MAYBE

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

Strict Trs:
  { a__U11(X1, X2, X3, X4) -> U11(X1, X2, X3, X4)
  , a__U11(tt(), N, X, XS) ->
    a__U12(a__splitAt(mark(N), mark(XS)), X)
  , a__U12(X1, X2) -> U12(X1, X2)
  , a__U12(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS))
  , a__splitAt(X1, X2) -> splitAt(X1, X2)
  , a__splitAt(s(N), cons(X, XS)) -> a__U11(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, X4)) -> a__U11(mark(X1), X2, X3, X4)
  , mark(U12(X1, X2)) -> a__U12(mark(X1), X2)
  , mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2))
  , mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2))
  , mark(snd(X)) -> a__snd(mark(X))
  , mark(and(X1, X2)) -> a__and(mark(X1), X2)
  , mark(fst(X)) -> a__fst(mark(X))
  , mark(head(X)) -> a__head(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__afterNth(N, XS) -> a__snd(a__splitAt(mark(N), mark(XS)))
  , a__afterNth(X1, X2) -> afterNth(X1, X2)
  , a__snd(X) -> snd(X)
  , a__snd(pair(X, Y)) -> mark(Y)
  , a__and(X1, X2) -> and(X1, X2)
  , a__and(tt(), X) -> mark(X)
  , a__fst(X) -> fst(X)
  , a__fst(pair(X, Y)) -> mark(X)
  , a__head(X) -> head(X)
  , a__head(cons(N, XS)) -> mark(N)
  , a__natsFrom(N) -> cons(mark(N), natsFrom(s(N)))
  , a__natsFrom(X) -> natsFrom(X)
  , a__sel(N, XS) -> a__head(a__afterNth(mark(N), mark(XS)))
  , a__sel(X1, X2) -> sel(X1, X2)
  , a__tail(X) -> tail(X)
  , a__tail(cons(N, XS)) -> mark(XS)
  , a__take(N, XS) -> a__fst(a__splitAt(mark(N), mark(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, X4) -> c_1()
  , a__U11^#(tt(), N, X, XS) ->
    c_2(a__U12^#(a__splitAt(mark(N), mark(XS)), X),
        a__splitAt^#(mark(N), mark(XS)),
        mark^#(N),
        mark^#(XS))
  , a__U12^#(X1, X2) -> c_3()
  , a__U12^#(pair(YS, ZS), X) -> c_4(mark^#(X), mark^#(ZS))
  , a__splitAt^#(X1, X2) -> c_5()
  , a__splitAt^#(s(N), cons(X, XS)) -> c_6(a__U11^#(tt(), N, X, XS))
  , a__splitAt^#(0(), XS) -> c_7(mark^#(XS))
  , mark^#(tt()) -> c_8()
  , mark^#(pair(X1, X2)) -> c_9(mark^#(X1), mark^#(X2))
  , mark^#(cons(X1, X2)) -> c_10(mark^#(X1))
  , mark^#(natsFrom(X)) -> c_11(a__natsFrom^#(mark(X)), mark^#(X))
  , mark^#(s(X)) -> c_12(mark^#(X))
  , mark^#(0()) -> c_13()
  , mark^#(nil()) -> c_14()
  , mark^#(U11(X1, X2, X3, X4)) ->
    c_15(a__U11^#(mark(X1), X2, X3, X4), mark^#(X1))
  , mark^#(U12(X1, X2)) -> c_16(a__U12^#(mark(X1), X2), mark^#(X1))
  , mark^#(splitAt(X1, X2)) ->
    c_17(a__splitAt^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(afterNth(X1, X2)) ->
    c_18(a__afterNth^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(snd(X)) -> c_19(a__snd^#(mark(X)), mark^#(X))
  , mark^#(and(X1, X2)) -> c_20(a__and^#(mark(X1), X2), mark^#(X1))
  , mark^#(fst(X)) -> c_21(a__fst^#(mark(X)), mark^#(X))
