MAYBE

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

Strict Trs:
  { a__and(X1, X2) -> and(X1, X2)
  , a__and(tt(), T) -> mark(T)
  , mark(tt()) -> tt()
  , mark(0()) -> 0()
  , mark(s(X)) -> s(mark(X))
  , mark(length(X)) -> a__length(mark(X))
  , mark(zeros()) -> a__zeros()
  , mark(cons(X1, X2)) -> cons(mark(X1), X2)
  , mark(nil()) -> nil()
  , mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
  , mark(and(X1, X2)) -> a__and(mark(X1), mark(X2))
  , mark(isNatIList(X)) -> a__isNatIList(X)
  , mark(isNatList(X)) -> a__isNatList(X)
  , mark(isNat(X)) -> a__isNat(X)
  , mark(uTake1(X)) -> a__uTake1(mark(X))
  , mark(uTake2(X1, X2, X3, X4)) -> a__uTake2(mark(X1), X2, X3, X4)
  , mark(uLength(X1, X2)) -> a__uLength(mark(X1), X2)
  , a__isNatIList(IL) -> a__isNatList(IL)
  , a__isNatIList(X) -> isNatIList(X)
  , a__isNatIList(zeros()) -> tt()
  , a__isNatIList(cons(N, IL)) ->
    a__and(a__isNat(N), a__isNatIList(IL))
  , a__isNatList(X) -> isNatList(X)
  , a__isNatList(cons(N, L)) -> a__and(a__isNat(N), a__isNatList(L))
  , a__isNatList(nil()) -> tt()
  , a__isNatList(take(N, IL)) ->
    a__and(a__isNat(N), a__isNatIList(IL))
  , a__isNat(X) -> isNat(X)
  , a__isNat(0()) -> tt()
  , a__isNat(s(N)) -> a__isNat(N)
  , a__isNat(length(L)) -> a__isNatList(L)
  , a__zeros() -> zeros()
  , a__zeros() -> cons(0(), zeros())
  , a__take(X1, X2) -> take(X1, X2)
  , a__take(0(), IL) -> a__uTake1(a__isNatIList(IL))
  , a__take(s(M), cons(N, IL)) ->
    a__uTake2(a__and(a__isNat(M),
                     a__and(a__isNat(N), a__isNatIList(IL))),
              M,
              N,
              IL)
  , a__uTake1(X) -> uTake1(X)
  , a__uTake1(tt()) -> nil()
  , a__uTake2(X1, X2, X3, X4) -> uTake2(X1, X2, X3, X4)
  , a__uTake2(tt(), M, N, IL) -> cons(mark(N), take(M, IL))
  , a__length(X) -> length(X)
  , a__length(cons(N, L)) ->
    a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
  , a__uLength(X1, X2) -> uLength(X1, X2)
  , a__uLength(tt(), L) -> s(a__length(mark(L))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { a__and^#(X1, X2) -> c_1()
  , a__and^#(tt(), T) -> c_2(mark^#(T))
  , mark^#(tt()) -> c_3()
  , mark^#(0()) -> c_4()
  , mark^#(s(X)) -> c_5(mark^#(X))
  , mark^#(length(X)) -> c_6(a__length^#(mark(X)), mark^#(X))
  , mark^#(zeros()) -> c_7(a__zeros^#())
  , mark^#(cons(X1, X2)) -> c_8(mark^#(X1))
  , mark^#(nil()) -> c_9()
  , mark^#(take(X1, X2)) ->
    c_10(a__take^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(and(X1, X2)) ->
    c_11(a__and^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(isNatIList(X)) -> c_12(a__isNatIList^#(X))
  , mark^#(isNatList(X)) -> c_13(a__isNatList^#(X))
  , mark^#(isNat(X)) -> c_14(a__isNat^#(X))
  , mark^#(uTake1(X)) -> c_15(a__uTake1^#(mark(X)), mark^#(X))
  , mark^#(uTake2(X1, X2, X3, X4)) ->
    c_16(a__uTake2^#(mark(X1), X2, X3, X4), mark^#(X1))
  , mark^#(uLength(X1, X2)) ->
    c_17(a__uLength^#(mark(X1), X2), mark^#(X1))
  , a__length^#(X) -> c_39()
  , a__length^#(cons(N, L)) ->
    c_40(a__uLength^#(a__and(a__isNat(N), a__isNatList(L)), L),
         a__and^#(a__isNat(N), a__isNatList(L)),
         a__isNat^#(N),
         a__isNatList^#(L))
  , a__zeros^#() -> c_30()
  , a__zeros^#() -> c_31()
  , a__take^#(X1, X2) -> c_32()
  , a__take^#(0(), IL) ->
    c_33(a__uTake1^#(a__isNatIList(IL)), a__isNatIList^#(IL))
  , a__take^#(s(M), cons(N, IL)) ->
    c_34(a__uTake2^#(a__and(a__isNat(M),
                            a__and(a__isNat(N), a__isNatIList(IL))),
                     M,
                     N,
                     IL),
         a__and^#(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))),
         a__isNat^#(M),
         a__and^#(a__isNat(N), a__isNatIList(IL)),
         a__isNat^#(N),
         a__isNatIList^#(IL))
  , a__isNatIList^#(IL) -> c_18(a__isNatList^#(IL))
  , a__isNatIList^#(X) -> c_19()
  , a__isNatIList^#(zeros()) -> c_20()
  , a__isNatIList^#(cons(N, IL)) ->
    c_21(a__and^#(a__isNat(N), a__isNatIList(IL)),
         a__isNat^#(N),
         a__isNatIList^#(IL))
  , a__isNatList^#(X) -> c_22()
  , a__isNatList^#(cons(N, L)) ->
    c_23(a__and^#(a__isNat(N), a__isNatList(L)),
         a__isNat^#(N),
         a__isNatList^#(L))
  , a__isNatList^#(nil()) -> c_24()
  , a__isNatList^#(take(N, IL)) ->
    c_25(a__and^#(a__isNat(N), a__isNatIList(IL)),
         a__isNat^#(N),
         a__isNatIList^#(IL))
  , a__isNat^#(X) -> c_26()
  , a__isNat^#(0()) -> c_27()
  , a__isNat^#(s(N)) -> c_28(a__isNat^#(N))
  , a__isNat^#(length(L)) -> c_29(a__isNatList^#(L))
  , a__uTake1^#(X) -> c_35()
  , a__uTake1^#(tt()) -> c_36()
  , a__uTake2^#(X1, X2, X3, X4) -> c_37()
  , a__uTake2^#(tt(), M, N, IL) -> c_38(mark^#(N))
  , a__uLength^#(X1, X2) -> c_41()
  , a__uLength^#(tt(), L) -> c_42(a__length^#(mark(L)), mark^#(L)) }

and mark the set of starting terms.

