MAYBE

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

Strict Trs:
  { a__zeros() -> cons(0(), zeros())
  , a__zeros() -> zeros()
  , a__U11(X) -> U11(X)
  , a__U11(tt()) -> tt()
  , a__U21(X) -> U21(X)
  , a__U21(tt()) -> tt()
  , a__U31(X) -> U31(X)
  , a__U31(tt()) -> tt()
  , a__U41(X1, X2) -> U41(X1, X2)
  , a__U41(tt(), V2) -> a__U42(a__isNatIList(V2))
  , a__U42(X) -> U42(X)
  , a__U42(tt()) -> tt()
  , a__isNatIList(V) -> a__U31(a__isNatList(V))
  , a__isNatIList(X) -> isNatIList(X)
  , a__isNatIList(cons(V1, V2)) -> a__U41(a__isNat(V1), V2)
  , a__isNatIList(zeros()) -> tt()
  , a__U51(X1, X2) -> U51(X1, X2)
  , a__U51(tt(), V2) -> a__U52(a__isNatList(V2))
  , a__U52(X) -> U52(X)
  , a__U52(tt()) -> tt()
  , a__isNatList(X) -> isNatList(X)
  , a__isNatList(cons(V1, V2)) -> a__U51(a__isNat(V1), V2)
  , a__isNatList(nil()) -> tt()
  , a__U61(X1, X2, X3) -> U61(X1, X2, X3)
  , a__U61(tt(), L, N) -> a__U62(a__isNat(N), L)
  , a__U62(X1, X2) -> U62(X1, X2)
  , a__U62(tt(), L) -> s(a__length(mark(L)))
  , a__isNat(X) -> isNat(X)
  , a__isNat(0()) -> tt()
  , a__isNat(s(V1)) -> a__U21(a__isNat(V1))
  , a__isNat(length(V1)) -> a__U11(a__isNatList(V1))
  , a__length(X) -> length(X)
  , a__length(cons(N, L)) -> a__U61(a__isNatList(L), L, N)
  , a__length(nil()) -> 0()
  , mark(cons(X1, X2)) -> cons(mark(X1), X2)
  , mark(0()) -> 0()
  , mark(zeros()) -> a__zeros()
  , mark(tt()) -> tt()
  , mark(s(X)) -> s(mark(X))
  , mark(length(X)) -> a__length(mark(X))
  , mark(nil()) -> nil()
  , mark(U11(X)) -> a__U11(mark(X))
  , mark(U21(X)) -> a__U21(mark(X))
  , mark(U31(X)) -> a__U31(mark(X))
  , mark(U41(X1, X2)) -> a__U41(mark(X1), X2)
  , mark(U42(X)) -> a__U42(mark(X))
  , mark(isNatIList(X)) -> a__isNatIList(X)
  , mark(U51(X1, X2)) -> a__U51(mark(X1), X2)
  , mark(U52(X)) -> a__U52(mark(X))
  , mark(isNatList(X)) -> a__isNatList(X)
  , mark(U61(X1, X2, X3)) -> a__U61(mark(X1), X2, X3)
  , mark(U62(X1, X2)) -> a__U62(mark(X1), X2)
  , mark(isNat(X)) -> a__isNat(X) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { a__zeros^#() -> c_1()
  , a__zeros^#() -> c_2()
  , a__U11^#(X) -> c_3()
  , a__U11^#(tt()) -> c_4()
  , a__U21^#(X) -> c_5()
  , a__U21^#(tt()) -> c_6()
  , a__U31^#(X) -> c_7()
  , a__U31^#(tt()) -> c_8()
  , a__U41^#(X1, X2) -> c_9()
  , a__U41^#(tt(), V2) ->
    c_10(a__U42^#(a__isNatIList(V2)), a__isNatIList^#(V2))
  , a__U42^#(X) -> c_11()
  , a__U42^#(tt()) -> c_12()
  , a__isNatIList^#(V) ->
    c_13(a__U31^#(a__isNatList(V)), a__isNatList^#(V))
  , a__isNatIList^#(X) -> c_14()
  , a__isNatIList^#(cons(V1, V2)) ->
    c_15(a__U41^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__isNatIList^#(zeros()) -> c_16()
  , a__isNatList^#(X) -> c_21()
  , a__isNatList^#(cons(V1, V2)) ->
    c_22(a__U51^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__isNatList^#(nil()) -> c_23()
  , a__isNat^#(X) -> c_28()
  , a__isNat^#(0()) -> c_29()
  , a__isNat^#(s(V1)) -> c_30(a__U21^#(a__isNat(V1)), a__isNat^#(V1))
  , a__isNat^#(length(V1)) ->
    c_31(a__U11^#(a__isNatList(V1)), a__isNatList^#(V1))
  , a__U51^#(X1, X2) -> c_17()
  , a__U51^#(tt(), V2) ->
    c_18(a__U52^#(a__isNatList(V2)), a__isNatList^#(V2))
  , a__U52^#(X) -> c_19()
  , a__U52^#(tt()) -> c_20()
  , a__U61^#(X1, X2, X3) -> c_24()
  , a__U61^#(tt(), L, N) ->
    c_25(a__U62^#(a__isNat(N), L), a__isNat^#(N))
  , a__U62^#(X1, X2) -> c_26()
  , a__U62^#(tt(), L) -> c_27(a__length^#(mark(L)), mark^#(L))
  , a__length^#(X) -> c_32()
  , a__length^#(cons(N, L)) ->
    c_33(a__U61^#(a__isNatList(L), L, N), a__isNatList^#(L))
  , a__length^#(nil()) -> c_34()
  , mark^#(cons(X1, X2)) -> c_35(mark^#(X1))
  , mark^#(0()) -> c_36()
  , mark^#(zeros()) -> c_37(a__zeros^#())
  , mark^#(tt()) -> c_38()
  , mark^#(s(X)) -> c_39(mark^#(X))
  , mark^#(length(X)) -> c_40(a__length^#(mark(X)), mark^#(X))
  , mark^#(nil()) -> c_41()
  , mark^#(U11(X)) -> c_42(a__U11^#(mark(X)), mark^#(X))
  , mark^#(U21(X)) -> c_43(a__U21^#(mark(X)), mark^#(X))
  , mark^#(U31(X)) -> c_44(a__U31^#(mark(X)), mark^#(X))
  , mark^#(U41(X1, X2)) -> c_45(a__U41^#(mark(X1), X2), mark^#(X1))
  , mark^#(U42(X)) -> c_46(a__U42^#(mark(X)), mark^#(X))
  , mark^#(isNatIList(X)) -> c_47(a__isNatIList^#(X))
  , mark^#(U51(X1, X2)) -> c_48(a__U51^#(mark(X1), X2), mark^#(X1))
  , mark^#(U52(X)) -> c_49(a__U52^#(mark(X)), mark^#(X))
  , mark^#(isNatList(X)) -> c_50(a__isNatList^#(X))
  , mark^#(U61(X1, X2, X3)) ->
    c_51(a__U61^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U62(X1, X2)) -> c_52(a__U62^#(mark(X1), X2), mark^#(X1))
  , mark^#(isNat(X)) -> c_53(a__isNat^#(X)) }

and mark the set of starting terms.

