MAYBE

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

Strict Trs:
  { a__U101(X1, X2, X3) -> U101(X1, X2, X3)
  , a__U101(tt(), N, XS) -> a__fst(a__splitAt(mark(N), mark(XS)))
  , a__fst(X) -> fst(X)
  , a__fst(pair(X, Y)) -> a__U21(a__and(a__isLNat(X), isLNat(Y)), X)
  , a__splitAt(X1, X2) -> splitAt(X1, X2)
  , a__splitAt(s(N), cons(X, XS)) ->
    a__U81(a__and(a__isNatural(N), and(isNatural(X), isLNat(XS))),
           N,
           X,
           XS)
  , a__splitAt(0(), XS) -> a__U71(a__isLNat(XS), XS)
  , mark(tt()) -> tt()
  , 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(nil()) -> nil()
  , mark(isLNat(X)) -> a__isLNat(X)
  , mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2))
  , mark(fst(X)) -> a__fst(mark(X))
  , mark(snd(X)) -> a__snd(mark(X))
  , mark(tail(X)) -> a__tail(mark(X))
  , mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
  , mark(0()) -> 0()
  , mark(head(X)) -> a__head(mark(X))
  , mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2))
  , mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2))
  , mark(and(X1, X2)) -> a__and(mark(X1), X2)
  , mark(isNatural(X)) -> a__isNatural(X)
  , mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3)
  , mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
  , mark(U21(X1, X2)) -> a__U21(mark(X1), X2)
  , mark(U31(X1, X2)) -> a__U31(mark(X1), X2)
  , mark(U41(X1, X2)) -> a__U41(mark(X1), X2)
  , mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3)
  , mark(U61(X1, X2)) -> a__U61(mark(X1), X2)
  , mark(U71(X1, X2)) -> a__U71(mark(X1), X2)
  , mark(U81(X1, X2, X3, X4)) -> a__U81(mark(X1), X2, X3, X4)
  , mark(U82(X1, X2)) -> a__U82(mark(X1), X2)
  , mark(U91(X1, X2)) -> a__U91(mark(X1), X2)
  , mark(isPLNat(X)) -> a__isPLNat(X)
  , a__U11(X1, X2, X3) -> U11(X1, X2, X3)
  , a__U11(tt(), N, XS) -> a__snd(a__splitAt(mark(N), mark(XS)))
  , a__snd(X) -> snd(X)
  , a__snd(pair(X, Y)) -> a__U61(a__and(a__isLNat(X), isLNat(Y)), Y)
  , a__U21(X1, X2) -> U21(X1, X2)
  , a__U21(tt(), X) -> mark(X)
  , a__U31(X1, X2) -> U31(X1, X2)
  , a__U31(tt(), N) -> mark(N)
  , a__U41(X1, X2) -> U41(X1, X2)
  , a__U41(tt(), N) -> cons(mark(N), natsFrom(s(N)))
  , a__U51(X1, X2, X3) -> U51(X1, X2, X3)
  , a__U51(tt(), N, XS) -> a__head(a__afterNth(mark(N), mark(XS)))
  , a__head(X) -> head(X)
  , a__head(cons(N, XS)) ->
    a__U31(a__and(a__isNatural(N), isLNat(XS)), N)
  , a__afterNth(N, XS) ->
    a__U11(a__and(a__isNatural(N), isLNat(XS)), N, XS)
  , a__afterNth(X1, X2) -> afterNth(X1, X2)
  , a__U61(X1, X2) -> U61(X1, X2)
  , a__U61(tt(), Y) -> mark(Y)
  , a__U71(X1, X2) -> U71(X1, X2)
  , a__U71(tt(), XS) -> pair(nil(), mark(XS))
  , a__U81(X1, X2, X3, X4) -> U81(X1, X2, X3, X4)
  , a__U81(tt(), N, X, XS) ->
    a__U82(a__splitAt(mark(N), mark(XS)), X)
  , a__U82(X1, X2) -> U82(X1, X2)
  , a__U82(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS))
  , a__U91(X1, X2) -> U91(X1, X2)
  , a__U91(tt(), XS) -> mark(XS)
  , a__and(X1, X2) -> and(X1, X2)
  , a__and(tt(), X) -> mark(X)
  , a__isNatural(X) -> isNatural(X)
  , a__isNatural(s(V1)) -> a__isNatural(V1)
  , a__isNatural(0()) -> tt()
  , a__isNatural(head(V1)) -> a__isLNat(V1)
  , a__isNatural(sel(V1, V2)) -> a__and(a__isNatural(V1), isLNat(V2))
  , a__isLNat(X) -> isLNat(X)
  , a__isLNat(cons(V1, V2)) -> a__and(a__isNatural(V1), isLNat(V2))
  , a__isLNat(natsFrom(V1)) -> a__isNatural(V1)
  , a__isLNat(nil()) -> tt()
  , a__isLNat(afterNth(V1, V2)) ->
    a__and(a__isNatural(V1), isLNat(V2))
  , a__isLNat(fst(V1)) -> a__isPLNat(V1)
  , a__isLNat(snd(V1)) -> a__isPLNat(V1)
  , a__isLNat(tail(V1)) -> a__isLNat(V1)
  , a__isLNat(take(V1, V2)) -> a__and(a__isNatural(V1), isLNat(V2))
  , a__isPLNat(X) -> isPLNat(X)
  , a__isPLNat(pair(V1, V2)) -> a__and(a__isLNat(V1), isLNat(V2))
  , a__isPLNat(splitAt(V1, V2)) ->
    a__and(a__isNatural(V1), isLNat(V2))
  , a__natsFrom(N) -> a__U41(a__isNatural(N), N)
  , a__natsFrom(X) -> natsFrom(X)
  , a__sel(N, XS) ->
    a__U51(a__and(a__isNatural(N), isLNat(XS)), N, XS)
  , a__sel(X1, X2) -> sel(X1, X2)
  , a__tail(X) -> tail(X)
  , a__tail(cons(N, XS)) ->
    a__U91(a__and(a__isNatural(N), isLNat(XS)), XS)
  , a__take(N, XS) ->
    a__U101(a__and(a__isNatural(N), isLNat(XS)), N, XS)
  , a__take(X1, X2) -> take(X1, X2) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { a__U101^#(X1, X2, X3) -> c_1()
  , a__U101^#(tt(), N, XS) ->
    c_2(a__fst^#(a__splitAt(mark(N), mark(XS))),
        a__splitAt^#(mark(N), mark(XS)),
        mark^#(N),
        mark^#(XS))
  , a__fst^#(X) -> c_3()
  , a__fst^#(pair(X, Y)) ->
    c_4(a__U21^#(a__and(a__isLNat(X), isLNat(Y)), X),
        a__and^#(a__isLNat(X), isLNat(Y)),
        a__isLNat^#(X))
  , a__splitAt^#(X1, X2) -> c_5()
  , a__splitAt^#(s(N), cons(X, XS)) ->
    c_6(a__U81^#(a__and(a__isNatural(N),
                        and(isNatural(X), isLNat(XS))),
                 N,
                 X,
                 XS),
        a__and^#(a__isNatural(N), and(isNatural(X), isLNat(XS))),
        a__isNatural^#(N))
  , a__splitAt^#(0(), XS) ->
    c_7(a__U71^#(a__isLNat(XS), XS), a__isLNat^#(XS))
  , mark^#(tt()) -> c_8()
  , mark^#(cons(X1, X2)) -> c_9(mark^#(X1))
  , mark^#(natsFrom(X)) -> c_10(a__natsFrom^#(mark(X)), mark^#(X))
  , mark^#(s(X)) -> c_11(mark^#(X))
  , mark^#(pair(X1, X2)) -> c_12(mark^#(X1), mark^#(X2))
