark(XS) , a__snd(X) -> snd(X) , a__snd(pair(XS, YS)) -> mark(YS) , a__splitAt(X1, X2) -> splitAt(X1, X2) , a__splitAt(s(N), cons(X, XS)) -> a__u(a__splitAt(mark(N), mark(XS)), N, X, XS) , a__splitAt(0(), XS) -> pair(nil(), mark(XS)) , a__u(X1, X2, X3, X4) -> u(X1, X2, X3, X4) , a__u(pair(YS, ZS), N, X, XS) -> pair(cons(mark(X), YS), mark(ZS)) , a__head(X) -> head(X) , a__head(cons(N, XS)) -> mark(N) , a__tail(X) -> tail(X) , a__tail(cons(N, XS)) -> mark(XS) , a__sel(N, XS) -> a__head(a__afterNth(mark(N), mark(XS))) , a__sel(X1, X2) -> sel(X1, X2) , a__afterNth(N, XS) -> a__snd(a__splitAt(mark(N), mark(XS))) , a__afterNth(X1, X2) -> afterNth(X1, X2) , a__take(N, XS) -> a__fst(a__splitAt(mark(N), mark(XS))) , a__take(X1, X2) -> take(X1, X2) } 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__natsFrom^#(N) -> c_1(mark^#(N), N) , a__natsFrom^#(X) -> c_2(X) , mark^#(cons(X1, X2)) -> c_3(mark^#(X1), X2) , mark^#(natsFrom(X)) -> c_4(a__natsFrom^#(mark(X))) , mark^#(s(X)) -> c_5(mark^#(X)) , mark^#(pair(X1, X2)) -> c_6(mark^#(X1), mark^#(X2)) , mark^#(0()) -> c_7() , mark^#(nil()) -> c_8() , mark^#(fst(X)) -> c_9(a__fst^#(mark(X))) , mark^#(snd(X)) -> c_10(a__snd^#(mark(X))) , mark^#(splitAt(X1, X2)) -> c_11(a__splitAt^#(mark(X1), mark(X2))) , mark^#(u(X1, X2, X3, X4)) -> c_12(a__u^#(mark(X1), X2, X3, X4)) , mark^#(head(X)) -> c_13(a__head^#(mark(X))) , mark^#(tail(X)) -> c_14(a__tail^#(mark(X))) , mark^#(sel(X1, X2)) -> c_15(a__sel^#(mark(X1), mark(X2))) , mark^#(afterNth(X1, X2)) -> c_16(a__afterNth^#(mark(X1), mark(X2))) , mark^#(take(X1, X2)) -> c_17(a__take^#(mark(X1), mark(X2))) , a__fst^#(X) -> c_18(X) , a__fst^#(pair(XS, YS)) -> c_19(mark^#(XS)) , a__snd^#(X) -> c_20(X) , a__snd^#(pair(XS, YS)) -> c_21(mark^#(YS)) , a__splitAt^#(X1, X2) -> c_22(X1, X2) , a__splitAt^#(s(N), cons(X, XS)) -> c_23(a__u^#(a__splitAt(mark(N), mark(XS)), N, X, XS)) , a__splitAt^#(0(), XS) -> c_24(mark^#(XS)) , a__u^#(X1, X2, X3, X4) -> c_25(X1, X2, X3, X4) , a__u^#(pair(YS, ZS), N, X, XS) -> c_26(mark^#(X), YS, mark^#(ZS)) , a__head^#(X) -> c_27(X) , a__head^#(cons(N, XS)) -> c_28(mark^#(N)) , a__tail^#(X) -> c_29(X) , a__tail^#(cons(N, XS)) -> c_30(mark^#(XS)) , a__sel^#(N, XS) -> c_31(a__head^#(a__afterNth(mark(N), mark(XS)))) , a__sel^#(X1, X2) -> c_32(X1, X2) , a__afterNth^#(N, XS) -> c_33(a__snd^#(a__splitAt(mark(N), mark(XS)))) , a__afterNth^#(X1, X2) -> c_34(X1, X2) , a__take^#(N, XS) -> c_35(a__fst^#(a__splitAt(mark(N), mark(XS)))) , a__take^#(X1, X2) -> c_36(X1, X2) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { a__natsFrom^#(N) -> c_1(mark^#(N), N) , a__natsFrom^#(X) -> c_2(X) , mark^#(cons(X1, X2)) -> c_3(mark^#(X1), X2) , mark^#(natsFrom(X)) -> c_4(a__natsFrom^#(mark(X))) , mark^#(s(X)) -> c_5(mark^#(X)) , mark^#(pair(X1, X2)) -> c_6(mark^#(X1), mark^#(X2)) , mark^#(0()) -> c_7() , mark^#(nil()) -> c_8() , mark^#(fst(X)) -> c_9(a__fst^#(mark(X))) , mark^#(snd(X)) -> c_10(a__snd^#(mark(X))) , mark^#(splitAt(X1, X2)) -> c_11(a__splitAt^#(mark(X1), mark(X2))) , mark^#(u(X1, X2, X3, X4)) -> c_12(a__u^#(mark(X1), X2, X3, X4)) , mark^#(head(X)) -> c_13(a__head^#(mark(X))) , mark^#(tail(X)) -> c_14(a__tail^#(mark(X))) , mark^#(sel(X1, X2)) -> c_15(a__sel^#(mark(X1), mark(X2))) , mark^#(afterNth(X1, X2)) -> c_16(a__afterNth^#(mark(X1), mark(X2))) , mark^#(take(X1, X2)) -> c_17(a__take^#(mark(X1), mark(X2))) , a__fst^#(X) -> c_18(X) , a__fst^#(pair(XS, YS)) -> c_19(mark^#(XS)) , a__snd^#(X) -> c_20(X) , a__snd^#(pair(XS, YS)) -> c_21(mark^#(YS)) , a__splitAt^#(X1, X2) -> c_22(X1, X2) , a__splitAt^#(s(N), cons(X, XS)) -> c_23(a__u^#(a__splitAt(mark(N), mark(XS)), N, X, XS)) , a__splitAt^#(0(), XS) -> c_24(mark^#(XS)) , a__u^#(X1, X2, X3, X4) -> c_25(X1, X2, X3, X4) , a__u^#(pair(YS, ZS), N, X, XS) -> c_26(mark^#(X), YS, mark^#(ZS)) , a__head^#(X) -> c_27(X) , a__head^#(cons(N, XS)) -> c_28(mark^#(N)) , a__tail^#(X) -> c_29(X) , a__tail^#(cons(N, XS)) -> c_30(mark^#(XS)) , a__sel^#(N, XS) -> c_31(a__head^#(a__afterNth(mark(N), mark(XS)))) , a__sel^#(X1, X2) -> c_32(X1, X2) , a__afterNth^#(N, XS) -> c_33(a__snd^#(a__splitAt(mark(N), mark(XS)))) , a__afterNth^#(X1, X2) -> c_34(X1, X2) , a__take^#(N, XS) -> c_35(a__fst^#(a__splitAt(mark(N), mark(XS)))) , a__take^#(X1, X2) -> c_36(X1, X2) } Strict Trs: { a__natsFrom(N) -> cons(mark(N), natsFrom(s(N))) , a__natsFrom(X) -> natsFrom(X) , mark(cons(X1, X2)) -> cons(mark(X1), X2) , mark(natsFrom(X)) -> a__natsFrom(mark(X)) , mark(s(X)) -> s(mark(X)) , mark(pair(X1, X2)) -> pair(mark(X1), mark(X2)) , mark(0()) -> 0() , mark(nil()) -> nil() , mark(fst(X)) -> a__fst(mark(X)) , mark(snd(X)) -> a__snd(mark(X)) , mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2)) , mark(u(X1, X2, X3, X4)) -> a__u(mark(X1), X2, X3, X4) , mark(head(X)) -> a__head(mark(X)) , mark(tail(X)) -> a__tail(mark(X)) , mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) , mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2)) , mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) , a__fst(X) -> fst(X) , a__fst(pair(XS, YS)) -> mark(XS) , a__snd(X) -> snd(X) , a__snd(pair(XS, YS)) -> mark(YS) , a__splitAt(X1, X2) -> splitAt(X1, X2) , a__splitAt(s(N), cons(X, XS)) -> a__u(a__splitAt(mark(N), mark(XS)), N, X, XS) , a__splitAt(0(), XS) -> pair(nil(), mark(XS)) , a__u(X1, X2, X3, X4) -> u(X1, X2, X3, X4) , a__u(pair(YS, ZS), N, X, XS) -> pair(cons(mark(X), YS), mark(ZS)) , a__head(X) -> head(X) , a__head(cons(N, XS)) -> mark(N) , a__tail(X) -> tail(X) , a__tail(cons(N, XS)) -> mark(XS) , a__sel(N, XS) -> a__head(a__afterNth(mark(N), mark(XS))) , a__sel(X1, X2) -> sel(X1, X2) , a__afterNth(N, XS) -> a__snd(a__splitAt(mark(N), mark(XS))) , a__afterNth(X1, X2) -> afterNth(X1, X2) , a__take(N, XS) -> a__fst(a__splitAt(mark(N), mark(XS))) , a__take(X1, X2) -> take(X1, X2) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {7,8} by