MAYBE

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict Trs:
  { a__U11(X1, X2, X3) -> U11(X1, X2, X3)
  , a__U11(tt(), N, XS) -> a__U12(tt(), N, XS)
  , a__U12(X1, X2, X3) -> U12(X1, X2, X3)
  , a__U12(tt(), N, XS) -> a__snd(a__splitAt(mark(N), mark(XS)))
  , a__snd(X) -> snd(X)
  , a__snd(pair(X, Y)) -> a__U51(tt(), Y)
  , a__splitAt(X1, X2) -> splitAt(X1, X2)
  , a__splitAt(s(N), cons(X, XS)) -> a__U61(tt(), N, X, XS)
  , a__splitAt(0(), XS) -> pair(nil(), mark(XS))
  , mark(tt()) -> tt()
  , mark(pair(X1, X2)) -> pair(mark(X1), mark(X2))
  , mark(cons(X1, X2)) -> cons(mark(X1), X2)
  , mark(natsFrom(X)) -> a__natsFrom(mark(X))
  , mark(s(X)) -> s(mark(X))
  , mark(0()) -> 0()
  , mark(nil()) -> nil()
  , mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
  , mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3)
  , mark(snd(X)) -> a__snd(mark(X))
  , mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2))
  , mark(U21(X1, X2)) -> a__U21(mark(X1), X2)
  , mark(U22(X1, X2)) -> a__U22(mark(X1), X2)
  , mark(U31(X1, X2)) -> a__U31(mark(X1), X2)
  , mark(U32(X1, X2)) -> a__U32(mark(X1), X2)
  , mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3)
  , mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3)
  , mark(head(X)) -> a__head(mark(X))
  , mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2))
  , mark(U51(X1, X2)) -> a__U51(mark(X1), X2)
  , mark(U52(X1, X2)) -> a__U52(mark(X1), X2)
  , mark(U61(X1, X2, X3, X4)) -> a__U61(mark(X1), X2, X3, X4)
  , mark(U62(X1, X2, X3, X4)) -> a__U62(mark(X1), X2, X3, X4)
  , mark(U63(X1, X2, X3, X4)) -> a__U63(mark(X1), X2, X3, X4)
  , mark(U64(X1, X2)) -> a__U64(mark(X1), X2)
  , mark(U71(X1, X2)) -> a__U71(mark(X1), X2)
  , mark(U72(X1, X2)) -> a__U72(mark(X1), X2)
  , mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3)
  , mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3)
  , mark(fst(X)) -> a__fst(mark(X))
  , mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2))
  , mark(tail(X)) -> a__tail(mark(X))
  , mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
  , a__U21(X1, X2) -> U21(X1, X2)
  , a__U21(tt(), X) -> a__U22(tt(), X)
  , a__U22(X1, X2) -> U22(X1, X2)
  , a__U22(tt(), X) -> mark(X)
  , a__U31(X1, X2) -> U31(X1, X2)
  , a__U31(tt(), N) -> a__U32(tt(), N)
  , a__U32(X1, X2) -> U32(X1, X2)
  , a__U32(tt(), N) -> mark(N)
  , a__U41(X1, X2, X3) -> U41(X1, X2, X3)
  , a__U41(tt(), N, XS) -> a__U42(tt(), N, XS)
  , a__U42(X1, X2, X3) -> U42(X1, X2, X3)
  , a__U42(tt(), N, XS) -> a__head(a__afterNth(mark(N), mark(XS)))
  , a__head(X) -> head(X)
  , a__head(cons(N, XS)) -> a__U31(tt(), N)
  , a__afterNth(N, XS) -> a__U11(tt(), N, XS)
  , a__afterNth(X1, X2) -> afterNth(X1, X2)
  , a__U51(X1, X2) -> U51(X1, X2)
  , a__U51(tt(), Y) -> a__U52(tt(), Y)
  , a__U52(X1, X2) -> U52(X1, X2)
  , a__U52(tt(), Y) -> mark(Y)
  , a__U61(X1, X2, X3, X4) -> U61(X1, X2, X3, X4)
  , a__U61(tt(), N, X, XS) -> a__U62(tt(), N, X, XS)
  , a__U62(X1, X2, X3, X4) -> U62(X1, X2, X3, X4)
  , a__U62(tt(), N, X, XS) -> a__U63(tt(), N, X, XS)
  , a__U63(X1, X2, X3, X4) -> U63(X1, X2, X3, X4)
  , a__U63(tt(), N, X, XS) ->
    a__U64(a__splitAt(mark(N), mark(XS)), X)
  , a__U64(X1, X2) -> U64(X1, X2)
  , a__U64(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS))
  , a__U71(X1, X2) -> U71(X1, X2)
  , a__U71(tt(), XS) -> a__U72(tt(), XS)
  , a__U72(X1, X2) -> U72(X1, X2)
  , a__U72(tt(), XS) -> mark(XS)
  , a__U81(X1, X2, X3) -> U81(X1, X2, X3)
  , a__U81(tt(), N, XS) -> a__U82(tt(), N, XS)
  , a__U82(X1, X2, X3) -> U82(X1, X2, X3)
  , a__U82(tt(), N, XS) -> a__fst(a__splitAt(mark(N), mark(XS)))
  , a__fst(X) -> fst(X)
  , a__fst(pair(X, Y)) -> a__U21(tt(), X)
  , a__natsFrom(N) -> cons(mark(N), natsFrom(s(N)))
  , a__natsFrom(X) -> natsFrom(X)
  , a__sel(N, XS) -> a__U41(tt(), N, XS)
  , a__sel(X1, X2) -> sel(X1, X2)
  , a__tail(X) -> tail(X)
  , a__tail(cons(N, XS)) -> a__U71(tt(), XS)
  , a__take(N, XS) -> a__U81(tt(), N, XS)
  , a__take(X1, X2) -> take(X1, X2) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { a__U11^#(X1, X2, X3) -> c_1()
  , a__U11^#(tt(), N, XS) -> c_2(a__U12^#(tt(), N, XS))
  , a__U12^#(X1, X2, X3) -> c_3()
  , a__U12^#(tt(), N, XS) ->
    c_4(a__snd^#(a__splitAt(mark(N), mark(XS))),
        a__splitAt^#(mark(N), mark(XS)),
        mark^#(N),
        mark^#(XS))
  , a__snd^#(X) -> c_5()
  , a__snd^#(pair(X, Y)) -> c_6(a__U51^#(tt(), Y))
  , a__splitAt^#(X1, X2) -> c_7()
  , a__splitAt^#(s(N), cons(X, XS)) -> c_8(a__U61^#(tt(), N, X, XS))
  , a__splitAt^#(0(), XS) -> c_9(mark^#(XS))
  , mark^#(tt()) -> c_10()