  , mark^#(nil()) -> c_13()
  , mark^#(isLNat(X)) -> c_14(a__isLNat^#(X))
  , mark^#(afterNth(X1, X2)) ->
    c_15(a__afterNth^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(fst(X)) -> c_16(a__fst^#(mark(X)), mark^#(X))
  , mark^#(snd(X)) -> c_17(a__snd^#(mark(X)), mark^#(X))
  , mark^#(tail(X)) -> c_18(a__tail^#(mark(X)), mark^#(X))
  , mark^#(take(X1, X2)) ->
    c_19(a__take^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(0()) -> c_20()
  , mark^#(head(X)) -> c_21(a__head^#(mark(X)), mark^#(X))
  , mark^#(sel(X1, X2)) ->
    c_22(a__sel^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(splitAt(X1, X2)) ->
    c_23(a__splitAt^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(and(X1, X2)) -> c_24(a__and^#(mark(X1), X2), mark^#(X1))
  , mark^#(isNatural(X)) -> c_25(a__isNatural^#(X))
  , mark^#(U101(X1, X2, X3)) ->
    c_26(a__U101^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U11(X1, X2, X3)) ->
    c_27(a__U11^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U21(X1, X2)) -> c_28(a__U21^#(mark(X1), X2), mark^#(X1))
  , mark^#(U31(X1, X2)) -> c_29(a__U31^#(mark(X1), X2), mark^#(X1))
  , mark^#(U41(X1, X2)) -> c_30(a__U41^#(mark(X1), X2), mark^#(X1))
  , mark^#(U51(X1, X2, X3)) ->
    c_31(a__U51^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U61(X1, X2)) -> c_32(a__U61^#(mark(X1), X2), mark^#(X1))
  , mark^#(U71(X1, X2)) -> c_33(a__U71^#(mark(X1), X2), mark^#(X1))
  , mark^#(U81(X1, X2, X3, X4)) ->
    c_34(a__U81^#(mark(X1), X2, X3, X4), mark^#(X1))
  , mark^#(U82(X1, X2)) -> c_35(a__U82^#(mark(X1), X2), mark^#(X1))
  , mark^#(U91(X1, X2)) -> c_36(a__U91^#(mark(X1), X2), mark^#(X1))
  , mark^#(isPLNat(X)) -> c_37(a__isPLNat^#(X))
  , a__U21^#(X1, X2) -> c_42()
  , a__U21^#(tt(), X) -> c_43(mark^#(X))
  , a__and^#(X1, X2) -> c_64()
  , a__and^#(tt(), X) -> c_65(mark^#(X))
  , a__isLNat^#(X) -> c_71()
  , a__isLNat^#(cons(V1, V2)) ->
    c_72(a__and^#(a__isNatural(V1), isLNat(V2)), a__isNatural^#(V1))
  , a__isLNat^#(natsFrom(V1)) -> c_73(a__isNatural^#(V1))
  , a__isLNat^#(nil()) -> c_74()
  , a__isLNat^#(afterNth(V1, V2)) ->
    c_75(a__and^#(a__isNatural(V1), isLNat(V2)), a__isNatural^#(V1))
  , a__isLNat^#(fst(V1)) -> c_76(a__isPLNat^#(V1))
  , a__isLNat^#(snd(V1)) -> c_77(a__isPLNat^#(V1))
  , a__isLNat^#(tail(V1)) -> c_78(a__isLNat^#(V1))
  , a__isLNat^#(take(V1, V2)) ->
    c_79(a__and^#(a__isNatural(V1), isLNat(V2)), a__isNatural^#(V1))
  , a__U81^#(X1, X2, X3, X4) -> c_58()
  , a__U81^#(tt(), N, X, XS) ->
    c_59(a__U82^#(a__splitAt(mark(N), mark(XS)), X),
         a__splitAt^#(mark(N), mark(XS)),
         mark^#(N),
         mark^#(XS))
  , a__isNatural^#(X) -> c_66()
  , a__isNatural^#(s(V1)) -> c_67(a__isNatural^#(V1))
  , a__isNatural^#(0()) -> c_68()
  , a__isNatural^#(head(V1)) -> c_69(a__isLNat^#(V1))
  , a__isNatural^#(sel(V1, V2)) ->
    c_70(a__and^#(a__isNatural(V1), isLNat(V2)), a__isNatural^#(V1))
  , a__U71^#(X1, X2) -> c_56()
  , a__U71^#(tt(), XS) -> c_57(mark^#(XS))
  , a__natsFrom^#(N) ->
    c_83(a__U41^#(a__isNatural(N), N), a__isNatural^#(N))
  , a__natsFrom^#(X) -> c_84()
  , a__afterNth^#(N, XS) ->
    c_52(a__U11^#(a__and(a__isNatural(N), isLNat(XS)), N, XS),
         a__and^#(a__isNatural(N), isLNat(XS)),
         a__isNatural^#(N))
  , a__afterNth^#(X1, X2) -> c_53()
  , a__snd^#(X) -> c_40()
  , a__snd^#(pair(X, Y)) ->
    c_41(a__U61^#(a__and(a__isLNat(X), isLNat(Y)), Y),
         a__and^#(a__isLNat(X), isLNat(Y)),
         a__isLNat^#(X))
  , a__tail^#(X) -> c_87()
  , a__tail^#(cons(N, XS)) ->
    c_88(a__U91^#(a__and(a__isNatural(N), isLNat(XS)), XS),
         a__and^#(a__isNatural(N), isLNat(XS)),
         a__isNatural^#(N))
  , a__take^#(N, XS) ->
    c_89(a__U101^#(a__and(a__isNatural(N), isLNat(XS)), N, XS),
         a__and^#(a__isNatural(N), isLNat(XS)),
         a__isNatural^#(N))
  , a__take^#(X1, X2) -> c_90()
  , a__head^#(X) -> c_50()
  , a__head^#(cons(N, XS)) ->
    c_51(a__U31^#(a__and(a__isNatural(N), isLNat(XS)), N),
         a__and^#(a__isNatural(N), isLNat(XS)),
         a__isNatural^#(N))
  , a__sel^#(N, XS) ->
    c_85(a__U51^#(a__and(a__isNatural(N), isLNat(XS)), N, XS),
         a__and^#(a__isNatural(N), isLNat(XS)),
         a__isNatural^#(N))
  , a__sel^#(X1, X2) -> c_86()
  , a__U11^#(X1, X2, X3) -> c_38()
  , a__U11^#(tt(), N, XS) ->
    c_39(a__snd^#(a__splitAt(mark(N), mark(XS))),
         a__splitAt^#(mark(N), mark(XS)),
         mark^#(N),
         mark^#(XS))
  , a__U31^#(X1, X2) -> c_44()
  , a__U31^#(tt(), N) -> c_45(mark^#(N))
  , a__U41^#(X1, X2) -> c_46()
  , a__U41^#(tt(), N) -> c_47(mark^#(N))
  , a__U51^#(X1, X2, X3) -> c_48()
  , a__U51^#(tt(), N, XS) ->
    c_49(a__head^#(a__afterNth(mark(N), mark(XS))),
         a__afterNth^#(mark(N), mark(XS)),
         mark^#(N),
         mark^#(XS))
  , a__U61^#(X1, X2) -> c_54()
  , a__U61^#(tt(), Y) -> c_55(mark^#(Y))
  , a__U82^#(X1, X2) -> c_60()
  , a__U82^#(pair(YS, ZS), X) -> c_61(mark^#(X), mark^#(ZS))
  , a__U91^#(X1, X2) -> c_62()
  , a__U91^#(tt(), XS) -> c_63(mark^#(XS))
  , a__isPLNat^#(X) -> c_80()
  , a__isPLNat^#(pair(V1, V2)) ->
    c_81(a__and^#(a__isLNat(V1), isLNat(V2)), a__isLNat^#(V1))
  , a__isPLNat^#(splitAt(V1, V2)) ->
    c_82(a__and^#(a__isNatural(V1), isLNat(V2)), a__isNatural^#(V1)) }

and mark the set of starting terms.

