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(X1, X2) -> U11(X1, X2)
  , a__U11(tt(), V1) -> a__U12(a__isNatList(V1))
  , a__U12(X) -> U12(X)
  , a__U12(tt()) -> tt()
  , a__isNatList(X) -> isNatList(X)
  , a__isNatList(cons(V1, V2)) ->
    a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
  , a__isNatList(nil()) -> tt()
  , a__U21(X1, X2) -> U21(X1, X2)
  , a__U21(tt(), V1) -> a__U22(a__isNat(V1))
  , a__U22(X) -> U22(X)
  , a__U22(tt()) -> tt()
  , a__isNat(X) -> isNat(X)
  , a__isNat(0()) -> tt()
  , a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1)
  , a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1)
  , a__U31(X1, X2) -> U31(X1, X2)
  , a__U31(tt(), V) -> a__U32(a__isNatList(V))
  , a__U32(X) -> U32(X)
  , a__U32(tt()) -> tt()
  , a__U41(X1, X2, X3) -> U41(X1, X2, X3)
  , a__U41(tt(), V1, V2) -> a__U42(a__isNat(V1), V2)
  , a__U42(X1, X2) -> U42(X1, X2)
  , a__U42(tt(), V2) -> a__U43(a__isNatIList(V2))
  , a__U43(X) -> U43(X)
  , a__U43(tt()) -> tt()
  , a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V)
  , a__isNatIList(X) -> isNatIList(X)
  , a__isNatIList(cons(V1, V2)) ->
    a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
  , a__isNatIList(zeros()) -> tt()
  , a__U51(X1, X2, X3) -> U51(X1, X2, X3)
  , a__U51(tt(), V1, V2) -> a__U52(a__isNat(V1), V2)
  , a__U52(X1, X2) -> U52(X1, X2)
  , a__U52(tt(), V2) -> a__U53(a__isNatList(V2))
  , a__U53(X) -> U53(X)
  , a__U53(tt()) -> tt()
  , a__U61(X1, X2) -> U61(X1, X2)
  , a__U61(tt(), L) -> s(a__length(mark(L)))
  , a__length(X) -> length(X)
  , a__length(cons(N, L)) ->
    a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)),
                  and(isNat(N), isNatKind(N))),
           L)
  , 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(isNatIListKind(X)) -> a__isNatIListKind(X)
  , mark(nil()) -> nil()
  , mark(and(X1, X2)) -> a__and(mark(X1), X2)
  , mark(isNat(X)) -> a__isNat(X)
  , mark(isNatKind(X)) -> a__isNatKind(X)
  , mark(U11(X1, X2)) -> a__U11(mark(X1), X2)
  , mark(U12(X)) -> a__U12(mark(X))
  , mark(isNatList(X)) -> a__isNatList(X)
  , mark(U21(X1, X2)) -> a__U21(mark(X1), X2)
  , mark(U22(X)) -> a__U22(mark(X))
  , mark(U31(X1, X2)) -> a__U31(mark(X1), X2)
  , mark(U32(X)) -> a__U32(mark(X))
  , mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3)
  , mark(U42(X1, X2)) -> a__U42(mark(X1), X2)
  , mark(U43(X)) -> a__U43(mark(X))
  , mark(isNatIList(X)) -> a__isNatIList(X)
  , mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3)
  , mark(U52(X1, X2)) -> a__U52(mark(X1), X2)
  , mark(U53(X)) -> a__U53(mark(X))
  , mark(U61(X1, X2)) -> a__U61(mark(X1), X2)
  , a__and(X1, X2) -> and(X1, X2)
  , a__and(tt(), X) -> mark(X)
  , a__isNatIListKind(X) -> isNatIListKind(X)
  , a__isNatIListKind(cons(V1, V2)) ->
    a__and(a__isNatKind(V1), isNatIListKind(V2))
  , a__isNatIListKind(zeros()) -> tt()
  , a__isNatIListKind(nil()) -> tt()
  , a__isNatKind(X) -> isNatKind(X)
  , a__isNatKind(0()) -> tt()
  , a__isNatKind(s(V1)) -> a__isNatKind(V1)
  , a__isNatKind(length(V1)) -> a__isNatIListKind(V1) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { a__zeros^#() -> c_1()
  , a__zeros^#() -> c_2()
  , a__U11^#(X1, X2) -> c_3()
  , a__U11^#(tt(), V1) ->
    c_4(a__U12^#(a__isNatList(V1)), a__isNatList^#(V1))
  , a__U12^#(X) -> c_5()
  , a__U12^#(tt()) -> c_6()
  , a__isNatList^#(X) -> c_7()
  , a__isNatList^#(cons(V1, V2)) ->
    c_8(a__U51^#(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2),
        a__and^#(a__isNatKind(V1), isNatIListKind(V2)),
        a__isNatKind^#(V1))
  , a__isNatList^#(nil()) -> c_9()
  , a__U51^#(X1, X2, X3) -> c_32()
  , a__U51^#(tt(), V1, V2) ->
    c_33(a__U52^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__and^#(X1, X2) -> c_69()
  , a__and^#(tt(), X) -> c_70(mark^#(X))
  , a__isNatKind^#(X) -> c_75()
  , a__isNatKind^#(0()) -> c_76()
  , a__isNatKind^#(s(V1)) -> c_77(a__isNatKind^#(V1))
  , a__isNatKind^#(length(V1)) -> c_78(a__isNatIListKind^#(V1))
  , a__U21^#(X1, X2) -> c_10()
  , a__U21^#(tt(), V1) ->
    c_11(a__U22^#(a__isNat(V1)), a__isNat^#(V1))
  , a__U22^#(X) -> c_12()
  , a__U22^#(tt()) -> c_13()
  , a__isNat^#(X) -> c_14()
  , a__isNat^#(0()) -> c_15()
  , a__isNat^#(s(V1)) ->
    c_16(a__U21^#(a__isNatKind(V1), V1), a__isNatKind^#(V1))
  , a__isNat^#(length(V1)) ->
    c_17(a__U11^#(a__isNatIListKind(V1), V1), a__isNatIListKind^#(V1))
  , a__isNatIListKind^#(X) -> c_71()
  , a__isNatIListKind^#(cons(V1, V2)) ->
    c_72(a__and^#(a__isNatKind(V1), isNatIListKind(V2)),
         a__isNatKind^#(V1))
  , a__isNatIListKind^#(zeros()) -> c_73()
  , a__isNatIListKind^#(nil()) -> c_74()
  , a__U31^#(X1, X2) -> c_18()
  , a__U31^#(tt(), V) ->
    c_19(a__U32^#(a__isNatList(V)), a__isNatList^#(V))
  , a__U32^#(X) -> c_20()
  , a__U32^#(tt()) -> c_21()
  , a__U41^#(X1, X2, X3) -> c_22()
  , a__U41^#(tt(), V1, V2) ->
    c_23(a__U42^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__U42^#(X1, X2) -> c_24()
  , a__U42^#(tt(), V2) ->
    c_25(a__U43^#(a__isNatIList(V2)), a__isNatIList^#(V2))
  , a__U43^#(X) -> c_26()
  , a__U43^#(tt()) -> c_27()
  , a__isNatIList^#(V) ->
    c_28(a__U31^#(a__isNatIListKind(V), V), a__isNatIListKind^#(V))
  , a__isNatIList^#(X) -> c_29()
  , a__isNatIList^#(cons(V1, V2)) ->
    c_30(a__U41^#(a__and(a__isNatKind(V1), isNatIListKind(V2)),
                  V1,
                  V2),
         a__and^#(a__isNatKind(V1), isNatIListKind(V2)),
         a__isNatKind^#(V1))
  , a__isNatIList^#(zeros()) -> c_31()
  , a__U52^#(X1, X2) -> c_34()
  , a__U52^#(tt(), V2) ->
    c_35(a__U53^#(a__isNatList(V2)), a__isNatList^#(V2))
  , a__U53^#(X) -> c_36()
  , a__U53^#(tt()) -> c_37()
  , a__U61^#(X1, X2) -> c_38()
  , a__U61^#(tt(), L) -> c_39(a__length^#(mark(L)), mark^#(L))
  , a__length^#(X) -> c_40()
  , a__length^#(cons(N, L)) ->
    c_41(a__U61^#(a__and(a__and(a__isNatList(L), isNatIListKind(L)),
                         and(isNat(N), isNatKind(N))),
                  L),
         a__and^#(a__and(a__isNatList(L), isNatIListKind(L)),
                  and(isNat(N), isNatKind(N))),
         a__and^#(a__isNatList(L), isNatIListKind(L)),
         a__isNatList^#(L))
  , a__length^#(nil()) -> c_42()
  , mark^#(cons(X1, X2)) -> c_43(mark^#(X1))
  , mark^#(0()) -> c_44()
  , mark^#(zeros()) -> c_45(a__zeros^#())
  , mark^#(tt()) -> c_46()
  , mark^#(s(X)) -> c_47(mark^#(X))
  , mark^#(length(X)) -> c_48(a__length^#(mark(X)), mark^#(X))
  , mark^#(isNatIListKind(X)) -> c_49(a__isNatIListKind^#(X))