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

Strict DPs:
  { a__zeros^#() -> c_1()
  , a__zeros^#() -> c_2()
  , a__U11^#(X) -> c_3()
  , a__U11^#(tt()) -> c_4()
  , a__U21^#(X) -> c_5()
  , a__U21^#(tt()) -> c_6()
  , a__U31^#(X) -> c_7()
  , a__U31^#(tt()) -> c_8()
  , a__U41^#(X1, X2) -> c_9()
  , a__U41^#(tt(), V2) ->
    c_10(a__U42^#(a__isNatIList(V2)), a__isNatIList^#(V2))
  , a__U42^#(X) -> c_11()
  , a__U42^#(tt()) -> c_12()
  , a__isNatIList^#(V) ->
    c_13(a__U31^#(a__isNatList(V)), a__isNatList^#(V))
  , a__isNatIList^#(X) -> c_14()
  , a__isNatIList^#(cons(V1, V2)) ->
    c_15(a__U41^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__isNatIList^#(zeros()) -> c_16()
  , a__isNatList^#(X) -> c_21()
  , a__isNatList^#(cons(V1, V2)) ->
    c_22(a__U51^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__isNatList^#(nil()) -> c_23()
  , a__isNat^#(X) -> c_28()
  , a__isNat^#(0()) -> c_29()
  , a__isNat^#(s(V1)) -> c_30(a__U21^#(a__isNat(V1)), a__isNat^#(V1))
  , a__isNat^#(length(V1)) ->
    c_31(a__U11^#(a__isNatList(V1)), a__isNatList^#(V1))
  , a__U51^#(X1, X2) -> c_17()
  , a__U51^#(tt(), V2) ->
    c_18(a__U52^#(a__isNatList(V2)), a__isNatList^#(V2))
  , a__U52^#(X) -> c_19()
  , a__U52^#(tt()) -> c_20()
  , a__U61^#(X1, X2, X3) -> c_24()
  , a__U61^#(tt(), L, N) ->
    c_25(a__U62^#(a__isNat(N), L), a__isNat^#(N))
  , a__U62^#(X1, X2) -> c_26()
  , a__U62^#(tt(), L) -> c_27(a__length^#(mark(L)), mark^#(L))
  , a__length^#(X) -> c_32()
  , a__length^#(cons(N, L)) ->
    c_33(a__U61^#(a__isNatList(L), L, N), a__isNatList^#(L))
  , a__length^#(nil()) -> c_34()
  , mark^#(cons(X1, X2)) -> c_35(mark^#(X1))
  , mark^#(0()) -> c_36()
  , mark^#(zeros()) -> c_37(a__zeros^#())
  , mark^#(tt()) -> c_38()
  , mark^#(s(X)) -> c_39(mark^#(X))
  , mark^#(length(X)) -> c_40(a__length^#(mark(X)), mark^#(X))
  , mark^#(nil()) -> c_41()
  , mark^#(U11(X)) -> c_42(a__U11^#(mark(X)), mark^#(X))
  , mark^#(U21(X)) -> c_43(a__U21^#(mark(X)), mark^#(X))
  , mark^#(U31(X)) -> c_44(a__U31^#(mark(X)), mark^#(X))
  , mark^#(U41(X1, X2)) -> c_45(a__U41^#(mark(X1), X2), mark^#(X1))
  , mark^#(U42(X)) -> c_46(a__U42^#(mark(X)), mark^#(X))
  , mark^#(isNatIList(X)) -> c_47(a__isNatIList^#(X))
  , mark^#(U51(X1, X2)) -> c_48(a__U51^#(mark(X1), X2), mark^#(X1))
  , mark^#(U52(X)) -> c_49(a__U52^#(mark(X)), mark^#(X))
  , mark^#(isNatList(X)) -> c_50(a__isNatList^#(X))
  , mark^#(U61(X1, X2, X3)) ->
    c_51(a__U61^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U62(X1, X2)) -> c_52(a__U62^#(mark(X1), X2), mark^#(X1))
  , mark^#(isNat(X)) -> c_53(a__isNat^#(X)) }
Weak Trs:
  { a__zeros() -> cons(0(), zeros())
  , a__zeros() -> zeros()
  , a__U11(X) -> U11(X)
  , a__U11(tt()) -> tt()
  , a__U21(X) -> U21(X)
  , a__U21(tt()) -> tt()
  , a__U31(X) -> U31(X)
  , a__U31(tt()) -> tt()
  , a__U41(X1, X2) -> U41(X1, X2)
  , a__U41(tt(), V2) -> a__U42(a__isNatIList(V2))
  , a__U42(X) -> U42(X)
  , a__U42(tt()) -> tt()
  , a__isNatIList(V) -> a__U31(a__isNatList(V))
  , a__isNatIList(X) -> isNatIList(X)
  , a__isNatIList(cons(V1, V2)) -> a__U41(a__isNat(V1), V2)
  , a__isNatIList(zeros()) -> tt()
  , a__U51(X1, X2) -> U51(X1, X2)
  , a__U51(tt(), V2) -> a__U52(a__isNatList(V2))
  , a__U52(X) -> U52(X)
  , a__U52(tt()) -> tt()
  , a__isNatList(X) -> isNatList(X)
  , a__isNatList(cons(V1, V2)) -> a__U51(a__isNat(V1), V2)
  , a__isNatList(nil()) -> tt()
  , a__U61(X1, X2, X3) -> U61(X1, X2, X3)
  , a__U61(tt(), L, N) -> a__U62(a__isNat(N), L)
  , a__U62(X1, X2) -> U62(X1, X2)
  , a__U62(tt(), L) -> s(a__length(mark(L)))
  , a__isNat(X) -> isNat(X)
  , a__isNat(0()) -> tt()
  , a__isNat(s(V1)) -> a__U21(a__isNat(V1))
  , a__isNat(length(V1)) -> a__U11(a__isNatList(V1))
  , a__length(X) -> length(X)
  , a__length(cons(N, L)) -> a__U61(a__isNatList(L), L, N)
  , a__length(nil()) -> 0()
  , mark(cons(X1, X2)) -> cons(mark(X1), X2)
  , mark(0()) -> 0()
  , mark(zeros()) -> a__zeros()
  , mark(tt()) -> tt()
  , mark(s(X)) -> s(mark(X))
  , mark(length(X)) -> a__length(mark(X))
  , mark(nil()) -> nil()
  , mark(U11(X)) -> a__U11(mark(X))
  , mark(U21(X)) -> a__U21(mark(X))
  , mark(U31(X)) -> a__U31(mark(X))
  , mark(U41(X1, X2)) -> a__U41(mark(X1), X2)
  , mark(U42(X)) -> a__U42(mark(X))
  , mark(isNatIList(X)) -> a__isNatIList(X)
  , mark(U51(X1, X2)) -> a__U51(mark(X1), X2)
  , mark(U52(X)) -> a__U52(mark(X))
  , mark(isNatList(X)) -> a__isNatList(X)
  , mark(U61(X1, X2, X3)) -> a__U61(mark(X1), X2, X3)
  , mark(U62(X1, X2)) -> a__U62(mark(X1), X2)
  , mark(isNat(X)) -> a__isNat(X) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of
{1,2,3,4,5,6,7,8,9,11,12,14,16,17,19,20,21,24,26,27,28,30,32,34,36,38,41}
by applications of
Pre({1,2,3,4,5,6,7,8,9,11,12,14,16,17,19,20,21,24,26,27,28,30,32,34,36,38,41})
=
{10,13,15,18,22,23,25,29,31,33,35,37,39,40,42,43,44,45,46,47,48,49,50,51,52,53}.
Here rules are labeled as follows:

  DPs:
    { 1: a__zeros^#() -> c_1()
    , 2: a__zeros^#() -> c_2()
    , 3: a__U11^#(X) -> c_3()
    , 4: a__U11^#(tt()) -> c_4()
    , 5: a__U21^#(X) -> c_5()
    , 6: a__U21^#(tt()) -> c_6()
    , 7: a__U31^#(X) -> c_7()
    , 8: a__U31^#(tt()) -> c_8()
    , 9: a__U41^#(X1, X2) -> c_9()
    , 10: a__U41^#(tt(), V2) ->
          c_10(a__U42^#(a__isNatIList(V2)), a__isNatIList^#(V2))
    , 11: a__U42^#(X) -> c_11()
    , 12: a__U42^#(tt()) -> c_12()
    , 13: a__isNatIList^#(V) ->
          c_13(a__U31^#(a__isNatList(V)), a__isNatList^#(V))
    , 14: a__isNatIList^#(X) -> c_14()
    , 15: a__isNatIList^#(cons(V1, V2)) ->
          c_15(a__U41^#(a__isNat(V1), V2), a__isNat^#(V1))
    , 16: a__isNatIList^#(zeros()) -> c_16()
    , 17: a__isNatList^#(X) -> c_21()
    , 18: a__isNatList^#(cons(V1, V2)) ->
          c_22(a__U51^#(a__isNat(V1), V2), a__isNat^#(V1))
    , 19: a__isNatList^#(nil()) -> c_23()
    , 20: a__isNat^#(X) -> c_28()
    , 21: a__isNat^#(0()) -> c_29()
    , 22: a__isNat^#(s(V1)) ->
          c_30(a__U21^#(a__isNat(V1)), a__isNat^#(V1))
    , 23: a__isNat^#(length(V1)) ->
          c_31(a__U11^#(a__isNatList(V1)), a__isNatList^#(V1))
    , 24: a__U51^#(X1, X2) -> c_17()
    , 25: a__U51^#(tt(), V2) ->
          c_18(a__U52^#(a__isNatList(V2)), a__isNatList^#(V2))
    , 26: a__U52^#(X) -> c_19()
    , 27: a__U52^#(tt()) -> c_20()
    , 28: a__U61^#(X1, X2, X3) -> c_24()
    , 29: a__U61^#(tt(), L, N) ->
          c_25(a__U62^#(a__isNat(N), L), a__isNat^#(N))
    , 30: a__U62^#(X1, X2) -> c_26()
    , 31: a__U62^#(tt(), L) -> c_27(a__length^#(mark(L)), mark^#(L))
    , 32: a__length^#(X) -> c_32()
    , 33: a__length^#(cons(N, L)) ->
          c_33(a__U61^#(a__isNatList(L), L, N), a__isNatList^#(L))
    , 34: a__length^#(nil()) -> c_34()
    , 35: mark^#(cons(X1, X2)) -> c_35(mark^#(X1))
    , 36: mark^#(0()) -> c_36()
    , 37: mark^#(zeros()) -> c_37(a__zeros^#())
    , 38: mark^#(tt()) -> c_38()
    , 39: mark^#(s(X)) -> c_39(mark^#(X))
    , 40: mark^#(length(X)) -> c_40(a__length^#(mark(X)), mark^#(X))
    , 41: mark^#(nil()) -> c_41()
    , 42: mark^#(U11(X)) -> c_42(a__U11^#(mark(X)), mark^#(X))
    , 43: mark^#(U21(X)) -> c_43(a__U21^#(mark(X)), mark^#(X))
    , 44: mark^#(U31(X)) -> c_44(a__U31^#(mark(X)), mark^#(X))
    , 45: mark^#(U41(X1, X2)) ->
          c_45(a__U41^#(mark(X1), X2), mark^#(X1))
    , 46: mark^#(U42(X)) -> c_46(a__U42^#(mark(X)), mark^#(X))
    , 47: mark^#(isNatIList(X)) -> c_47(a__isNatIList^#(X))
    , 48: mark^#(U51(X1, X2)) ->
          c_48(a__U51^#(mark(X1), X2), mark^#(X1))
    , 49: mark^#(U52(X)) -> c_49(a__U52^#(mark(X)), mark^#(X))
    , 50: mark^#(isNatList(X)) -> c_50(a__isNatList^#(X))
    , 51: mark^#(U61(X1, X2, X3)) ->
          c_51(a__U61^#(mark(X1), X2, X3), mark^#(X1))
    , 52: mark^#(U62(X1, X2)) ->
          c_52(a__U62^#(mark(X1), X2), mark^#(X1))
    , 53: mark^#(isNat(X)) -> c_53(a__isNat^#(X)) }