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

Strict DPs:
  { a__U101^#(X1, X2, X3) -> c_1()
  , a__U101^#(tt(), N, XS) ->
    c_2(a__fst^#(a__splitAt(mark(N), mark(XS))),
        a__splitAt^#(mark(N), mark(XS)),
        mark^#(N),
        mark^#(XS))
  , a__fst^#(X) -> c_3()
  , a__fst^#(pair(X, Y)) ->
    c_4(a__U21^#(a__and(a__isLNat(X), isLNat(Y)), X),
        a__and^#(a__isLNat(X), isLNat(Y)),
        a__isLNat^#(X))
  , a__splitAt^#(X1, X2) -> c_5()
  , a__splitAt^#(s(N), cons(X, XS)) ->
    c_6(a__U81^#(a__and(a__isNatural(N),
                        and(isNatural(X), isLNat(XS))),
                 N,
                 X,
                 XS),
        a__and^#(a__isNatural(N), and(isNatural(X), isLNat(XS))),
        a__isNatural^#(N))
  , a__splitAt^#(0(), XS) ->
    c_7(a__U71^#(a__isLNat(XS), XS), a__isLNat^#(XS))
  , mark^#(tt()) -> c_8()
  , mark^#(cons(X1, X2)) -> c_9(mark^#(X1))
  , mark^#(natsFrom(X)) -> c_10(a__natsFrom^#(mark(X)), mark^#(X))
  , mark^#(s(X)) -> c_11(mark^#(X))
  , mark^#(pair(X1, X2)) -> c_12(mark^#(X1), mark^#(X2))
  , mark^#(nil()) -> c_13()
  , mark^#(isLNat(X)) -> c_14(a__isLNat^#(X))
  , mark^#(afterNth(X1, X2)) ->
    c_15(a__afterNth^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(fst(X)) -> c_16(a__fst^#(mark(X)), mark^#(X))
  , mark^#(snd(X)) -> c_17(a__snd^#(mark(X)), mark^#(X))
  , mark^#(tail(X)) -> c_18(a__tail^#(mark(X)), mark^#(X))
  , mark^#(take(X1, X2)) ->
    c_19(a__take^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(0()) -> c_20()
  , mark^#(head(X)) -> c_21(a__head^#(mark(X)), mark^#(X))
  , mark^#(sel(X1, X2)) ->
    c_22(a__sel^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(splitAt(X1, X2)) ->
    c_23(a__splitAt^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(and(X1, X2)) -> c_24(a__and^#(mark(X1), X2), mark^#(X1))
  , mark^#(isNatural(X)) -> c_25(a__isNatural^#(X))
  , mark^#(U101(X1, X2, X3)) ->
    c_26(a__U101^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U11(X1, X2, X3)) ->
    c_27(a__U11^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U21(X1, X2)) -> c_28(a__U21^#(mark(X1), X2), mark^#(X1))
  , mark^#(U31(X1, X2)) -> c_29(a__U31^#(mark(X1), X2), mark^#(X1))
  , mark^#(U41(X1, X2)) -> c_30(a__U41^#(mark(X1), X2), mark^#(X1))
  , mark^#(U51(X1, X2, X3)) ->
    c_31(a__U51^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U61(X1, X2)) -> c_32(a__U61^#(mark(X1), X2), mark^#(X1))
  , mark^#(U71(X1, X2)) -> c_33(a__U71^#(mark(X1), X2), mark^#(X1))
  , mark^#(U81(X1, X2, X3, X4)) ->
    c_34(a__U81^#(mark(X1), X2, X3, X4), mark^#(X1))
  , mark^#(U82(X1, X2)) -> c_35(a__U82^#(mark(X1), X2), mark^#(X1))
  , mark^#(U91(X1, X2)) -> c_36(a__U91^#(mark(X1), X2), mark^#(X1))
  , mark^#(isPLNat(X)) -> c_37(a__isPLNat^#(X))
  , a__U21^#(X1, X2) -> c_42()
  , a__U21^#(tt(), X) -> c_43(mark^#(X))
  , a__and^#(X1, X2) -> c_64()
  , a__and^#(tt(), X) -> c_65(mark^#(X))
  , a__isLNat^#(X) -> c_71()
  , a__isLNat^#(cons(V1, V2)) ->
    c_72(a__and^#(a__isNatural(V1), isLNat(V2)), a__isNatural^#(V1))
  , a__isLNat^#(natsFrom(V1)) -> c_73(a__isNatural^#(V1))
  , a__isLNat^#(nil()) -> c_74()
  , a__isLNat^#(afterNth(V1, V2)) ->
    c_75(a__and^#(a__isNatural(V1), isLNat(V2)), a__isNatural^#(V1))
  , a__isLNat^#(fst(V1)) -> c_76(a__isPLNat^#(V1))
  , a__isLNat^#(snd(V1)) -> c_77(a__isPLNat^#(V1))
  , a__isLNat^#(tail(V1)) -> c_78(a__isLNat^#(V1))
  , a__isLNat^#(take(V1, V2)) ->
    c_79(a__and^#(a__isNatural(V1), isLNat(V2)), a__isNatural^#(V1))
  , a__U81^#(X1, X2, X3, X4) -> c_58()
  , a__U81^#(tt(), N, X, XS) ->
    c_59(a__U82^#(a__splitAt(mark(N), mark(XS)), X),
         a__splitAt^#(mark(N), mark(XS)),
         mark^#(N),
         mark^#(XS))
  , a__isNatural^#(X) -> c_66()
  , a__isNatural^#(s(V1)) -> c_67(a__isNatural^#(V1))
  , a__isNatural^#(0()) -> c_68()
  , a__isNatural^#(head(V1)) -> c_69(a__isLNat^#(V1))
  , a__isNatural^#(sel(V1, V2)) ->
    c_70(a__and^#(a__isNatural(V1), isLNat(V2)), a__isNatural^#(V1))
  , a__U71^#(X1, X2) -> c_56()
  , a__U71^#(tt(), XS) -> c_57(mark^#(XS))
  , a__natsFrom^#(N) ->
    c_83(a__U41^#(a__isNatural(N), N), a__isNatural^#(N))
  , a__natsFrom^#(X) -> c_84()
  , a__afterNth^#(N, XS) ->
    c_52(a__U11^#(a__and(a__isNatural(N), isLNat(XS)), N, XS),
         a__and^#(a__isNatural(N), isLNat(XS)),
         a__isNatural^#(N))
  , a__afterNth^#(X1, X2) -> c_53()
  , a__snd^#(X) -> c_40()
  , a__snd^#(pair(X, Y)) ->
    c_41(a__U61^#(a__and(a__isLNat(X), isLNat(Y)), Y),
         a__and^#(a__isLNat(X), isLNat(Y)),
         a__isLNat^#(X))
  , a__tail^#(X) -> c_87()
  , a__tail^#(cons(N, XS)) ->
    c_88(a__U91^#(a__and(a__isNatural(N), isLNat(XS)), XS),
         a__and^#(a__isNatural(N), isLNat(XS)),
         a__isNatural^#(N))
  , a__take^#(N, XS) ->
    c_89(a__U101^#(a__and(a__isNatural(N), isLNat(XS)), N, XS),
         a__and^#(a__isNatural(N), isLNat(XS)),
         a__isNatural^#(N))
  , a__take^#(X1, X2) -> c_90()
  , a__head^#(X) -> c_50()
  , a__head^#(cons(N, XS)) ->
    c_51(a__U31^#(a__and(a__isNatural(N), isLNat(XS)), N),
         a__and^#(a__isNatural(N), isLNat(XS)),
         a__isNatural^#(N))
  , a__sel^#(N, XS) ->
    c_85(a__U51^#(a__and(a__isNatural(N), isLNat(XS)), N, XS),
         a__and^#(a__isNatural(N), isLNat(XS)),
         a__isNatural^#(N))
  , a__sel^#(X1, X2) -> c_86()
  , a__U11^#(X1, X2, X3) -> c_38()
  , a__U11^#(tt(), N, XS) ->
    c_39(a__snd^#(a__splitAt(mark(N), mark(XS))),
         a__splitAt^#(mark(N), mark(XS)),
         mark^#(N),
         mark^#(XS))
  , a__U31^#(X1, X2) -> c_44()
  , a__U31^#(tt(), N) -> c_45(mark^#(N))
  , a__U41^#(X1, X2) -> c_46()
  , a__U41^#(tt(), N) -> c_47(mark^#(N))
  , a__U51^#(X1, X2, X3) -> c_48()
  , a__U51^#(tt(), N, XS) ->
    c_49(a__head^#(a__afterNth(mark(N), mark(XS))),
         a__afterNth^#(mark(N), mark(XS)),
         mark^#(N),
         mark^#(XS))
  , a__U61^#(X1, X2) -> c_54()
  , a__U61^#(tt(), Y) -> c_55(mark^#(Y))
  , a__U82^#(X1, X2) -> c_60()
  , a__U82^#(pair(YS, ZS), X) -> c_61(mark^#(X), mark^#(ZS))
  , a__U91^#(X1, X2) -> c_62()
  , a__U91^#(tt(), XS) -> c_63(mark^#(XS))
  , a__isPLNat^#(X) -> c_80()
  , a__isPLNat^#(pair(V1, V2)) ->
    c_81(a__and^#(a__isLNat(V1), isLNat(V2)), a__isLNat^#(V1))
  , a__isPLNat^#(splitAt(V1, V2)) ->
    c_82(a__and^#(a__isNatural(V1), isLNat(V2)), a__isNatural^#(V1)) }
Weak Trs:
  { a__U101(X1, X2, X3) -> U101(X1, X2, X3)
  , a__U101(tt(), N, XS) -> a__fst(a__splitAt(mark(N), mark(XS)))
  , a__fst(X) -> fst(X)
  , a__fst(pair(X, Y)) -> a__U21(a__and(a__isLNat(X), isLNat(Y)), X)