  , mark^#(head(X)) -> c_22(a__head^#(mark(X)), mark^#(X))
  , mark^#(sel(X1, X2)) ->
    c_23(a__sel^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(tail(X)) -> c_24(a__tail^#(mark(X)), mark^#(X))
  , mark^#(take(X1, X2)) ->
    c_25(a__take^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , a__natsFrom^#(N) -> c_36(mark^#(N))
  , a__natsFrom^#(X) -> c_37()
  , a__afterNth^#(N, XS) ->
    c_26(a__snd^#(a__splitAt(mark(N), mark(XS))),
         a__splitAt^#(mark(N), mark(XS)),
         mark^#(N),
         mark^#(XS))
  , a__afterNth^#(X1, X2) -> c_27()
  , a__snd^#(X) -> c_28()
  , a__snd^#(pair(X, Y)) -> c_29(mark^#(Y))
  , a__and^#(X1, X2) -> c_30()
  , a__and^#(tt(), X) -> c_31(mark^#(X))
  , a__fst^#(X) -> c_32()
  , a__fst^#(pair(X, Y)) -> c_33(mark^#(X))
  , a__head^#(X) -> c_34()
  , a__head^#(cons(N, XS)) -> c_35(mark^#(N))
  , a__sel^#(N, XS) ->
    c_38(a__head^#(a__afterNth(mark(N), mark(XS))),
         a__afterNth^#(mark(N), mark(XS)),
         mark^#(N),
         mark^#(XS))
  , a__sel^#(X1, X2) -> c_39()
  , a__tail^#(X) -> c_40()
  , a__tail^#(cons(N, XS)) -> c_41(mark^#(XS))
  , a__take^#(N, XS) ->
    c_42(a__fst^#(a__splitAt(mark(N), mark(XS))),
         a__splitAt^#(mark(N), mark(XS)),
         mark^#(N),
         mark^#(XS))
  , a__take^#(X1, X2) -> c_43() }

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, X4) -> c_1()
  , a__U11^#(tt(), N, X, XS) ->
    c_2(a__U12^#(a__splitAt(mark(N), mark(XS)), X),
        a__splitAt^#(mark(N), mark(XS)),
        mark^#(N),
        mark^#(XS))
  , a__U12^#(X1, X2) -> c_3()
  , a__U12^#(pair(YS, ZS), X) -> c_4(mark^#(X), mark^#(ZS))
  , a__splitAt^#(X1, X2) -> c_5()
  , a__splitAt^#(s(N), cons(X, XS)) -> c_6(a__U11^#(tt(), N, X, XS))
  , a__splitAt^#(0(), XS) -> c_7(mark^#(XS))
  , mark^#(tt()) -> c_8()
  , mark^#(pair(X1, X2)) -> c_9(mark^#(X1), mark^#(X2))
  , mark^#(cons(X1, X2)) -> c_10(mark^#(X1))
  , mark^#(natsFrom(X)) -> c_11(a__natsFrom^#(mark(X)), mark^#(X))
  , mark^#(s(X)) -> c_12(mark^#(X))
  , mark^#(0()) -> c_13()
  , mark^#(nil()) -> c_14()
  , mark^#(U11(X1, X2, X3, X4)) ->
    c_15(a__U11^#(mark(X1), X2, X3, X4), mark^#(X1))
  , mark^#(U12(X1, X2)) -> c_16(a__U12^#(mark(X1), X2), mark^#(X1))
  , mark^#(splitAt(X1, X2)) ->
    c_17(a__splitAt^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(afterNth(X1, X2)) ->
    c_18(a__afterNth^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(snd(X)) -> c_19(a__snd^#(mark(X)), mark^#(X))
  , mark^#(and(X1, X2)) -> c_20(a__and^#(mark(X1), X2), mark^#(X1))
  , mark^#(fst(X)) -> c_21(a__fst^#(mark(X)), mark^#(X))
  , mark^#(head(X)) -> c_22(a__head^#(mark(X)), mark^#(X))
  , mark^#(sel(X1, X2)) ->
    c_23(a__sel^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(tail(X)) -> c_24(a__tail^#(mark(X)), mark^#(X))
  , mark^#(take(X1, X2)) ->