applications of Pre({7,8}) = {1,2,3,5,6,18,19,20,21,22,24,25,26,27,28,29,30,32,34,36}. Here rules are labeled as follows: DPs: { 1: a__natsFrom^#(N) -> c_1(mark^#(N), N) , 2: a__natsFrom^#(X) -> c_2(X) , 3: mark^#(cons(X1, X2)) -> c_3(mark^#(X1), X2) , 4: mark^#(natsFrom(X)) -> c_4(a__natsFrom^#(mark(X))) , 5: mark^#(s(X)) -> c_5(mark^#(X)) , 6: mark^#(pair(X1, X2)) -> c_6(mark^#(X1), mark^#(X2)) , 7: mark^#(0()) -> c_7() , 8: mark^#(nil()) -> c_8() , 9: mark^#(fst(X)) -> c_9(a__fst^#(mark(X))) , 10: mark^#(snd(X)) -> c_10(a__snd^#(mark(X))) , 11: mark^#(splitAt(X1, X2)) -> c_11(a__splitAt^#(mark(X1), mark(X2))) , 12: mark^#(u(X1, X2, X3, X4)) -> c_12(a__u^#(mark(X1), X2, X3, X4)) , 13: mark^#(head(X)) -> c_13(a__head^#(mark(X))) , 14: mark^#(tail(X)) -> c_14(a__tail^#(mark(X))) , 15: mark^#(sel(X1, X2)) -> c_15(a__sel^#(mark(X1), mark(X2))) , 16: mark^#(afterNth(X1, X2)) -> c_16(a__afterNth^#(mark(X1), mark(X2))) , 17: mark^#(take(X1, X2)) -> c_17(a__take^#(mark(X1), mark(X2))) , 18: a__fst^#(X) -> c_18(X) , 19: a__fst^#(pair(XS, YS)) -> c_19(mark^#(XS)) , 20: a__snd^#(X) -> c_20(X) , 21: a__snd^#(pair(XS, YS)) -> c_21(mark^#(YS)) , 22: a__splitAt^#(X1, X2) -> c_22(X1, X2) , 23: a__splitAt^#(s(N), cons(X, XS)) -> c_23(a__u^#(a__splitAt(mark(N), mark(XS)), N, X, XS)) , 24: a__splitAt^#(0(), XS) -> c_24(mark^#(XS)) , 25: a__u^#(X1, X2, X3, X4) -> c_25(X1, X2, X3, X4) , 26: a__u^#(pair(YS, ZS), N, X, XS) -> c_26(mark^#(X), YS, mark^#(ZS)) , 27: a__head^#(X) -> c_27(X) , 28: a__head^#(cons(N, XS)) -> c_28(mark^#(N)) , 29: a__tail^#(X) -> c_29(X) , 30: a__tail^#(cons(N, XS)) -> c_30(mark^#(XS)) , 31: a__sel^#(N, XS) -> c_31(a__head^#(a__afterNth(mark(N), mark(XS)))) , 32: a__sel^#(X1, X2) -> c_32(X1, X2) , 33: a__afterNth^#(N, XS) -> c_33(a__snd^#(a__splitAt(mark(N), mark(XS)))) , 34: a__afterNth^#(X1, X2) -> c_34(X1, X2) , 35: a__take^#(N, XS) -> c_35(a__fst^#(a__splitAt(mark(N), mark(XS)))) , 36: a__take^#(X1, X2) -> c_36(X1, X2) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { a__natsFrom^#(N) -> c_1(mark^#(N), N) , a__natsFrom^#(X) -> c_2(X) , mark^#(cons(X1, X2)) -> c_3(mark^#(X1), X2) , mark^#(natsFrom(X)) -> c_4(a__natsFrom^#(mark(X))) , mark^#(s(X)) -> c_5(mark^#(X)) , mark^#(pair(X1, X2)) -> c_6(mark^#(X1), mark^#(X2)) , mark^#(fst(X)) -> c_9(a__fst^#(mark(X))) , mark^#(snd(X)) -> c_10(a__snd^#(mark(X))) , mark^#(splitAt(X1, X2)) -> c_11(a__splitAt^#(mark(X1), mark(X2))) , mark^#(u(X1, X2, X3, X4)) -> c_12(a__u^#(mark(X1), X2, X3, X4)) , mark^#(head(X)) -> c_13(a__head