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

Strict DPs:
  { a__and^#(X1, X2) -> c_1()
  , a__and^#(tt(), T) -> c_2(mark^#(T))
  , mark^#(tt()) -> c_3()
  , mark^#(0()) -> c_4()
  , mark^#(s(X)) -> c_5(mark^#(X))
  , mark^#(length(X)) -> c_6(a__length^#(mark(X)), mark^#(X))
  , mark^#(zeros()) -> c_7(a__zeros^#())
  , mark^#(cons(X1, X2)) -> c_8(mark^#(X1))
  , mark^#(nil()) -> c_9()
  , mark^#(take(X1, X2)) ->
    c_10(a__take^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(and(X1, X2)) ->
    c_11(a__and^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(isNatIList(X)) -> c_12(a__isNatIList^#(X))
  , mark^#(isNatList(X)) -> c_13(a__isNatList^#(X))
  , mark^#(isNat(X)) -> c_14(a__isNat^#(X))
  , mark^#(uTake1(X)) -> c_15(a__uTake1^#(mark(X)), mark^#(X))
  , mark^#(uTake2(X1, X2, X3, X4)) ->
    c_16(a__uTake2^#(mark(X1), X2, X3, X4), mark^#(X1))
  , mark^#(uLength(X1, X2)) ->
    c_17(a__uLength^#(mark(X1), X2), mark^#(X1))
  , a__length^#(X) -> c_39()
  , a__length^#(cons(N, L)) ->
    c_40(a__uLength^#(a__and(a__isNat(N), a__isNatList(L)), L),
         a__and^#(a__isNat(N), a__isNatList(L)),
         a__isNat^#(N),
         a__isNatList^#(L))
  , a__zeros^#() -> c_30()
  , a__zeros^#() -> c_31()
  , a__take^#(X1, X2) -> c_32()
  , a__take^#(0(), IL) ->
    c_33(a__uTake1^#(a__isNatIList(IL)), a__isNatIList^#(IL))
  , a__take^#(s(M), cons(N, IL)) ->
    c_34(a__uTake2^#(a__and(a__isNat(M),
                            a__and(a__isNat(N), a__isNatIList(IL))),
                     M,
                     N,
                     IL),
         a__and^#(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))),
         a__isNat^#(M),
         a__and^#(a__isNat(N), a__isNatIList(IL)),
         a__isNat^#(N),
         a__isNatIList^#(IL))
  , a__isNatIList^#(IL) -> c_18(a__isNatList^#(IL))
  , a__isNatIList^#(X) -> c_19()
  , a__isNatIList^#(zeros()) -> c_20()
  , a__isNatIList^#(cons(N, IL)) ->
    c_21(a__and^#(a__isNat(N), a__isNatIList(IL)),
         a__isNat^#(N),
         a__isNatIList^#(IL))
  , a__isNatList^#(X) -> c_22()
  , a__isNatList^#(cons(N, L)) ->
    c_23(a__and^#(a__isNat(N), a__isNatList(L)),
         a__isNat^#(N),
         a__isNatList^#(L))
  , a__isNatList^#(nil()) -> c_24()
  , a__isNatList^#(take(N, IL)) ->
    c_25(a__and^#(a__isNat(N), a__isNatIList(IL)),
         a__isNat^#(N),
         a__isNatIList^#(IL))
  , a__isNat^#(X) -> c_26()
  , a__isNat^#(0()) -> c_27()
  , a__isNat^#(s(N)) -> c_28(a__isNat^#(N))
  , a__isNat^#(length(L)) -> c_29(a__isNatList^#(L))
  , a__uTake1^#(X) -> c_35()
  , a__uTake1^#(tt()) -> c_36()
  , a__uTake2^#(X1, X2, X3, X4) -> c_37()
  , a__uTake2^#(tt(), M, N, IL) -> c_38(mark^#(N))
  , a__uLength^#(X1, X2) -> c_41()
  , a__uLength^#(tt(), L) -> c_42(a__length^#(mark(L)), mark^#(L)) }
Weak Trs:
  { a__and(X1, X2) -> and(X1, X2)
  , a__and(tt(), T) -> mark(T)
  , mark(tt()) -> tt()
  , mark(0()) -> 0()
  , mark(s(X)) -> s(mark(X))
  , mark(length(X)) -> a__length(mark(X))
  , mark(zeros()) -> a__zeros()
  , mark(cons(X1, X2)) -> cons(mark(X1), X2)
  , mark(nil()) -> nil()
  , mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
  , mark(and(X1, X2)) -> a__and(mark(X1), mark(X2))
  , mark(isNatIList(X)) -> a__isNatIList(X)
  , mark(isNatList(X)) -> a__isNatList(X)
  , mark(isNat(X)) -> a__isNat(X)
  , mark(uTake1(X)) -> a__uTake1(mark(X))
  , mark(uTake2(X1, X2, X3, X4)) -> a__uTake2(mark(X1), X2, X3, X4)
  , mark(uLength(X1, X2)) -> a__uLength(mark(X1), X2)
  , a__isNatIList(IL) -> a__isNatList(IL)
  , a__isNatIList(X) -> isNatIList(X)
  , a__isNatIList(zeros()) -> tt()
  , a__isNatIList(cons(N, IL)) ->
    a__and(a__isNat(N), a__isNatIList(IL))
  , a__isNatList(X) -> isNatList(X)
  , a__isNatList(cons(N, L)) -> a__and(a__isNat(N), a__isNatList(L))
  , a__isNatList(nil()) -> tt()
  , a__isNatList(take(N, IL)) ->
    a__and(a__isNat(N), a__isNatIList(IL))
  , a__isNat(X) -> isNat(X)
  , a__isNat(0()) -> tt()
  , a__isNat(s(N)) -> a__isNat(N)
  , a__isNat(length(L)) -> a__isNatList(L)
  , a__zeros() -> zeros()
  , a__zeros() -> cons(0(), zeros())
  , a__take(X1, X2) -> take(X1, X2)
  , a__take(0(), IL) -> a__uTake1(a__isNatIList(IL))
  , a__take(s(M), cons(N, IL)) ->
    a__uTake2(a__and(a__isNat(M),
                     a__and(a__isNat(N), a__isNatIList(IL))),
              M,
              N,
              IL)
  , a__uTake1(X) -> uTake1(X)
  , a__uTake1(tt()) -> nil()
  , a__uTake2(X1, X2, X3, X4) -> uTake2(X1, X2, X3, X4)
  , a__uTake2(tt(), M, N, IL) -> cons(mark(N), take(M, IL))
  , a__length(X) -> length(X)
  , a__length(cons(N, L)) ->
    a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
  , a__uLength(X1, X2) -> uLength(X1, X2)
  , a__uLength(tt(), L) -> s(a__length(mark(L))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