    c_25(a__take^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , a__natsFrom^#(N) -> c_36(mark^#(N))
  , a__natsFrom^#(X) -> c_37()
  , a__afterNth^#(N, XS) ->
    c_26(a__snd^#(a__splitAt(mark(N), mark(XS))),
         a__splitAt^#(mark(N), mark(XS)),
         mark^#(N),
         mark^#(XS))
  , a__afterNth^#(X1, X2) -> c_27()
  , a__snd^#(X) -> c_28()
  , a__snd^#(pair(X, Y)) -> c_29(mark^#(Y))
  , a__and^#(X1, X2) -> c_30()
  , a__and^#(tt(), X) -> c_31(mark^#(X))
  , a__fst^#(X) -> c_32()
  , a__fst^#(pair(X, Y)) -> c_33(mark^#(X))
  , a__head^#(X) -> c_34()
  , a__head^#(cons(N, XS)) -> c_35(mark^#(N))
  , a__sel^#(N, XS) ->
    c_38(a__head^#(a__afterNth(mark(N), mark(XS))),
         a__afterNth^#(mark(N), mark(XS)),
         mark^#(N),
         mark^#(XS))
  , a__sel^#(X1, X2) -> c_39()
  , a__tail^#(X) -> c_40()
  , a__tail^#(cons(N, XS)) -> c_41(mark^#(XS))
  , a__take^#(N, XS) ->
    c_42(a__fst^#(a__splitAt(mark(N), mark(XS))),
         a__splitAt^#(mark(N), mark(XS)),
         mark^#(N),
         mark^#(XS))
  , a__take^#(X1, X2) -> c_43() }
Weak Trs:
  { a__U11(X1, X2, X3, X4) -> U11(X1, X2, X3, X4)
  , a__U11(tt(), N, X, XS) ->
    a__U12(a__splitAt(mark(N), mark(XS)), X)
  , a__U12(X1, X2) -> U12(X1, X2)
  , a__U12(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS))
  , a__splitAt(X1, X2) -> splitAt(X1, X2)
  , a__splitAt(s(N), cons(X, XS)) -> a__U11(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, X4)) -> a__U11(mark(X1), X2, X3, X4)
  , mark(U12(X1, X2)) -> a__U12(mark(X1), X2)
  , mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2))
  , mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2))
  , mark(snd(X)) -> a__snd(mark(X))
  , mark(and(X1, X2)) -> a__and(mark(X1), X2)
  , mark(fst(X)) -> a__fst(mark(X))
  , mark(head(X)) -> a__head(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__afterNth(N, XS) -> a__snd(a__splitAt(mark(N), mark(XS)))
  , a__afterNth(X1, X2) -> afterNth(X1, X2)
  , a__snd(X) -> snd(X)
  , a__snd(pair(X, Y)) -> mark(Y)
  , a__and(X1, X2) -> and(X1, X2)
  , a__and(tt(), X) -> mark(X)
  , a__fst(X) -> fst(X)
  , a__fst(pair(X, Y)) -> mark(X)
  , a__head(X) -> head(X)
  , a__head(cons(N, XS)) -> mark(N)
  , a__natsFrom(N) -> cons(mark(N), natsFrom(s(N)))
  , a__natsFrom(X) -> natsFrom(X)
  , a__sel(N, XS) -> a__head(a__afterNth(mark(N), mark(XS)))
  , a__sel(X1, X2) -> sel(X1, X2)
  , a__tail(X) -> tail(X)
  , a__tail(cons(N, XS)) -> mark(XS)
  , a__take(N, XS) -> a__fst(a__splitAt(mark(N), mark(XS)))
  , a__take(X1, X2) -> take(X1, X2) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of
{1,3,5,8,13,14,27,29,30,32,34,36,39,40,43} by applications of
Pre({1,3,5,8,13,14,27,29,30,32,34,36,39,40,43}) =
{2,4,6,7,9,10,11,12,15,16,17,18,19,20,21,22,23,24,25,26,28,31,33,35,37,38,41,42}.