  , mark^#(pair(X1, X2)) -> c_11(mark^#(X1), mark^#(X2))
  , mark^#(cons(X1, X2)) -> c_12(mark^#(X1))
  , mark^#(natsFrom(X)) -> c_13(a__natsFrom^#(mark(X)), mark^#(X))
  , mark^#(s(X)) -> c_14(mark^#(X))
  , mark^#(0()) -> c_15()
  , mark^#(nil()) -> c_16()
  , mark^#(U11(X1, X2, X3)) ->
    c_17(a__U11^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U12(X1, X2, X3)) ->
    c_18(a__U12^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(snd(X)) -> c_19(a__snd^#(mark(X)), mark^#(X))
  , mark^#(splitAt(X1, X2)) ->
    c_20(a__splitAt^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(U21(X1, X2)) -> c_21(a__U21^#(mark(X1), X2), mark^#(X1))
  , mark^#(U22(X1, X2)) -> c_22(a__U22^#(mark(X1), X2), mark^#(X1))
  , mark^#(U31(X1, X2)) -> c_23(a__U31^#(mark(X1), X2), mark^#(X1))
  , mark^#(U32(X1, X2)) -> c_24(a__U32^#(mark(X1), X2), mark^#(X1))
  , mark^#(U41(X1, X2, X3)) ->
    c_25(a__U41^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U42(X1, X2, X3)) ->
    c_26(a__U42^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(head(X)) -> c_27(a__head^#(mark(X)), mark^#(X))
  , mark^#(afterNth(X1, X2)) ->
    c_28(a__afterNth^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(U51(X1, X2)) -> c_29(a__U51^#(mark(X1), X2), mark^#(X1))
  , mark^#(U52(X1, X2)) -> c_30(a__U52^#(mark(X1), X2), mark^#(X1))
  , mark^#(U61(X1, X2, X3, X4)) ->
    c_31(a__U61^#(mark(X1), X2, X3, X4), mark^#(X1))
  , mark^#(U62(X1, X2, X3, X4)) ->
    c_32(a__U62^#(mark(X1), X2, X3, X4), mark^#(X1))
  , mark^#(U63(X1, X2, X3, X4)) ->
    c_33(a__U63^#(mark(X1), X2, X3, X4), mark^#(X1))
  , mark^#(U64(X1, X2)) -> c_34(a__U64^#(mark(X1), X2), mark^#(X1))
  , mark^#(U71(X1, X2)) -> c_35(a__U71^#(mark(X1), X2), mark^#(X1))
  , mark^#(U72(X1, X2)) -> c_36(a__U72^#(mark(X1), X2), mark^#(X1))
  , mark^#(U81(X1, X2, X3)) ->
    c_37(a__U81^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U82(X1, X2, X3)) ->
    c_38(a__U82^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(fst(X)) -> c_39(a__fst^#(mark(X)), mark^#(X))
  , mark^#(sel(X1, X2)) ->
    c_40(a__sel^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(tail(X)) -> c_41(a__tail^#(mark(X)), mark^#(X))
  , mark^#(take(X1, X2)) ->
    c_42(a__take^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , a__U51^#(X1, X2) -> c_59()
  , a__U51^#(tt(), Y) -> c_60(a__U52^#(tt(), Y))
  , a__U61^#(X1, X2, X3, X4) -> c_63()
  , a__U61^#(tt(), N, X, XS) -> c_64(a__U62^#(tt(), N, X, XS))
  , a__natsFrom^#(N) -> c_81(mark^#(N))
  , a__natsFrom^#(X) -> c_82()
  , a__U21^#(X1, X2) -> c_43()
  , a__U21^#(tt(), X) -> c_44(a__U22^#(tt(), X))
  , a__U22^#(X1, X2) -> c_45()
  , a__U22^#(tt(), X) -> c_46(mark^#(X))
  , a__U31^#(X1, X2) -> c_47()
  , a__U31^#(tt(), N) -> c_48(a__U32^#(tt(), N))
  , a__U32^#(X1, X2) -> c_49()
  , a__U32^#(tt(), N) -> c_50(mark^#(N))
  , a__U41^#(X1, X2, X3) -> c_51()
  , a__U41^#(tt(), N, XS) -> c_52(a__U42^#(tt(), N, XS))
  , a__U42^#(X1, X2, X3) -> c_53()
  , a__U42^#(tt(), N, XS) ->
    c_54(a__head^#(a__afterNth(mark(N), mark(XS))),
         a__afterNth^#(mark(N), mark(XS)),
         mark^#(N),
         mark^#(XS))
  , a__head^#(X) -> c_55()
  , a__head^#(cons(N, XS)) -> c_56(a__U31^#(tt(), N))
  , a__afterNth^#(N, XS) -> c_57(a__U11^#(tt(), N, XS))
  , a__afterNth^#(X1, X2) -> c_58()
  , a__U52^#(X1, X2) -> c_61()
  , a__U52^#(tt(), Y) -> c_62(mark^#(Y))
  , a__U62^#(X1, X2, X3, X4) -> c_65()
  , a__U62^#(tt(), N, X, XS) -> c_66(a__U63^#(tt(), N, X, XS))
  , a__U63^#(X1, X2, X3, X4) -> c_67()
  , a__U63^#(tt(), N, X, XS) ->
    c_68(a__U64^#(a__splitAt(mark(N), mark(XS)), X),
         a__splitAt^#(mark(N), mark(XS)),
         mark^#(N),
         mark^#(XS))
  , a__U64^#(X1, X2) -> c_69()
  , a__U64^#(pair(YS, ZS), X) -> c_70(mark^#(X), mark^#(ZS))
  , a__U71^#(X1, X2) -> c_71()
  , a__U71^#(tt(), XS) -> c_72(a__U72^#(tt(), XS))
  , a__U72^#(X1, X2) -> c_73()
  , a__U72^#(tt(), XS) -> c_74(mark^#(XS))
  , a__U81^#(X1, X2, X3) -> c_75()
  , a__U81^#(tt(), N, XS) -> c_76(a__U82^#(tt(), N, XS))
  , a__U82^#(X1, X2, X3) -> c_77()
  , a__U82^#(tt(), N, XS) ->
    c_78(a__fst^#(a__splitAt(mark(N), mark(XS))),
         a__splitAt^#(mark(N), mark(XS)),
         mark^#(N),
         mark^#(XS))
  , a__fst^#(X) -> c_79()
  , a__fst^#(pair(X, Y)) -> c_80(a__U21^#(tt(), X))
  , a__sel^#(N, XS) -> c_83(a__U41^#(tt(), N, XS))
  , a__sel^#(X1, X2) -> c_84()
  , a__tail^#(X) -> c_85()
  , a__tail^#(cons(N, XS)) -> c_86(a__U71^#(tt(), XS))
  , a__take^#(N, XS) -> c_87(a__U81^#(tt(), N, XS))
  , a__take^#(X1, X2) -> c_88() }

and mark the set of starting terms.