  , mark^#(nil()) -> c_50()
  , mark^#(and(X1, X2)) -> c_51(a__and^#(mark(X1), X2), mark^#(X1))
  , mark^#(isNat(X)) -> c_52(a__isNat^#(X))
  , mark^#(isNatKind(X)) -> c_53(a__isNatKind^#(X))
  , mark^#(U11(X1, X2)) -> c_54(a__U11^#(mark(X1), X2), mark^#(X1))
  , mark^#(U12(X)) -> c_55(a__U12^#(mark(X)), mark^#(X))
  , mark^#(isNatList(X)) -> c_56(a__isNatList^#(X))
  , mark^#(U21(X1, X2)) -> c_57(a__U21^#(mark(X1), X2), mark^#(X1))
  , mark^#(U22(X)) -> c_58(a__U22^#(mark(X)), mark^#(X))
  , mark^#(U31(X1, X2)) -> c_59(a__U31^#(mark(X1), X2), mark^#(X1))
  , mark^#(U32(X)) -> c_60(a__U32^#(mark(X)), mark^#(X))
  , mark^#(U41(X1, X2, X3)) ->
    c_61(a__U41^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U42(X1, X2)) -> c_62(a__U42^#(mark(X1), X2), mark^#(X1))
  , mark^#(U43(X)) -> c_63(a__U43^#(mark(X)), mark^#(X))
  , mark^#(isNatIList(X)) -> c_64(a__isNatIList^#(X))
  , mark^#(U51(X1, X2, X3)) ->
    c_65(a__U51^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U52(X1, X2)) -> c_66(a__U52^#(mark(X1), X2), mark^#(X1))
  , mark^#(U53(X)) -> c_67(a__U53^#(mark(X)), mark^#(X))
  , mark^#(U61(X1, X2)) -> c_68(a__U61^#(mark(X1), X2), mark^#(X1)) }

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^#(X1, X2) -> c_3()
  , a__U11^#(tt(), V1) ->
    c_4(a__U12^#(a__isNatList(V1)), a__isNatList^#(V1))
  , a__U12^#(X) -> c_5()
  , a__U12^#(tt()) -> c_6()
  , a__isNatList^#(X) -> c_7()
  , a__isNatList^#(cons(V1, V2)) ->
    c_8(a__U51^#(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2),
        a__and^#(a__isNatKind(V1), isNatIListKind(V2)),
        a__isNatKind^#(V1))
  , a__isNatList^#(nil()) -> c_9()
  , a__U51^#(X1, X2, X3) -> c_32()
  , a__U51^#(tt(), V1, V2) ->
    c_33(a__U52^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__and^#(X1, X2) -> c_69()
  , a__and^#(tt(), X) -> c_70(mark^#(X))
  , a__isNatKind^#(X) -> c_75()
  , a__isNatKind^#(0()) -> c_76()
  , a__isNatKind^#(s(V1)) -> c_77(a__isNatKind^#(V1))
  , a__isNatKind^#(length(V1)) -> c_78(a__isNatIListKind^#(V1))
  , a__U21^#(X1, X2) -> c_10()
  , a__U21^#(tt(), V1) ->
    c_11(a__U22^#(a__isNat(V1)), a__isNat^#(V1))
  , a__U22^#(X) -> c_12()
  , a__U22^#(tt()) -> c_13()
  , a__isNat^#(X) -> c_14()
  , a__isNat^#(0()) -> c_15()
  , a__isNat^#(s(V1)) ->
    c_16(a__U21^#(a__isNatKind(V1), V1), a__isNatKind^#(V1))
  , a__isNat^#(length(V1)) ->
    c_17(a__U11^#(a__isNatIListKind(V1), V1), a__isNatIListKind^#(V1))
  , a__isNatIListKind^#(X) -> c_71()
  , a__isNatIListKind^#(cons(V1, V2)) ->
    c_72(a__and^#(a__isNatKind(V1), isNatIListKind(V2)),
         a__isNatKind^#(V1))
  , a__isNatIListKind^#(zeros()) -> c_73()
  , a__isNatIListKind^#(nil()) -> c_74()
  , a__U31^#(X1, X2) -> c_18()
  , a__U31^#(tt(), V) ->
    c_19(a__U32^#(a__isNatList(V)), a__isNatList^#(V))
  , a__U32^#(X) -> c_20()
  , a__U32^#(tt()) -> c_21()
  , a__U41^#(X1, X2, X3) -> c_22()
  , a__U41^#(tt(), V1, V2) ->
    c_23(a__U42^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__U42^#(X1, X2) -> c_24()
  , a__U42^#(tt(), V2) ->
    c_25(a__U43^#(a__isNatIList(V2)), a__isNatIList^#(V2))
  , a__U43^#(X) -> c_26()
  , a__U43^#(tt()) -> c_27()
  , a__isNatIList^#(V) ->
    c_28(a__U31^#(a__isNatIListKind(V), V), a__isNatIListKind^#(V))
  , a__isNatIList^#(X) -> c_29()
  , a__isNatIList^#(cons(V1, V2)) ->
    c_30(a__U41^#(a__and(a__isNatKind(V1), isNatIListKind(V2)),
                  V1,
                  V2),
         a__and^#(a__isNatKind(V1), isNatIListKind(V2)),
         a__isNatKind^#(V1))
  , a__isNatIList^#(zeros()) -> c_31()
  , a__U52^#(X1, X2) -> c_34()
  , a__U52^#(tt(), V2) ->
    c_35(a__U53^#(a__isNatList(V2)), a__isNatList^#(V2))
  , a__U53^#(X) -> c_36()
  , a__U53^#(tt()) -> c_37()
  , a__U61^#(X1, X2) -> c_38()
  , a__U61^#(tt(), L) -> c_39(a__length^#(mark(L)), mark^#(L))
  , a__length^#(X) -> c_40()
  , a__length^#(cons(N, L)) ->
    c_41(a__U61^#(a__and(a__and(a__isNatList(L), isNatIListKind(L)),
                         and(isNat(N), isNatKind(N))),
                  L),
         a__and^#(a__and(a__isNatList(L), isNatIListKind(L)),
                  and(isNat(N), isNatKind(N))),
         a__and^#(a__isNatList(L), isNatIListKind(L)),
         a__isNatList^#(L))
  , a__length^#(nil()) -> c_42()
  , mark^#(cons(X1, X2)) -> c_43(mark^#(X1))
  , mark^#(0()) -> c_44()
  , mark^#(zeros()) -> c_45(a__zeros^#())
  , mark^#(tt()) -> c_46()
  , mark^#(s(X)) -> c_47(mark^#(X))
  , mark^#(length(X)) -> c_48(a__length^#(mark(X)), mark^#(X))
  , mark^#(isNatIListKind(X)) -> c_49(a__isNatIListKind^#(X))
  , mark^#(nil()) -> c_50()
  , mark^#(and(X1, X2)) -> c_51(a__and^#(mark(X1), X2), mark^#(X1))
  , mark^#(isNat(X)) -> c_52(a__isNat^#(X))
  , mark^#(isNatKind(X)) -> c_53(a__isNatKind^#(X))
  , mark^#(U11(X1, X2)) -> c_54(a__U11^#(mark(X1), X2), mark^#(X1))
  , mark^#(U12(X)) -> c_55(a__U12^#(mark(X)), mark^#(X))
  , mark^#(isNatList(X)) -> c_56(a__isNatList^#(X))
  , mark^#(U21(X1, X2)) -> c_57(a__U21^#(mark(X1), X2), mark^#(X1))
  , mark^#(U22(X)) -> c_58(a__U22^#(mark(X)), mark^#(X))
  , mark^#(U31(X1, X2)) -> c_59(a__U31^#(mark(X1), X2), mark^#(X1))
  , mark^#(U32(X)) -> c_60(a__U32^#(mark(X)), mark^#(X))
  , mark^#(U41(X1, X2, X3)) ->
    c_61(a__U41^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U42(X1, X2)) -> c_62(a__U42^#(mark(X1), X2), mark^#(X1))
  , mark^#(U43(X)) -> c_63(a__U43^#(mark(X)), mark^#(X))
  , mark^#(isNatIList(X)) -> c_64(a__isNatIList^#(X))
  , mark^#(U51(X1, X2, X3)) ->
    c_65(a__U51^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U52(X1, X2)) -> c_66(a__U52^#(mark(X1), X2), mark^#(X1))
  , mark^#(U53(X)) -> c_67(a__U53^#(mark(X)), mark^#(X))
  , mark^#(U61(X1, X2)) -> c_68(a__U61^#(mark(X1), X2), mark^#(X1)) }
Weak Trs:
  { a__zeros() -> cons(0(), zeros())
  , a__zeros() -> zeros()
  , a__U11(X1, X2) -> U11(X1, X2)
  , a__U11(tt(), V1) -> a__U12(a__isNatList(V1))
  , a__U12(X) -> U12(X)
  , a__U12(tt()) -> tt()
  , a__isNatList(X) -> isNatList(X)
  , a__isNatList(cons(V1, V2)) ->
    a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
  , a__isNatList(nil()) -> tt()
  , a__U21(X1, X2) -> U21(X1, X2)
  , a__U21(tt(), V1) -> a__U22(a__isNat(V1))
  , a__U22(X) -> U22(X)
  , a__U22(tt()) -> tt()
  , a__isNat(X) -> isNat(X)
  , a__isNat(0()) -> tt()
  , a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1)
  , a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1)
  , a__U31(X1, X2) -> U31(X1, X2)
  , a__U31(tt(), V) -> a__U32(a__isNatList(V))