Here rules are labeled as follows:

  DPs:
    { 1: a__U11^#(X1, X2, X3, X4) -> c_1()
    , 2: a__U11^#(tt(), N, X, XS) ->
         c_2(a__U12^#(a__splitAt(mark(N), mark(XS)), X),
             a__splitAt^#(mark(N), mark(XS)),
             mark^#(N),
             mark^#(XS))
    , 3: a__U12^#(X1, X2) -> c_3()
    , 4: a__U12^#(pair(YS, ZS), X) -> c_4(mark^#(X), mark^#(ZS))
    , 5: a__splitAt^#(X1, X2) -> c_5()
    , 6: a__splitAt^#(s(N), cons(X, XS)) ->
         c_6(a__U11^#(tt(), N, X, XS))
    , 7: a__splitAt^#(0(), XS) -> c_7(mark^#(XS))
    , 8: mark^#(tt()) -> c_8()
    , 9: mark^#(pair(X1, X2)) -> c_9(mark^#(X1), mark^#(X2))
    , 10: mark^#(cons(X1, X2)) -> c_10(mark^#(X1))
    , 11: mark^#(natsFrom(X)) ->
          c_11(a__natsFrom^#(mark(X)), mark^#(X))
    , 12: mark^#(s(X)) -> c_12(mark^#(X))
    , 13: mark^#(0()) -> c_13()
    , 14: mark^#(nil()) -> c_14()
    , 15: mark^#(U11(X1, X2, X3, X4)) ->
          c_15(a__U11^#(mark(X1), X2, X3, X4), mark^#(X1))
    , 16: mark^#(U12(X1, X2)) ->
          c_16(a__U12^#(mark(X1), X2), mark^#(X1))
    , 17: mark^#(splitAt(X1, X2)) ->
          c_17(a__splitAt^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
    , 18: mark^#(afterNth(X1, X2)) ->
          c_18(a__afterNth^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
    , 19: mark^#(snd(X)) -> c_19(a__snd^#(mark(X)), mark^#(X))
    , 20: mark^#(and(X1, X2)) ->
          c_20(a__and^#(mark(X1), X2), mark^#(X1))
    , 21: mark^#(fst(X)) -> c_21(a__fst^#(mark(X)), mark^#(X))
    , 22: mark^#(head(X)) -> c_22(a__head^#(mark(X)), mark^#(X))
    , 23: mark^#(sel(X1, X2)) ->
          c_23(a__sel^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
    , 24: mark^#(tail(X)) -> c_24(a__tail^#(mark(X)), mark^#(X))
    , 25: mark^#(take(X1, X2)) ->
          c_25(a__take^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
    , 26: a__natsFrom^#(N) -> c_36(mark^#(N))
    , 27: a__natsFrom^#(X) -> c_37()
    , 28: a__afterNth^#(N, XS) ->
          c_26(a__snd^#(a__splitAt(mark(N), mark(XS))),
               a__splitAt^#(mark(N), mark(XS)),
               mark^#(N),
               mark^#(XS))
    , 29: a__afterNth^#(X1, X2) -> c_27()
    , 30: a__snd^#(X) -> c_28()
    , 31: a__snd^#(pair(X, Y)) -> c_29(mark^#(Y))
    , 32: a__and^#(X1, X2) -> c_30()
    , 33: a__and^#(tt(), X) -> c_31(mark^#(X))
    , 34: a__fst^#(X) -> c_32()
    , 35: a__fst^#(pair(X, Y)) -> c_33(mark^#(X))
    , 36: a__head^#(X) -> c_34()
    , 37: a__head^#(cons(N, XS)) -> c_35(mark^#(N))
    , 38: a__sel^#(N, XS) ->
          c_38(a__head^#(a__afterNth(mark(N), mark(XS))),
               a__afterNth^#(mark(N), mark(XS)),
               mark^#(N),
               mark^#(XS))
    , 39: a__sel^#(X1, X2) -> c_39()
    , 40: a__tail^#(X) -> c_40()
    , 41: a__tail^#(cons(N, XS)) -> c_41(mark^#(XS))