  DPs:
    { 1: a__and^#(X1, X2) -> c_1()
    , 2: a__and^#(tt(), T) -> c_2(mark^#(T))
    , 3: mark^#(tt()) -> c_3()
    , 4: mark^#(0()) -> c_4()
    , 5: mark^#(s(X)) -> c_5(mark^#(X))
    , 6: mark^#(length(X)) -> c_6(a__length^#(mark(X)), mark^#(X))
    , 7: mark^#(zeros()) -> c_7(a__zeros^#())
    , 8: mark^#(cons(X1, X2)) -> c_8(mark^#(X1))
    , 9: mark^#(nil()) -> c_9()
    , 10: mark^#(take(X1, X2)) ->
          c_10(a__take^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
    , 11: mark^#(and(X1, X2)) ->
          c_11(a__and^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
    , 12: mark^#(isNatIList(X)) -> c_12(a__isNatIList^#(X))
    , 13: mark^#(isNatList(X)) -> c_13(a__isNatList^#(X))
    , 14: mark^#(isNat(X)) -> c_14(a__isNat^#(X))
    , 15: mark^#(uTake1(X)) -> c_15(a__uTake1^#(mark(X)), mark^#(X))
    , 16: mark^#(uTake2(X1, X2, X3, X4)) ->
          c_16(a__uTake2^#(mark(X1), X2, X3, X4), mark^#(X1))
    , 17: mark^#(uLength(X1, X2)) ->
          c_17(a__uLength^#(mark(X1), X2), mark^#(X1))
    , 18: a__length^#(X) -> c_39()
    , 19: a__length^#(cons(N, L)) ->
          c_40(a__uLength^#(a__and(a__isNat(N), a__isNatList(L)), L),
               a__and^#(a__isNat(N), a__isNatList(L)),
               a__isNat^#(N),
               a__isNatList^#(L))
    , 20: a__zeros^#() -> c_30()
    , 21: a__zeros^#() -> c_31()
    , 22: a__take^#(X1, X2) -> c_32()
    , 23: a__take^#(0(), IL) ->
          c_33(a__uTake1^#(a__isNatIList(IL)), a__isNatIList^#(IL))
    , 24: a__take^#(s(M), cons(N, IL)) ->
          c_34(a__uTake2^#(a__and(a__isNat(M),
                                  a__and(a__isNat(N), a__isNatIList(IL))),
                           M,
                           N,
                           IL),
               a__and^#(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))),
               a__isNat^#(M),
               a__and^#(a__isNat(N), a__isNatIList(IL)),
               a__isNat^#(N),
               a__isNatIList^#(IL))
    , 25: a__isNatIList^#(IL) -> c_18(a__isNatList^#(IL))
    , 26: a__isNatIList^#(X) -> c_19()
    , 27: a__isNatIList^#(zeros()) -> c_20()
    , 28: a__isNatIList^#(cons(N, IL)) ->
          c_21(a__and^#(a__isNat(N), a__isNatIList(IL)),
               a__isNat^#(N),
               a__isNatIList^#(IL))
    , 29: a__isNatList^#(X) -> c_22()
    , 30: a__isNatList^#(cons(N, L)) ->
          c_23(a__and^#(a__isNat(N), a__isNatList(L)),
               a__isNat^#(N),
               a__isNatList^#(L))
    , 31: a__isNatList^#(nil()) -> c_24()
    , 32: a__isNatList^#(take(N, IL)) ->
          c_25(a__and^#(a__isNat(N), a__isNatIList(IL)),
               a__isNat^#(N),
               a__isNatIList^#(IL))
    , 33: a__isNat^#(X) -> c_26()
    , 34: a__isNat^#(0()) -> c_27()
    , 35: a__isNat^#(s(N)) -> c_28(a__isNat^#(N))
    , 36: a__isNat^#(length(L)) -> c_29(a__isNatList^#(L))
    , 37: a__uTake1^#(X) -> c_35()
    , 38: a__uTake1^#(tt()) -> c_36()
    , 39: a__uTake2^#(X1, X2, X3, X4) -> c_37()
    , 40: a__uTake2^#(tt(), M, N, IL) -> c_38(mark^#(N))
    , 41: a__uLength^#(X1, X2) -> c_41()
    , 42: a__uLength^#(tt(), L) ->
          c_42(a__length^#(mark(L)), mark^#(L)) }