  , a__U32(X) -> U32(X)
  , a__U32(tt()) -> tt()
  , a__U41(X1, X2, X3) -> U41(X1, X2, X3)
  , a__U41(tt(), V1, V2) -> a__U42(a__isNat(V1), V2)
  , a__U42(X1, X2) -> U42(X1, X2)
  , a__U42(tt(), V2) -> a__U43(a__isNatIList(V2))
  , a__U43(X) -> U43(X)
  , a__U43(tt()) -> tt()
  , a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V)
  , a__isNatIList(X) -> isNatIList(X)
  , a__isNatIList(cons(V1, V2)) ->
    a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
  , a__isNatIList(zeros()) -> tt()
  , a__U51(X1, X2, X3) -> U51(X1, X2, X3)
  , a__U51(tt(), V1, V2) -> a__U52(a__isNat(V1), V2)
  , a__U52(X1, X2) -> U52(X1, X2)
  , a__U52(tt(), V2) -> a__U53(a__isNatList(V2))
  , a__U53(X) -> U53(X)
  , a__U53(tt()) -> tt()
  , a__U61(X1, X2) -> U61(X1, X2)
  , a__U61(tt(), L) -> s(a__length(mark(L)))
  , a__length(X) -> length(X)
  , a__length(cons(N, L)) ->
    a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)),
                  and(isNat(N), isNatKind(N))),
           L)
  , 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(isNatIListKind(X)) -> a__isNatIListKind(X)
  , mark(nil()) -> nil()
  , mark(and(X1, X2)) -> a__and(mark(X1), X2)
  , mark(isNat(X)) -> a__isNat(X)
  , mark(isNatKind(X)) -> a__isNatKind(X)
  , mark(U11(X1, X2)) -> a__U11(mark(X1), X2)
  , mark(U12(X)) -> a__U12(mark(X))
  , mark(isNatList(X)) -> a__isNatList(X)
  , mark(U21(X1, X2)) -> a__U21(mark(X1), X2)
  , mark(U22(X)) -> a__U22(mark(X))
  , mark(U31(X1, X2)) -> a__U31(mark(X1), X2)
  , mark(U32(X)) -> a__U32(mark(X))
  , mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3)
  , mark(U42(X1, X2)) -> a__U42(mark(X1), X2)
  , mark(U43(X)) -> a__U43(mark(X))
  , mark(isNatIList(X)) -> a__isNatIList(X)
  , mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3)
  , mark(U52(X1, X2)) -> a__U52(mark(X1), X2)
  , mark(U53(X)) -> a__U53(mark(X))
  , mark(U61(X1, X2)) -> a__U61(mark(X1), X2)
  , a__and(X1, X2) -> and(X1, X2)
  , a__and(tt(), X) -> mark(X)
  , a__isNatIListKind(X) -> isNatIListKind(X)
  , a__isNatIListKind(cons(V1, V2)) ->
    a__and(a__isNatKind(V1), isNatIListKind(V2))
  , a__isNatIListKind(zeros()) -> tt()
  , a__isNatIListKind(nil()) -> tt()
  , a__isNatKind(X) -> isNatKind(X)
  , a__isNatKind(0()) -> tt()
  , a__isNatKind(s(V1)) -> a__isNatKind(V1)
  , a__isNatKind(length(V1)) -> a__isNatIListKind(V1) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

  DPs:
    { 1: a__zeros^#() -> c_1()
    , 2: a__zeros^#() -> c_2()
    , 3: a__U11^#(X1, X2) -> c_3()
    , 4: a__U11^#(tt(), V1) ->
         c_4(a__U12^#(a__isNatList(V1)), a__isNatList^#(V1))
    , 5: a__U12^#(X) -> c_5()
    , 6: a__U12^#(tt()) -> c_6()
    , 7: a__isNatList^#(X) -> c_7()
    , 8: a__isNatList^#(cons(V1, V2)) ->
         c_8(a__U51^#(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2),
             a__and^#(a__isNatKind(V1), isNatIListKind(V2)),
             a__isNatKind^#(V1))
    , 9: a__isNatList^#(nil()) -> c_9()
    , 10: a__U51^#(X1, X2, X3) -> c_32()
    , 11: a__U51^#(tt(), V1, V2) ->
          c_33(a__U52^#(a__isNat(V1), V2), a__isNat^#(V1))
    , 12: a__and^#(X1, X2) -> c_69()
    , 13: a__and^#(tt(), X) -> c_70(mark^#(X))
    , 14: a__isNatKind^#(X) -> c_75()
    , 15: a__isNatKind^#(0()) -> c_76()
    , 16: a__isNatKind^#(s(V1)) -> c_77(a__isNatKind^#(V1))
    , 17: a__isNatKind^#(length(V1)) -> c_78(a__isNatIListKind^#(V1))
    , 18: a__U21^#(X1, X2) -> c_10()
    , 19: a__U21^#(tt(), V1) ->
          c_11(a__U22^#(a__isNat(V1)), a__isNat^#(V1))
    , 20: a__U22^#(X) -> c_12()
    , 21: a__U22^#(tt()) -> c_13()
    , 22: a__isNat^#(X) -> c_14()
    , 23: a__isNat^#(0()) -> c_15()
    , 24: a__isNat^#(s(V1)) ->
          c_16(a__U21^#(a__isNatKind(V1), V1), a__isNatKind^#(V1))
    , 25: a__isNat^#(length(V1)) ->
          c_17(a__U11^#(a__isNatIListKind(V1), V1), a__isNatIListKind^#(V1))
    , 26: a__isNatIListKind^#(X) -> c_71()
    , 27: a__isNatIListKind^#(cons(V1, V2)) ->
          c_72(a__and^#(a__isNatKind(V1), isNatIListKind(V2)),
               a__isNatKind^#(V1))
    , 28: a__isNatIListKind^#(zeros()) -> c_73()
    , 29: a__isNatIListKind^#(nil()) -> c_74()
    , 30: a__U31^#(X1, X2) -> c_18()
    , 31: a__U31^#(tt(), V) ->
          c_19(a__U32^#(a__isNatList(V)), a__isNatList^#(V))
    , 32: a__U32^#(X) -> c_20()
    , 33: a__U32^#(tt()) -> c_21()
    , 34: a__U41^#(X1, X2, X3) -> c_22()
    , 35: a__U41^#(tt(), V1, V2) ->
          c_23(a__U42^#(a__isNat(V1), V2), a__isNat^#(V1))
    , 36: a__U42^#(X1, X2) -> c_24()
    , 37: a__U42^#(tt(), V2) ->
          c_25(a__U43^#(a__isNatIList(V2)), a__isNatIList^#(V2))
    , 38: a__U43^#(X) -> c_26()
    , 39: a__U43^#(tt()) -> c_27()
    , 40: a__isNatIList^#(V) ->
          c_28(a__U31^#(a__isNatIListKind(V), V), a__isNatIListKind^#(V))
    , 41: a__isNatIList^#(X) -> c_29()
    , 42: a__isNatIList^#(cons(V1, V2)) ->
          c_30(a__U41^#(a__and(a__isNatKind(V1), isNatIListKind(V2)),
                        V1,
                        V2),
               a__and^#(a__isNatKind(V1), isNatIListKind(V2)),
               a__isNatKind^#(V1))
    , 43: a__isNatIList^#(zeros()) -> c_31()
    , 44: a__U52^#(X1, X2) -> c_34()
    , 45: a__U52^#(tt(), V2) ->
          c_35(a__U53^#(a__isNatList(V2)), a__isNatList^#(V2))
    , 46: a__U53^#(X) -> c_36()
    , 47: a__U53^#(tt()) -> c_37()
    , 48: a__U61^#(X1, X2) -> c_38()
    , 49: a__U61^#(tt(), L) -> c_39(a__length^#(mark(L)), mark^#(L))
    , 50: a__length^#(X) -> c_40()
    , 51: a__length^#(cons(N, L)) ->
          c_41(a__U61^#(a__and(a__and(a__isNatList(L), isNatIListKind(L)),
                               and(isNat(N), isNatKind(N))),
                        L),
               a__and^#(a__and(a__isNatList(L), isNatIListKind(L)),
                        and(isNat(N), isNatKind(N))),
               a__and^#(a__isNatList(L), isNatIListKind(L)),
               a__isNatList^#(L))
    , 52: a__length^#(nil()) -> c_42()
    , 53: mark^#(cons(X1, X2)) -> c_43(mark^#(X1))
    , 54: mark^#(0()) -> c_44()
    , 55: mark^#(zeros()) -> c_45(a__zeros^#())
    , 56: mark^#(tt()) -> c_46()
    , 57: mark^#(s(X)) -> c_47(mark^#(X))
    , 58: mark^#(length(X)) -> c_48(a__length^#(mark(X)), mark^#(X))
    , 59: mark^#(isNatIListKind(X)) -> c_49(a__isNatIListKind^#(X))
    , 60: mark^#(nil()) -> c_50()
    , 61: mark^#(and(X1, X2)) ->
          c_51(a__and^#(mark(X1), X2), mark^#(X1))
    , 62: mark^#(isNat(X)) -> c_52(a__isNat^#(X))