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { a__U11^#(X1, X2, X3) -> c_1()
  , a__U11^#(tt(), N, XS) -> c_2(a__U12^#(tt(), N, XS))
  , a__U12^#(X1, X2, X3) -> c_3()
  , a__U12^#(tt(), N, XS) ->
    c_4(a__snd^#(a__splitAt(mark(N), mark(XS))),
        a__splitAt^#(mark(N), mark(XS)),
        mark^#(N),
        mark^#(XS))
  , a__snd^#(X) -> c_5()
  , a__snd^#(pair(X, Y)) -> c_6(a__U51^#(tt(), Y))
  , a__splitAt^#(X1, X2) -> c_7()
  , a__splitAt^#(s(N), cons(X, XS)) -> c_8(a__U61^#(tt(), N, X, XS))
  , a__splitAt^#(0(), XS) -> c_9(mark^#(XS))
  , mark^#(tt()) -> c_10()
  , mark^#(pair(X1, X2)) -> c_11(mark^#(X1), mark^#(X2))
  , mark^#(cons(X1, X2)) -> c_12(mark^#(X1))
  , mark^#(natsFrom(X)) -> c_13(a__natsFrom^#(mark(X)), mark^#(X))
  , mark^#(s(X)) -> c_14(mark^#(X))
  , mark^#(0()) -> c_15()
  , mark^#(nil()) -> c_16()
  , mark^#(U11(X1, X2, X3)) ->
    c_17(a__U11^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U12(X1, X2, X3)) ->
    c_18(a__U12^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(snd(X)) -> c_19(a__snd^#(mark(X)), mark^#(X))
  , mark^#(splitAt(X1, X2)) ->
    c_20(a__splitAt^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(U21(X1, X2)) -> c_21(a__U21^#(mark(X1), X2), mark^#(X1))
  , mark^#(U22(X1, X2)) -> c_22(a__U22^#(mark(X1), X2), mark^#(X1))
  , mark^#(U31(X1, X2)) -> c_23(a__U31^#(mark(X1), X2), mark^#(X1))
  , mark^#(U32(X1, X2)) -> c_24(a__U32^#(mark(X1), X2), mark^#(X1))
  , mark^#(U41(X1, X2, X3)) ->
    c_25(a__U41^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U42(X1, X2, X3)) ->
    c_26(a__U42^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(head(X)) -> c_27(a__head^#(mark(X)), mark^#(X))
  , mark^#(afterNth(X1, X2)) ->
    c_28(a__afterNth^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(U51(X1, X2)) -> c_29(a__U51^#(mark(X1), X2), mark^#(X1))
  , mark^#(U52(X1, X2)) -> c_30(a__U52^#(mark(X1), X2), mark^#(X1))
  , mark^#(U61(X1, X2, X3, X4)) ->
    c_31(a__U61^#(mark(X1), X2, X3, X4), mark^#(X1))
  , mark^#(U62(X1, X2, X3, X4)) ->
    c_32(a__U62^#(mark(X1), X2, X3, X4), mark^#(X1))
  , mark^#(U63(X1, X2, X3, X4)) ->
    c_33(a__U63^#(mark(X1), X2, X3, X4), mark^#(X1))
  , mark^#(U64(X1, X2)) -> c_34(a__U64^#(mark(X1), X2), mark^#(X1))
  , mark^#(U71(X1, X2)) -> c_35(a__U71^#(mark(X1), X2), mark^#(X1))
  , mark^#(U72(X1, X2)) -> c_36(a__U72^#(mark(X1), X2), mark^#(X1))
  , mark^#(U81(X1, X2, X3)) ->
    c_37(a__U81^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U82(X1, X2, X3)) ->
    c_38(a__U82^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(fst(X)) -> c_39(a__fst^#(mark(X)), mark^#(X))
  , mark^#(sel(X1, X2)) ->
    c_40(a__sel^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(tail(X)) -> c_41(a__tail^#(mark(X)), mark^#(X))
  , mark^#(take(X1, X2)) ->
    c_42(a__take^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , a__U51^#(X1, X2) -> c_59()
  , a__U51^#(tt(), Y) -> c_60(a__U52^#(tt(), Y))
  , a__U61^#(X1, X2, X3, X4) -> c_63()
  , a__U61^#(tt(), N, X, XS) -> c_64(a__U62^#(tt(), N, X, XS))
  , a__natsFrom^#(N) -> c_81(mark^#(N))
  , a__natsFrom^#(X) -> c_82()
  , a__U21^#(X1, X2) -> c_43()
  , a__U21^#(tt(), X) -> c_44(a__U22^#(tt(), X))
  , a__U22^#(X1, X2) -> c_45()
  , a__U22^#(tt(), X) -> c_46(mark^#(X))
  , a__U31^#(X1, X2) -> c_47()
  , a__U31^#(tt(), N) -> c_48(a__U32^#(tt(), N))
  , a__U32^#(X1, X2) -> c_49()
  , a__U32^#(tt(), N) -> c_50(mark^#(N))
  , a__U41^#(X1, X2, X3) -> c_51()
  , a__U41^#(tt(), N, XS) -> c_52(a__U42^#(tt(), N, XS))
  , a__U42^#(X1, X2, X3) -> c_53()
  , a__U42^#(tt(), N, XS) ->
    c_54(a__head^#(a__afterNth(mark(N), mark(XS))),
         a__afterNth^#(mark(N), mark(XS)),
         mark^#(N),
         mark^#(XS))
  , a__head^#(X) -> c_55()
  , a__head^#(cons(N, XS)) -> c_56(a__U31^#(tt(), N))
  , a__afterNth^#(N, XS) -> c_57(a__U11^#(tt(), N, XS))
  , a__afterNth^#(X1, X2) -> c_58()
  , a__U52^#(X1, X2) -> c_61()
  , a__U52^#(tt(), Y) -> c_62(mark^#(Y))
  , a__U62^#(X1, X2, X3, X4) -> c_65()
  , a__U62^#(tt(), N, X, XS) -> c_66(a__U63^#(tt(), N, X, XS))
  , a__U63^#(X1, X2, X3, X4) -> c_67()
  , a__U63^#(tt(), N, X, XS) ->
    c_68(a__U64^#(a__splitAt(mark(N), mark(XS)), X),
         a__splitAt^#(mark(N), mark(XS)),
         mark^#(N),
         mark^#(XS))
  , a__U64^#(X1, X2) -> c_69()
  , a__U64^#(pair(YS, ZS), X) -> c_70(mark^#(X), mark^#(ZS))
  , a__U71^#(X1, X2) -> c_71()
  , a__U71^#(tt(), XS) -> c_72(a__U72^#(tt(), XS))
  , a__U72^#(X1, X2) -> c_73()
  , a__U72^#(tt(), XS) -> c_74(mark^#(XS))
  , a__U81^#(X1, X2, X3) -> c_75()
  , a__U81^#(tt(), N, XS) -> c_76(a__U82^#(tt(), N, XS))
  , a__U82^#(X1, X2, X3) -> c_77()
  , a__U82^#(tt(), N, XS) ->
    c_78(a__fst^#(a__splitAt(mark(N), mark(XS))),
         a__splitAt^#(mark(N), mark(XS)),
         mark^#(N),
         mark^#(XS))
  , a__fst^#(X) -> c_79()
  , a__fst^#(pair(X, Y)) -> c_80(a__U21^#(tt(), X))
  , a__sel^#(N, XS) -> c_83(a__U41^#(tt(), N, XS))
  , a__sel^#(X1, X2) -> c_84()
  , a__tail^#(X) -> c_85()
  , a__tail^#(cons(N, XS)) -> c_86(a__U71^#(tt(), XS))
  , a__take^#(N, XS) -> c_87(a__U81^#(tt(), N, XS))
  , a__take^#(X1, X2) -> c_88() }
Weak Trs:
  { a__U11(X1, X2, X3) -> U11(X1, X2, X3)
  , a__U11(tt(), N, XS) -> a__U12(tt(), N, XS)
  , a__U12(X1, X2, X3) -> U12(X1, X2, X3)
  , a__U12(tt(), N, XS) -> a__snd(a__splitAt(mark(N), mark(XS)))
  , a__snd(X) -> snd(X)
  , a__snd(pair(X, Y)) -> a__U51(tt(), Y)
  , a__splitAt(X1, X2) -> splitAt(X1, X2)
  , a__splitAt(s(N), cons(X, XS)) -> a__U61(tt(), N, X, XS)
  , a__splitAt(0(), XS) -> pair(nil(), mark(XS))
  , mark(tt()) -> tt()