  , a__splitAt(X1, X2) -> splitAt(X1, X2)
  , a__splitAt(s(N), cons(X, XS)) ->
    a__U81(a__and(a__isNatural(N), and(isNatural(X), isLNat(XS))),
           N,
           X,
           XS)
  , a__splitAt(0(), XS) -> a__U71(a__isLNat(XS), XS)
  , mark(tt()) -> tt()
  , 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(nil()) -> nil()
  , mark(isLNat(X)) -> a__isLNat(X)
  , mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2))
  , mark(fst(X)) -> a__fst(mark(X))
  , mark(snd(X)) -> a__snd(mark(X))
  , mark(tail(X)) -> a__tail(mark(X))
  , mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
  , mark(0()) -> 0()
  , mark(head(X)) -> a__head(mark(X))
  , mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2))
  , mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2))
  , mark(and(X1, X2)) -> a__and(mark(X1), X2)
  , mark(isNatural(X)) -> a__isNatural(X)
  , mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3)
  , mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
  , mark(U21(X1, X2)) -> a__U21(mark(X1), X2)
  , mark(U31(X1, X2)) -> a__U31(mark(X1), X2)
  , mark(U41(X1, X2)) -> a__U41(mark(X1), X2)
  , mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3)
  , mark(U61(X1, X2)) -> a__U61(mark(X1), X2)
  , mark(U71(X1, X2)) -> a__U71(mark(X1), X2)
  , mark(U81(X1, X2, X3, X4)) -> a__U81(mark(X1), X2, X3, X4)
  , mark(U82(X1, X2)) -> a__U82(mark(X1), X2)
  , mark(U91(X1, X2)) -> a__U91(mark(X1), X2)
  , mark(isPLNat(X)) -> a__isPLNat(X)
  , a__U11(X1, X2, X3) -> U11(X1, X2, X3)
  , a__U11(tt(), N, XS) -> a__snd(a__splitAt(mark(N), mark(XS)))
  , a__snd(X) -> snd(X)
  , a__snd(pair(X, Y)) -> a__U61(a__and(a__isLNat(X), isLNat(Y)), Y)
  , a__U21(X1, X2) -> U21(X1, X2)
  , a__U21(tt(), X) -> mark(X)
  , a__U31(X1, X2) -> U31(X1, X2)
  , a__U31(tt(), N) -> mark(N)
  , a__U41(X1, X2) -> U41(X1, X2)
  , a__U41(tt(), N) -> cons(mark(N), natsFrom(s(N)))
  , a__U51(X1, X2, X3) -> U51(X1, X2, X3)
  , a__U51(tt(), N, XS) -> a__head(a__afterNth(mark(N), mark(XS)))
  , a__head(X) -> head(X)
  , a__head(cons(N, XS)) ->
    a__U31(a__and(a__isNatural(N), isLNat(XS)), N)
  , a__afterNth(N, XS) ->
    a__U11(a__and(a__isNatural(N), isLNat(XS)), N, XS)
  , a__afterNth(X1, X2) -> afterNth(X1, X2)
  , a__U61(X1, X2) -> U61(X1, X2)
  , a__U61(tt(), Y) -> mark(Y)
  , a__U71(X1, X2) -> U71(X1, X2)
  , a__U71(tt(), XS) -> pair(nil(), mark(XS))
  , a__U81(X1, X2, X3, X4) -> U81(X1, X2, X3, X4)
  , a__U81(tt(), N, X, XS) ->
    a__U82(a__splitAt(mark(N), mark(XS)), X)
  , a__U82(X1, X2) -> U82(X1, X2)
  , a__U82(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS))
  , a__U91(X1, X2) -> U91(X1, X2)
  , a__U91(tt(), XS) -> mark(XS)
  , a__and(X1, X2) -> and(X1, X2)
  , a__and(tt(), X) -> mark(X)
  , a__isNatural(X) -> isNatural(X)
  , a__isNatural(s(V1)) -> a__isNatural(V1)
  , a__isNatural(0()) -> tt()
  , a__isNatural(head(V1)) -> a__isLNat(V1)
  , a__isNatural(sel(V1, V2)) -> a__and(a__isNatural(V1), isLNat(V2))
  , a__isLNat(X) -> isLNat(X)
  , a__isLNat(cons(V1, V2)) -> a__and(a__isNatural(V1), isLNat(V2))
  , a__isLNat(natsFrom(V1)) -> a__isNatural(V1)
  , a__isLNat(nil()) -> tt()
  , a__isLNat(afterNth(V1, V2)) ->
    a__and(a__isNatural(V1), isLNat(V2))
  , a__isLNat(fst(V1)) -> a__isPLNat(V1)
  , a__isLNat(snd(V1)) -> a__isPLNat(V1)
  , a__isLNat(tail(V1)) -> a__isLNat(V1)
  , a__isLNat(take(V1, V2)) -> a__and(a__isNatural(V1), isLNat(V2))
  , a__isPLNat(X) -> isPLNat(X)
  , a__isPLNat(pair(V1, V2)) -> a__and(a__isLNat(V1), isLNat(V2))
  , a__isPLNat(splitAt(V1, V2)) ->
    a__and(a__isNatural(V1), isLNat(V2))
  , a__natsFrom(N) -> a__U41(a__isNatural(N), N)
  , a__natsFrom(X) -> natsFrom(X)
  , a__sel(N, XS) ->
    a__U51(a__and(a__isNatural(N), isLNat(XS)), N, XS)
  , a__sel(X1, X2) -> sel(X1, X2)
  , a__tail(X) -> tail(X)
  , a__tail(cons(N, XS)) ->
    a__U91(a__and(a__isNatural(N), isLNat(XS)), XS)
  , a__take(N, XS) ->
    a__U101(a__and(a__isNatural(N), isLNat(XS)), N, XS)
  , a__take(X1, X2) -> take(X1, X2) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of
{1,3,5,8,13,20,38,40,42,45,51,53,55,58,61,63,64,66,69,70,73,74,76,78,80,82,84,86,88}
by applications of
Pre({1,3,5,8,13,20,38,40,42,45,51,53,55,58,61,63,64,66,69,70,73,74,76,78,80,82,84,86,88})
=
{2,4,6,7,9,10,11,12,14,15,16,17,18,19,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,39,41,43,44,46,47,48,49,50,52,54,56,57,59,60,62,65,67,68,71,72,75,77,79,81,83,85,87,89,90}.
Here rules are labeled as follows:

  DPs:
    { 1: a__U101^#(X1, X2, X3) -> c_1()
    , 2: a__U101^#(tt(), N, XS) ->
         c_2(a__fst^#(a__splitAt(mark(N), mark(XS))),
             a__splitAt^#(mark(N), mark(XS)),
             mark^#(N),
             mark^#(XS))
    , 3: a__fst^#(X) -> c_3()
    , 4: a__fst^#(pair(X, Y)) ->
         c_4(a__U21^#(a__and(a__isLNat(X), isLNat(Y)), X),
             a__and^#(a__isLNat(X), isLNat(Y)),
             a__isLNat^#(X))
    , 5: a__splitAt^#(X1, X2) -> c_5()
    , 6: a__splitAt^#(s(N), cons(X, XS)) ->
         c_6(a__U81^#(a__and(a__isNatural(N),
                             and(isNatural(X), isLNat(XS))),
                      N,
                      X,
                      XS),
             a__and^#(a__isNatural(N), and(isNatural(X), isLNat(XS))),
             a__isNatural^#(N))
    , 7: a__splitAt^#(0(), XS) ->
         c_7(a__U71^#(a__isLNat(XS), XS), a__isLNat^#(XS))
    , 8: mark^#(tt()) -> c_8()
    , 9: mark^#(cons(X1, X2)) -> c_9(mark^#(X1))
    , 10: mark^#(natsFrom(X)) ->
          c_10(a__natsFrom^#(mark(X)), mark^#(X))
    , 11: mark^#(s(X)) -> c_11(mark^#(X))
    , 12: mark^#(pair(X1, X2)) -> c_12(mark^#(X1), mark^#(X2))
    , 13: mark^#(nil()) -> c_13()
    , 14: mark^#(isLNat(X)) -> c_14(a__isLNat^#(X))
    , 15: mark^#(afterNth(X1, X2)) ->
          c_15(a__afterNth^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
    , 16: mark^#(fst(X)) -> c_16(a__fst^#(mark(X)), mark^#(X))
    , 17: mark^#(snd(X)) -> c_17(a__snd^#(mark(X)), mark^#(X))
    , 18: mark^#(tail(X)) -> c_18(a__tail^#(mark(X)), mark^#(X))
    , 19: mark^#(take(X1, X2)) ->
          c_19(a__take^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
    , 20: mark^#(0()) -> c_20()
    , 21: mark^#(head(X)) -> c_21(a__head^#(mark(X)), mark^#(X))
    , 22: mark^#(sel(X1, X2)) ->
          c_22(a__sel^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
    , 23: mark^#(splitAt(X1, X2)) ->
          c_23(a__splitAt^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
    , 24: mark^#(and(X1, X2)) ->
          c_24(a__and^#(mark(X1), X2), mark^#(X1))
    , 25: mark^#(isNatural(X)) -> c_25(a__isNatural^#(X))