    , 42: a__take^#(N, XS) ->
          c_42(a__fst^#(a__splitAt(mark(N), mark(XS))),
               a__splitAt^#(mark(N), mark(XS)),
               mark^#(N),
               mark^#(XS))
    , 43: a__take^#(X1, X2) -> c_43() }

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

Strict DPs:
  { a__U11^#(tt(), N, X, XS) ->
    c_2(a__U12^#(a__splitAt(mark(N), mark(XS)), X),
        a__splitAt^#(mark(N), mark(XS)),
        mark^#(N),
        mark^#(XS))
  , a__U12^#(pair(YS, ZS), X) -> c_4(mark^#(X), mark^#(ZS))
  , a__splitAt^#(s(N), cons(X, XS)) -> c_6(a__U11^#(tt(), N, X, XS))
  , a__splitAt^#(0(), XS) -> c_7(mark^#(XS))
  , mark^#(pair(X1, X2)) -> c_9(mark^#(X1), mark^#(X2))
  , mark^#(cons(X1, X2)) -> c_10(mark^#(X1))
  , mark^#(natsFrom(X)) -> c_11(a__natsFrom^#(mark(X)), mark^#(X))
  , mark^#(s(X)) -> c_12(mark^#(X))
  , mark^#(U11(X1, X2, X3, X4)) ->
    c_15(a__U11^#(mark(X1), X2, X3, X4), mark^#(X1))
  , mark^#(U12(X1, X2)) -> c_16(a__U12^#(mark(X1), X2), mark^#(X1))
  , mark^#(splitAt(X1, X2)) ->
    c_17(a__splitAt^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(afterNth(X1, X2)) ->
    c_18(a__afterNth^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(snd(X)) -> c_19(a__snd^#(mark(X)), mark^#(X))
  , mark^#(and(X1, X2)) -> c_20(a__and^#(mark(X1), X2), mark^#(X1))
  , mark^#(fst(X)) -> c_21(a__fst^#(mark(X)), mark^#(X))
  , mark^#(head(X)) -> c_22(a__head^#(mark(X)), mark^#(X))
  , mark^#(sel(X1, X2)) ->
    c_23(a__sel^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(tail(X)) -> c_24(a__tail^#(mark(X)), mark^#(X))
  , mark^#(take(X1, X2)) ->
    c_25(a__take^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , a__natsFrom^#(N) -> c_36(mark^#(N))
  , a__afterNth^#(N, XS) ->
    c_26(a__snd^#(a__splitAt(mark(N), mark(XS))),
         a__splitAt^#(mark(N), mark(XS)),
         mark^#(N),
         mark^#(XS))
  , a__snd^#(pair(X, Y)) -> c_29(mark^#(Y))
  , a__and^#(tt(), X) -> c_31(mark^#(X))
  , a__fst^#(pair(X, Y)) -> c_33(mark^#(X))
  , a__head^#(cons(N, XS)) -> c_35(mark^#(N))
  , a__sel^#(N, XS) ->
    c_38(a__head^#(a__afterNth(mark(N), mark(XS))),
         a__afterNth^#(mark(N), mark(XS)),
         mark^#(N),
         mark^#(XS))
  , a__tail^#(cons(N, XS)) -> c_41(mark^#(XS))
  , a__take^#(N, XS) ->
    c_42(a__fst^#(a__splitAt(mark(N), mark(XS))),
         a__splitAt^#(mark(N), mark(XS)),
         mark^#(N),
         mark^#(XS)) }
Weak DPs:
  { a__U11^#(X1, X2, X3, X4) -> c_1()
  , a__U12^#(X1, X2) -> c_3()
  , a__splitAt^#(X1, X2) -> c_5()
  , mark^#(tt()) -> c_8()
  , mark^#(0()) -> c_13()
  , mark^#(nil()) -> c_14()
  , a__natsFrom^#(X) -> c_37()
  , a__afterNth^#(X1, X2) -> c_27()
  , a__snd^#(X) -> c_28()
  , a__and^#(X1, X2) -> c_30()
  , a__fst^#(X) -> c_32()
  , a__head^#(X) -> c_34()
  , a__sel^#(X1, X2) -> c_39()