  , mark(pair(X1, X2)) -> pair(mark(X1), mark(X2))
  , mark(cons(X1, X2)) -> cons(mark(X1), X2)
  , mark(natsFrom(X)) -> a__natsFrom(mark(X))
  , mark(s(X)) -> s(mark(X))
  , mark(0()) -> 0()
  , mark(nil()) -> nil()
  , mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
  , mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3)
  , mark(snd(X)) -> a__snd(mark(X))
  , mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2))
  , mark(U21(X1, X2)) -> a__U21(mark(X1), X2)
  , mark(U22(X1, X2)) -> a__U22(mark(X1), X2)
  , mark(U31(X1, X2)) -> a__U31(mark(X1), X2)
  , mark(U32(X1, X2)) -> a__U32(mark(X1), X2)
  , mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3)
  , mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3)
  , mark(head(X)) -> a__head(mark(X))
  , mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2))
  , mark(U51(X1, X2)) -> a__U51(mark(X1), X2)
  , mark(U52(X1, X2)) -> a__U52(mark(X1), X2)
  , mark(U61(X1, X2, X3, X4)) -> a__U61(mark(X1), X2, X3, X4)
  , mark(U62(X1, X2, X3, X4)) -> a__U62(mark(X1), X2, X3, X4)
  , mark(U63(X1, X2, X3, X4)) -> a__U63(mark(X1), X2, X3, X4)
  , mark(U64(X1, X2)) -> a__U64(mark(X1), X2)
  , mark(U71(X1, X2)) -> a__U71(mark(X1), X2)
  , mark(U72(X1, X2)) -> a__U72(mark(X1), X2)
  , mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3)
  , mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3)
  , mark(fst(X)) -> a__fst(mark(X))
  , mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2))
  , mark(tail(X)) -> a__tail(mark(X))
  , mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
  , a__U21(X1, X2) -> U21(X1, X2)
  , a__U21(tt(), X) -> a__U22(tt(), X)
  , a__U22(X1, X2) -> U22(X1, X2)
  , a__U22(tt(), X) -> mark(X)
  , a__U31(X1, X2) -> U31(X1, X2)
  , a__U31(tt(), N) -> a__U32(tt(), N)
  , a__U32(X1, X2) -> U32(X1, X2)
  , a__U32(tt(), N) -> mark(N)
  , a__U41(X1, X2, X3) -> U41(X1, X2, X3)
  , a__U41(tt(), N, XS) -> a__U42(tt(), N, XS)
  , a__U42(X1, X2, X3) -> U42(X1, X2, X3)
  , a__U42(tt(), N, XS) -> a__head(a__afterNth(mark(N), mark(XS)))
  , a__head(X) -> head(X)
  , a__head(cons(N, XS)) -> a__U31(tt(), N)
  , a__afterNth(N, XS) -> a__U11(tt(), N, XS)
  , a__afterNth(X1, X2) -> afterNth(X1, X2)
  , a__U51(X1, X2) -> U51(X1, X2)
  , a__U51(tt(), Y) -> a__U52(tt(), Y)
  , a__U52(X1, X2) -> U52(X1, X2)
  , a__U52(tt(), Y) -> mark(Y)
  , a__U61(X1, X2, X3, X4) -> U61(X1, X2, X3, X4)
  , a__U61(tt(), N, X, XS) -> a__U62(tt(), N, X, XS)
  , a__U62(X1, X2, X3, X4) -> U62(X1, X2, X3, X4)
  , a__U62(tt(), N, X, XS) -> a__U63(tt(), N, X, XS)
  , a__U63(X1, X2, X3, X4) -> U63(X1, X2, X3, X4)
  , a__U63(tt(), N, X, XS) ->
    a__U64(a__splitAt(mark(N), mark(XS)), X)
  , a__U64(X1, X2) -> U64(X1, X2)
  , a__U64(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS))
  , a__U71(X1, X2) -> U71(X1, X2)
  , a__U71(tt(), XS) -> a__U72(tt(), XS)
  , a__U72(X1, X2) -> U72(X1, X2)
  , a__U72(tt(), XS) -> mark(XS)
  , a__U81(X1, X2, X3) -> U81(X1, X2, X3)
  , a__U81(tt(), N, XS) -> a__U82(tt(), N, XS)
  , a__U82(X1, X2, X3) -> U82(X1, X2, X3)
  , a__U82(tt(), N, XS) -> a__fst(a__splitAt(mark(N), mark(XS)))
  , a__fst(X) -> fst(X)
  , a__fst(pair(X, Y)) -> a__U21(tt(), X)
  , a__natsFrom(N) -> cons(mark(N), natsFrom(s(N)))
  , a__natsFrom(X) -> natsFrom(X)
  , a__sel(N, XS) -> a__U41(tt(), N, XS)
  , a__sel(X1, X2) -> sel(X1, X2)
  , a__tail(X) -> tail(X)
  , a__tail(cons(N, XS)) -> a__U71(tt(), XS)
  , a__take(N, XS) -> a__U81(tt(), N, XS)
  , a__take(X1, X2) -> take(X1, X2) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of
{1,3,5,7,10,15,16,43,45,48,49,51,53,55,57,59,61,64,65,67,69,71,73,75,77,79,81,84,85,88}
by applications of
Pre({1,3,5,7,10,15,16,43,45,48,49,51,53,55,57,59,61,64,65,67,69,71,73,75,77,79,81,84,85,88})
=
{2,4,6,8,9,11,12,13,14,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,44,46,47,50,52,54,56,58,60,62,63,66,68,70,72,74,76,78,80,82,83,86,87}.
Here rules are labeled as follows:

  DPs:
    { 1: a__U11^#(X1, X2, X3) -> c_1()
    , 2: a__U11^#(tt(), N, XS) -> c_2(a__U12^#(tt(), N, XS))
    , 3: a__U12^#(X1, X2, X3) -> c_3()
    , 4: a__U12^#(tt(), N, XS) ->
         c_4(a__snd^#(a__splitAt(mark(N), mark(XS))),
             a__splitAt^#(mark(N), mark(XS)),
             mark^#(N),
             mark^#(XS))
    , 5: a__snd^#(X) -> c_5()
    , 6: a__snd^#(pair(X, Y)) -> c_6(a__U51^#(tt(), Y))
    , 7: a__splitAt^#(X1, X2) -> c_7()
    , 8: a__splitAt^#(s(N), cons(X, XS)) ->
         c_8(a__U61^#(tt(), N, X, XS))
    , 9: a__splitAt^#(0(), XS) -> c_9(mark^#(XS))
    , 10: mark^#(tt()) -> c_10()
    , 11: mark^#(pair(X1, X2)) -> c_11(mark^#(X1), mark^#(X2))
    , 12: mark^#(cons(X1, X2)) -> c_12(mark^#(X1))
    , 13: mark^#(natsFrom(X)) ->
          c_13(a__natsFrom^#(mark(X)), mark^#(X))
    , 14: mark^#(s(X)) -> c_14(mark^#(X))
    , 15: mark^#(0()) -> c_15()
    , 16: mark^#(nil()) -> c_16()
    , 17: mark^#(U11(X1, X2, X3)) ->
          c_17(a__U11^#(mark(X1), X2, X3), mark^#(X1))
    , 18: mark^#(U12(X1, X2, X3)) ->
          c_18(a__U12^#(mark(X1), X2, X3), mark^#(X1))
    , 19: mark^#(snd(X)) -> c_19(a__snd^#(mark(X)), mark^#(X))
    , 20: mark^#(splitAt(X1, X2)) ->
          c_20(a__splitAt^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
    , 21: mark^#(U21(X1, X2)) ->
          c_21(a__U21^#(mark(X1), X2), mark^#(X1))
    , 22: mark^#(U22(X1, X2)) ->
          c_22(a__U22^#(mark(X1), X2), mark^#(X1))
    , 23: mark^#(U31(X1, X2)) ->
          c_23(a__U31^#(mark(X1), X2), mark^#(X1))
    , 24: mark^#(U32(X1, X2)) ->
          c_24(a__U32^#(mark(X1), X2), mark^#(X1))
    , 25: mark^#(U41(X1, X2, X3)) ->