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

Strict DPs:
  { a__and^#(tt(), T) -> c_2(mark^#(T))
  , mark^#(s(X)) -> c_5(mark^#(X))
  , mark^#(length(X)) -> c_6(a__length^#(mark(X)), mark^#(X))
  , mark^#(zeros()) -> c_7(a__zeros^#())
  , mark^#(cons(X1, X2)) -> c_8(mark^#(X1))
  , mark^#(take(X1, X2)) ->
    c_10(a__take^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(and(X1, X2)) ->
    c_11(a__and^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(isNatIList(X)) -> c_12(a__isNatIList^#(X))
  , mark^#(isNatList(X)) -> c_13(a__isNatList^#(X))
  , mark^#(isNat(X)) -> c_14(a__isNat^#(X))
  , mark^#(uTake1(X)) -> c_15(a__uTake1^#(mark(X)), mark^#(X))
  , mark^#(uTake2(X1, X2, X3, X4)) ->
    c_16(a__uTake2^#(mark(X1), X2, X3, X4), mark^#(X1))
  , mark^#(uLength(X1, X2)) ->
    c_17(a__uLength^#(mark(X1), X2), mark^#(X1))
  , a__length^#(cons(N, L)) ->
    c_40(a__uLength^#(a__and(a__isNat(N), a__isNatList(L)), L),
         a__and^#(a__isNat(N), a__isNatList(L)),
         a__isNat^#(N),
         a__isNatList^#(L))
  , a__take^#(0(), IL) ->
    c_33(a__uTake1^#(a__isNatIList(IL)), a__isNatIList^#(IL))
  , a__take^#(s(M), cons(N, IL)) ->
    c_34(a__uTake2^#(a__and(a__isNat(M),
                            a__and(a__isNat(N), a__isNatIList(IL))),
                     M,
                     N,
                     IL),
         a__and^#(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))),
         a__isNat^#(M),
         a__and^#(a__isNat(N), a__isNatIList(IL)),
         a__isNat^#(N),
         a__isNatIList^#(IL))
  , a__isNatIList^#(IL) -> c_18(a__isNatList^#(IL))
  , a__isNatIList^#(cons(N, IL)) ->
    c_21(a__and^#(a__isNat(N), a__isNatIList(IL)),
         a__isNat^#(N),
         a__isNatIList^#(IL))
  , a__isNatList^#(cons(N, L)) ->
    c_23(a__and^#(a__isNat(N), a__isNatList(L)),
         a__isNat^#(N),
         a__isNatList^#(L))
  , a__isNatList^#(take(N, IL)) ->
    c_25(a__and^#(a__isNat(N), a__isNatIList(IL)),
         a__isNat^#(N),
         a__isNatIList^#(IL))
  , a__isNat^#(s(N)) -> c_28(a__isNat^#(N))
  , a__isNat^#(length(L)) -> c_29(a__isNatList^#(L))
  , a__uTake2^#(tt(), M, N, IL) -> c_38(mark^#(N))
  , a__uLength^#(tt(), L) -> c_42(a__length^#(mark(L)), mark^#(L)) }
Weak DPs:
  { a__and^#(X1, X2) -> c_1()
  , mark^#(tt()) -> c_3()
  , mark^#(0()) -> c_4()
  , mark^#(nil()) -> c_9()
  , a__length^#(X) -> c_39()
  , a__zeros^#() -> c_30()
  , a__zeros^#() -> c_31()
  , a__take^#(X1, X2) -> c_32()
  , a__isNatIList^#(X) -> c_19()
  , a__isNatIList^#(zeros()) -> c_20()
  , a__isNatList^#(X) -> c_22()
  , a__isNatList^#(nil()) -> c_24()
  , a__isNat^#(X) -> c_26()
  , a__isNat^#(0()) -> c_27()
  , a__uTake1^#(X) -> c_35()
  , a__uTake1^#(tt()) -> c_36()
  , a__uTake2^#(X1, X2, X3, X4) -> c_37()
  , a__uLength^#(X1, X2) -> c_41() }
Weak Trs:
  { a__and(X1, X2) -> and(X1, X2)
  , a__and(tt(), T) -> mark(T)
  , mark(tt()) -> tt()
  , mark(0()) -> 0()
  , mark(s(X)) -> s(mark(X))
  , mark(length(X)) -> a__length(mark(X))
  , mark(zeros()) -> a__zeros()
  , mark(cons(X1, X2)) -> cons(mark(X1), X2)
  , mark(nil()) -> nil()
  , mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
  , mark(and(X1, X2)) -> a__and(mark(X1), mark(X2))
  , mark(isNatIList(X)) -> a__isNatIList(X)
  , mark(isNatList(X)) -> a__isNatList(X)
  , mark(isNat(X)) -> a__isNat(X)
  , mark(uTake1(X)) -> a__uTake1(mark(X))
  , mark(uTake2(X1, X2, X3, X4)) -> a__uTake2(mark(X1), X2, X3, X4)
  , mark(uLength(X1, X2)) -> a__uLength(mark(X1), X2)
  , a__isNatIList(IL) -> a__isNatList(IL)
  , a__isNatIList(X) -> isNatIList(X)
  , a__isNatIList(zeros()) -> tt()
  , a__isNatIList(cons(N, IL)) ->
    a__and(a__isNat(N), a__isNatIList(IL))
  , a__isNatList(X) -> isNatList(X)
  , a__isNatList(cons(N, L)) -> a__and(a__isNat(N), a__isNatList(L))
  , a__isNatList(nil()) -> tt()
  , a__isNatList(take(N, IL)) ->
    a__and(a__isNat(N), a__isNatIList(IL))
  , a__isNat(X) -> isNat(X)
  , a__isNat(0()) -> tt()
  , a__isNat(s(N)) -> a__isNat(N)
  , a__isNat(length(L)) -> a__isNatList(L)
  , a__zeros() -> zeros()
  , a__zeros() -> cons(0(), zeros())
  , a__take(X1, X2) -> take(X1, X2)
  , a__take(0(), IL) -> a__uTake1(a__isNatIList(IL))
  , a__take(s(M), cons(N, IL)) ->
    a__uTake2(a__and(a__isNat(M),
                     a__and(a__isNat(N), a__isNatIList(IL))),
              M,
              N,
              IL)
  , a__uTake1(X) -> uTake1(X)
  , a__uTake1(tt()) -> nil()
  , a__uTake2(X1, X2, X3, X4) -> uTake2(X1, X2, X3, X4)
  , a__uTake2(tt(), M, N, IL) -> cons(mark(N), take(M, IL))
  , a__length(X) -> length(X)
  , a__length(cons(N, L)) ->
    a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
  , a__uLength(X1, X2) -> uLength(X1, X2)
  , a__uLength(tt(), L) -> s(a__length(mark(L))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of {4} by applications of
Pre({4}) = {1,2,3,5,6,7,11,12,13,23,24}. Here rules are labeled as
follows:

  DPs:
    { 1: a__and^#(tt(), T) -> c_2(mark^#(T))
    , 2: mark^#(s(X)) -> c_5(mark^#(X))
    , 3: mark^#(length(X)) -> c_6(a__length^#(mark(X)), mark^#(X))
    , 4: mark^#(zeros()) -> c_7(a__zeros^#())
    , 5: mark^#(cons(X1, X2)) -> c_8(mark^#(X1))
    , 6: mark^#(take(X1, X2)) ->
         c_10(a__take^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
    , 7: mark^#(and(X1, X2)) ->
         c_11(a__and^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
    , 8: mark^#(isNatIList(X)) -> c_12(a__isNatIList^#(X))
    , 9: mark^#(isNatList(X)) -> c_13(a__isNatList^#(X))
    , 10: mark^#(isNat(X)) -> c_14(a__isNat^#(X))
    , 11: mark^#(uTake1(X)) -> c_15(a__uTake1^#(mark(X)), mark^#(X))
    , 12: mark^#(uTake2(X1, X2, X3, X4)) ->
          c_16(a__uTake2^#(mark(X1), X2, X3, X4), mark^#(X1))
    , 13: mark^#(uLength(X1, X2)) ->
          c_17(a__uLength^#(mark(X1), X2), mark^#(X1))
    , 14: a__length^#(cons(N, L)) ->
          c_40(a__uLength^#(a__and(a__isNat(N), a__isNatList(L)), L),
               a__and^#(a__isNat(N), a__isNatList(L)),
               a__isNat^#(N),
               a__isNatList^#(L))
    , 15: a__take^#(0(), IL) ->
          c_33(a__uTake1^#(a__isNatIList(IL)), a__isNatIList^#(IL))
    , 16: a__take^#(s(M), cons(N, IL)) ->
          c_34(a__uTake2^#(a__and(a__isNat(M),
                                  a__and(a__isNat(N), a__isNatIList(IL))),
                           M,
                           N,
                           IL),
               a__and^#(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))),
               a__isNat^#(M),
               a__and^#(a__isNat(N), a__isNatIList(IL)),
               a__isNat^#(N),
               a__isNatIList^#(IL))
    , 17: a__isNatIList^#(IL) -> c_18(a__isNatList^#(IL))
    , 18: a__isNatIList^#(cons(N, IL)) ->
          c_21(a__and^#(a__isNat(N), a__isNatIList(IL)),
               a__isNat^#(N),
               a__isNatIList^#(IL))
    , 19: a__isNatList^#(cons(N, L)) ->
          c_23(a__and^#(a__isNat(N), a__isNatList(L)),
               a__isNat^#(N),
               a__isNatList^#(L))
    , 20: a__isNatList^#(take(N, IL)) ->
          c_25(a__and^#(a__isNat(N), a__isNatIList(IL)),
               a__isNat^#(N),
               a__isNatIList^#(IL))
    , 21: a__isNat^#(s(N)) -> c_28(a__isNat^#(N))
    , 22: a__isNat^#(length(L)) -> c_29(a__isNatList^#(L))
    , 23: a__uTake2^#(tt(), M, N, IL) -> c_38(mark^#(N))
    , 24: a__uLength^#(tt(), L) ->
          c_42(a__length^#(mark(L)), mark^#(L))
    , 25: a__and^#(X1, X2) -> c_1()
    , 26: mark^#(tt()) -> c_3()
    , 27: mark^#(0()) -> c_4()
    , 28: mark^#(nil()) -> c_9()
    , 29: a__length^#(X) -> c_39()
    , 30: a__zeros^#() -> c_30()
    , 31: a__zeros^#() -> c_31()
    , 32: a__take^#(X1, X2) -> c_32()
    , 33: a__isNatIList^#(X) -> c_19()
    , 34: a__isNatIList^#(zeros()) -> c_20()
    , 35: a__isNatList^#(X) -> c_22()
    , 36: a__isNatList^#(nil()) -> c_24()
    , 37: a__isNat^#(X) -> c_26()
    , 38: a__isNat^#(0()) -> c_27()
    , 39: a__uTake1^#(X) -> c_35()
    , 40: a__uTake1^#(tt()) -> c_36()
    , 41: a__uTake2^#(X1, X2, X3, X4) -> c_37()
    , 42: a__uLength^#(X1, X2) -> c_41() }