  , a__tail^#(X) -> c_40()
  , a__take^#(X1, X2) -> c_43() }
Weak Trs:
  { a__U11(X1, X2, X3, X4) -> U11(X1, X2, X3, X4)
  , a__U11(tt(), N, X, XS) ->
    a__U12(a__splitAt(mark(N), mark(XS)), X)
  , a__U12(X1, X2) -> U12(X1, X2)
  , a__U12(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS))
  , a__splitAt(X1, X2) -> splitAt(X1, X2)
  , a__splitAt(s(N), cons(X, XS)) -> a__U11(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, X4)) -> a__U11(mark(X1), X2, X3, X4)
  , mark(U12(X1, X2)) -> a__U12(mark(X1), X2)
  , mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2))
  , mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2))
  , mark(snd(X)) -> a__snd(mark(X))
  , mark(and(X1, X2)) -> a__and(mark(X1), X2)
  , mark(fst(X)) -> a__fst(mark(X))
  , mark(head(X)) -> a__head(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__afterNth(N, XS) -> a__snd(a__splitAt(mark(N), mark(XS)))
  , a__afterNth(X1, X2) -> afterNth(X1, X2)
  , a__snd(X) -> snd(X)
  , a__snd(pair(X, Y)) -> mark(Y)
  , a__and(X1, X2) -> and(X1, X2)
  , a__and(tt(), X) -> mark(X)
  , a__fst(X) -> fst(X)
  , a__fst(pair(X, Y)) -> mark(X)
  , a__head(X) -> head(X)
  , a__head(cons(N, XS)) -> mark(N)
  , a__natsFrom(N) -> cons(mark(N), natsFrom(s(N)))
  , a__natsFrom(X) -> natsFrom(X)
  , a__sel(N, XS) -> a__head(a__afterNth(mark(N), mark(XS)))
  , a__sel(X1, X2) -> sel(X1, X2)
  , a__tail(X) -> tail(X)
  , a__tail(cons(N, XS)) -> mark(XS)
  , a__take(N, XS) -> a__fst(a__splitAt(mark(N), mark(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, X4) -> c_1()
, a__U12^#(X1, X2) -> c_3()
, a__splitAt^#(X1, X2) -> c_5()
, mark^#(tt()) -> c_8()
, mark^#(0()) -> c_13()
, mark^#(nil()) -> c_14()
, a__natsFrom^#(X) -> c_37()
, a__afterNth^#(X1, X2) -> c_27()
, a__snd^#(X) -> c_28()
, a__and^#(X1, X2) -> c_30()
, a__fst^#(X) -> c_32()
, a__head^#(X) -> c_34()
, a__sel^#(X1, X2) -> c_39()
, a__tail^#(X) -> c_40()
, a__take^#(X1, X2) -> c_43() }

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

Strict DPs:
  { a__U11^#(tt(), N, X, XS) ->
    c_2(a__U12^#(a__splitAt(mark(N), mark(XS)), X),
        a__splitAt^#(mark(N), mark(XS)),
        mark^#(N),
        mark^#(XS))
  , a__U12^#(pair(YS, ZS), X) -> c_4(mark^#(X), mark^#(ZS))
  , a__splitAt^#(s(N), cons(X, XS)) -> c_6(a__U11^#(tt(), N, X, XS))
  , a__splitAt^#(0(), XS) -> c_7(mark^#(XS))
  , mark^#(pair(X1, X2)) -> c_9(mark^#(X1), mark^#(X2))
  , mark^#(cons(X1, X2)) -> c_10(mark^#(X1))
  , mark^#(natsFrom(X)) -> c_11(a__natsFrom^#(mark(X)), mark^#(X))
  , mark^#(s(X)) -> c_12(mark^#(X))
  , mark^#(U11(X1, X2, X3, X4)) ->
    c_15(a__U11^#(mark(X1), X2, X3, X4), mark^#(X1))
  , mark^#(U12(X1, X2)) -> c_16(a__U12^#(mark(X1), X2), mark^#(X1))
  , mark^#(splitAt(X1, X2)) ->
    c_17(a__splitAt^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(afterNth(X1, X2)) ->