          c_25(a__U41^#(mark(X1), X2, X3), mark^#(X1))
    , 26: mark^#(U42(X1, X2, X3)) ->
          c_26(a__U42^#(mark(X1), X2, X3), mark^#(X1))
    , 27: mark^#(head(X)) -> c_27(a__head^#(mark(X)), mark^#(X))
    , 28: mark^#(afterNth(X1, X2)) ->
          c_28(a__afterNth^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
    , 29: mark^#(U51(X1, X2)) ->
          c_29(a__U51^#(mark(X1), X2), mark^#(X1))
    , 30: mark^#(U52(X1, X2)) ->
          c_30(a__U52^#(mark(X1), X2), mark^#(X1))
    , 31: mark^#(U61(X1, X2, X3, X4)) ->
          c_31(a__U61^#(mark(X1), X2, X3, X4), mark^#(X1))
    , 32: mark^#(U62(X1, X2, X3, X4)) ->
          c_32(a__U62^#(mark(X1), X2, X3, X4), mark^#(X1))
    , 33: mark^#(U63(X1, X2, X3, X4)) ->
          c_33(a__U63^#(mark(X1), X2, X3, X4), mark^#(X1))
    , 34: mark^#(U64(X1, X2)) ->
          c_34(a__U64^#(mark(X1), X2), mark^#(X1))
    , 35: mark^#(U71(X1, X2)) ->
          c_35(a__U71^#(mark(X1), X2), mark^#(X1))
    , 36: mark^#(U72(X1, X2)) ->
          c_36(a__U72^#(mark(X1), X2), mark^#(X1))
    , 37: mark^#(U81(X1, X2, X3)) ->
          c_37(a__U81^#(mark(X1), X2, X3), mark^#(X1))
    , 38: mark^#(U82(X1, X2, X3)) ->
          c_38(a__U82^#(mark(X1), X2, X3), mark^#(X1))
    , 39: mark^#(fst(X)) -> c_39(a__fst^#(mark(X)), mark^#(X))
    , 40: mark^#(sel(X1, X2)) ->
          c_40(a__sel^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
    , 41: mark^#(tail(X)) -> c_41(a__tail^#(mark(X)), mark^#(X))
    , 42: mark^#(take(X1, X2)) ->
          c_42(a__take^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
    , 43: a__U51^#(X1, X2) -> c_59()
    , 44: a__U51^#(tt(), Y) -> c_60(a__U52^#(tt(), Y))
    , 45: a__U61^#(X1, X2, X3, X4) -> c_63()
    , 46: a__U61^#(tt(), N, X, XS) -> c_64(a__U62^#(tt(), N, X, XS))
    , 47: a__natsFrom^#(N) -> c_81(mark^#(N))
    , 48: a__natsFrom^#(X) -> c_82()
    , 49: a__U21^#(X1, X2) -> c_43()
    , 50: a__U21^#(tt(), X) -> c_44(a__U22^#(tt(), X))
    , 51: a__U22^#(X1, X2) -> c_45()
    , 52: a__U22^#(tt(), X) -> c_46(mark^#(X))
    , 53: a__U31^#(X1, X2) -> c_47()
    , 54: a__U31^#(tt(), N) -> c_48(a__U32^#(tt(), N))
    , 55: a__U32^#(X1, X2) -> c_49()
    , 56: a__U32^#(tt(), N) -> c_50(mark^#(N))
    , 57: a__U41^#(X1, X2, X3) -> c_51()
    , 58: a__U41^#(tt(), N, XS) -> c_52(a__U42^#(tt(), N, XS))
    , 59: a__U42^#(X1, X2, X3) -> c_53()
    , 60: a__U42^#(tt(), N, XS) ->
          c_54(a__head^#(a__afterNth(mark(N), mark(XS))),
               a__afterNth^#(mark(N), mark(XS)),
               mark^#(N),
               mark^#(XS))
    , 61: a__head^#(X) -> c_55()
    , 62: a__head^#(cons(N, XS)) -> c_56(a__U31^#(tt(), N))
    , 63: a__afterNth^#(N, XS) -> c_57(a__U11^#(tt(), N, XS))
    , 64: a__afterNth^#(X1, X2) -> c_58()
    , 65: a__U52^#(X1, X2) -> c_61()
    , 66: a__U52^#(tt(), Y) -> c_62(mark^#(Y))
    , 67: a__U62^#(X1, X2, X3, X4) -> c_65()
    , 68: a__U62^#(tt(), N, X, XS) -> c_66(a__U63^#(tt(), N, X, XS))
    , 69: a__U63^#(X1, X2, X3, X4) -> c_67()
    , 70: a__U63^#(tt(), N, X, XS) ->
          c_68(a__U64^#(a__splitAt(mark(N), mark(XS)), X),
               a__splitAt^#(mark(N), mark(XS)),
               mark^#(N),
               mark^#(XS))
    , 71: a__U64^#(X1, X2) -> c_69()
    , 72: a__U64^#(pair(YS, ZS), X) -> c_70(mark^#(X), mark^#(ZS))
    , 73: a__U71^#(X1, X2) -> c_71()
    , 74: a__U71^#(tt(), XS) -> c_72(a__U72^#(tt(), XS))
    , 75: a__U72^#(X1, X2) -> c_73()
    , 76: a__U72^#(tt(), XS) -> c_74(mark^#(XS))
    , 77: a__U81^#(X1, X2, X3) -> c_75()
    , 78: a__U81^#(tt(), N, XS) -> c_76(a__U82^#(tt(), N, XS))
    , 79: a__U82^#(X1, X2, X3) -> c_77()
    , 80: a__U82^#(tt(), N, XS) ->
          c_78(a__fst^#(a__splitAt(mark(N), mark(XS))),
               a__splitAt^#(mark(N), mark(XS)),
               mark^#(N),
               mark^#(XS))
    , 81: a__fst^#(X) -> c_79()
    , 82: a__fst^#(pair(X, Y)) -> c_80(a__U21^#(tt(), X))
    , 83: a__sel^#(N, XS) -> c_83(a__U41^#(tt(), N, XS))
    , 84: a__sel^#(X1, X2) -> c_84()
    , 85: a__tail^#(X) -> c_85()
    , 86: a__tail^#(cons(N, XS)) -> c_86(a__U71^#(tt(), XS))
    , 87: a__take^#(N, XS) -> c_87(a__U81^#(tt(), N, XS))
    , 88: a__take^#(X1, X2) -> c_88() }

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { a__U11^#(tt(), N, XS) -> c_2(a__U12^#(tt(), N, XS))
  , a__U12^#(tt(), N, XS) ->
    c_4(a__snd^#(a__splitAt(mark(N), mark(XS))),
        a__splitAt^#(mark(N), mark(XS)),
        mark^#(N),
        mark^#(XS))
  , a__snd^#(pair(X, Y)) -> c_6(a__U51^#(tt(), Y))
  , a__splitAt^#(s(N), cons(X, XS)) -> c_8(a__U61^#(tt(), N, X, XS))
  , a__splitAt^#(0(), XS) -> c_9(mark^#(XS))
  , mark^#(pair(X1, X2)) -> c_11(mark^#(X1), mark^#(X2))
  , mark^#(cons(X1, X2)) -> c_12(mark^#(X1))
  , mark^#(natsFrom(X)) -> c_13(a__natsFrom^#(mark(X)), mark^#(X))
  , mark^#(s(X)) -> c_14(mark^#(X))
  , mark^#(U11(X1, X2, X3)) ->
    c_17(a__U11^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U12(X1, X2, X3)) ->
    c_18(a__U12^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(snd(X)) -> c_19(a__snd^#(mark(X)), mark^#(X))
  , mark^#(splitAt(X1, X2)) ->
    c_20(a__splitAt^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(U21(X1, X2)) -> c_21(a__U21^#(mark(X1), X2), mark^#(X1))
  , mark^#(U22(X1, X2)) -> c_22(a__U22^#(mark(X1), X2), mark^#(X1))
  , mark^#(U31(X1, X2)) -> c_23(a__U31^#(mark(X1), X2), mark^#(X1))
  , mark^#(U32(X1, X2)) -> c_24(a__U32^#(mark(X1), X2), mark^#(X1))
  , mark^#(U41(X1, X2, X3)) ->
    c_25(a__U41^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U42(X1, X2, X3)) ->