    , 26: mark^#(U101(X1, X2, X3)) ->
          c_26(a__U101^#(mark(X1), X2, X3), mark^#(X1))
    , 27: mark^#(U11(X1, X2, X3)) ->
          c_27(a__U11^#(mark(X1), X2, X3), mark^#(X1))
    , 28: mark^#(U21(X1, X2)) ->
          c_28(a__U21^#(mark(X1), X2), mark^#(X1))
    , 29: mark^#(U31(X1, X2)) ->
          c_29(a__U31^#(mark(X1), X2), mark^#(X1))
    , 30: mark^#(U41(X1, X2)) ->
          c_30(a__U41^#(mark(X1), X2), mark^#(X1))
    , 31: mark^#(U51(X1, X2, X3)) ->
          c_31(a__U51^#(mark(X1), X2, X3), mark^#(X1))
    , 32: mark^#(U61(X1, X2)) ->
          c_32(a__U61^#(mark(X1), X2), mark^#(X1))
    , 33: mark^#(U71(X1, X2)) ->
          c_33(a__U71^#(mark(X1), X2), mark^#(X1))
    , 34: mark^#(U81(X1, X2, X3, X4)) ->
          c_34(a__U81^#(mark(X1), X2, X3, X4), mark^#(X1))
    , 35: mark^#(U82(X1, X2)) ->
          c_35(a__U82^#(mark(X1), X2), mark^#(X1))
    , 36: mark^#(U91(X1, X2)) ->
          c_36(a__U91^#(mark(X1), X2), mark^#(X1))
    , 37: mark^#(isPLNat(X)) -> c_37(a__isPLNat^#(X))
    , 38: a__U21^#(X1, X2) -> c_42()
    , 39: a__U21^#(tt(), X) -> c_43(mark^#(X))
    , 40: a__and^#(X1, X2) -> c_64()
    , 41: a__and^#(tt(), X) -> c_65(mark^#(X))
    , 42: a__isLNat^#(X) -> c_71()
    , 43: a__isLNat^#(cons(V1, V2)) ->
          c_72(a__and^#(a__isNatural(V1), isLNat(V2)), a__isNatural^#(V1))
    , 44: a__isLNat^#(natsFrom(V1)) -> c_73(a__isNatural^#(V1))
    , 45: a__isLNat^#(nil()) -> c_74()
    , 46: a__isLNat^#(afterNth(V1, V2)) ->
          c_75(a__and^#(a__isNatural(V1), isLNat(V2)), a__isNatural^#(V1))
    , 47: a__isLNat^#(fst(V1)) -> c_76(a__isPLNat^#(V1))
    , 48: a__isLNat^#(snd(V1)) -> c_77(a__isPLNat^#(V1))
    , 49: a__isLNat^#(tail(V1)) -> c_78(a__isLNat^#(V1))
    , 50: a__isLNat^#(take(V1, V2)) ->
          c_79(a__and^#(a__isNatural(V1), isLNat(V2)), a__isNatural^#(V1))
    , 51: a__U81^#(X1, X2, X3, X4) -> c_58()
    , 52: a__U81^#(tt(), N, X, XS) ->
          c_59(a__U82^#(a__splitAt(mark(N), mark(XS)), X),
               a__splitAt^#(mark(N), mark(XS)),
               mark^#(N),
               mark^#(XS))
    , 53: a__isNatural^#(X) -> c_66()
    , 54: a__isNatural^#(s(V1)) -> c_67(a__isNatural^#(V1))
    , 55: a__isNatural^#(0()) -> c_68()
    , 56: a__isNatural^#(head(V1)) -> c_69(a__isLNat^#(V1))
    , 57: a__isNatural^#(sel(V1, V2)) ->
          c_70(a__and^#(a__isNatural(V1), isLNat(V2)), a__isNatural^#(V1))
    , 58: a__U71^#(X1, X2) -> c_56()
    , 59: a__U71^#(tt(), XS) -> c_57(mark^#(XS))
    , 60: a__natsFrom^#(N) ->
          c_83(a__U41^#(a__isNatural(N), N), a__isNatural^#(N))
    , 61: a__natsFrom^#(X) -> c_84()
    , 62: a__afterNth^#(N, XS) ->
          c_52(a__U11^#(a__and(a__isNatural(N), isLNat(XS)), N, XS),
               a__and^#(a__isNatural(N), isLNat(XS)),
               a__isNatural^#(N))
    , 63: a__afterNth^#(X1, X2) -> c_53()
    , 64: a__snd^#(X) -> c_40()
    , 65: a__snd^#(pair(X, Y)) ->
          c_41(a__U61^#(a__and(a__isLNat(X), isLNat(Y)), Y),
               a__and^#(a__isLNat(X), isLNat(Y)),
               a__isLNat^#(X))
    , 66: a__tail^#(X) -> c_87()
    , 67: a__tail^#(cons(N, XS)) ->
          c_88(a__U91^#(a__and(a__isNatural(N), isLNat(XS)), XS),
               a__and^#(a__isNatural(N), isLNat(XS)),
               a__isNatural^#(N))
    , 68: a__take^#(N, XS) ->
          c_89(a__U101^#(a__and(a__isNatural(N), isLNat(XS)), N, XS),
               a__and^#(a__isNatural(N), isLNat(XS)),
               a__isNatural^#(N))
    , 69: a__take^#(X1, X2) -> c_90()
    , 70: a__head^#(X) -> c_50()
    , 71: a__head^#(cons(N, XS)) ->
          c_51(a__U31^#(a__and(a__isNatural(N), isLNat(XS)), N),
               a__and^#(a__isNatural(N), isLNat(XS)),
               a__isNatural^#(N))
    , 72: a__sel^#(N, XS) ->
          c_85(a__U51^#(a__and(a__isNatural(N), isLNat(XS)), N, XS),
               a__and^#(a__isNatural(N), isLNat(XS)),
               a__isNatural^#(N))
    , 73: a__sel^#(X1, X2) -> c_86()
    , 74: a__U11^#(X1, X2, X3) -> c_38()
    , 75: a__U11^#(tt(), N, XS) ->
          c_39(a__snd^#(a__splitAt(mark(N), mark(XS))),
               a__splitAt^#(mark(N), mark(XS)),
               mark^#(N),
               mark^#(XS))
    , 76: a__U31^#(X1, X2) -> c_44()
    , 77: a__U31^#(tt(), N) -> c_45(mark^#(N))
    , 78: a__U41^#(X1, X2) -> c_46()
    , 79: a__U41^#(tt(), N) -> c_47(mark^#(N))
    , 80: a__U51^#(X1, X2, X3) -> c_48()
    , 81: a__U51^#(tt(), N, XS) ->
          c_49(a__head^#(a__afterNth(mark(N), mark(XS))),
               a__afterNth^#(mark(N), mark(XS)),
               mark^#(N),
               mark^#(XS))
    , 82: a__U61^#(X1, X2) -> c_54()
    , 83: a__U61^#(tt(), Y) -> c_55(mark^#(Y))
    , 84: a__U82^#(X1, X2) -> c_60()
    , 85: a__U82^#(pair(YS, ZS), X) -> c_61(mark^#(X), mark^#(ZS))
    , 86: a__U91^#(X1, X2) -> c_62()
    , 87: a__U91^#(tt(), XS) -> c_63(mark^#(XS))
    , 88: a__isPLNat^#(X) -> c_80()
    , 89: a__isPLNat^#(pair(V1, V2)) ->
          c_81(a__and^#(a__isLNat(V1), isLNat(V2)), a__isLNat^#(V1))
    , 90: a__isPLNat^#(splitAt(V1, V2)) ->
          c_82(a__and^#(a__isNatural(V1), isLNat(V2)), a__isNatural^#(V1)) }

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

Strict DPs:
  { a__U101^#(tt(), N, XS) ->
    c_2(a__fst^#(a__splitAt(mark(N), mark(XS))),
        a__splitAt^#(mark(N), mark(XS)),
        mark^#(N),
        mark^#(XS))
  , a__fst^#(pair(X, Y)) ->
    c_4(a__U21^#(a__and(a__isLNat(X), isLNat(Y)), X),
        a__and^#(a__isLNat(X), isLNat(Y)),
        a__isLNat^#(X))
  , a__splitAt^#(s(N), cons(X, XS)) ->
    c_6(a__U81^#(a__and(a__isNatural(N),
                        and(isNatural(X), isLNat(XS))),
                 N,
                 X,
                 XS),
        a__and^#(a__isNatural(N), and(isNatural(X), isLNat(XS))),
        a__isNatural^#(N))
  , a__splitAt^#(0(), XS) ->
    c_7(a__U71^#(a__isLNat(XS), XS), a__isLNat^#(XS))
  , mark^#(cons(X1, X2)) -> c_9(mark^#(X1))
  , mark^#(natsFrom(X)) -> c_10(a__natsFrom^#(mark(X)), mark^#(X))
  , mark^#(s(X)) -> c_11(mark^#(X))
  , mark^#(pair(X1, X2)) -> c_12(mark^#(X1), mark^#(X2))
  , mark^#(isLNat(X)) -> c_14(a__isLNat^#(X))
  , mark^#(afterNth(X1, X2)) ->
    c_15(a__afterNth^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(fst(X)) -> c_16(a__fst^#(mark(X)), mark^#(X))
  , mark^#(snd(X)) -> c_17(a__snd^#(mark(X)), mark^#(X))
  , mark^#(tail(X)) -> c_18(a__tail^#(mark(X)), mark^#(X))
  , mark^#(take(X1, X2)) ->
    c_19(a__take^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(head(X)) -> c_21(a__head^#(mark(X)), mark^#(X))