    , 63: mark^#(isNatKind(X)) -> c_53(a__isNatKind^#(X))
    , 64: mark^#(U11(X1, X2)) ->
          c_54(a__U11^#(mark(X1), X2), mark^#(X1))
    , 65: mark^#(U12(X)) -> c_55(a__U12^#(mark(X)), mark^#(X))
    , 66: mark^#(isNatList(X)) -> c_56(a__isNatList^#(X))
    , 67: mark^#(U21(X1, X2)) ->
          c_57(a__U21^#(mark(X1), X2), mark^#(X1))
    , 68: mark^#(U22(X)) -> c_58(a__U22^#(mark(X)), mark^#(X))
    , 69: mark^#(U31(X1, X2)) ->
          c_59(a__U31^#(mark(X1), X2), mark^#(X1))
    , 70: mark^#(U32(X)) -> c_60(a__U32^#(mark(X)), mark^#(X))
    , 71: mark^#(U41(X1, X2, X3)) ->
          c_61(a__U41^#(mark(X1), X2, X3), mark^#(X1))
    , 72: mark^#(U42(X1, X2)) ->
          c_62(a__U42^#(mark(X1), X2), mark^#(X1))
    , 73: mark^#(U43(X)) -> c_63(a__U43^#(mark(X)), mark^#(X))
    , 74: mark^#(isNatIList(X)) -> c_64(a__isNatIList^#(X))
    , 75: mark^#(U51(X1, X2, X3)) ->
          c_65(a__U51^#(mark(X1), X2, X3), mark^#(X1))
    , 76: mark^#(U52(X1, X2)) ->
          c_66(a__U52^#(mark(X1), X2), mark^#(X1))
    , 77: mark^#(U53(X)) -> c_67(a__U53^#(mark(X)), mark^#(X))
    , 78: mark^#(U61(X1, X2)) ->
          c_68(a__U61^#(mark(X1), X2), mark^#(X1)) }

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

Strict DPs:
  { a__U11^#(tt(), V1) ->
    c_4(a__U12^#(a__isNatList(V1)), a__isNatList^#(V1))
  , a__isNatList^#(cons(V1, V2)) ->
    c_8(a__U51^#(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2),
        a__and^#(a__isNatKind(V1), isNatIListKind(V2)),
        a__isNatKind^#(V1))
  , a__U51^#(tt(), V1, V2) ->
    c_33(a__U52^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__and^#(tt(), X) -> c_70(mark^#(X))
  , a__isNatKind^#(s(V1)) -> c_77(a__isNatKind^#(V1))
  , a__isNatKind^#(length(V1)) -> c_78(a__isNatIListKind^#(V1))
  , a__U21^#(tt(), V1) ->
    c_11(a__U22^#(a__isNat(V1)), a__isNat^#(V1))
  , a__isNat^#(s(V1)) ->
    c_16(a__U21^#(a__isNatKind(V1), V1), a__isNatKind^#(V1))
  , a__isNat^#(length(V1)) ->
    c_17(a__U11^#(a__isNatIListKind(V1), V1), a__isNatIListKind^#(V1))
  , a__isNatIListKind^#(cons(V1, V2)) ->
    c_72(a__and^#(a__isNatKind(V1), isNatIListKind(V2)),
         a__isNatKind^#(V1))
  , a__U31^#(tt(), V) ->
    c_19(a__U32^#(a__isNatList(V)), a__isNatList^#(V))
  , a__U41^#(tt(), V1, V2) ->
    c_23(a__U42^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__U42^#(tt(), V2) ->
    c_25(a__U43^#(a__isNatIList(V2)), a__isNatIList^#(V2))
  , a__isNatIList^#(V) ->
    c_28(a__U31^#(a__isNatIListKind(V), V), a__isNatIListKind^#(V))
  , a__isNatIList^#(cons(V1, V2)) ->
    c_30(a__U41^#(a__and(a__isNatKind(V1), isNatIListKind(V2)),
                  V1,
                  V2),
         a__and^#(a__isNatKind(V1), isNatIListKind(V2)),
         a__isNatKind^#(V1))
  , a__U52^#(tt(), V2) ->
    c_35(a__U53^#(a__isNatList(V2)), a__isNatList^#(V2))
  , a__U61^#(tt(), L) -> c_39(a__length^#(mark(L)), mark^#(L))
  , a__length^#(cons(N, L)) ->
    c_41(a__U61^#(a__and(a__and(a__isNatList(L), isNatIListKind(L)),
                         and(isNat(N), isNatKind(N))),
                  L),
         a__and^#(a__and(a__isNatList(L), isNatIListKind(L)),
                  and(isNat(N), isNatKind(N))),
         a__and^#(a__isNatList(L), isNatIListKind(L)),
         a__isNatList^#(L))
  , mark^#(cons(X1, X2)) -> c_43(mark^#(X1))
  , mark^#(zeros()) -> c_45(a__zeros^#())
  , mark^#(s(X)) -> c_47(mark^#(X))
  , mark^#(length(X)) -> c_48(a__length^#(mark(X)), mark^#(X))
  , mark^#(isNatIListKind(X)) -> c_49(a__isNatIListKind^#(X))
  , mark^#(and(X1, X2)) -> c_51(a__and^#(mark(X1), X2), mark^#(X1))
  , mark^#(isNat(X)) -> c_52(a__isNat^#(X))
  , mark^#(isNatKind(X)) -> c_53(a__isNatKind^#(X))
  , mark^#(U11(X1, X2)) -> c_54(a__U11^#(mark(X1), X2), mark^#(X1))
  , mark^#(U12(X)) -> c_55(a__U12^#(mark(X)), mark^#(X))
  , mark^#(isNatList(X)) -> c_56(a__isNatList^#(X))
  , mark^#(U21(X1, X2)) -> c_57(a__U21^#(mark(X1), X2), mark^#(X1))
  , mark^#(U22(X)) -> c_58(a__U22^#(mark(X)), mark^#(X))
  , mark^#(U31(X1, X2)) -> c_59(a__U31^#(mark(X1), X2), mark^#(X1))
  , mark^#(U32(X)) -> c_60(a__U32^#(mark(X)), mark^#(X))
  , mark^#(U41(X1, X2, X3)) ->
    c_61(a__U41^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U42(X1, X2)) -> c_62(a__U42^#(mark(X1), X2), mark^#(X1))
  , mark^#(U43(X)) -> c_63(a__U43^#(mark(X)), mark^#(X))
  , mark^#(isNatIList(X)) -> c_64(a__isNatIList^#(X))
  , mark^#(U51(X1, X2, X3)) ->
    c_65(a__U51^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U52(X1, X2)) -> c_66(a__U52^#(mark(X1), X2), mark^#(X1))
  , mark^#(U53(X)) -> c_67(a__U53^#(mark(X)), mark^#(X))
  , mark^#(U61(X1, X2)) -> c_68(a__U61^#(mark(X1), X2), mark^#(X1)) }
Weak DPs:
  { a__zeros^#() -> c_1()
  , a__zeros^#() -> c_2()
  , a__U11^#(X1, X2) -> c_3()
  , a__U12^#(X) -> c_5()
  , a__U12^#(tt()) -> c_6()
  , a__isNatList^#(X) -> c_7()
  , a__isNatList^#(nil()) -> c_9()
  , a__U51^#(X1, X2, X3) -> c_32()
  , a__and^#(X1, X2) -> c_69()
  , a__isNatKind^#(X) -> c_75()
  , a__isNatKind^#(0()) -> c_76()
  , a__U21^#(X1, X2) -> c_10()
  , a__U22^#(X) -> c_12()
  , a__U22^#(tt()) -> c_13()
  , a__isNat^#(X) -> c_14()
  , a__isNat^#(0()) -> c_15()
  , a__isNatIListKind^#(X) -> c_71()
  , a__isNatIListKind^#(zeros()) -> c_73()
  , a__isNatIListKind^#(nil()) -> c_74()
  , a__U31^#(X1, X2) -> c_18()
  , a__U32^#(X) -> c_20()
  , a__U32^#(tt()) -> c_21()
  , a__U41^#(X1, X2, X3) -> c_22()
  , a__U42^#(X1, X2) -> c_24()
  , a__U43^#(X) -> c_26()
  , a__U43^#(tt()) -> c_27()
  , a__isNatIList^#(X) -> c_29()
  , a__isNatIList^#(zeros()) -> c_31()
  , a__U52^#(X1, X2) -> c_34()
  , a__U53^#(X) -> c_36()
  , a__U53^#(tt()) -> c_37()
  , a__U61^#(X1, X2) -> c_38()
  , a__length^#(X) -> c_40()
  , a__length^#(nil()) -> c_42()
  , mark^#(0()) -> c_44()
  , mark^#(tt()) -> c_46()
  , mark^#(nil()) -> c_50() }
Weak Trs:
  { a__zeros() -> cons(0(), zeros())
  , a__zeros() -> zeros()
  , a__U11(X1, X2) -> U11(X1, X2)
  , a__U11(tt(), V1) -> a__U12(a__isNatList(V1))
  , a__U12(X) -> U12(X)
  , a__U12(tt()) -> tt()
  , a__isNatList(X) -> isNatList(X)
  , a__isNatList(cons(V1, V2)) ->
    a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
  , a__isNatList(nil()) -> tt()
  , a__U21(X1, X2) -> U21(X1, X2)
  , a__U21(tt(), V1) -> a__U22(a__isNat(V1))
  , a__U22(X) -> U22(X)
  , a__U22(tt()) -> tt()
  , a__isNat(X) -> isNat(X)
  , a__isNat(0()) -> tt()
  , a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1)
  , a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1)