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

Strict DPs:
  { a__and^#(tt(), T) -> c_2(mark^#(T))
  , mark^#(s(X)) -> c_5(mark^#(X))
  , mark^#(length(X)) -> c_6(a__length^#(mark(X)), mark^#(X))
  , mark^#(cons(X1, X2)) -> c_8(mark^#(X1))
  , mark^#(take(X1, X2)) ->
    c_10(a__take^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(and(X1, X2)) ->
    c_11(a__and^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(isNatIList(X)) -> c_12(a__isNatIList^#(X))
  , mark^#(isNatList(X)) -> c_13(a__isNatList^#(X))
  , mark^#(isNat(X)) -> c_14(a__isNat^#(X))
  , mark^#(uTake1(X)) -> c_15(a__uTake1^#(mark(X)), mark^#(X))
  , mark^#(uTake2(X1, X2, X3, X4)) ->
    c_16(a__uTake2^#(mark(X1), X2, X3, X4), mark^#(X1))
  , mark^#(uLength(X1, X2)) ->
    c_17(a__uLength^#(mark(X1), X2), mark^#(X1))
  , a__length^#(cons(N, L)) ->
    c_40(a__uLength^#(a__and(a__isNat(N), a__isNatList(L)), L),
         a__and^#(a__isNat(N), a__isNatList(L)),
         a__isNat^#(N),
         a__isNatList^#(L))
  , a__take^#(0(), IL) ->
    c_33(a__uTake1^#(a__isNatIList(IL)), a__isNatIList^#(IL))
  , a__take^#(s(M), cons(N, IL)) ->
    c_34(a__uTake2^#(a__and(a__isNat(M),
                            a__and(a__isNat(N), a__isNatIList(IL))),
                     M,
                     N,
                     IL),
         a__and^#(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))),
         a__isNat^#(M),
         a__and^#(a__isNat(N), a__isNatIList(IL)),
         a__isNat^#(N),
         a__isNatIList^#(IL))
  , a__isNatIList^#(IL) -> c_18(a__isNatList^#(IL))
  , a__isNatIList^#(cons(N, IL)) ->
    c_21(a__and^#(a__isNat(N), a__isNatIList(IL)),
         a__isNat^#(N),
         a__isNatIList^#(IL))
  , a__isNatList^#(cons(N, L)) ->
    c_23(a__and^#(a__isNat(N), a__isNatList(L)),
         a__isNat^#(N),
         a__isNatList^#(L))
  , a__isNatList^#(take(N, IL)) ->
    c_25(a__and^#(a__isNat(N), a__isNatIList(IL)),
         a__isNat^#(N),
         a__isNatIList^#(IL))
  , a__isNat^#(s(N)) -> c_28(a__isNat^#(N))
  , a__isNat^#(length(L)) -> c_29(a__isNatList^#(L))
  , a__uTake2^#(tt(), M, N, IL) -> c_38(mark^#(N))
  , a__uLength^#(tt(), L) -> c_42(a__length^#(mark(L)), mark^#(L)) }
Weak DPs:
  { a__and^#(X1, X2) -> c_1()
  , mark^#(tt()) -> c_3()
  , mark^#(0()) -> c_4()
  , mark^#(zeros()) -> c_7(a__zeros^#())
  , mark^#(nil()) -> c_9()
  , a__length^#(X) -> c_39()
  , a__zeros^#() -> c_30()
  , a__zeros^#() -> c_31()
  , a__take^#(X1, X2) -> c_32()
  , a__isNatIList^#(X) -> c_19()
  , a__isNatIList^#(zeros()) -> c_20()
  , a__isNatList^#(X) -> c_22()
  , a__isNatList^#(nil()) -> c_24()
  , a__isNat^#(X) -> c_26()
  , a__isNat^#(0()) -> c_27()
  , a__uTake1^#(X) -> c_35()
  , a__uTake1^#(tt()) -> c_36()
  , a__uTake2^#(X1, X2, X3, X4) -> c_37()
  , a__uLength^#(X1, X2) -> c_41() }
Weak Trs:
  { a__and(X1, X2) -> and(X1, X2)
  , a__and(tt(), T) -> mark(T)
  , mark(tt()) -> tt()
  , mark(0()) -> 0()
  , mark(s(X)) -> s(mark(X))
  , mark(length(X)) -> a__length(mark(X))
  , mark(zeros()) -> a__zeros()
  , mark(cons(X1, X2)) -> cons(mark(X1), X2)
  , mark(nil()) -> nil()
  , mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
  , mark(and(X1, X2)) -> a__and(mark(X1), mark(X2))
  , mark(isNatIList(X)) -> a__isNatIList(X)
  , mark(isNatList(X)) -> a__isNatList(X)
  , mark(isNat(X)) -> a__isNat(X)
  , mark(uTake1(X)) -> a__uTake1(mark(X))
  , mark(uTake2(X1, X2, X3, X4)) -> a__uTake2(mark(X1), X2, X3, X4)
  , mark(uLength(X1, X2)) -> a__uLength(mark(X1), X2)
  , a__isNatIList(IL) -> a__isNatList(IL)
  , a__isNatIList(X) -> isNatIList(X)
  , a__isNatIList(zeros()) -> tt()
  , a__isNatIList(cons(N, IL)) ->
    a__and(a__isNat(N), a__isNatIList(IL))
  , a__isNatList(X) -> isNatList(X)
  , a__isNatList(cons(N, L)) -> a__and(a__isNat(N), a__isNatList(L))
  , a__isNatList(nil()) -> tt()
  , a__isNatList(take(N, IL)) ->
    a__and(a__isNat(N), a__isNatIList(IL))
  , a__isNat(X) -> isNat(X)
  , a__isNat(0()) -> tt()
  , a__isNat(s(N)) -> a__isNat(N)
  , a__isNat(length(L)) -> a__isNatList(L)
  , a__zeros() -> zeros()
  , a__zeros() -> cons(0(), zeros())
  , a__take(X1, X2) -> take(X1, X2)
  , a__take(0(), IL) -> a__uTake1(a__isNatIList(IL))
  , a__take(s(M), cons(N, IL)) ->
    a__uTake2(a__and(a__isNat(M),
                     a__and(a__isNat(N), a__isNatIList(IL))),
              M,
              N,
              IL)
  , a__uTake1(X) -> uTake1(X)
  , a__uTake1(tt()) -> nil()
  , a__uTake2(X1, X2, X3, X4) -> uTake2(X1, X2, X3, X4)
  , a__uTake2(tt(), M, N, IL) -> cons(mark(N), take(M, IL))
  , a__length(X) -> length(X)
  , a__length(cons(N, L)) ->
    a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
  , a__uLength(X1, X2) -> uLength(X1, X2)
  , a__uLength(tt(), L) -> s(a__length(mark(L))) }
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__and^#(X1, X2) -> c_1()
, mark^#(tt()) -> c_3()
, mark^#(0()) -> c_4()
, mark^#(zeros()) -> c_7(a__zeros^#())
, mark^#(nil()) -> c_9()
, a__length^#(X) -> c_39()
, a__zeros^#() -> c_30()
, a__zeros^#() -> c_31()
, a__take^#(X1, X2) -> c_32()
, a__isNatIList^#(X) -> c_19()
, a__isNatIList^#(zeros()) -> c_20()
, a__isNatList^#(X) -> c_22()
, a__isNatList^#(nil()) -> c_24()
, a__isNat^#(X) -> c_26()
, a__isNat^#(0()) -> c_27()
, a__uTake1^#(X) -> c_35()
, a__uTake1^#(tt()) -> c_36()
, a__uTake2^#(X1, X2, X3, X4) -> c_37()
, a__uLength^#(X1, X2) -> c_41() }