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

Strict DPs:
  { a__U41^#(tt(), V2) ->
    c_10(a__U42^#(a__isNatIList(V2)), a__isNatIList^#(V2))
  , a__isNatIList^#(V) ->
    c_13(a__U31^#(a__isNatList(V)), a__isNatList^#(V))
  , a__isNatIList^#(cons(V1, V2)) ->
    c_15(a__U41^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__isNatList^#(cons(V1, V2)) ->
    c_22(a__U51^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__isNat^#(s(V1)) -> c_30(a__U21^#(a__isNat(V1)), a__isNat^#(V1))
  , a__isNat^#(length(V1)) ->
    c_31(a__U11^#(a__isNatList(V1)), a__isNatList^#(V1))
  , a__U51^#(tt(), V2) ->
    c_18(a__U52^#(a__isNatList(V2)), a__isNatList^#(V2))
  , a__U61^#(tt(), L, N) ->
    c_25(a__U62^#(a__isNat(N), L), a__isNat^#(N))
  , a__U62^#(tt(), L) -> c_27(a__length^#(mark(L)), mark^#(L))
  , a__length^#(cons(N, L)) ->
    c_33(a__U61^#(a__isNatList(L), L, N), a__isNatList^#(L))
  , mark^#(cons(X1, X2)) -> c_35(mark^#(X1))
  , mark^#(zeros()) -> c_37(a__zeros^#())
  , mark^#(s(X)) -> c_39(mark^#(X))
  , mark^#(length(X)) -> c_40(a__length^#(mark(X)), mark^#(X))
  , mark^#(U11(X)) -> c_42(a__U11^#(mark(X)), mark^#(X))
  , mark^#(U21(X)) -> c_43(a__U21^#(mark(X)), mark^#(X))
  , mark^#(U31(X)) -> c_44(a__U31^#(mark(X)), mark^#(X))
  , mark^#(U41(X1, X2)) -> c_45(a__U41^#(mark(X1), X2), mark^#(X1))
  , mark^#(U42(X)) -> c_46(a__U42^#(mark(X)), mark^#(X))
  , mark^#(isNatIList(X)) -> c_47(a__isNatIList^#(X))
  , mark^#(U51(X1, X2)) -> c_48(a__U51^#(mark(X1), X2), mark^#(X1))
  , mark^#(U52(X)) -> c_49(a__U52^#(mark(X)), mark^#(X))
  , mark^#(isNatList(X)) -> c_50(a__isNatList^#(X))
  , mark^#(U61(X1, X2, X3)) ->
    c_51(a__U61^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U62(X1, X2)) -> c_52(a__U62^#(mark(X1), X2), mark^#(X1))
  , mark^#(isNat(X)) -> c_53(a__isNat^#(X)) }
Weak DPs:
  { a__zeros^#() -> c_1()
  , a__zeros^#() -> c_2()
  , a__U11^#(X) -> c_3()
  , a__U11^#(tt()) -> c_4()
  , a__U21^#(X) -> c_5()
  , a__U21^#(tt()) -> c_6()
  , a__U31^#(X) -> c_7()
  , a__U31^#(tt()) -> c_8()
  , a__U41^#(X1, X2) -> c_9()
  , a__U42^#(X) -> c_11()
  , a__U42^#(tt()) -> c_12()
  , a__isNatIList^#(X) -> c_14()
  , a__isNatIList^#(zeros()) -> c_16()
  , a__isNatList^#(X) -> c_21()
  , a__isNatList^#(nil()) -> c_23()
  , a__isNat^#(X) -> c_28()
  , a__isNat^#(0()) -> c_29()
  , a__U51^#(X1, X2) -> c_17()
  , a__U52^#(X) -> c_19()
  , a__U52^#(tt()) -> c_20()
  , a__U61^#(X1, X2, X3) -> c_24()
  , a__U62^#(X1, X2) -> c_26()
  , a__length^#(X) -> c_32()
  , a__length^#(nil()) -> c_34()
  , mark^#(0()) -> c_36()
  , mark^#(tt()) -> c_38()
  , mark^#(nil()) -> c_41() }
Weak Trs:
  { a__zeros() -> cons(0(), zeros())
  , a__zeros() -> zeros()
  , a__U11(X) -> U11(X)
  , a__U11(tt()) -> tt()
  , a__U21(X) -> U21(X)
  , a__U21(tt()) -> tt()
  , a__U31(X) -> U31(X)
  , a__U31(tt()) -> tt()
  , a__U41(X1, X2) -> U41(X1, X2)
  , a__U41(tt(), V2) -> a__U42(a__isNatIList(V2))
  , a__U42(X) -> U42(X)
  , a__U42(tt()) -> tt()
  , a__isNatIList(V) -> a__U31(a__isNatList(V))
  , a__isNatIList(X) -> isNatIList(X)
  , a__isNatIList(cons(V1, V2)) -> a__U41(a__isNat(V1), V2)
  , a__isNatIList(zeros()) -> tt()
  , a__U51(X1, X2) -> U51(X1, X2)
  , a__U51(tt(), V2) -> a__U52(a__isNatList(V2))
  , a__U52(X) -> U52(X)
  , a__U52(tt()) -> tt()
  , a__isNatList(X) -> isNatList(X)
  , a__isNatList(cons(V1, V2)) -> a__U51(a__isNat(V1), V2)
  , a__isNatList(nil()) -> tt()
  , a__U61(X1, X2, X3) -> U61(X1, X2, X3)
  , a__U61(tt(), L, N) -> a__U62(a__isNat(N), L)
  , a__U62(X1, X2) -> U62(X1, X2)
  , a__U62(tt(), L) -> s(a__length(mark(L)))
  , a__isNat(X) -> isNat(X)
  , a__isNat(0()) -> tt()
  , a__isNat(s(V1)) -> a__U21(a__isNat(V1))
  , a__isNat(length(V1)) -> a__U11(a__isNatList(V1))
  , a__length(X) -> length(X)
  , a__length(cons(N, L)) -> a__U61(a__isNatList(L), L, N)
  , a__length(nil()) -> 0()
  , mark(cons(X1, X2)) -> cons(mark(X1), X2)
  , mark(0()) -> 0()
  , mark(zeros()) -> a__zeros()
  , mark(tt()) -> tt()
  , mark(s(X)) -> s(mark(X))
  , mark(length(X)) -> a__length(mark(X))
  , mark(nil()) -> nil()
  , mark(U11(X)) -> a__U11(mark(X))
  , mark(U21(X)) -> a__U21(mark(X))
  , mark(U31(X)) -> a__U31(mark(X))
  , mark(U41(X1, X2)) -> a__U41(mark(X1), X2)
  , mark(U42(X)) -> a__U42(mark(X))
  , mark(isNatIList(X)) -> a__isNatIList(X)
  , mark(U51(X1, X2)) -> a__U51(mark(X1), X2)
  , mark(U52(X)) -> a__U52(mark(X))
  , mark(isNatList(X)) -> a__isNatList(X)
  , mark(U61(X1, X2, X3)) -> a__U61(mark(X1), X2, X3)
  , mark(U62(X1, X2)) -> a__U62(mark(X1), X2)
  , mark(isNat(X)) -> a__isNat(X) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of {12} by applications of
Pre({12}) = {9,11,13,14,15,16,17,18,19,21,22,24,25}. Here rules are
labeled as follows:

  DPs:
    { 1: a__U41^#(tt(), V2) ->
         c_10(a__U42^#(a__isNatIList(V2)), a__isNatIList^#(V2))
    , 2: a__isNatIList^#(V) ->
         c_13(a__U31^#(a__isNatList(V)), a__isNatList^#(V))
    , 3: a__isNatIList^#(cons(V1, V2)) ->
         c_15(a__U41^#(a__isNat(V1), V2), a__isNat^#(V1))
    , 4: a__isNatList^#(cons(V1, V2)) ->
         c_22(a__U51^#(a__isNat(V1), V2), a__isNat^#(V1))
    , 5: a__isNat^#(s(V1)) ->
         c_30(a__U21^#(a__isNat(V1)), a__isNat^#(V1))
    , 6: a__isNat^#(length(V1)) ->
         c_31(a__U11^#(a__isNatList(V1)), a__isNatList^#(V1))
    , 7: a__U51^#(tt(), V2) ->
         c_18(a__U52^#(a__isNatList(V2)), a__isNatList^#(V2))
    , 8: a__U61^#(tt(), L, N) ->
         c_25(a__U62^#(a__isNat(N), L), a__isNat^#(N))
    , 9: a__U62^#(tt(), L) -> c_27(a__length^#(mark(L)), mark^#(L))
    , 10: a__length^#(cons(N, L)) ->
          c_33(a__U61^#(a__isNatList(L), L, N), a__isNatList^#(L))
    , 11: mark^#(cons(X1, X2)) -> c_35(mark^#(X1))
    , 12: mark^#(zeros()) -> c_37(a__zeros^#())
    , 13: mark^#(s(X)) -> c_39(mark^#(X))
    , 14: mark^#(length(X)) -> c_40(a__length^#(mark(X)), mark^#(X))
    , 15: mark^#(U11(X)) -> c_42(a__U11^#(mark(X)), mark^#(X))
    , 16: mark^#(U21(X)) -> c_43(a__U21^#(mark(X)), mark^#(X))
    , 17: mark^#(U31(X)) -> c_44(a__U31^#(mark(X)), mark^#(X))
    , 18: mark^#(U41(X1, X2)) ->
          c_45(a__U41^#(mark(X1), X2), mark^#(X1))
    , 19: mark^#(U42(X)) -> c_46(a__U42^#(mark(X)), mark^#(X))
    , 20: mark^#(isNatIList(X)) -> c_47(a__isNatIList^#(X))
    , 21: mark^#(U51(X1, X2)) ->
          c_48(a__U51^#(mark(X1), X2), mark^#(X1))
    , 22: mark^#(U52(X)) -> c_49(a__U52^#(mark(X)), mark^#(X))
    , 23: mark^#(isNatList(X)) -> c_50(a__isNatList^#(X))
    , 24: mark^#(U61(X1, X2, X3)) ->
          c_51(a__U61^#(mark(X1), X2, X3), mark^#(X1))
    , 25: mark^#(U62(X1, X2)) ->
          c_52(a__U62^#(mark(X1), X2), mark^#(X1))
    , 26: mark^#(isNat(X)) -> c_53(a__isNat^#(X))
    , 27: a__zeros^#() -> c_1()
    , 28: a__zeros^#() -> c_2()
    , 29: a__U11^#(X) -> c_3()
    , 30: a__U11^#(tt()) -> c_4()
    , 31: a__U21^#(X) -> c_5()
    , 32: a__U21^#(tt()) -> c_6()
    , 33: a__U31^#(X) -> c_7()
    , 34: a__U31^#(tt()) -> c_8()
    , 35: a__U41^#(X1, X2) -> c_9()
    , 36: a__U42^#(X) -> c_11()
    , 37: a__U42^#(tt()) -> c_12()
    , 38: a__isNatIList^#(X) -> c_14()
    , 39: a__isNatIList^#(zeros()) -> c_16()
    , 40: a__isNatList^#(X) -> c_21()
    , 41: a__isNatList^#(nil()) -> c_23()
    , 42: a__isNat^#(X) -> c_28()
    , 43: a__isNat^#(0()) -> c_29()
    , 44: a__U51^#(X1, X2) -> c_17()
    , 45: a__U52^#(X) -> c_19()
    , 46: a__U52^#(tt()) -> c_20()
    , 47: a__U61^#(X1, X2, X3) -> c_24()
    , 48: a__U62^#(X1, X2) -> c_26()
    , 49: a__length^#(X) -> c_32()
    , 50: a__length^#(nil()) -> c_34()
    , 51: mark^#(0()) -> c_36()
    , 52: mark^#(tt()) -> c_38()
    , 53: mark^#(nil()) -> c_41() }