  , mark^#(sel(X1, X2)) ->
    c_22(a__sel^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(splitAt(X1, X2)) ->
    c_23(a__splitAt^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(and(X1, X2)) -> c_24(a__and^#(mark(X1), X2), mark^#(X1))
  , mark^#(isNatural(X)) -> c_25(a__isNatural^#(X))
  , mark^#(U101(X1, X2, X3)) ->
    c_26(a__U101^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U11(X1, X2, X3)) ->
    c_27(a__U11^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U21(X1, X2)) -> c_28(a__U21^#(mark(X1), X2), mark^#(X1))
  , mark^#(U31(X1, X2)) -> c_29(a__U31^#(mark(X1), X2), mark^#(X1))
  , mark^#(U41(X1, X2)) -> c_30(a__U41^#(mark(X1), X2), mark^#(X1))
  , mark^#(U51(X1, X2, X3)) ->
    c_31(a__U51^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U61(X1, X2)) -> c_32(a__U61^#(mark(X1), X2), mark^#(X1))
  , mark^#(U71(X1, X2)) -> c_33(a__U71^#(mark(X1), X2), mark^#(X1))
  , mark^#(U81(X1, X2, X3, X4)) ->
    c_34(a__U81^#(mark(X1), X2, X3, X4), mark^#(X1))
  , mark^#(U82(X1, X2)) -> c_35(a__U82^#(mark(X1), X2), mark^#(X1))
  , mark^#(U91(X1, X2)) -> c_36(a__U91^#(mark(X1), X2), mark^#(X1))
  , mark^#(isPLNat(X)) -> c_37(a__isPLNat^#(X))
  , a__U21^#(tt(), X) -> c_43(mark^#(X))
  , a__and^#(tt(), X) -> c_65(mark^#(X))
  , a__isLNat^#(cons(V1, V2)) ->
    c_72(a__and^#(a__isNatural(V1), isLNat(V2)), a__isNatural^#(V1))
  , a__isLNat^#(natsFrom(V1)) -> c_73(a__isNatural^#(V1))
  , a__isLNat^#(afterNth(V1, V2)) ->
    c_75(a__and^#(a__isNatural(V1), isLNat(V2)), a__isNatural^#(V1))
  , a__isLNat^#(fst(V1)) -> c_76(a__isPLNat^#(V1))
  , a__isLNat^#(snd(V1)) -> c_77(a__isPLNat^#(V1))
  , a__isLNat^#(tail(V1)) -> c_78(a__isLNat^#(V1))
  , a__isLNat^#(take(V1, V2)) ->
    c_79(a__and^#(a__isNatural(V1), isLNat(V2)), a__isNatural^#(V1))
  , a__U81^#(tt(), N, X, XS) ->
    c_59(a__U82^#(a__splitAt(mark(N), mark(XS)), X),
         a__splitAt^#(mark(N), mark(XS)),
         mark^#(N),
         mark^#(XS))
  , a__isNatural^#(s(V1)) -> c_67(a__isNatural^#(V1))
  , a__isNatural^#(head(V1)) -> c_69(a__isLNat^#(V1))
  , a__isNatural^#(sel(V1, V2)) ->
    c_70(a__and^#(a__isNatural(V1), isLNat(V2)), a__isNatural^#(V1))
  , a__U71^#(tt(), XS) -> c_57(mark^#(XS))
  , a__natsFrom^#(N) ->
    c_83(a__U41^#(a__isNatural(N), N), a__isNatural^#(N))
  , a__afterNth^#(N, XS) ->
    c_52(a__U11^#(a__and(a__isNatural(N), isLNat(XS)), N, XS),
         a__and^#(a__isNatural(N), isLNat(XS)),
         a__isNatural^#(N))
  , a__snd^#(pair(X, Y)) ->
    c_41(a__U61^#(a__and(a__isLNat(X), isLNat(Y)), Y),
         a__and^#(a__isLNat(X), isLNat(Y)),
         a__isLNat^#(X))
  , a__tail^#(cons(N, XS)) ->
    c_88(a__U91^#(a__and(a__isNatural(N), isLNat(XS)), XS),
         a__and^#(a__isNatural(N), isLNat(XS)),
         a__isNatural^#(N))
  , a__take^#(N, XS) ->
    c_89(a__U101^#(a__and(a__isNatural(N), isLNat(XS)), N, XS),
         a__and^#(a__isNatural(N), isLNat(XS)),
         a__isNatural^#(N))
  , a__head^#(cons(N, XS)) ->
    c_51(a__U31^#(a__and(a__isNatural(N), isLNat(XS)), N),
         a__and^#(a__isNatural(N), isLNat(XS)),
         a__isNatural^#(N))
  , a__sel^#(N, XS) ->
    c_85(a__U51^#(a__and(a__isNatural(N), isLNat(XS)), N, XS),
         a__and^#(a__isNatural(N), isLNat(XS)),
         a__isNatural^#(N))
  , a__U11^#(tt(), N, XS) ->
    c_39(a__snd^#(a__splitAt(mark(N), mark(XS))),
         a__splitAt^#(mark(N), mark(XS)),
         mark^#(N),
         mark^#(XS))
  , a__U31^#(tt(), N) -> c_45(mark^#(N))
  , a__U41^#(tt(), N) -> c_47(mark^#(N))
  , a__U51^#(tt(), N, XS) ->
    c_49(a__head^#(a__afterNth(mark(N), mark(XS))),
         a__afterNth^#(mark(N), mark(XS)),
         mark^#(N),
         mark^#(XS))
  , a__U61^#(tt(), Y) -> c_55(mark^#(Y))
  , a__U82^#(pair(YS, ZS), X) -> c_61(mark^#(X), mark^#(ZS))
  , a__U91^#(tt(), XS) -> c_63(mark^#(XS))
  , a__isPLNat^#(pair(V1, V2)) ->
    c_81(a__and^#(a__isLNat(V1), isLNat(V2)), a__isLNat^#(V1))
  , a__isPLNat^#(splitAt(V1, V2)) ->
    c_82(a__and^#(a__isNatural(V1), isLNat(V2)), a__isNatural^#(V1)) }
Weak DPs:
  { a__U101^#(X1, X2, X3) -> c_1()
  , a__fst^#(X) -> c_3()
  , a__splitAt^#(X1, X2) -> c_5()
  , mark^#(tt()) -> c_8()
  , mark^#(nil()) -> c_13()
  , mark^#(0()) -> c_20()
  , a__U21^#(X1, X2) -> c_42()
  , a__and^#(X1, X2) -> c_64()
  , a__isLNat^#(X) -> c_71()
  , a__isLNat^#(nil()) -> c_74()
  , a__U81^#(X1, X2, X3, X4) -> c_58()
  , a__isNatural^#(X) -> c_66()
  , a__isNatural^#(0()) -> c_68()
  , a__U71^#(X1, X2) -> c_56()
  , a__natsFrom^#(X) -> c_84()
  , a__afterNth^#(X1, X2) -> c_53()
  , a__snd^#(X) -> c_40()
  , a__tail^#(X) -> c_87()
  , a__take^#(X1, X2) -> c_90()
  , a__head^#(X) -> c_50()
  , a__sel^#(X1, X2) -> c_86()
  , a__U11^#(X1, X2, X3) -> c_38()
  , a__U31^#(X1, X2) -> c_44()
  , a__U41^#(X1, X2) -> c_46()
  , a__U51^#(X1, X2, X3) -> c_48()
  , a__U61^#(X1, X2) -> c_54()
  , a__U82^#(X1, X2) -> c_60()
  , a__U91^#(X1, X2) -> c_62()
  , a__isPLNat^#(X) -> c_80() }
Weak Trs:
  { a__U101(X1, X2, X3) -> U101(X1, X2, X3)
  , a__U101(tt(), N, XS) -> a__fst(a__splitAt(mark(N), mark(XS)))
  , a__fst(X) -> fst(X)
  , a__fst(pair(X, Y)) -> a__U21(a__and(a__isLNat(X), isLNat(Y)), X)
  , a__splitAt(X1, X2) -> splitAt(X1, X2)
  , a__splitAt(s(N), cons(X, XS)) ->
    a__U81(a__and(a__isNatural(N), and(isNatural(X), isLNat(XS))),
           N,
           X,
           XS)
  , a__splitAt(0(), XS) -> a__U71(a__isLNat(XS), XS)
  , mark(tt()) -> tt()
  , 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(nil()) -> nil()
  , mark(isLNat(X)) -> a__isLNat(X)
  , mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2))
  , mark(fst(X)) -> a__fst(mark(X))
  , mark(snd(X)) -> a__snd(mark(X))
  , mark(tail(X)) -> a__tail(mark(X))
  , mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
  , mark(0()) -> 0()
  , mark(head(X)) -> a__head(mark(X))
  , mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2))
  , mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2))
  , mark(and(X1, X2)) -> a__and(mark(X1), X2)
  , mark(isNatural(X)) -> a__isNatural(X)
  , mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3)
  , mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
  , mark(U21(X1, X2)) -> a__U21(mark(X1), X2)
  , mark(U31(X1, X2)) -> a__U31(mark(X1), X2)
  , mark(U41(X1, X2)) -> a__U41(mark(X1), X2)
  , mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3)
  , mark(U61(X1, X2)) -> a__U61(mark(X1), X2)
  , mark(U71(X1, X2)) -> a__U71(mark(X1), X2)
  , mark(U81(X1, X2, X3, X4)) -> a__U81(mark(X1), X2, X3, X4)
  , mark(U82(X1, X2)) -> a__U82(mark(X1), X2)