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

Strict DPs:
  { a__and^#(tt(), T) -> c_2(mark^#(T))
  , mark^#(s(X)) -> c_5(mark^#(X))
  , mark^#(length(X)) -> c_6(a__length^#(mark(X)), mark^#(X))
  , mark^#(cons(X1, X2)) -> c_8(mark^#(X1))
  , mark^#(take(X1, X2)) ->
    c_10(a__take^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(and(X1, X2)) ->
    c_11(a__and^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(isNatIList(X)) -> c_12(a__isNatIList^#(X))
  , mark^#(isNatList(X)) -> c_13(a__isNatList^#(X))
  , mark^#(isNat(X)) -> c_14(a__isNat^#(X))
  , mark^#(uTake1(X)) -> c_15(a__uTake1^#(mark(X)), mark^#(X))
  , mark^#(uTake2(X1, X2, X3, X4)) ->
    c_16(a__uTake2^#(mark(X1), X2, X3, X4), mark^#(X1))
  , mark^#(uLength(X1, X2)) ->
    c_17(a__uLength^#(mark(X1), X2), mark^#(X1))
  , a__length^#(cons(N, L)) ->
    c_40(a__uLength^#(a__and(a__isNat(N), a__isNatList(L)), L),
         a__and^#(a__isNat(N), a__isNatList(L)),
         a__isNat^#(N),
         a__isNatList^#(L))
  , a__take^#(0(), IL) ->
    c_33(a__uTake1^#(a__isNatIList(IL)), a__isNatIList^#(IL))
  , a__take^#(s(M), cons(N, IL)) ->
    c_34(a__uTake2^#(a__and(a__isNat(M),
                            a__and(a__isNat(N), a__isNatIList(IL))),
                     M,
                     N,
                     IL),
         a__and^#(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))),
         a__isNat^#(M),
         a__and^#(a__isNat(N), a__isNatIList(IL)),
         a__isNat^#(N),
         a__isNatIList^#(IL))
  , a__isNatIList^#(IL) -> c_18(a__isNatList^#(IL))
  , a__isNatIList^#(cons(N, IL)) ->
    c_21(a__and^#(a__isNat(N), a__isNatIList(IL)),
         a__isNat^#(N),
         a__isNatIList^#(IL))
  , a__isNatList^#(cons(N, L)) ->
    c_23(a__and^#(a__isNat(N), a__isNatList(L)),
         a__isNat^#(N),
         a__isNatList^#(L))
  , a__isNatList^#(take(N, IL)) ->
    c_25(a__and^#(a__isNat(N), a__isNatIList(IL)),
         a__isNat^#(N),
         a__isNatIList^#(IL))
  , a__isNat^#(s(N)) -> c_28(a__isNat^#(N))
  , a__isNat^#(length(L)) -> c_29(a__isNatList^#(L))
  , a__uTake2^#(tt(), M, N, IL) -> c_38(mark^#(N))
  , a__uLength^#(tt(), L) -> c_42(a__length^#(mark(L)), mark^#(L)) }
Weak Trs:
  { a__and(X1, X2) -> and(X1, X2)
  , a__and(tt(), T) -> mark(T)
  , mark(tt()) -> tt()
  , mark(0()) -> 0()
  , mark(s(X)) -> s(mark(X))
  , mark(length(X)) -> a__length(mark(X))
  , mark(zeros()) -> a__zeros()
  , mark(cons(X1, X2)) -> cons(mark(X1), X2)
  , mark(nil()) -> nil()
  , mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
  , mark(and(X1, X2)) -> a__and(mark(X1), mark(X2))
  , mark(isNatIList(X)) -> a__isNatIList(X)
  , mark(isNatList(X)) -> a__isNatList(X)
  , mark(isNat(X)) -> a__isNat(X)
  , mark(uTake1(X)) -> a__uTake1(mark(X))
  , mark(uTake2(X1, X2, X3, X4)) -> a__uTake2(mark(X1), X2, X3, X4)
  , mark(uLength(X1, X2)) -> a__uLength(mark(X1), X2)
  , a__isNatIList(IL) -> a__isNatList(IL)
  , a__isNatIList(X) -> isNatIList(X)
  , a__isNatIList(zeros()) -> tt()
  , a__isNatIList(cons(N, IL)) ->
    a__and(a__isNat(N), a__isNatIList(IL))
  , a__isNatList(X) -> isNatList(X)
  , a__isNatList(cons(N, L)) -> a__and(a__isNat(N), a__isNatList(L))
  , a__isNatList(nil()) -> tt()
  , a__isNatList(take(N, IL)) ->
    a__and(a__isNat(N), a__isNatIList(IL))
  , a__isNat(X) -> isNat(X)
  , a__isNat(0()) -> tt()
  , a__isNat(s(N)) -> a__isNat(N)
  , a__isNat(length(L)) -> a__isNatList(L)
  , a__zeros() -> zeros()
  , a__zeros() -> cons(0(), zeros())
  , a__take(X1, X2) -> take(X1, X2)
  , a__take(0(), IL) -> a__uTake1(a__isNatIList(IL))
  , a__take(s(M), cons(N, IL)) ->
    a__uTake2(a__and(a__isNat(M),
                     a__and(a__isNat(N), a__isNatIList(IL))),
              M,
              N,
              IL)
  , a__uTake1(X) -> uTake1(X)
  , a__uTake1(tt()) -> nil()
  , a__uTake2(X1, X2, X3, X4) -> uTake2(X1, X2, X3, X4)
  , a__uTake2(tt(), M, N, IL) -> cons(mark(N), take(M, IL))
  , a__length(X) -> length(X)
  , a__length(cons(N, L)) ->
    a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
  , a__uLength(X1, X2) -> uLength(X1, X2)
  , a__uLength(tt(), L) -> s(a__length(mark(L))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