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

Strict DPs:
  { a__U41^#(tt(), V2) ->
    c_10(a__U42^#(a__isNatIList(V2)), a__isNatIList^#(V2))
  , a__isNatIList^#(V) ->
    c_13(a__U31^#(a__isNatList(V)), a__isNatList^#(V))
  , a__isNatIList^#(cons(V1, V2)) ->
    c_15(a__U41^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__isNatList^#(cons(V1, V2)) ->
    c_22(a__U51^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__isNat^#(s(V1)) -> c_30(a__U21^#(a__isNat(V1)), a__isNat^#(V1))
  , a__isNat^#(length(V1)) ->
    c_31(a__U11^#(a__isNatList(V1)), a__isNatList^#(V1))
  , a__U51^#(tt(), V2) ->
    c_18(a__U52^#(a__isNatList(V2)), a__isNatList^#(V2))
  , a__U61^#(tt(), L, N) ->
    c_25(a__U62^#(a__isNat(N), L), a__isNat^#(N))
  , a__U62^#(tt(), L) -> c_27(a__length^#(mark(L)), mark^#(L))
  , a__length^#(cons(N, L)) ->
    c_33(a__U61^#(a__isNatList(L), L, N), a__isNatList^#(L))
  , mark^#(cons(X1, X2)) -> c_35(mark^#(X1))
  , mark^#(s(X)) -> c_39(mark^#(X))
  , mark^#(length(X)) -> c_40(a__length^#(mark(X)), mark^#(X))
  , mark^#(U11(X)) -> c_42(a__U11^#(mark(X)), mark^#(X))
  , mark^#(U21(X)) -> c_43(a__U21^#(mark(X)), mark^#(X))
  , mark^#(U31(X)) -> c_44(a__U31^#(mark(X)), mark^#(X))
  , mark^#(U41(X1, X2)) -> c_45(a__U41^#(mark(X1), X2), mark^#(X1))
  , mark^#(U42(X)) -> c_46(a__U42^#(mark(X)), mark^#(X))
  , mark^#(isNatIList(X)) -> c_47(a__isNatIList^#(X))
  , mark^#(U51(X1, X2)) -> c_48(a__U51^#(mark(X1), X2), mark^#(X1))
  , mark^#(U52(X)) -> c_49(a__U52^#(mark(X)), mark^#(X))
  , mark^#(isNatList(X)) -> c_50(a__isNatList^#(X))
  , mark^#(U61(X1, X2, X3)) ->
    c_51(a__U61^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U62(X1, X2)) -> c_52(a__U62^#(mark(X1), X2), mark^#(X1))
  , mark^#(isNat(X)) -> c_53(a__isNat^#(X)) }
Weak DPs:
  { a__zeros^#() -> c_1()
  , a__zeros^#() -> c_2()
  , a__U11^#(X) -> c_3()
  , a__U11^#(tt()) -> c_4()
  , a__U21^#(X) -> c_5()
  , a__U21^#(tt()) -> c_6()
  , a__U31^#(X) -> c_7()
  , a__U31^#(tt()) -> c_8()
  , a__U41^#(X1, X2) -> c_9()
  , a__U42^#(X) -> c_11()
  , a__U42^#(tt()) -> c_12()
  , a__isNatIList^#(X) -> c_14()
  , a__isNatIList^#(zeros()) -> c_16()
  , a__isNatList^#(X) -> c_21()
  , a__isNatList^#(nil()) -> c_23()
  , a__isNat^#(X) -> c_28()
  , a__isNat^#(0()) -> c_29()
  , a__U51^#(X1, X2) -> c_17()
  , a__U52^#(X) -> c_19()
  , a__U52^#(tt()) -> c_20()
  , a__U61^#(X1, X2, X3) -> c_24()
  , a__U62^#(X1, X2) -> c_26()
  , a__length^#(X) -> c_32()
  , a__length^#(nil()) -> c_34()
  , mark^#(0()) -> c_36()
  , mark^#(zeros()) -> c_37(a__zeros^#())
  , mark^#(tt()) -> c_38()
  , mark^#(nil()) -> c_41() }
Weak Trs:
  { a__zeros() -> cons(0(), zeros())
  , a__zeros() -> zeros()
  , a__U11(X) -> U11(X)
  , a__U11(tt()) -> tt()
  , a__U21(X) -> U21(X)
  , a__U21(tt()) -> tt()
  , a__U31(X) -> U31(X)
  , a__U31(tt()) -> tt()
  , a__U41(X1, X2) -> U41(X1, X2)
  , a__U41(tt(), V2) -> a__U42(a__isNatIList(V2))
  , a__U42(X) -> U42(X)
  , a__U42(tt()) -> tt()
  , a__isNatIList(V) -> a__U31(a__isNatList(V))
  , a__isNatIList(X) -> isNatIList(X)
  , a__isNatIList(cons(V1, V2)) -> a__U41(a__isNat(V1), V2)
  , a__isNatIList(zeros()) -> tt()
  , a__U51(X1, X2) -> U51(X1, X2)
  , a__U51(tt(), V2) -> a__U52(a__isNatList(V2))
  , a__U52(X) -> U52(X)
  , a__U52(tt()) -> tt()
  , a__isNatList(X) -> isNatList(X)
  , a__isNatList(cons(V1, V2)) -> a__U51(a__isNat(V1), V2)
  , a__isNatList(nil()) -> tt()
  , a__U61(X1, X2, X3) -> U61(X1, X2, X3)
  , a__U61(tt(), L, N) -> a__U62(a__isNat(N), L)
  , a__U62(X1, X2) -> U62(X1, X2)
  , a__U62(tt(), L) -> s(a__length(mark(L)))
  , a__isNat(X) -> isNat(X)
  , a__isNat(0()) -> tt()
  , a__isNat(s(V1)) -> a__U21(a__isNat(V1))
  , a__isNat(length(V1)) -> a__U11(a__isNatList(V1))
  , a__length(X) -> length(X)
  , a__length(cons(N, L)) -> a__U61(a__isNatList(L), L, N)
  , a__length(nil()) -> 0()
  , mark(cons(X1, X2)) -> cons(mark(X1), X2)
  , mark(0()) -> 0()
  , mark(zeros()) -> a__zeros()
  , mark(tt()) -> tt()
  , mark(s(X)) -> s(mark(X))
  , mark(length(X)) -> a__length(mark(X))
  , mark(nil()) -> nil()
  , mark(U11(X)) -> a__U11(mark(X))
  , mark(U21(X)) -> a__U21(mark(X))
  , mark(U31(X)) -> a__U31(mark(X))
  , mark(U41(X1, X2)) -> a__U41(mark(X1), X2)
  , mark(U42(X)) -> a__U42(mark(X))
  , mark(isNatIList(X)) -> a__isNatIList(X)
  , mark(U51(X1, X2)) -> a__U51(mark(X1), X2)
  , mark(U52(X)) -> a__U52(mark(X))
  , mark(isNatList(X)) -> a__isNatList(X)
  , mark(U61(X1, X2, X3)) -> a__U61(mark(X1), X2, X3)
  , mark(U62(X1, X2)) -> a__U62(mark(X1), X2)
  , mark(isNat(X)) -> a__isNat(X) }
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__zeros^#() -> c_1()
, a__zeros^#() -> c_2()
, a__U11^#(X) -> c_3()
, a__U11^#(tt()) -> c_4()
, a__U21^#(X) -> c_5()
, a__U21^#(tt()) -> c_6()
, a__U31^#(X) -> c_7()
, a__U31^#(tt()) -> c_8()
, a__U41^#(X1, X2) -> c_9()
, a__U42^#(X) -> c_11()
, a__U42^#(tt()) -> c_12()
, a__isNatIList^#(X) -> c_14()
, a__isNatIList^#(zeros()) -> c_16()
, a__isNatList^#(X) -> c_21()
, a__isNatList^#(nil()) -> c_23()
, a__isNat^#(X) -> c_28()
, a__isNat^#(0()) -> c_29()
, a__U51^#(X1, X2) -> c_17()
, a__U52^#(X) -> c_19()
, a__U52^#(tt()) -> c_20()
, a__U61^#(X1, X2, X3) -> c_24()
, a__U62^#(X1, X2) -> c_26()
, a__length^#(X) -> c_32()
, a__length^#(nil()) -> c_34()
, mark^#(0()) -> c_36()
, mark^#(zeros()) -> c_37(a__zeros^#())
, mark^#(tt()) -> c_38()
, mark^#(nil()) -> c_41() }