    c_26(a__U42^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(head(X)) -> c_27(a__head^#(mark(X)), mark^#(X))
  , mark^#(afterNth(X1, X2)) ->
    c_28(a__afterNth^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(U51(X1, X2)) -> c_29(a__U51^#(mark(X1), X2), mark^#(X1))
  , mark^#(U52(X1, X2)) -> c_30(a__U52^#(mark(X1), X2), mark^#(X1))
  , mark^#(U61(X1, X2, X3, X4)) ->
    c_31(a__U61^#(mark(X1), X2, X3, X4), mark^#(X1))
  , mark^#(U62(X1, X2, X3, X4)) ->
    c_32(a__U62^#(mark(X1), X2, X3, X4), mark^#(X1))
  , mark^#(U63(X1, X2, X3, X4)) ->
    c_33(a__U63^#(mark(X1), X2, X3, X4), mark^#(X1))
  , mark^#(U64(X1, X2)) -> c_34(a__U64^#(mark(X1), X2), mark^#(X1))
  , mark^#(U71(X1, X2)) -> c_35(a__U71^#(mark(X1), X2), mark^#(X1))
  , mark^#(U72(X1, X2)) -> c_36(a__U72^#(mark(X1), X2), mark^#(X1))
  , mark^#(U81(X1, X2, X3)) ->
    c_37(a__U81^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U82(X1, X2, X3)) ->
    c_38(a__U82^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(fst(X)) -> c_39(a__fst^#(mark(X)), mark^#(X))
  , mark^#(sel(X1, X2)) ->
    c_40(a__sel^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(tail(X)) -> c_41(a__tail^#(mark(X)), mark^#(X))
  , mark^#(take(X1, X2)) ->
    c_42(a__take^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , a__U51^#(tt(), Y) -> c_60(a__U52^#(tt(), Y))
  , a__U61^#(tt(), N, X, XS) -> c_64(a__U62^#(tt(), N, X, XS))
  , a__natsFrom^#(N) -> c_81(mark^#(N))
  , a__U21^#(tt(), X) -> c_44(a__U22^#(tt(), X))
  , a__U22^#(tt(), X) -> c_46(mark^#(X))
  , a__U31^#(tt(), N) -> c_48(a__U32^#(tt(), N))
  , a__U32^#(tt(), N) -> c_50(mark^#(N))
  , a__U41^#(tt(), N, XS) -> c_52(a__U42^#(tt(), N, XS))
  , a__U42^#(tt(), N, XS) ->
    c_54(a__head^#(a__afterNth(mark(N), mark(XS))),
         a__afterNth^#(mark(N), mark(XS)),
         mark^#(N),
         mark^#(XS))
  , a__head^#(cons(N, XS)) -> c_56(a__U31^#(tt(), N))
  , a__afterNth^#(N, XS) -> c_57(a__U11^#(tt(), N, XS))
  , a__U52^#(tt(), Y) -> c_62(mark^#(Y))
  , a__U62^#(tt(), N, X, XS) -> c_66(a__U63^#(tt(), N, X, XS))
  , a__U63^#(tt(), N, X, XS) ->
    c_68(a__U64^#(a__splitAt(mark(N), mark(XS)), X),
         a__splitAt^#(mark(N), mark(XS)),
         mark^#(N),
         mark^#(XS))
  , a__U64^#(pair(YS, ZS), X) -> c_70(mark^#(X), mark^#(ZS))
  , a__U71^#(tt(), XS) -> c_72(a__U72^#(tt(), XS))
  , a__U72^#(tt(), XS) -> c_74(mark^#(XS))
  , a__U81^#(tt(), N, XS) -> c_76(a__U82^#(tt(), N, XS))
  , a__U82^#(tt(), N, XS) ->
    c_78(a__fst^#(a__splitAt(mark(N), mark(XS))),
         a__splitAt^#(mark(N), mark(XS)),
         mark^#(N),
         mark^#(XS))
  , a__fst^#(pair(X, Y)) -> c_80(a__U21^#(tt(), X))
  , a__sel^#(N, XS) -> c_83(a__U41^#(tt(), N, XS))
  , a__tail^#(cons(N, XS)) -> c_86(a__U71^#(tt(), XS))
  , a__take^#(N, XS) -> c_87(a__U81^#(tt(), N, XS)) }
Weak DPs:
  { a__U11^#(X1, X2, X3) -> c_1()
  , a__U12^#(X1, X2, X3) -> c_3()
  , a__snd^#(X) -> c_5()
  , a__splitAt^#(X1, X2) -> c_7()
  , mark^#(tt()) -> c_10()
  , mark^#(0()) -> c_15()
  , mark^#(nil()) -> c_16()
  , a__U51^#(X1, X2) -> c_59()
  , a__U61^#(X1, X2, X3, X4) -> c_63()
  , a__natsFrom^#(X) -> c_82()
  , a__U21^#(X1, X2) -> c_43()
  , a__U22^#(X1, X2) -> c_45()
  , a__U31^#(X1, X2) -> c_47()
  , a__U32^#(X1, X2) -> c_49()
  , a__U41^#(X1, X2, X3) -> c_51()
  , a__U42^#(X1, X2, X3) -> c_53()
  , a__head^#(X) -> c_55()
  , a__afterNth^#(X1, X2) -> c_58()
  , a__U52^#(X1, X2) -> c_61()
  , a__U62^#(X1, X2, X3, X4) -> c_65()
  , a__U63^#(X1, X2, X3, X4) -> c_67()
  , a__U64^#(X1, X2) -> c_69()
  , a__U71^#(X1, X2) -> c_71()
  , a__U72^#(X1, X2) -> c_73()
  , a__U81^#(X1, X2, X3) -> c_75()
  , a__U82^#(X1, X2, X3) -> c_77()
  , a__fst^#(X) -> c_79()
  , a__sel^#(X1, X2) -> c_84()
  , a__tail^#(X) -> c_85()
  , a__take^#(X1, X2) -> c_88() }
Weak Trs:
  { a__U11(X1, X2, X3) -> U11(X1, X2, X3)
  , a__U11(tt(), N, XS) -> a__U12(tt(), N, XS)
  , a__U12(X1, X2, X3) -> U12(X1, X2, X3)
  , a__U12(tt(), N, XS) -> a__snd(a__splitAt(mark(N), mark(XS)))
  , a__snd(X) -> snd(X)
  , a__snd(pair(X, Y)) -> a__U51(tt(), Y)
  , a__splitAt(X1, X2) -> splitAt(X1, X2)
  , a__splitAt(s(N), cons(X, XS)) -> a__U61(tt(), N, X, XS)
  , a__splitAt(0(), XS) -> pair(nil(), mark(XS))
  , mark(tt()) -> tt()
  , mark(pair(X1, X2)) -> pair(mark(X1), mark(X2))
  , mark(cons(X1, X2)) -> cons(mark(X1), X2)
  , mark(natsFrom(X)) -> a__natsFrom(mark(X))
  , mark(s(X)) -> s(mark(X))
  , mark(0()) -> 0()
  , mark(nil()) -> nil()
  , mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
  , mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3)
  , mark(snd(X)) -> a__snd(mark(X))
  , mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2))
  , mark(U21(X1, X2)) -> a__U21(mark(X1), X2)
  , mark(U22(X1, X2)) -> a__U22(mark(X1), X2)
  , mark(U31(X1, X2)) -> a__U31(mark(X1), X2)
  , mark(U32(X1, X2)) -> a__U32(mark(X1), X2)
  , mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3)
  , mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3)
  , mark(head(X)) -> a__head(mark(X))
  , mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2))
  , mark(U51(X1, X2)) -> a__U51(mark(X1), X2)
  , mark(U52(X1, X2)) -> a__U52(mark(X1), X2)
  , mark(U61(X1, X2, X3, X4)) -> a__U61(mark(X1), X2, X3, X4)
  , mark(U62(X1, X2, X3, X4)) -> a__U62(mark(X1), X2, X3, X4)
  , mark(U63(X1, X2, X3, X4)) -> a__U63(mark(X1), X2, X3, X4)
  , mark(U64(X1, X2)) -> a__U64(mark(X1), X2)
  , mark(U71(X1, X2)) -> a__U71(mark(X1), X2)
  , mark(U72(X1, X2)) -> a__U72(mark(X1), X2)
  , mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3)
  , mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3)
  , mark(fst(X)) -> a__fst(mark(X))
  , mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2))
  , mark(tail(X)) -> a__tail(mark(X))
  , mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
  , a__U21(X1, X2) -> U21(X1, X2)
  , a__U21(tt(), X) -> a__U22(tt(), X)
  , a__U22(X1, X2) -> U22(X1, X2)
  , a__U22(tt(), X) -> mark(X)
  , a__U31(X1, X2) -> U31(X1, X2)
  , a__U31(tt(), N) -> a__U32(tt(), N)
  , a__U32(X1, X2) -> U32(X1, X2)
  , a__U32(tt(), N) -> mark(N)
  , a__U41(X1, X2, X3) -> U41(X1, X2, X3)
  , a__U41(tt(), N, XS) -> a__U42(tt(), N, XS)
  , a__U42(X1, X2, X3) -> U42(X1, X2, X3)
  , a__U42(tt(), N, XS) -> a__head(a__afterNth(mark(N), mark(XS)))
  , a__head(X) -> head(X)
  , a__head(cons(N, XS)) -> a__U31(tt(), N)
  , a__afterNth(N, XS) -> a__U11(tt(), N, XS)
  , a__afterNth(X1, X2) -> afterNth(X1, X2)
  , a__U51(X1, X2) -> U51(X1, X2)
  , a__U51(tt(), Y) -> a__U52(tt(), Y)
  , a__U52(X1, X2) -> U52(X1, X2)
  , a__U52(tt(), Y) -> mark(Y)
  , a__U61(X1, X2, X3, X4) -> U61(X1, X2, X3, X4)
  , a__U61(tt(), N, X, XS) -> a__U62(tt(), N, X, XS)
  , a__U62(X1, X2, X3, X4) -> U62(X1, X2, X3, X4)
  , a__U62(tt(), N, X, XS) -> a__U63(tt(), N, X, XS)
  , a__U63(X1, X2, X3, X4) -> U63(X1, X2, X3, X4)
  , a__U63(tt(), N, X, XS) ->
    a__U64(a__splitAt(mark(N), mark(XS)), X)
  , a__U64(X1, X2) -> U64(X1, X2)
  , a__U64(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS))
  , a__U71(X1, X2) -> U71(X1, X2)
  , a__U71(tt(), XS) -> a__U72(tt(), XS)
  , a__U72(X1, X2) -> U72(X1, X2)
  , a__U72(tt(), XS) -> mark(XS)
  , a__U81(X1, X2, X3) -> U81(X1, X2, X3)
  , a__U81(tt(), N, XS) -> a__U82(tt(), N, XS)
  , a__U82(X1, X2, X3) -> U82(X1, X2, X3)
  , a__U82(tt(), N, XS) -> a__fst(a__splitAt(mark(N), mark(XS)))
  , a__fst(X) -> fst(X)
  , a__fst(pair(X, Y)) -> a__U21(tt(), X)
  , a__natsFrom(N) -> cons(mark(N), natsFrom(s(N)))
  , a__natsFrom(X) -> natsFrom(X)
  , a__sel(N, XS) -> a__U41(tt(), N, XS)
  , a__sel(X1, X2) -> sel(X1, X2)
  , a__tail(X) -> tail(X)
  , a__tail(cons(N, XS)) -> a__U71(tt(), XS)
  , a__take(N, XS) -> a__U81(tt(), N, XS)
  , a__take(X1, X2) -> take(X1, X2) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.

{ a__U11^#(X1, X2, X3) -> c_1()
, a__U12^#(X1, X2, X3) -> c_3()
, a__snd^#(X) -> c_5()
, a__splitAt^#(X1, X2) -> c_7()
, mark^#(tt()) -> c_10()
, mark^#(0()) -> c_15()
, mark^#(nil()) -> c_16()
, a__U51^#(X1, X2) -> c_59()
, a__U61^#(X1, X2, X3, X4) -> c_63()
, a__natsFrom^#(X) -> c_82()
, a__U21^#(X1, X2) -> c_43()
, a__U22^#(X1, X2) -> c_45()
, a__U31^#(X1, X2) -> c_47()
, a__U32^#(X1, X2) -> c_49()
, a__U41^#(X1, X2, X3) -> c_51()
, a__U42^#(X1, X2, X3) -> c_53()
, a__head^#(X) -> c_55()
, a__afterNth^#(X1, X2) -> c_58()
, a__U52^#(X1, X2) -> c_61()
, a__U62^#(X1, X2, X3, X4) -> c_65()
, a__U63^#(X1, X2, X3, X4) -> c_67()
, a__U64^#(X1, X2) -> c_69()
, a__U71^#(X1, X2) -> c_71()
, a__U72^#(X1, X2) -> c_73()
, a__U81^#(X1, X2, X3) -> c_75()
, a__U82^#(X1, X2, X3) -> c_77()
, a__fst^#(X) -> c_79()
, a__sel^#(X1, X2) -> c_84()
, a__tail^#(X) -> c_85()
, a__take^#(X1, X2) -> c_88() }

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { a__U11^#(tt(), N, XS) -> c_2(a__U12^#(tt(), N, XS))
  , a__U12^#(tt(), N, XS) ->
    c_4(a__snd^#(a__splitAt(mark(N), mark(XS))),
        a__splitAt^#(mark(N), mark(XS)),
        mark^#(N),
        mark^#(XS))
  , a__snd^#(pair(X, Y)) -> c_6(a__U51^#(tt(), Y))
  , a__splitAt^#(s(N), cons(X, XS)) -> c_8(a__U61^#(tt(), N, X, XS))
  , a__splitAt^#(0(), XS) -> c_9(mark^#(XS))
  , mark^#(pair(X1, X2)) -> c_11(mark^#(X1), mark^#(X2))
  , mark^#(cons(X1, X2)) -> c_12(mark^#(X1))
  , mark^#(natsFrom(X)) -> c_13(a__natsFrom^#(mark(X)), mark^#(X))
  , mark^#(s(X)) -> c_14(mark^#(X))
  , mark^#(U11(X1, X2, X3)) ->
    c_17(a__U11^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U12(X1, X2, X3)) ->
    c_18(a__U12^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(snd(X)) -> c_19(a__snd^#(mark(X)), mark^#(X))
  , mark^#(splitAt(X1, X2)) ->
    c_20(a__splitAt^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(U21(X1, X2)) -> c_21(a__U21^#(mark(X1), X2), mark^#(X1))
  , mark^#(U22(X1, X2)) -> c_22(a__U22^#(mark(X1), X2), mark^#(X1))
  , mark^#(U31(X1, X2)) -> c_23(a__U31^#(mark(X1), X2), mark^#(X1))
  , mark^#(U32(X1, X2)) -> c_24(a__U32^#(mark(X1), X2), mark^#(X1))
  , mark^#(U41(X1, X2, X3)) ->
    c_25(a__U41^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U42(X1, X2, X3)) ->
    c_26(a__U42^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(head(X)) -> c_27(a__head^#(mark(X)), mark^#(X))
  , mark^#(afterNth(X1, X2)) ->
    c_28(a__afterNth^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(U51(X1, X2)) -> c_29(a__U51^#(mark(X1), X2), mark^#(X1))
  , mark^#(U52(X1, X2)) -> c_30(a__U52^#(mark(X1), X2), mark^#(X1))
  , mark^#(U61(X1, X2, X3, X4)) ->
    c_31(a__U61^#(mark(X1), X2, X3, X4), mark^#(X1))
  , mark^#(U62(X1, X2, X3, X4)) ->
    c_32(a__U62^#(mark(X1), X2, X3, X4), mark^#(X1))
  , mark^#(U63(X1, X2, X3, X4)) ->
    c_33(a__U63^#(mark(X1), X2, X3, X4), mark^#(X1))
  , mark^#(U64(X1, X2)) -> c_34(a__U64^#(mark(X1), X2), mark^#(X1))
  , mark^#(U71(X1, X2)) -> c_35(a__U71^#(mark(X1), X2), mark^#(X1))
  , mark^#(U72(X1, X2)) -> c_36(a__U72^#(mark(X1), X2), mark^#(X1))
  , mark^#(U81(X1, X2, X3)) ->
    c_37(a__U81^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U82(X1, X2, X3)) ->
    c_38(a__U82^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(fst(X)) -> c_39(a__fst^#(mark(X)), mark^#(X))
  , mark^#(sel(X1, X2)) ->
    c_40(a__sel^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(tail(X)) -> c_41(a__tail^#(mark(X)), mark^#(X))
  , mark^#(take(X1, X2)) ->