  , mark(U91(X1, X2)) -> a__U91(mark(X1), X2)
  , mark(isPLNat(X)) -> a__isPLNat(X)
  , a__U11(X1, X2, X3) -> U11(X1, X2, X3)
  , a__U11(tt(), N, XS) -> a__snd(a__splitAt(mark(N), mark(XS)))
  , a__snd(X) -> snd(X)
  , a__snd(pair(X, Y)) -> a__U61(a__and(a__isLNat(X), isLNat(Y)), Y)
  , a__U21(X1, X2) -> U21(X1, X2)
  , a__U21(tt(), X) -> mark(X)
  , a__U31(X1, X2) -> U31(X1, X2)
  , a__U31(tt(), N) -> mark(N)
  , a__U41(X1, X2) -> U41(X1, X2)
  , a__U41(tt(), N) -> cons(mark(N), natsFrom(s(N)))
  , a__U51(X1, X2, X3) -> U51(X1, X2, X3)
  , a__U51(tt(), N, XS) -> a__head(a__afterNth(mark(N), mark(XS)))
  , a__head(X) -> head(X)
  , a__head(cons(N, XS)) ->
    a__U31(a__and(a__isNatural(N), isLNat(XS)), N)
  , a__afterNth(N, XS) ->
    a__U11(a__and(a__isNatural(N), isLNat(XS)), N, XS)
  , a__afterNth(X1, X2) -> afterNth(X1, X2)
  , a__U61(X1, X2) -> U61(X1, X2)
  , a__U61(tt(), Y) -> mark(Y)
  , a__U71(X1, X2) -> U71(X1, X2)
  , a__U71(tt(), XS) -> pair(nil(), mark(XS))
  , a__U81(X1, X2, X3, X4) -> U81(X1, X2, X3, X4)
  , a__U81(tt(), N, X, XS) ->
    a__U82(a__splitAt(mark(N), mark(XS)), X)
  , a__U82(X1, X2) -> U82(X1, X2)
  , a__U82(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS))
  , a__U91(X1, X2) -> U91(X1, X2)
  , a__U91(tt(), XS) -> mark(XS)
  , a__and(X1, X2) -> and(X1, X2)
  , a__and(tt(), X) -> mark(X)
  , a__isNatural(X) -> isNatural(X)
  , a__isNatural(s(V1)) -> a__isNatural(V1)
  , a__isNatural(0()) -> tt()
  , a__isNatural(head(V1)) -> a__isLNat(V1)
  , a__isNatural(sel(V1, V2)) -> a__and(a__isNatural(V1), isLNat(V2))
  , a__isLNat(X) -> isLNat(X)
  , a__isLNat(cons(V1, V2)) -> a__and(a__isNatural(V1), isLNat(V2))
  , a__isLNat(natsFrom(V1)) -> a__isNatural(V1)
  , a__isLNat(nil()) -> tt()
  , a__isLNat(afterNth(V1, V2)) ->
    a__and(a__isNatural(V1), isLNat(V2))
  , a__isLNat(fst(V1)) -> a__isPLNat(V1)
  , a__isLNat(snd(V1)) -> a__isPLNat(V1)
  , a__isLNat(tail(V1)) -> a__isLNat(V1)
  , a__isLNat(take(V1, V2)) -> a__and(a__isNatural(V1), isLNat(V2))
  , a__isPLNat(X) -> isPLNat(X)
  , a__isPLNat(pair(V1, V2)) -> a__and(a__isLNat(V1), isLNat(V2))
  , a__isPLNat(splitAt(V1, V2)) ->
    a__and(a__isNatural(V1), isLNat(V2))
  , a__natsFrom(N) -> a__U41(a__isNatural(N), N)
  , a__natsFrom(X) -> natsFrom(X)
  , a__sel(N, XS) ->
    a__U51(a__and(a__isNatural(N), isLNat(XS)), N, XS)
  , a__sel(X1, X2) -> sel(X1, X2)
  , a__tail(X) -> tail(X)
  , a__tail(cons(N, XS)) ->
    a__U91(a__and(a__isNatural(N), isLNat(XS)), XS)
  , a__take(N, XS) ->
    a__U101(a__and(a__isNatural(N), isLNat(XS)), 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__U101^#(X1, X2, X3) -> c_1()
, a__fst^#(X) -> c_3()
, a__splitAt^#(X1, X2) -> c_5()
, mark^#(tt()) -> c_8()
, mark^#(nil()) -> c_13()
, mark^#(0()) -> c_20()
, a__U21^#(X1, X2) -> c_42()
, a__and^#(X1, X2) -> c_64()
, a__isLNat^#(X) -> c_71()
, a__isLNat^#(nil()) -> c_74()
, a__U81^#(X1, X2, X3, X4) -> c_58()
, a__isNatural^#(X) -> c_66()
, a__isNatural^#(0()) -> c_68()
, a__U71^#(X1, X2) -> c_56()
, a__natsFrom^#(X) -> c_84()
, a__afterNth^#(X1, X2) -> c_53()
, a__snd^#(X) -> c_40()
, a__tail^#(X) -> c_87()
, a__take^#(X1, X2) -> c_90()
, a__head^#(X) -> c_50()
, a__sel^#(X1, X2) -> c_86()
, a__U11^#(X1, X2, X3) -> c_38()
, a__U31^#(X1, X2) -> c_44()
, a__U41^#(X1, X2) -> c_46()
, a__U51^#(X1, X2, X3) -> c_48()
, a__U61^#(X1, X2) -> c_54()
, a__U82^#(X1, X2) -> c_60()
, a__U91^#(X1, X2) -> c_62()
, a__isPLNat^#(X) -> c_80() }

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

Strict DPs:
  { a__U101^#(tt(), N, XS) ->
    c_2(a__fst^#(a__splitAt(mark(N), mark(XS))),
        a__splitAt^#(mark(N), mark(XS)),
        mark^#(N),
        mark^#(XS))
  , a__fst^#(pair(X, Y)) ->
    c_4(a__U21^#(a__and(a__isLNat(X), isLNat(Y)), X),
        a__and^#(a__isLNat(X), isLNat(Y)),
        a__isLNat^#(X))
  , a__splitAt^#(s(N), cons(X, XS)) ->
    c_6(a__U81^#(a__and(a__isNatural(N),
                        and(isNatural(X), isLNat(XS))),
                 N,
                 X,
                 XS),
        a__and^#(a__isNatural(N), and(isNatural(X), isLNat(XS))),
        a__isNatural^#(N))
  , a__splitAt^#(0(), XS) ->
    c_7(a__U71^#(a__isLNat(XS), XS), a__isLNat^#(XS))
  , mark^#(cons(X1, X2)) -> c_9(mark^#(X1))
  , mark^#(natsFrom(X)) -> c_10(a__natsFrom^#(mark(X)), mark^#(X))
  , mark^#(s(X)) -> c_11(mark^#(X))
  , mark^#(pair(X1, X2)) -> c_12(mark^#(X1), mark^#(X2))
  , mark^#(isLNat(X)) -> c_14(a__isLNat^#(X))
  , mark^#(afterNth(X1, X2)) ->
    c_15(a__afterNth^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(fst(X)) -> c_16(a__fst^#(mark(X)), mark^#(X))
  , mark^#(snd(X)) -> c_17(a__snd^#(mark(X)), mark^#(X))
  , mark^#(tail(X)) -> c_18(a__tail^#(mark(X)), mark^#(X))
  , mark^#(take(X1, X2)) ->
    c_19(a__take^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(head(X)) -> c_21(a__head^#(mark(X)), mark^#(X))
  , mark^#(sel(X1, X2)) ->
    c_22(a__sel^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(splitAt(X1, X2)) ->
    c_23(a__splitAt^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(and(X1, X2)) -> c_24(a__and^#(mark(X1), X2), mark^#(X1))
  , mark^#(isNatural(X)) -> c_25(a__isNatural^#(X))
  , mark^#(U101(X1, X2, X3)) ->
    c_26(a__U101^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U11(X1, X2, X3)) ->
    c_27(a__U11^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U21(X1, X2)) -> c_28(a__U21^#(mark(X1), X2), mark^#(X1))
  , mark^#(U31(X1, X2)) -> c_29(a__U31^#(mark(X1), X2), mark^#(X1))
  , mark^#(U41(X1, X2)) -> c_30(a__U41^#(mark(X1), X2), mark^#(X1))
  , mark^#(U51(X1, X2, X3)) ->
    c_31(a__U51^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U61(X1, X2)) -> c_32(a__U61^#(mark(X1), X2), mark^#(X1))
  , mark^#(U71(X1, X2)) -> c_33(a__U71^#(mark(X1), X2), mark^#(X1))
  , mark^#(U81(X1, X2, X3, X4)) ->
    c_34(a__U81^#(mark(X1), X2, X3, X4), mark^#(X1))
  , mark^#(U82(X1, X2)) -> c_35(a__U82^#(mark(X1), X2), mark^#(X1))
  , mark^#(U91(X1, X2)) -> c_36(a__U91^#(mark(X1), X2), mark^#(X1))
  , mark^#(isPLNat(X)) -> c_37(a__isPLNat^#(X))
  , a__U21^#(tt(), X) -> c_43(mark^#(X))
  , a__and^#(tt(), X) -> c_65(mark^#(X))
  , a__isLNat^#(cons(V1, V2)) ->
    c_72(a__and^#(a__isNatural(V1), isLNat(V2)), a__isNatural^#(V1))
  , a__isLNat^#(natsFrom(V1)) -> c_73(a__isNatural^#(V1))
  , a__isLNat^#(afterNth(V1, V2)) ->
    c_75(a__and^#(a__isNatural(V1), isLNat(V2)), a__isNatural^#(V1))
  , a__isLNat^#(fst(V1)) -> c_76(a__isPLNat^#(V1))
  , a__isLNat^#(snd(V1)) -> c_77(a__isPLNat^#(V1))
  , a__isLNat^#(tail(V1)) -> c_78(a__isLNat^#(V1))
  , a__isLNat^#(take(V1, V2)) ->