  , a__U31(X1, X2) -> U31(X1, X2)
  , a__U31(tt(), V) -> a__U32(a__isNatList(V))
  , a__U32(X) -> U32(X)
  , a__U32(tt()) -> tt()
  , a__U41(X1, X2, X3) -> U41(X1, X2, X3)
  , a__U41(tt(), V1, V2) -> a__U42(a__isNat(V1), V2)
  , a__U42(X1, X2) -> U42(X1, X2)
  , a__U42(tt(), V2) -> a__U43(a__isNatIList(V2))
  , a__U43(X) -> U43(X)
  , a__U43(tt()) -> tt()
  , a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V)
  , a__isNatIList(X) -> isNatIList(X)
  , a__isNatIList(cons(V1, V2)) ->
    a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
  , a__isNatIList(zeros()) -> tt()
  , a__U51(X1, X2, X3) -> U51(X1, X2, X3)
  , a__U51(tt(), V1, V2) -> a__U52(a__isNat(V1), V2)
  , a__U52(X1, X2) -> U52(X1, X2)
  , a__U52(tt(), V2) -> a__U53(a__isNatList(V2))
  , a__U53(X) -> U53(X)
  , a__U53(tt()) -> tt()
  , a__U61(X1, X2) -> U61(X1, X2)
  , a__U61(tt(), L) -> s(a__length(mark(L)))
  , a__length(X) -> length(X)
  , a__length(cons(N, L)) ->
    a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)),
                  and(isNat(N), isNatKind(N))),
           L)
  , 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(isNatIListKind(X)) -> a__isNatIListKind(X)
  , mark(nil()) -> nil()
  , mark(and(X1, X2)) -> a__and(mark(X1), X2)
  , mark(isNat(X)) -> a__isNat(X)
  , mark(isNatKind(X)) -> a__isNatKind(X)
  , mark(U11(X1, X2)) -> a__U11(mark(X1), X2)
  , mark(U12(X)) -> a__U12(mark(X))
  , mark(isNatList(X)) -> a__isNatList(X)
  , mark(U21(X1, X2)) -> a__U21(mark(X1), X2)
  , mark(U22(X)) -> a__U22(mark(X))
  , mark(U31(X1, X2)) -> a__U31(mark(X1), X2)
  , mark(U32(X)) -> a__U32(mark(X))
  , mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3)
  , mark(U42(X1, X2)) -> a__U42(mark(X1), X2)
  , mark(U43(X)) -> a__U43(mark(X))
  , mark(isNatIList(X)) -> a__isNatIList(X)
  , mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3)
  , mark(U52(X1, X2)) -> a__U52(mark(X1), X2)
  , mark(U53(X)) -> a__U53(mark(X))
  , mark(U61(X1, X2)) -> a__U61(mark(X1), X2)
  , a__and(X1, X2) -> and(X1, X2)
  , a__and(tt(), X) -> mark(X)
  , a__isNatIListKind(X) -> isNatIListKind(X)
  , a__isNatIListKind(cons(V1, V2)) ->
    a__and(a__isNatKind(V1), isNatIListKind(V2))
  , a__isNatIListKind(zeros()) -> tt()
  , a__isNatIListKind(nil()) -> tt()
  , a__isNatKind(X) -> isNatKind(X)
  , a__isNatKind(0()) -> tt()
  , a__isNatKind(s(V1)) -> a__isNatKind(V1)
  , a__isNatKind(length(V1)) -> a__isNatIListKind(V1) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of {20} by applications of
Pre({20}) =
{4,17,19,21,22,24,27,28,30,31,32,33,34,35,36,38,39,40,41}. Here
rules are labeled as follows:

  DPs:
    { 1: a__U11^#(tt(), V1) ->
         c_4(a__U12^#(a__isNatList(V1)), a__isNatList^#(V1))
    , 2: a__isNatList^#(cons(V1, V2)) ->
         c_8(a__U51^#(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2),
             a__and^#(a__isNatKind(V1), isNatIListKind(V2)),
             a__isNatKind^#(V1))
    , 3: a__U51^#(tt(), V1, V2) ->
         c_33(a__U52^#(a__isNat(V1), V2), a__isNat^#(V1))
    , 4: a__and^#(tt(), X) -> c_70(mark^#(X))
    , 5: a__isNatKind^#(s(V1)) -> c_77(a__isNatKind^#(V1))
    , 6: a__isNatKind^#(length(V1)) -> c_78(a__isNatIListKind^#(V1))
    , 7: a__U21^#(tt(), V1) ->
         c_11(a__U22^#(a__isNat(V1)), a__isNat^#(V1))
    , 8: a__isNat^#(s(V1)) ->
         c_16(a__U21^#(a__isNatKind(V1), V1), a__isNatKind^#(V1))
    , 9: a__isNat^#(length(V1)) ->
         c_17(a__U11^#(a__isNatIListKind(V1), V1), a__isNatIListKind^#(V1))
    , 10: a__isNatIListKind^#(cons(V1, V2)) ->
          c_72(a__and^#(a__isNatKind(V1), isNatIListKind(V2)),
               a__isNatKind^#(V1))
    , 11: a__U31^#(tt(), V) ->
          c_19(a__U32^#(a__isNatList(V)), a__isNatList^#(V))
    , 12: a__U41^#(tt(), V1, V2) ->
          c_23(a__U42^#(a__isNat(V1), V2), a__isNat^#(V1))
    , 13: a__U42^#(tt(), V2) ->
          c_25(a__U43^#(a__isNatIList(V2)), a__isNatIList^#(V2))
    , 14: a__isNatIList^#(V) ->
          c_28(a__U31^#(a__isNatIListKind(V), V), a__isNatIListKind^#(V))
    , 15: a__isNatIList^#(cons(V1, V2)) ->
          c_30(a__U41^#(a__and(a__isNatKind(V1), isNatIListKind(V2)),
                        V1,
                        V2),
               a__and^#(a__isNatKind(V1), isNatIListKind(V2)),
               a__isNatKind^#(V1))
    , 16: a__U52^#(tt(), V2) ->
          c_35(a__U53^#(a__isNatList(V2)), a__isNatList^#(V2))
    , 17: a__U61^#(tt(), L) -> c_39(a__length^#(mark(L)), mark^#(L))
    , 18: a__length^#(cons(N, L)) ->
          c_41(a__U61^#(a__and(a__and(a__isNatList(L), isNatIListKind(L)),
                               and(isNat(N), isNatKind(N))),
                        L),
               a__and^#(a__and(a__isNatList(L), isNatIListKind(L)),
                        and(isNat(N), isNatKind(N))),
               a__and^#(a__isNatList(L), isNatIListKind(L)),
               a__isNatList^#(L))
    , 19: mark^#(cons(X1, X2)) -> c_43(mark^#(X1))
    , 20: mark^#(zeros()) -> c_45(a__zeros^#())
    , 21: mark^#(s(X)) -> c_47(mark^#(X))
    , 22: mark^#(length(X)) -> c_48(a__length^#(mark(X)), mark^#(X))
    , 23: mark^#(isNatIListKind(X)) -> c_49(a__isNatIListKind^#(X))
    , 24: mark^#(and(X1, X2)) ->
          c_51(a__and^#(mark(X1), X2), mark^#(X1))
    , 25: mark^#(isNat(X)) -> c_52(a__isNat^#(X))
    , 26: mark^#(isNatKind(X)) -> c_53(a__isNatKind^#(X))
    , 27: mark^#(U11(X1, X2)) ->
          c_54(a__U11^#(mark(X1), X2), mark^#(X1))
    , 28: mark^#(U12(X)) -> c_55(a__U12^#(mark(X)), mark^#(X))
    , 29: mark^#(isNatList(X)) -> c_56(a__isNatList^#(X))
    , 30: mark^#(U21(X1, X2)) ->
          c_57(a__U21^#(mark(X1), X2), mark^#(X1))
    , 31: mark^#(U22(X)) -> c_58(a__U22^#(mark(X)), mark^#(X))
    , 32: mark^#(U31(X1, X2)) ->
          c_59(a__U31^#(mark(X1), X2), mark^#(X1))
    , 33: mark^#(U32(X)) -> c_60(a__U32^#(mark(X)), mark^#(X))
    , 34: mark^#(U41(X1, X2, X3)) ->
          c_61(a__U41^#(mark(X1), X2, X3), mark^#(X1))
    , 35: mark^#(U42(X1, X2)) ->
          c_62(a__U42^#(mark(X1), X2), mark^#(X1))
    , 36: mark^#(U43(X)) -> c_63(a__U43^#(mark(X)), mark^#(X))
    , 37: mark^#(isNatIList(X)) -> c_64(a__isNatIList^#(X))
    , 38: mark^#(U51(X1, X2, X3)) ->
          c_65(a__U51^#(mark(X1), X2, X3), mark^#(X1))
    , 39: mark^#(U52(X1, X2)) ->
          c_66(a__U52^#(mark(X1), X2), mark^#(X1))
    , 40: mark^#(U53(X)) -> c_67(a__U53^#(mark(X)), mark^#(X))
    , 41: mark^#(U61(X1, X2)) ->
          c_68(a__U61^#(mark(X1), X2), mark^#(X1))