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

Strict DPs:
  { a__U41^#(tt(), V2) ->
    c_10(a__U42^#(a__isNatIList(V2)), a__isNatIList^#(V2))
  , a__isNatIList^#(V) ->
    c_13(a__U31^#(a__isNatList(V)), a__isNatList^#(V))
  , a__isNatIList^#(cons(V1, V2)) ->
    c_15(a__U41^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__isNatList^#(cons(V1, V2)) ->
    c_22(a__U51^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__isNat^#(s(V1)) -> c_30(a__U21^#(a__isNat(V1)), a__isNat^#(V1))
  , a__isNat^#(length(V1)) ->
    c_31(a__U11^#(a__isNatList(V1)), a__isNatList^#(V1))
  , a__U51^#(tt(), V2) ->
    c_18(a__U52^#(a__isNatList(V2)), a__isNatList^#(V2))
  , a__U61^#(tt(), L, N) ->
    c_25(a__U62^#(a__isNat(N), L), a__isNat^#(N))
  , a__U62^#(tt(), L) -> c_27(a__length^#(mark(L)), mark^#(L))
  , a__length^#(cons(N, L)) ->
    c_33(a__U61^#(a__isNatList(L), L, N), a__isNatList^#(L))
  , mark^#(cons(X1, X2)) -> c_35(mark^#(X1))
  , mark^#(s(X)) -> c_39(mark^#(X))
  , mark^#(length(X)) -> c_40(a__length^#(mark(X)), mark^#(X))
  , mark^#(U11(X)) -> c_42(a__U11^#(mark(X)), mark^#(X))
  , mark^#(U21(X)) -> c_43(a__U21^#(mark(X)), mark^#(X))
  , mark^#(U31(X)) -> c_44(a__U31^#(mark(X)), mark^#(X))
  , mark^#(U41(X1, X2)) -> c_45(a__U41^#(mark(X1), X2), mark^#(X1))
  , mark^#(U42(X)) -> c_46(a__U42^#(mark(X)), mark^#(X))
  , mark^#(isNatIList(X)) -> c_47(a__isNatIList^#(X))
  , mark^#(U51(X1, X2)) -> c_48(a__U51^#(mark(X1), X2), mark^#(X1))
  , mark^#(U52(X)) -> c_49(a__U52^#(mark(X)), mark^#(X))
  , mark^#(isNatList(X)) -> c_50(a__isNatList^#(X))
  , mark^#(U61(X1, X2, X3)) ->
    c_51(a__U61^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U62(X1, X2)) -> c_52(a__U62^#(mark(X1), X2), mark^#(X1))
  , mark^#(isNat(X)) -> c_53(a__isNat^#(X)) }
Weak Trs:
  { a__zeros() -> cons(0(), zeros())
  , a__zeros() -> zeros()
  , a__U11(X) -> U11(X)
  , a__U11(tt()) -> tt()
  , a__U21(X) -> U21(X)
  , a__U21(tt()) -> tt()
  , a__U31(X) -> U31(X)
  , a__U31(tt()) -> tt()
  , a__U41(X1, X2) -> U41(X1, X2)
  , a__U41(tt(), V2) -> a__U42(a__isNatIList(V2))
  , a__U42(X) -> U42(X)
  , a__U42(tt()) -> tt()
  , a__isNatIList(V) -> a__U31(a__isNatList(V))
  , a__isNatIList(X) -> isNatIList(X)
  , a__isNatIList(cons(V1, V2)) -> a__U41(a__isNat(V1), V2)
  , a__isNatIList(zeros()) -> tt()
  , a__U51(X1, X2) -> U51(X1, X2)
  , a__U51(tt(), V2) -> a__U52(a__isNatList(V2))
  , a__U52(X) -> U52(X)
  , a__U52(tt()) -> tt()
  , a__isNatList(X) -> isNatList(X)
  , a__isNatList(cons(V1, V2)) -> a__U51(a__isNat(V1), V2)
  , a__isNatList(nil()) -> tt()
  , a__U61(X1, X2, X3) -> U61(X1, X2, X3)
  , a__U61(tt(), L, N) -> a__U62(a__isNat(N), L)
  , a__U62(X1, X2) -> U62(X1, X2)
  , a__U62(tt(), L) -> s(a__length(mark(L)))
  , a__isNat(X) -> isNat(X)
  , a__isNat(0()) -> tt()
  , a__isNat(s(V1)) -> a__U21(a__isNat(V1))
  , a__isNat(length(V1)) -> a__U11(a__isNatList(V1))
  , a__length(X) -> length(X)
  , a__length(cons(N, L)) -> a__U61(a__isNatList(L), L, N)
  , a__length(nil()) -> 0()
  , mark(cons(X1, X2)) -> cons(mark(X1), X2)
  , mark(0()) -> 0()
  , mark(zeros()) -> a__zeros()
  , mark(tt()) -> tt()
  , mark(s(X)) -> s(mark(X))
  , mark(length(X)) -> a__length(mark(X))
  , mark(nil()) -> nil()
  , mark(U11(X)) -> a__U11(mark(X))
  , mark(U21(X)) -> a__U21(mark(X))
  , mark(U31(X)) -> a__U31(mark(X))
  , mark(U41(X1, X2)) -> a__U41(mark(X1), X2)
  , mark(U42(X)) -> a__U42(mark(X))
  , mark(isNatIList(X)) -> a__isNatIList(X)
  , mark(U51(X1, X2)) -> a__U51(mark(X1), X2)
  , mark(U52(X)) -> a__U52(mark(X))
  , mark(isNatList(X)) -> a__isNatList(X)
  , mark(U61(X1, X2, X3)) -> a__U61(mark(X1), X2, X3)
  , mark(U62(X1, X2)) -> a__U62(mark(X1), X2)
  , mark(isNat(X)) -> a__isNat(X) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

  { a__U41^#(tt(), V2) ->
    c_10(a__U42^#(a__isNatIList(V2)), a__isNatIList^#(V2))
  , a__isNatIList^#(V) ->
    c_13(a__U31^#(a__isNatList(V)), a__isNatList^#(V))
  , a__isNat^#(s(V1)) -> c_30(a__U21^#(a__isNat(V1)), a__isNat^#(V1))
  , a__isNat^#(length(V1)) ->
    c_31(a__U11^#(a__isNatList(V1)), a__isNatList^#(V1))
  , a__U51^#(tt(), V2) ->
    c_18(a__U52^#(a__isNatList(V2)), a__isNatList^#(V2))
  , mark^#(U11(X)) -> c_42(a__U11^#(mark(X)), mark^#(X))
  , mark^#(U21(X)) -> c_43(a__U21^#(mark(X)), mark^#(X))
  , mark^#(U31(X)) -> c_44(a__U31^#(mark(X)), mark^#(X))
  , mark^#(U42(X)) -> c_46(a__U42^#(mark(X)), mark^#(X))
  , mark^#(U52(X)) -> c_49(a__U52^#(mark(X)), mark^#(X)) }