    c_79(a__and^#(a__isNatural(V1), isLNat(V2)), a__isNatural^#(V1))
  , a__U81^#(tt(), N, X, XS) ->
    c_59(a__U82^#(a__splitAt(mark(N), mark(XS)), X),
         a__splitAt^#(mark(N), mark(XS)),
         mark^#(N),
         mark^#(XS))
  , a__isNatural^#(s(V1)) -> c_67(a__isNatural^#(V1))
  , a__isNatural^#(head(V1)) -> c_69(a__isLNat^#(V1))
  , a__isNatural^#(sel(V1, V2)) ->
    c_70(a__and^#(a__isNatural(V1), isLNat(V2)), a__isNatural^#(V1))
  , a__U71^#(tt(), XS) -> c_57(mark^#(XS))
  , a__natsFrom^#(N) ->
    c_83(a__U41^#(a__isNatural(N), N), a__isNatural^#(N))
  , a__afterNth^#(N, XS) ->
    c_52(a__U11^#(a__and(a__isNatural(N), isLNat(XS)), N, XS),
         a__and^#(a__isNatural(N), isLNat(XS)),
         a__isNatural^#(N))
  , a__snd^#(pair(X, Y)) ->
    c_41(a__U61^#(a__and(a__isLNat(X), isLNat(Y)), Y),
         a__and^#(a__isLNat(X), isLNat(Y)),
         a__isLNat^#(X))
  , a__tail^#(cons(N, XS)) ->
    c_88(a__U91^#(a__and(a__isNatural(N), isLNat(XS)), XS),
         a__and^#(a__isNatural(N), isLNat(XS)),
         a__isNatural^#(N))
  , a__take^#(N, XS) ->
    c_89(a__U101^#(a__and(a__isNatural(N), isLNat(XS)), N, XS),
         a__and^#(a__isNatural(N), isLNat(XS)),
         a__isNatural^#(N))
  , a__head^#(cons(N, XS)) ->
    c_51(a__U31^#(a__and(a__isNatural(N), isLNat(XS)), N),
         a__and^#(a__isNatural(N), isLNat(XS)),
         a__isNatural^#(N))
  , a__sel^#(N, XS) ->
    c_85(a__U51^#(a__and(a__isNatural(N), isLNat(XS)), N, XS),
         a__and^#(a__isNatural(N), isLNat(XS)),
         a__isNatural^#(N))
  , a__U11^#(tt(), N, XS) ->
    c_39(a__snd^#(a__splitAt(mark(N), mark(XS))),
         a__splitAt^#(mark(N), mark(XS)),
         mark^#(N),
         mark^#(XS))
  , a__U31^#(tt(), N) -> c_45(mark^#(N))
  , a__U41^#(tt(), N) -> c_47(mark^#(N))
  , a__U51^#(tt(), N, XS) ->
    c_49(a__head^#(a__afterNth(mark(N), mark(XS))),
         a__afterNth^#(mark(N), mark(XS)),
         mark^#(N),
         mark^#(XS))
  , a__U61^#(tt(), Y) -> c_55(mark^#(Y))
  , a__U82^#(pair(YS, ZS), X) -> c_61(mark^#(X), mark^#(ZS))
  , a__U91^#(tt(), XS) -> c_63(mark^#(XS))
  , a__isPLNat^#(pair(V1, V2)) ->
    c_81(a__and^#(a__isLNat(V1), isLNat(V2)), a__isLNat^#(V1))
  , a__isPLNat^#(splitAt(V1, V2)) ->
    c_82(a__and^#(a__isNatural(V1), isLNat(V2)), a__isNatural^#(V1)) }
Weak Trs:
  { a__U101(X1, X2, X3) -> U101(X1, X2, X3)
  , a__U101(tt(), N, XS) -> a__fst(a__splitAt(mark(N), mark(XS)))
  , a__fst(X) -> fst(X)
  , a__fst(pair(X, Y)) -> a__U21(a__and(a__isLNat(X), isLNat(Y)), X)
  , a__splitAt(X1, X2) -> splitAt(X1, X2)
  , a__splitAt(s(N), cons(X, XS)) ->
    a__U81(a__and(a__isNatural(N), and(isNatural(X), isLNat(XS))),
           N,
           X,
           XS)
  , a__splitAt(0(), XS) -> a__U71(a__isLNat(XS), XS)
  , mark(tt()) -> tt()
  , 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(nil()) -> nil()
  , mark(isLNat(X)) -> a__isLNat(X)
  , mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2))
  , mark(fst(X)) -> a__fst(mark(X))
  , mark(snd(X)) -> a__snd(mark(X))
  , mark(tail(X)) -> a__tail(mark(X))
  , mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
  , mark(0()) -> 0()
  , mark(head(X)) -> a__head(mark(X))
  , mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2))
  , mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2))
  , mark(and(X1, X2)) -> a__and(mark(X1), X2)
  , mark(isNatural(X)) -> a__isNatural(X)
  , mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3)
  , mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
  , mark(U21(X1, X2)) -> a__U21(mark(X1), X2)
  , mark(U31(X1, X2)) -> a__U31(mark(X1), X2)
  , mark(U41(X1, X2)) -> a__U41(mark(X1), X2)
  , mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3)
  , mark(U61(X1, X2)) -> a__U61(mark(X1), X2)
  , mark(U71(X1, X2)) -> a__U71(mark(X1), X2)
  , mark(U81(X1, X2, X3, X4)) -> a__U81(mark(X1), X2, X3, X4)
  , mark(U82(X1, X2)) -> a__U82(mark(X1), X2)
  , mark(U91(X1, X2)) -> a__U91(mark(X1), X2)
  , mark(isPLNat(X)) -> a__isPLNat(X)
  , a__U11(X1, X2, X3) -> U11(X1, X2, X3)
  , a__U11(tt(), N, XS) -> a__snd(a__splitAt(mark(N), mark(XS)))
  , a__snd(X) -> snd(X)
  , a__snd(pair(X, Y)) -> a__U61(a__and(a__isLNat(X), isLNat(Y)), Y)
  , a__U21(X1, X2) -> U21(X1, X2)
  , a__U21(tt(), X) -> mark(X)
  , a__U31(X1, X2) -> U31(X1, X2)
  , a__U31(tt(), N) -> mark(N)
  , a__U41(X1, X2) -> U41(X1, X2)
  , a__U41(tt(), N) -> cons(mark(N), natsFrom(s(N)))
  , a__U51(X1, X2, X3) -> U51(X1, X2, X3)
  , a__U51(tt(), N, XS) -> a__head(a__afterNth(mark(N), mark(XS)))
  , a__head(X) -> head(X)
  , a__head(cons(N, XS)) ->
    a__U31(a__and(a__isNatural(N), isLNat(XS)), N)
  , a__afterNth(N, XS) ->
    a__U11(a__and(a__isNatural(N), isLNat(XS)), N, XS)
  , a__afterNth(X1, X2) -> afterNth(X1, X2)
  , a__U61(X1, X2) -> U61(X1, X2)
  , a__U61(tt(), Y) -> mark(Y)
  , a__U71(X1, X2) -> U71(X1, X2)
  , a__U71(tt(), XS) -> pair(nil(), mark(XS))
  , a__U81(X1, X2, X3, X4) -> U81(X1, X2, X3, X4)
  , a__U81(tt(), N, X, XS) ->
    a__U82(a__splitAt(mark(N), mark(XS)), X)
  , a__U82(X1, X2) -> U82(X1, X2)
  , a__U82(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS))
  , a__U91(X1, X2) -> U91(X1, X2)
  , a__U91(tt(), XS) -> mark(XS)
  , a__and(X1, X2) -> and(X1, X2)
  , a__and(tt(), X) -> mark(X)
  , a__isNatural(X) -> isNatural(X)
  , a__isNatural(s(V1)) -> a__isNatural(V1)
  , a__isNatural(0()) -> tt()
  , a__isNatural(head(V1)) -> a__isLNat(V1)
  , a__isNatural(sel(V1, V2)) -> a__and(a__isNatural(V1), isLNat(V2))
  , a__isLNat(X) -> isLNat(X)
  , a__isLNat(cons(V1, V2)) -> a__and(a__isNatural(V1), isLNat(V2))
  , a__isLNat(natsFrom(V1)) -> a__isNatural(V1)
  , a__isLNat(nil()) -> tt()
  , a__isLNat(afterNth(V1, V2)) ->
    a__and(a__isNatural(V1), isLNat(V2))
  , a__isLNat(fst(V1)) -> a__isPLNat(V1)
  , a__isLNat(snd(V1)) -> a__isPLNat(V1)
  , a__isLNat(tail(V1)) -> a__isLNat(V1)
  , a__isLNat(take(V1, V2)) -> a__and(a__isNatural(V1), isLNat(V2))
  , a__isPLNat(X) -> isPLNat(X)
  , a__isPLNat(pair(V1, V2)) -> a__and(a__isLNat(V1), isLNat(V2))
  , a__isPLNat(splitAt(V1, V2)) ->
    a__and(a__isNatural(V1), isLNat(V2))
  , a__natsFrom(N) -> a__U41(a__isNatural(N), N)
  , a__natsFrom(X) -> natsFrom(X)
  , a__sel(N, XS) ->
    a__U51(a__and(a__isNatural(N), isLNat(XS)), N, XS)
  , a__sel(X1, X2) -> sel(X1, X2)
  , a__tail(X) -> tail(X)
  , a__tail(cons(N, XS)) ->
    a__U91(a__and(a__isNatural(N), isLNat(XS)), XS)
  , a__take(N, XS) ->
    a__U101(a__and(a__isNatural(N), isLNat(XS)), N, XS)
  , a__take(X1, X2) -> take(X1, X2) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..