    c_42(a__take^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , a__U51^#(tt(), Y) -> c_60(a__U52^#(tt(), Y))
  , a__U61^#(tt(), N, X, XS) -> c_64(a__U62^#(tt(), N, X, XS))
  , a__natsFrom^#(N) -> c_81(mark^#(N))
  , a__U21^#(tt(), X) -> c_44(a__U22^#(tt(), X))
  , a__U22^#(tt(), X) -> c_46(mark^#(X))
  , a__U31^#(tt(), N) -> c_48(a__U32^#(tt(), N))
  , a__U32^#(tt(), N) -> c_50(mark^#(N))
  , a__U41^#(tt(), N, XS) -> c_52(a__U42^#(tt(), N, XS))
  , a__U42^#(tt(), N, XS) ->
    c_54(a__head^#(a__afterNth(mark(N), mark(XS))),
         a__afterNth^#(mark(N), mark(XS)),
         mark^#(N),
         mark^#(XS))
  , a__head^#(cons(N, XS)) -> c_56(a__U31^#(tt(), N))
  , a__afterNth^#(N, XS) -> c_57(a__U11^#(tt(), N, XS))
  , a__U52^#(tt(), Y) -> c_62(mark^#(Y))
  , a__U62^#(tt(), N, X, XS) -> c_66(a__U63^#(tt(), N, X, XS))
  , a__U63^#(tt(), N, X, XS) ->
    c_68(a__U64^#(a__splitAt(mark(N), mark(XS)), X),
         a__splitAt^#(mark(N), mark(XS)),
         mark^#(N),
         mark^#(XS))
  , a__U64^#(pair(YS, ZS), X) -> c_70(mark^#(X), mark^#(ZS))
  , a__U71^#(tt(), XS) -> c_72(a__U72^#(tt(), XS))
  , a__U72^#(tt(), XS) -> c_74(mark^#(XS))
  , a__U81^#(tt(), N, XS) -> c_76(a__U82^#(tt(), N, XS))
  , a__U82^#(tt(), N, XS) ->
    c_78(a__fst^#(a__splitAt(mark(N), mark(XS))),
         a__splitAt^#(mark(N), mark(XS)),
         mark^#(N),
         mark^#(XS))
  , a__fst^#(pair(X, Y)) -> c_80(a__U21^#(tt(), X))
  , a__sel^#(N, XS) -> c_83(a__U41^#(tt(), N, XS))
  , a__tail^#(cons(N, XS)) -> c_86(a__U71^#(tt(), XS))
  , a__take^#(N, XS) -> c_87(a__U81^#(tt(), N, XS)) }
Weak Trs:
  { a__U11(X1, X2, X3) -> U11(X1, X2, X3)
  , a__U11(tt(), N, XS) -> a__U12(tt(), N, XS)
  , a__U12(X1, X2, X3) -> U12(X1, X2, X3)
  , a__U12(tt(), N, XS) -> a__snd(a__splitAt(mark(N), mark(XS)))
  , a__snd(X) -> snd(X)
  , a__snd(pair(X, Y)) -> a__U51(tt(), Y)
  , a__splitAt(X1, X2) -> splitAt(X1, X2)
  , a__splitAt(s(N), cons(X, XS)) -> a__U61(tt(), N, X, XS)
  , a__splitAt(0(), XS) -> pair(nil(), mark(XS))
  , mark(tt()) -> tt()
  , mark(pair(X1, X2)) -> pair(mark(X1), mark(X2))
  , mark(cons(X1, X2)) -> cons(mark(X1), X2)
  , mark(natsFrom(X)) -> a__natsFrom(mark(X))
  , mark(s(X)) -> s(mark(X))
  , mark(0()) -> 0()
  , mark(nil()) -> nil()
  , mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
  , mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3)
  , mark(snd(X)) -> a__snd(mark(X))
  , mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2))
  , mark(U21(X1, X2)) -> a__U21(mark(X1), X2)
  , mark(U22(X1, X2)) -> a__U22(mark(X1), X2)
  , mark(U31(X1, X2)) -> a__U31(mark(X1), X2)
  , mark(U32(X1, X2)) -> a__U32(mark(X1), X2)
  , mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3)
  , mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3)
  , mark(head(X)) -> a__head(mark(X))
  , mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2))
  , mark(U51(X1, X2)) -> a__U51(mark(X1), X2)
  , mark(U52(X1, X2)) -> a__U52(mark(X1), X2)
  , mark(U61(X1, X2, X3, X4)) -> a__U61(mark(X1), X2, X3, X4)
  , mark(U62(X1, X2, X3, X4)) -> a__U62(mark(X1), X2, X3, X4)
  , mark(U63(X1, X2, X3, X4)) -> a__U63(mark(X1), X2, X3, X4)
  , mark(U64(X1, X2)) -> a__U64(mark(X1), X2)
  , mark(U71(X1, X2)) -> a__U71(mark(X1), X2)
  , mark(U72(X1, X2)) -> a__U72(mark(X1), X2)
  , mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3)
  , mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3)
  , mark(fst(X)) -> a__fst(mark(X))
  , mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2))
  , mark(tail(X)) -> a__tail(mark(X))
  , mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
  , a__U21(X1, X2) -> U21(X1, X2)
  , a__U21(tt(), X) -> a__U22(tt(), X)
  , a__U22(X1, X2) -> U22(X1, X2)
  , a__U22(tt(), X) -> mark(X)
  , a__U31(X1, X2) -> U31(X1, X2)
  , a__U31(tt(), N) -> a__U32(tt(), N)
  , a__U32(X1, X2) -> U32(X1, X2)
  , a__U32(tt(), N) -> mark(N)
  , a__U41(X1, X2, X3) -> U41(X1, X2, X3)
  , a__U41(tt(), N, XS) -> a__U42(tt(), N, XS)
  , a__U42(X1, X2, X3) -> U42(X1, X2, X3)
  , a__U42(tt(), N, XS) -> a__head(a__afterNth(mark(N), mark(XS)))
  , a__head(X) -> head(X)
  , a__head(cons(N, XS)) -> a__U31(tt(), N)
  , a__afterNth(N, XS) -> a__U11(tt(), N, XS)
  , a__afterNth(X1, X2) -> afterNth(X1, X2)
  , a__U51(X1, X2) -> U51(X1, X2)
  , a__U51(tt(), Y) -> a__U52(tt(), Y)
  , a__U52(X1, X2) -> U52(X1, X2)
  , a__U52(tt(), Y) -> mark(Y)
  , a__U61(X1, X2, X3, X4) -> U61(X1, X2, X3, X4)
  , a__U61(tt(), N, X, XS) -> a__U62(tt(), N, X, XS)
  , a__U62(X1, X2, X3, X4) -> U62(X1, X2, X3, X4)
  , a__U62(tt(), N, X, XS) -> a__U63(tt(), N, X, XS)
  , a__U63(X1, X2, X3, X4) -> U63(X1, X2, X3, X4)
  , a__U63(tt(), N, X, XS) ->
    a__U64(a__splitAt(mark(N), mark(XS)), X)
  , a__U64(X1, X2) -> U64(X1, X2)
  , a__U64(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS))
  , a__U71(X1, X2) -> U71(X1, X2)
  , a__U71(tt(), XS) -> a__U72(tt(), XS)
  , a__U72(X1, X2) -> U72(X1, X2)
  , a__U72(tt(), XS) -> mark(XS)
  , a__U81(X1, X2, X3) -> U81(X1, X2, X3)
  , a__U81(tt(), N, XS) -> a__U82(tt(), N, XS)
  , a__U82(X1, X2, X3) -> U82(X1, X2, X3)
  , a__U82(tt(), N, XS) -> a__fst(a__splitAt(mark(N), mark(XS)))
  , a__fst(X) -> fst(X)
  , a__fst(pair(X, Y)) -> a__U21(tt(), X)
  , a__natsFrom(N) -> cons(mark(N), natsFrom(s(N)))
  , a__natsFrom(X) -> natsFrom(X)
  , a__sel(N, XS) -> a__U41(tt(), N, XS)
  , a__sel(X1, X2) -> sel(X1, X2)
  , a__tail(X) -> tail(X)
  , a__tail(cons(N, XS)) -> a__U71(tt(), XS)
  , a__take(N, XS) -> a__U81(tt(), N, XS)
  , a__take(X1, X2) -> take(X1, X2) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..