Due to missing edges in the dependency-graph, the right-hand sides
of following rules could be simplified:

  { mark^#(uTake1(X)) -> c_15(a__uTake1^#(mark(X)), mark^#(X))
  , a__take^#(0(), IL) ->
    c_33(a__uTake1^#(a__isNatIList(IL)), a__isNatIList^#(IL)) }

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

Strict DPs:
  { a__and^#(tt(), T) -> c_1(mark^#(T))
  , mark^#(s(X)) -> c_2(mark^#(X))
  , mark^#(length(X)) -> c_3(a__length^#(mark(X)), mark^#(X))
  , mark^#(cons(X1, X2)) -> c_4(mark^#(X1))
  , mark^#(take(X1, X2)) ->
    c_5(a__take^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(and(X1, X2)) ->
    c_6(a__and^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(isNatIList(X)) -> c_7(a__isNatIList^#(X))
  , mark^#(isNatList(X)) -> c_8(a__isNatList^#(X))
  , mark^#(isNat(X)) -> c_9(a__isNat^#(X))
  , mark^#(uTake1(X)) -> c_10(mark^#(X))
  , mark^#(uTake2(X1, X2, X3, X4)) ->
    c_11(a__uTake2^#(mark(X1), X2, X3, X4), mark^#(X1))
  , mark^#(uLength(X1, X2)) ->
    c_12(a__uLength^#(mark(X1), X2), mark^#(X1))
  , a__length^#(cons(N, L)) ->
    c_13(a__uLength^#(a__and(a__isNat(N), a__isNatList(L)), L),
         a__and^#(a__isNat(N), a__isNatList(L)),
         a__isNat^#(N),
         a__isNatList^#(L))
  , a__take^#(0(), IL) -> c_14(a__isNatIList^#(IL))
  , a__take^#(s(M), cons(N, IL)) ->
    c_15(a__uTake2^#(a__and(a__isNat(M),
                            a__and(a__isNat(N), a__isNatIList(IL))),
                     M,
                     N,
                     IL),
         a__and^#(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))),
         a__isNat^#(M),
         a__and^#(a__isNat(N), a__isNatIList(IL)),
         a__isNat^#(N),
         a__isNatIList^#(IL))
  , a__isNatIList^#(IL) -> c_16(a__isNatList^#(IL))
  , a__isNatIList^#(cons(N, IL)) ->
    c_17(a__and^#(a__isNat(N), a__isNatIList(IL)),
         a__isNat^#(N),
         a__isNatIList^#(IL))
  , a__isNatList^#(cons(N, L)) ->
    c_18(a__and^#(a__isNat(N), a__isNatList(L)),
         a__isNat^#(N),
         a__isNatList^#(L))
  , a__isNatList^#(take(N, IL)) ->
    c_19(a__and^#(a__isNat(N), a__isNatIList(IL)),
         a__isNat^#(N),
         a__isNatIList^#(IL))
  , a__isNat^#(s(N)) -> c_20(a__isNat^#(N))
  , a__isNat^#(length(L)) -> c_21(a__isNatList^#(L))
  , a__uTake2^#(tt(), M, N, IL) -> c_22(mark^#(N))
  , a__uLength^#(tt(), L) -> c_23(a__length^#(mark(L)), mark^#(L)) }
Weak Trs:
  { a__and(X1, X2) -> and(X1, X2)
  , a__and(tt(), T) -> mark(T)
  , mark(tt()) -> tt()
  , mark(0()) -> 0()
  , mark(s(X)) -> s(mark(X))
  , mark(length(X)) -> a__length(mark(X))
  , mark(zeros()) -> a__zeros()
  , mark(cons(X1, X2)) -> cons(mark(X1), X2)
  , mark(nil()) -> nil()
  , mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
  , mark(and(X1, X2)) -> a__and(mark(X1), mark(X2))
  , mark(isNatIList(X)) -> a__isNatIList(X)
  , mark(isNatList(X)) -> a__isNatList(X)
  , mark(isNat(X)) -> a__isNat(X)
  , mark(uTake1(X)) -> a__uTake1(mark(X))
  , mark(uTake2(X1, X2, X3, X4)) -> a__uTake2(mark(X1), X2, X3, X4)
  , mark(uLength(X1, X2)) -> a__uLength(mark(X1), X2)
  , a__isNatIList(IL) -> a__isNatList(IL)
  , a__isNatIList(X) -> isNatIList(X)
  , a__isNatIList(zeros()) -> tt()
  , a__isNatIList(cons(N, IL)) ->
    a__and(a__isNat(N), a__isNatIList(IL))
  , a__isNatList(X) -> isNatList(X)
  , a__isNatList(cons(N, L)) -> a__and(a__isNat(N), a__isNatList(L))
  , a__isNatList(nil()) -> tt()
  , a__isNatList(take(N, IL)) ->
    a__and(a__isNat(N), a__isNatIList(IL))
  , a__isNat(X) -> isNat(X)
  , a__isNat(0()) -> tt()
  , a__isNat(s(N)) -> a__isNat(N)
  , a__isNat(length(L)) -> a__isNatList(L)
  , a__zeros() -> zeros()
  , a__zeros() -> cons(0(), zeros())
  , a__take(X1, X2) -> take(X1, X2)
  , a__take(0(), IL) -> a__uTake1(a__isNatIList(IL))
  , a__take(s(M), cons(N, IL)) ->
    a__uTake2(a__and(a__isNat(M),
                     a__and(a__isNat(N), a__isNatIList(IL))),
              M,
              N,
              IL)
  , a__uTake1(X) -> uTake1(X)
  , a__uTake1(tt()) -> nil()
  , a__uTake2(X1, X2, X3, X4) -> uTake2(X1, X2, X3, X4)
  , a__uTake2(tt(), M, N, IL) -> cons(mark(N), take(M, IL))
  , a__length(X) -> length(X)
  , a__length(cons(N, L)) ->
    a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
  , a__uLength(X1, X2) -> uLength(X1, X2)
  , a__uLength(tt(), L) -> s(a__length(mark(L))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..