    c_18(a__afterNth^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(snd(X)) -> c_19(a__snd^#(mark(X)), mark^#(X))
  , mark^#(and(X1, X2)) -> c_20(a__and^#(mark(X1), X2), mark^#(X1))
  , mark^#(fst(X)) -> c_21(a__fst^#(mark(X)), mark^#(X))
  , mark^#(head(X)) -> c_22(a__head^#(mark(X)), mark^#(X))
  , mark^#(sel(X1, X2)) ->
    c_23(a__sel^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(tail(X)) -> c_24(a__tail^#(mark(X)), mark^#(X))
  , mark^#(take(X1, X2)) ->
    c_25(a__take^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , a__natsFrom^#(N) -> c_36(mark^#(N))
  , a__afterNth^#(N, XS) ->
    c_26(a__snd^#(a__splitAt(mark(N), mark(XS))),
         a__splitAt^#(mark(N), mark(XS)),
         mark^#(N),
         mark^#(XS))
  , a__snd^#(pair(X, Y)) -> c_29(mark^#(Y))
  , a__and^#(tt(), X) -> c_31(mark^#(X))
  , a__fst^#(pair(X, Y)) -> c_33(mark^#(X))
  , a__head^#(cons(N, XS)) -> c_35(mark^#(N))
  , a__sel^#(N, XS) ->
    c_38(a__head^#(a__afterNth(mark(N), mark(XS))),
         a__afterNth^#(mark(N), mark(XS)),
         mark^#(N),
         mark^#(XS))
  , a__tail^#(cons(N, XS)) -> c_41(mark^#(XS))
  , a__take^#(N, XS) ->
    c_42(a__fst^#(a__splitAt(mark(N), mark(XS))),
         a__splitAt^#(mark(N), mark(XS)),
         mark^#(N),
         mark^#(XS)) }
Weak Trs:
  { a__U11(X1, X2, X3, X4) -> U11(X1, X2, X3, X4)
  , a__U11(tt(), N, X, XS) ->
    a__U12(a__splitAt(mark(N), mark(XS)), X)
  , a__U12(X1, X2) -> U12(X1, X2)
  , a__U12(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS))
  , a__splitAt(X1, X2) -> splitAt(X1, X2)
  , a__splitAt(s(N), cons(X, XS)) -> a__U11(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, X4)) -> a__U11(mark(X1), X2, X3, X4)
  , mark(U12(X1, X2)) -> a__U12(mark(X1), X2)
  , mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2))
  , mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2))
  , mark(snd(X)) -> a__snd(mark(X))
  , mark(and(X1, X2)) -> a__and(mark(X1), X2)
  , mark(fst(X)) -> a__fst(mark(X))
  , mark(head(X)) -> a__head(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__afterNth(N, XS) -> a__snd(a__splitAt(mark(N), mark(XS)))
  , a__afterNth(X1, X2) -> afterNth(X1, X2)
  , a__snd(X) -> snd(X)
  , a__snd(pair(X, Y)) -> mark(Y)
  , a__and(X1, X2) -> and(X1, X2)
  , a__and(tt(), X) -> mark(X)
  , a__fst(X) -> fst(X)
  , a__fst(pair(X, Y)) -> mark(X)
  , a__head(X) -> head(X)
  , a__head(cons(N, XS)) -> mark(N)
  , a__natsFrom(N) -> cons(mark(N), natsFrom(s(N)))
  , a__natsFrom(X) -> natsFrom(X)
  , a__sel(N, XS) -> a__head(a__afterNth(mark(N), mark(XS)))
  , a__sel(X1, X2) -> sel(X1, X2)
  , a__tail(X) -> tail(X)
  , a__tail(cons(N, XS)) -> mark(XS)
  , a__take(N, XS) -> a__fst(a__splitAt(mark(N), mark(XS)))
  , a__take(X1, X2) -> take(X1, X2) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..