^#(mark(X))) , mark^#(tail(X)) -> c_14(a__tail^#(mark(X))) , mark^#(sel(X1, X2)) -> c_15(a__sel^#(mark(X1), mark(X2))) , mark^#(afterNth(X1, X2)) -> c_16(a__afterNth^#(mark(X1), mark(X2))) , mark^#(take(X1, X2)) -> c_17(a__take^#(mark(X1), mark(X2))) , a__fst^#(X) -> c_18(X) , a__fst^#(pair(XS, YS)) -> c_19(mark^#(XS)) , a__snd^#(X) -> c_20(X) , a__snd^#(pair(XS, YS)) -> c_21(mark^#(YS)) , a__splitAt^#(X1, X2) -> c_22(X1, X2) , a__splitAt^#(s(N), cons(X, XS)) -> c_23(a__u^#(a__splitAt(mark(N), mark(XS)), N, X, XS)) , a__splitAt^#(0(), XS) -> c_24(mark^#(XS)) , a__u^#(X1, X2, X3, X4) -> c_25(X1, X2, X3, X4) , a__u^#(pair(YS, ZS), N, X, XS) -> c_26(mark^#(X), YS, mark^#(ZS)) , a__head^#(X) -> c_27(X) , a__head^#(cons(N, XS)) -> c_28(mark^#(N)) , a__tail^#(X) -> c_29(X) , a__tail^#(cons(N, XS)) -> c_30(mark^#(XS)) , a__sel^#(N, XS) -> c_31(a__head^#(a__afterNth(mark(N), mark(XS)))) , a__sel^#(X1, X2) -> c_32(X1, X2) , a__afterNth^#(N, XS) -> c_33(a__snd^#(a__splitAt(mark(N), mark(XS)))) , a__afterNth^#(X1, X2) -> c_34(X1, X2) , a__take^#(N, XS) -> c_35(a__fst^#(a__splitAt(mark(N), mark(XS)))) , a__take^#(X1, X2) -> c_36(X1, X2) } Strict Trs: { a__natsFrom(N) -> cons(mark(N), natsFrom(s(N))) , a__natsFrom(X) -> natsFrom(X) , mark(cons(X1, X2)) -> cons(mark(X1), X2) , mark(natsFrom(X)) -> a__natsFrom(mark(X)) , mark(s(X)) -> s(mark(X)) , mark(pair(X1, X2)) -> pair(mark(X1), mark(X2)) , mark(0()) -> 0() , mark(nil()) -> nil() , mark(fst(X)) -> a__fst(mark(X)) , mark(snd(X)) -> a__snd(mark(X)) , mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2)) , mark(u(X1, X2, X3, X4)) -> a__u(mark(X1), X2, X3, X4) , mark(head(X)) -> a__head(mark(X)) , mark(tail(X)) -> a__tail(mark(X)) , mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) , mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2)) , mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) , a__fst(X) -> fst(X) , a__fst(pair(XS, YS)) -> mark(XS) , a__snd(X) -> snd(X) , a__snd(pair(XS, YS)) -> mark(YS) , a__splitAt(X1, X2) -> splitAt(X1, X2) , a__splitAt(s(N), cons(X, XS)) -> a__u(a__splitAt(mark(N), mark(XS)), N, X, XS) , a__splitAt(0(), XS) -> pair(nil(), mark(XS)) , a__u(X1, X2, X3, X4) -> u(X1, X2, X3, X4) , a__u(pair(YS, ZS), N, X, XS) -> pair(cons(mark(X), YS), mark(ZS)) , a__head(X) -> head(X) , a__head(cons(N, XS)) -> mark(N) , a__tail(X) -> tail(X) , a__tail(cons(N, XS)) -> mark(XS) , a__sel(N, XS) -> a__head(a__afterNth(mark(N), mark(XS))) , a__sel(X1, X2) -> sel(X1, X2) , a__afterNth(N, XS) -> a__snd(a__splitAt(mark(N), mark(XS))) , a__afterNth(X1, X2) -> afterNth(X1, X2) , a__take(N, XS) -> a__fst(a__splitAt(mark(N), mark(XS))) , a__take(X1, X2) -> take(X1, X2) } Weak DPs: { mark^#(0()) -> c_7() , mark^#(nil()) -> c_8() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..