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

Strict DPs:
  { a__U41^#(tt(), V2) -> c_1(a__isNatIList^#(V2))
  , a__isNatIList^#(V) -> c_2(a__isNatList^#(V))
  , a__isNatIList^#(cons(V1, V2)) ->
    c_3(a__U41^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__isNatList^#(cons(V1, V2)) ->
    c_4(a__U51^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__isNat^#(s(V1)) -> c_5(a__isNat^#(V1))
  , a__isNat^#(length(V1)) -> c_6(a__isNatList^#(V1))
  , a__U51^#(tt(), V2) -> c_7(a__isNatList^#(V2))
  , a__U61^#(tt(), L, N) ->
    c_8(a__U62^#(a__isNat(N), L), a__isNat^#(N))
  , a__U62^#(tt(), L) -> c_9(a__length^#(mark(L)), mark^#(L))
  , a__length^#(cons(N, L)) ->
    c_10(a__U61^#(a__isNatList(L), L, N), a__isNatList^#(L))
  , mark^#(cons(X1, X2)) -> c_11(mark^#(X1))
  , mark^#(s(X)) -> c_12(mark^#(X))
  , mark^#(length(X)) -> c_13(a__length^#(mark(X)), mark^#(X))
  , mark^#(U11(X)) -> c_14(mark^#(X))
  , mark^#(U21(X)) -> c_15(mark^#(X))
  , mark^#(U31(X)) -> c_16(mark^#(X))
  , mark^#(U41(X1, X2)) -> c_17(a__U41^#(mark(X1), X2), mark^#(X1))
  , mark^#(U42(X)) -> c_18(mark^#(X))
  , mark^#(isNatIList(X)) -> c_19(a__isNatIList^#(X))
  , mark^#(U51(X1, X2)) -> c_20(a__U51^#(mark(X1), X2), mark^#(X1))
  , mark^#(U52(X)) -> c_21(mark^#(X))
  , mark^#(isNatList(X)) -> c_22(a__isNatList^#(X))
  , mark^#(U61(X1, X2, X3)) ->
    c_23(a__U61^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U62(X1, X2)) -> c_24(a__U62^#(mark(X1), X2), mark^#(X1))
  , mark^#(isNat(X)) -> c_25(a__isNat^#(X)) }
Weak Trs:
  { a__zeros() -> cons(0(), zeros())
  , a__zeros() -> zeros()
  , a__U11(X) -> U11(X)
  , a__U11(tt()) -> tt()
  , a__U21(X) -> U21(X)
  , a__U21(tt()) -> tt()
  , a__U31(X) -> U31(X)
  , a__U31(tt()) -> tt()
  , a__U41(X1, X2) -> U41(X1, X2)
  , a__U41(tt(), V2) -> a__U42(a__isNatIList(V2))
  , a__U42(X) -> U42(X)
  , a__U42(tt()) -> tt()
  , a__isNatIList(V) -> a__U31(a__isNatList(V))
  , a__isNatIList(X) -> isNatIList(X)
  , a__isNatIList(cons(V1, V2)) -> a__U41(a__isNat(V1), V2)
  , a__isNatIList(zeros()) -> tt()
  , a__U51(X1, X2) -> U51(X1, X2)
  , a__U51(tt(), V2) -> a__U52(a__isNatList(V2))
  , a__U52(X) -> U52(X)
  , a__U52(tt()) -> tt()
  , a__isNatList(X) -> isNatList(X)
  , a__isNatList(cons(V1, V2)) -> a__U51(a__isNat(V1), V2)
  , a__isNatList(nil()) -> tt()
  , a__U61(X1, X2, X3) -> U61(X1, X2, X3)
  , a__U61(tt(), L, N) -> a__U62(a__isNat(N), L)
  , a__U62(X1, X2) -> U62(X1, X2)
  , a__U62(tt(), L) -> s(a__length(mark(L)))
  , a__isNat(X) -> isNat(X)
  , a__isNat(0()) -> tt()
  , a__isNat(s(V1)) -> a__U21(a__isNat(V1))
  , a__isNat(length(V1)) -> a__U11(a__isNatList(V1))
  , a__length(X) -> length(X)
  , a__length(cons(N, L)) -> a__U61(a__isNatList(L), L, N)
  , a__length(nil()) -> 0()
  , mark(cons(X1, X2)) -> cons(mark(X1), X2)
  , mark(0()) -> 0()
  , mark(zeros()) -> a__zeros()
  , mark(tt()) -> tt()
  , mark(s(X)) -> s(mark(X))
  , mark(length(X)) -> a__length(mark(X))
  , mark(nil()) -> nil()
  , mark(U11(X)) -> a__U11(mark(X))
  , mark(U21(X)) -> a__U21(mark(X))
  , mark(U31(X)) -> a__U31(mark(X))
  , mark(U41(X1, X2)) -> a__U41(mark(X1), X2)
  , mark(U42(X)) -> a__U42(mark(X))
  , mark(isNatIList(X)) -> a__isNatIList(X)
  , mark(U51(X1, X2)) -> a__U51(mark(X1), X2)
  , mark(U52(X)) -> a__U52(mark(X))
  , mark(isNatList(X)) -> a__isNatList(X)
  , mark(U61(X1, X2, X3)) -> a__U61(mark(X1), X2, X3)
  , mark(U62(X1, X2)) -> a__U62(mark(X1), X2)
  , mark(isNat(X)) -> a__isNat(X) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

None of the processors succeeded.

Details of failed attempt(s):
-----------------------------
1) 'matrices' failed due to the following reason:
   
   None of the processors succeeded.
   
   Details of failed attempt(s):
   -----------------------------
   1) 'matrix interpretation of dimension 4' failed due to the
      following reason:
      
      Following exception was raised:
        stack overflow
   
   2) 'matrix interpretation of dimension 3' failed due to the
      following reason:
      
      The input cannot be shown compatible
   
   3) 'matrix interpretation of dimension 3' failed due to the
      following reason:
      
      The input cannot be shown compatible
   
   4) 'matrix interpretation of dimension 2' failed due to the
      following reason:
      
      The input cannot be shown compatible
   
   5) 'matrix interpretation of dimension 2' failed due to the
      following reason:
      
      The input cannot be shown compatible
   
   6) 'matrix interpretation of dimension 1' failed due to the
      following reason:
      
      The input cannot be shown compatible
   

2) 'empty' failed due to the following reason:
   
   Empty strict component of the problem is NOT empty.


Arrrr..