    , 42: a__zeros^#() -> c_1()
    , 43: a__zeros^#() -> c_2()
    , 44: a__U11^#(X1, X2) -> c_3()
    , 45: a__U12^#(X) -> c_5()
    , 46: a__U12^#(tt()) -> c_6()
    , 47: a__isNatList^#(X) -> c_7()
    , 48: a__isNatList^#(nil()) -> c_9()
    , 49: a__U51^#(X1, X2, X3) -> c_32()
    , 50: a__and^#(X1, X2) -> c_69()
    , 51: a__isNatKind^#(X) -> c_75()
    , 52: a__isNatKind^#(0()) -> c_76()
    , 53: a__U21^#(X1, X2) -> c_10()
    , 54: a__U22^#(X) -> c_12()
    , 55: a__U22^#(tt()) -> c_13()
    , 56: a__isNat^#(X) -> c_14()
    , 57: a__isNat^#(0()) -> c_15()
    , 58: a__isNatIListKind^#(X) -> c_71()
    , 59: a__isNatIListKind^#(zeros()) -> c_73()
    , 60: a__isNatIListKind^#(nil()) -> c_74()
    , 61: a__U31^#(X1, X2) -> c_18()
    , 62: a__U32^#(X) -> c_20()
    , 63: a__U32^#(tt()) -> c_21()
    , 64: a__U41^#(X1, X2, X3) -> c_22()
    , 65: a__U42^#(X1, X2) -> c_24()
    , 66: a__U43^#(X) -> c_26()
    , 67: a__U43^#(tt()) -> c_27()
    , 68: a__isNatIList^#(X) -> c_29()
    , 69: a__isNatIList^#(zeros()) -> c_31()
    , 70: a__U52^#(X1, X2) -> c_34()
    , 71: a__U53^#(X) -> c_36()
    , 72: a__U53^#(tt()) -> c_37()
    , 73: a__U61^#(X1, X2) -> c_38()
    , 74: a__length^#(X) -> c_40()
    , 75: a__length^#(nil()) -> c_42()
    , 76: mark^#(0()) -> c_44()
    , 77: mark^#(tt()) -> c_46()
    , 78: mark^#(nil()) -> c_50() }

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

Strict DPs:
  { a__U11^#(tt(), V1) ->
    c_4(a__U12^#(a__isNatList(V1)), a__isNatList^#(V1))
  , a__isNatList^#(cons(V1, V2)) ->
    c_8(a__U51^#(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2),
        a__and^#(a__isNatKind(V1), isNatIListKind(V2)),
        a__isNatKind^#(V1))
  , a__U51^#(tt(), V1, V2) ->
    c_33(a__U52^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__and^#(tt(), X) -> c_70(mark^#(X))
  , a__isNatKind^#(s(V1)) -> c_77(a__isNatKind^#(V1))
  , a__isNatKind^#(length(V1)) -> c_78(a__isNatIListKind^#(V1))
  , a__U21^#(tt(), V1) ->
    c_11(a__U22^#(a__isNat(V1)), a__isNat^#(V1))
  , a__isNat^#(s(V1)) ->
    c_16(a__U21^#(a__isNatKind(V1), V1), a__isNatKind^#(V1))
  , a__isNat^#(length(V1)) ->
    c_17(a__U11^#(a__isNatIListKind(V1), V1), a__isNatIListKind^#(V1))
  , a__isNatIListKind^#(cons(V1, V2)) ->
    c_72(a__and^#(a__isNatKind(V1), isNatIListKind(V2)),
         a__isNatKind^#(V1))
  , a__U31^#(tt(), V) ->
    c_19(a__U32^#(a__isNatList(V)), a__isNatList^#(V))
  , a__U41^#(tt(), V1, V2) ->
    c_23(a__U42^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__U42^#(tt(), V2) ->
    c_25(a__U43^#(a__isNatIList(V2)), a__isNatIList^#(V2))
  , a__isNatIList^#(V) ->
    c_28(a__U31^#(a__isNatIListKind(V), V), a__isNatIListKind^#(V))
  , a__isNatIList^#(cons(V1, V2)) ->
    c_30(a__U41^#(a__and(a__isNatKind(V1), isNatIListKind(V2)),
                  V1,
                  V2),
         a__and^#(a__isNatKind(V1), isNatIListKind(V2)),
         a__isNatKind^#(V1))
  , a__U52^#(tt(), V2) ->
    c_35(a__U53^#(a__isNatList(V2)), a__isNatList^#(V2))
  , a__U61^#(tt(), L) -> c_39(a__length^#(mark(L)), mark^#(L))
  , a__length^#(cons(N, L)) ->
    c_41(a__U61^#(a__and(a__and(a__isNatList(L), isNatIListKind(L)),
                         and(isNat(N), isNatKind(N))),
                  L),
         a__and^#(a__and(a__isNatList(L), isNatIListKind(L)),
                  and(isNat(N), isNatKind(N))),
         a__and^#(a__isNatList(L), isNatIListKind(L)),
         a__isNatList^#(L))
  , mark^#(cons(X1, X2)) -> c_43(mark^#(X1))
  , mark^#(s(X)) -> c_47(mark^#(X))
  , mark^#(length(X)) -> c_48(a__length^#(mark(X)), mark^#(X))
  , mark^#(isNatIListKind(X)) -> c_49(a__isNatIListKind^#(X))
  , mark^#(and(X1, X2)) -> c_51(a__and^#(mark(X1), X2), mark^#(X1))
  , mark^#(isNat(X)) -> c_52(a__isNat^#(X))
  , mark^#(isNatKind(X)) -> c_53(a__isNatKind^#(X))
  , mark^#(U11(X1, X2)) -> c_54(a__U11^#(mark(X1), X2), mark^#(X1))
  , mark^#(U12(X)) -> c_55(a__U12^#(mark(X)), mark^#(X))
  , mark^#(isNatList(X)) -> c_56(a__isNatList^#(X))
  , mark^#(U21(X1, X2)) -> c_57(a__U21^#(mark(X1), X2), mark^#(X1))
  , mark^#(U22(X)) -> c_58(a__U22^#(mark(X)), mark^#(X))
  , mark^#(U31(X1, X2)) -> c_59(a__U31^#(mark(X1), X2), mark^#(X1))
  , mark^#(U32(X)) -> c_60(a__U32^#(mark(X)), mark^#(X))
  , mark^#(U41(X1, X2, X3)) ->
    c_61(a__U41^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U42(X1, X2)) -> c_62(a__U42^#(mark(X1), X2), mark^#(X1))
  , mark^#(U43(X)) -> c_63(a__U43^#(mark(X)), mark^#(X))
  , mark^#(isNatIList(X)) -> c_64(a__isNatIList^#(X))
  , mark^#(U51(X1, X2, X3)) ->
    c_65(a__U51^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U52(X1, X2)) -> c_66(a__U52^#(mark(X1), X2), mark^#(X1))
  , mark^#(U53(X)) -> c_67(a__U53^#(mark(X)), mark^#(X))
  , mark^#(U61(X1, X2)) -> c_68(a__U61^#(mark(X1), X2), mark^#(X1)) }
Weak DPs:
  { a__zeros^#() -> c_1()
  , a__zeros^#() -> c_2()
  , a__U11^#(X1, X2) -> c_3()
  , a__U12^#(X) -> c_5()
  , a__U12^#(tt()) -> c_6()
  , a__isNatList^#(X) -> c_7()
  , a__isNatList^#(nil()) -> c_9()
  , a__U51^#(X1, X2, X3) -> c_32()
  , a__and^#(X1, X2) -> c_69()
  , a__isNatKind^#(X) -> c_75()
  , a__isNatKind^#(0()) -> c_76()
  , a__U21^#(X1, X2) -> c_10()
  , a__U22^#(X) -> c_12()
  , a__U22^#(tt()) -> c_13()
  , a__isNat^#(X) -> c_14()
  , a__isNat^#(0()) -> c_15()
  , a__isNatIListKind^#(X) -> c_71()
  , a__isNatIListKind^#(zeros()) -> c_73()
  , a__isNatIListKind^#(nil()) -> c_74()
  , a__U31^#(X1, X2) -> c_18()
  , a__U32^#(X) -> c_20()
  , a__U32^#(tt()) -> c_21()
  , a__U41^#(X1, X2, X3) -> c_22()
  , a__U42^#(X1, X2) -> c_24()
  , a__U43^#(X) -> c_26()
  , a__U43^#(tt()) -> c_27()
  , a__isNatIList^#(X) -> c_29()
  , a__isNatIList^#(zeros()) -> c_31()
  , a__U52^#(X1, X2) -> c_34()
  , a__U53^#(X) -> c_36()
  , a__U53^#(tt()) -> c_37()
  , a__U61^#(X1, X2) -> c_38()
  , a__length^#(X) -> c_40()
  , a__length^#(nil()) -> c_42()
  , mark^#(0()) -> c_44()
  , mark^#(zeros()) -> c_45(a__zeros^#())
  , mark^#(tt()) -> c_46()
  , mark^#(nil()) -> c_50() }
Weak Trs:
  { a__zeros() -> cons(0(), zeros())
  , a__zeros() -> zeros()
  , a__U11(X1, X2) -> U11(X1, X2)
  , a__U11(tt(), V1) -> a__U12(a__isNatList(V1))
  , a__U12(X) -> U12(X)
  , a__U12(tt()) -> tt()
  , a__isNatList(X) -> isNatList(X)
  , a__isNatList(cons(V1, V2)) ->
    a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
  , a__isNatList(nil()) -> tt()
  , a__U21(X1, X2) -> U21(X1, X2)
  , a__U21(tt(), V1) -> a__U22(a__isNat(V1))
  , a__U22(X) -> U22(X)
  , a__U22(tt()) -> tt()
  , a__isNat(X) -> isNat(X)
  , a__isNat(0()) -> tt()
  , a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1)
  , a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1)
  , a__U31(X1, X2) -> U31(X1, X2)
  , a__U31(tt(), V) -> a__U32(a__isNatList(V))
  , a__U32(X) -> U32(X)
  , a__U32(tt()) -> tt()
  , a__U41(X1, X2, X3) -> U41(X1, X2, X3)
  , a__U41(tt(), V1, V2) -> a__U42(a__isNat(V1), V2)
  , a__U42(X1, X2) -> U42(X1, X2)
  , a__U42(tt(), V2) -> a__U43(a__isNatIList(V2))
  , a__U43(X) -> U43(X)
  , a__U43(tt()) -> tt()
  , a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V)
  , a__isNatIList(X) -> isNatIList(X)
  , a__isNatIList(cons(V1, V2)) ->
    a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
  , a__isNatIList(zeros()) -> tt()
  , a__U51(X1, X2, X3) -> U51(X1, X2, X3)
  , a__U51(tt(), V1, V2) -> a__U52(a__isNat(V1), V2)
  , a__U52(X1, X2) -> U52(X1, X2)
  , a__U52(tt(), V2) -> a__U53(a__isNatList(V2))
  , a__U53(X) -> U53(X)
  , a__U53(tt()) -> tt()
  , a__U61(X1, X2) -> U61(X1, X2)
  , a__U61(tt(), L) -> s(a__length(mark(L)))
  , a__length(X) -> length(X)
  , a__length(cons(N, L)) ->
    a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)),
                  and(isNat(N), isNatKind(N))),
           L)
  , 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(isNatIListKind(X)) -> a__isNatIListKind(X)
  , mark(nil()) -> nil()
  , mark(and(X1, X2)) -> a__and(mark(X1), X2)
  , mark(isNat(X)) -> a__isNat(X)
  , mark(isNatKind(X)) -> a__isNatKind(X)
  , mark(U11(X1, X2)) -> a__U11(mark(X1), X2)
  , mark(U12(X)) -> a__U12(mark(X))
  , mark(isNatList(X)) -> a__isNatList(X)
  , mark(U21(X1, X2)) -> a__U21(mark(X1), X2)
  , mark(U22(X)) -> a__U22(mark(X))
  , mark(U31(X1, X2)) -> a__U31(mark(X1), X2)
  , mark(U32(X)) -> a__U32(mark(X))
  , mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3)
  , mark(U42(X1, X2)) -> a__U42(mark(X1), X2)
  , mark(U43(X)) -> a__U43(mark(X))
  , mark(isNatIList(X)) -> a__isNatIList(X)
  , mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3)
  , mark(U52(X1, X2)) -> a__U52(mark(X1), X2)
  , mark(U53(X)) -> a__U53(mark(X))
  , mark(U61(X1, X2)) -> a__U61(mark(X1), X2)
  , a__and(X1, X2) -> and(X1, X2)
  , a__and(tt(), X) -> mark(X)
  , a__isNatIListKind(X) -> isNatIListKind(X)
  , a__isNatIListKind(cons(V1, V2)) ->
    a__and(a__isNatKind(V1), isNatIListKind(V2))
  , a__isNatIListKind(zeros()) -> tt()
  , a__isNatIListKind(nil()) -> tt()
  , a__isNatKind(X) -> isNatKind(X)
  , a__isNatKind(0()) -> tt()
  , a__isNatKind(s(V1)) -> a__isNatKind(V1)
  , a__isNatKind(length(V1)) -> a__isNatIListKind(V1) }
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^#(X1, X2) -> c_3()
, a__U12^#(X) -> c_5()
, a__U12^#(tt()) -> c_6()
, a__isNatList^#(X) -> c_7()
, a__isNatList^#(nil()) -> c_9()
, a__U51^#(X1, X2, X3) -> c_32()
, a__and^#(X1, X2) -> c_69()
, a__isNatKind^#(X) -> c_75()
, a__isNatKind^#(0()) -> c_76()
, a__U21^#(X1, X2) -> c_10()
, a__U22^#(X) -> c_12()
, a__U22^#(tt()) -> c_13()
, a__isNat^#(X) -> c_14()
, a__isNat^#(0()) -> c_15()
, a__isNatIListKind^#(X) -> c_71()
, a__isNatIListKind^#(zeros()) -> c_73()
, a__isNatIListKind^#(nil()) -> c_74()
, a__U31^#(X1, X2) -> c_18()
, a__U32^#(X) -> c_20()
, a__U32^#(tt()) -> c_21()
, a__U41^#(X1, X2, X3) -> c_22()
, a__U42^#(X1, X2) -> c_24()
, a__U43^#(X) -> c_26()
, a__U43^#(tt()) -> c_27()
, a__isNatIList^#(X) -> c_29()
, a__isNatIList^#(zeros()) -> c_31()
, a__U52^#(X1, X2) -> c_34()
, a__U53^#(X) -> c_36()
, a__U53^#(tt()) -> c_37()
, a__U61^#(X1, X2) -> c_38()
, a__length^#(X) -> c_40()
, a__length^#(nil()) -> c_42()
, mark^#(0()) -> c_44()
, mark^#(zeros()) -> c_45(a__zeros^#())
, mark^#(tt()) -> c_46()
, mark^#(nil()) -> c_50() }

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

Strict DPs:
  { a__U11^#(tt(), V1) ->
    c_4(a__U12^#(a__isNatList(V1)), a__isNatList^#(V1))
  , a__isNatList^#(cons(V1, V2)) ->
    c_8(a__U51^#(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2),
        a__and^#(a__isNatKind(V1), isNatIListKind(V2)),
        a__isNatKind^#(V1))
  , a__U51^#(tt(), V1, V2) ->
    c_33(a__U52^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__and^#(tt(), X) -> c_70(mark^#(X))
  , a__isNatKind^#(s(V1)) -> c_77(a__isNatKind^#(V1))
  , a__isNatKind^#(length(V1)) -> c_78(a__isNatIListKind^#(V1))
  , a__U21^#(tt(), V1) ->
    c_11(a__U22^#(a__isNat(V1)), a__isNat^#(V1))
  , a__isNat^#(s(V1)) ->
    c_16(a__U21^#(a__isNatKind(V1), V1), a__isNatKind^#(V1))
  , a__isNat^#(length(V1)) ->
    c_17(a__U11^#(a__isNatIListKind(V1), V1), a__isNatIListKind^#(V1))
  , a__isNatIListKind^#(cons(V1, V2)) ->
    c_72(a__and^#(a__isNatKind(V1), isNatIListKind(V2)),
         a__isNatKind^#(V1))
  , a__U31^#(tt(), V) ->
    c_19(a__U32^#(a__isNatList(V)), a__isNatList^#(V))
  , a__U41^#(tt(), V1, V2) ->
    c_23(a__U42^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__U42^#(tt(), V2) ->
    c_25(a__U43^#(a__isNatIList(V2)), a__isNatIList^#(V2))
  , a__isNatIList^#(V) ->
    c_28(a__U31^#(a__isNatIListKind(V), V), a__isNatIListKind^#(V))
  , a__isNatIList^#(cons(V1, V2)) ->
    c_30(a__U41^#(a__and(a__isNatKind(V1), isNatIListKind(V2)),
                  V1,
                  V2),
         a__and^#(a__isNatKind(V1), isNatIListKind(V2)),
         a__isNatKind^#(V1))
  , a__U52^#(tt(), V2) ->
    c_35(a__U53^#(a__isNatList(V2)), a__isNatList^#(V2))
  , a__U61^#(tt(), L) -> c_39(a__length^#(mark(L)), mark^#(L))
  , a__length^#(cons(N, L)) ->
    c_41(a__U61^#(a__and(a__and(a__isNatList(L), isNatIListKind(L)),
                         and(isNat(N), isNatKind(N))),
                  L),
         a__and^#(a__and(a__isNatList(L), isNatIListKind(L)),
                  and(isNat(N), isNatKind(N))),
         a__and^#(a__isNatList(L), isNatIListKind(L)),
         a__isNatList^#(L))
  , mark^#(cons(X1, X2)) -> c_43(mark^#(X1))
  , mark^#(s(X)) -> c_47(mark^#(X))
  , mark^#(length(X)) -> c_48(a__length^#(mark(X)), mark^#(X))
  , mark^#(isNatIListKind(X)) -> c_49(a__isNatIListKind^#(X))
  , mark^#(and(X1, X2)) -> c_51(a__and^#(mark(X1), X2), mark^#(X1))
  , mark^#(isNat(X)) -> c_52(a__isNat^#(X))
  , mark^#(isNatKind(X)) -> c_53(a__isNatKind^#(X))
  , mark^#(U11(X1, X2)) -> c_54(a__U11^#(mark(X1), X2), mark^#(X1))
  , mark^#(U12(X)) -> c_55(a__U12^#(mark(X)), mark^#(X))
  , mark^#(isNatList(X)) -> c_56(a__isNatList^#(X))
  , mark^#(U21(X1, X2)) -> c_57(a__U21^#(mark(X1), X2), mark^#(X1))
  , mark^#(U22(X)) -> c_58(a__U22^#(mark(X)), mark^#(X))
  , mark^#(U31(X1, X2)) -> c_59(a__U31^#(mark(X1), X2), mark^#(X1))
  , mark^#(U32(X)) -> c_60(a__U32^#(mark(X)), mark^#(X))
  , mark^#(U41(X1, X2, X3)) ->
    c_61(a__U41^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U42(X1, X2)) -> c_62(a__U42^#(mark(X1), X2), mark^#(X1))
  , mark^#(U43(X)) -> c_63(a__U43^#(mark(X)), mark^#(X))
  , mark^#(isNatIList(X)) -> c_64(a__isNatIList^#(X))
  , mark^#(U51(X1, X2, X3)) ->
    c_65(a__U51^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U52(X1, X2)) -> c_66(a__U52^#(mark(X1), X2), mark^#(X1))
  , mark^#(U53(X)) -> c_67(a__U53^#(mark(X)), mark^#(X))
  , mark^#(U61(X1, X2)) -> c_68(a__U61^#(mark(X1), X2), mark^#(X1)) }
Weak Trs:
  { a__zeros() -> cons(0(), zeros())
  , a__zeros() -> zeros()
  , a__U11(X1, X2) -> U11(X1, X2)
  , a__U11(tt(), V1) -> a__U12(a__isNatList(V1))
  , a__U12(X) -> U12(X)
  , a__U12(tt()) -> tt()
  , a__isNatList(X) -> isNatList(X)
  , a__isNatList(cons(V1, V2)) ->
    a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
  , a__isNatList(nil()) -> tt()
  , a__U21(X1, X2) -> U21(X1, X2)
  , a__U21(tt(), V1) -> a__U22(a__isNat(V1))
  , a__U22(X) -> U22(X)
  , a__U22(tt()) -> tt()
  , a__isNat(X) -> isNat(X)
  , a__isNat(0()) -> tt()
  , a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1)
  , a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1)
  , a__U31(X1, X2) -> U31(X1, X2)
  , a__U31(tt(), V) -> a__U32(a__isNatList(V))
  , a__U32(X) -> U32(X)
  , a__U32(tt()) -> tt()
  , a__U41(X1, X2, X3) -> U41(X1, X2, X3)
  , a__U41(tt(), V1, V2) -> a__U42(a__isNat(V1), V2)
  , a__U42(X1, X2) -> U42(X1, X2)
  , a__U42(tt(), V2) -> a__U43(a__isNatIList(V2))
  , a__U43(X) -> U43(X)
  , a__U43(tt()) -> tt()
  , a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V)
  , a__isNatIList(X) -> isNatIList(X)
  , a__isNatIList(cons(V1, V2)) ->
    a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
  , a__isNatIList(zeros()) -> tt()
  , a__U51(X1, X2, X3) -> U51(X1, X2, X3)
  , a__U51(tt(), V1, V2) -> a__U52(a__isNat(V1), V2)
  , a__U52(X1, X2) -> U52(X1, X2)
  , a__U52(tt(), V2) -> a__U53(a__isNatList(V2))
  , a__U53(X) -> U53(X)
  , a__U53(tt()) -> tt()
  , a__U61(X1, X2) -> U61(X1, X2)
  , a__U61(tt(), L) -> s(a__length(mark(L)))
  , a__length(X) -> length(X)
  , a__length(cons(N, L)) ->
    a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)),
                  and(isNat(N), isNatKind(N))),
           L)
  , 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(isNatIListKind(X)) -> a__isNatIListKind(X)
  , mark(nil()) -> nil()
  , mark(and(X1, X2)) -> a__and(mark(X1), X2)
  , mark(isNat(X)) -> a__isNat(X)
  , mark(isNatKind(X)) -> a__isNatKind(X)
  , mark(U11(X1, X2)) -> a__U11(mark(X1), X2)
  , mark(U12(X)) -> a__U12(mark(X))
  , mark(isNatList(X)) -> a__isNatList(X)
  , mark(U21(X1, X2)) -> a__U21(mark(X1), X2)
  , mark(U22(X)) -> a__U22(mark(X))
  , mark(U31(X1, X2)) -> a__U31(mark(X1), X2)
  , mark(U32(X)) -> a__U32(mark(X))
  , mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3)
  , mark(U42(X1, X2)) -> a__U42(mark(X1), X2)
  , mark(U43(X)) -> a__U43(mark(X))
  , mark(isNatIList(X)) -> a__isNatIList(X)
  , mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3)
  , mark(U52(X1, X2)) -> a__U52(mark(X1), X2)
  , mark(U53(X)) -> a__U53(mark(X))
  , mark(U61(X1, X2)) -> a__U61(mark(X1), X2)
  , a__and(X1, X2) -> and(X1, X2)
  , a__and(tt(), X) -> mark(X)
  , a__isNatIListKind(X) -> isNatIListKind(X)
  , a__isNatIListKind(cons(V1, V2)) ->
    a__and(a__isNatKind(V1), isNatIListKind(V2))
  , a__isNatIListKind(zeros()) -> tt()
  , a__isNatIListKind(nil()) -> tt()
  , a__isNatKind(X) -> isNatKind(X)
  , a__isNatKind(0()) -> tt()
  , a__isNatKind(s(V1)) -> a__isNatKind(V1)
  , a__isNatKind(length(V1)) -> a__isNatIListKind(V1) }
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__U11^#(tt(), V1) ->
    c_4(a__U12^#(a__isNatList(V1)), a__isNatList^#(V1))
  , a__U21^#(tt(), V1) ->
    c_11(a__U22^#(a__isNat(V1)), a__isNat^#(V1))
  , a__U31^#(tt(), V) ->
    c_19(a__U32^#(a__isNatList(V)), a__isNatList^#(V))
  , a__U42^#(tt(), V2) ->
    c_25(a__U43^#(a__isNatIList(V2)), a__isNatIList^#(V2))
  , a__U52^#(tt(), V2) ->
    c_35(a__U53^#(a__isNatList(V2)), a__isNatList^#(V2))
  , mark^#(U12(X)) -> c_55(a__U12^#(mark(X)), mark^#(X))
  , mark^#(U22(X)) -> c_58(a__U22^#(mark(X)), mark^#(X))
  , mark^#(U32(X)) -> c_60(a__U32^#(mark(X)), mark^#(X))
  , mark^#(U43(X)) -> c_63(a__U43^#(mark(X)), mark^#(X))
  , mark^#(U53(X)) -> c_67(a__U53^#(mark(X)), mark^#(X)) }

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

Strict DPs:
  { a__U11^#(tt(), V1) -> c_1(a__isNatList^#(V1))
  , a__isNatList^#(cons(V1, V2)) ->
    c_2(a__U51^#(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2),
        a__and^#(a__isNatKind(V1), isNatIListKind(V2)),
        a__isNatKind^#(V1))
  , a__U51^#(tt(), V1, V2) ->
    c_3(a__U52^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__and^#(tt(), X) -> c_4(mark^#(X))
  , a__isNatKind^#(s(V1)) -> c_5(a__isNatKind^#(V1))
  , a__isNatKind^#(length(V1)) -> c_6(a__isNatIListKind^#(V1))
  , a__U21^#(tt(), V1) -> c_7(a__isNat^#(V1))
  , a__isNat^#(s(V1)) ->
    c_8(a__U21^#(a__isNatKind(V1), V1), a__isNatKind^#(V1))
  , a__isNat^#(length(V1)) ->
    c_9(a__U11^#(a__isNatIListKind(V1), V1), a__isNatIListKind^#(V1))
  , a__isNatIListKind^#(cons(V1, V2)) ->
    c_10(a__and^#(a__isNatKind(V1), isNatIListKind(V2)),
         a__isNatKind^#(V1))
  , a__U31^#(tt(), V) -> c_11(a__isNatList^#(V))
  , a__U41^#(tt(), V1, V2) ->
    c_12(a__U42^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__U42^#(tt(), V2) -> c_13(a__isNatIList^#(V2))
  , a__isNatIList^#(V) ->
    c_14(a__U31^#(a__isNatIListKind(V), V), a__isNatIListKind^#(V))
  , a__isNatIList^#(cons(V1, V2)) ->
    c_15(a__U41^#(a__and(a__isNatKind(V1), isNatIListKind(V2)),
                  V1,
                  V2),
         a__and^#(a__isNatKind(V1), isNatIListKind(V2)),
         a__isNatKind^#(V1))
  , a__U52^#(tt(), V2) -> c_16(a__isNatList^#(V2))
  , a__U61^#(tt(), L) -> c_17(a__length^#(mark(L)), mark^#(L))
  , a__length^#(cons(N, L)) ->
    c_18(a__U61^#(a__and(a__and(a__isNatList(L), isNatIListKind(L)),
                         and(isNat(N), isNatKind(N))),
                  L),
         a__and^#(a__and(a__isNatList(L), isNatIListKind(L)),
                  and(isNat(N), isNatKind(N))),
         a__and^#(a__isNatList(L), isNatIListKind(L)),
         a__isNatList^#(L))
  , mark^#(cons(X1, X2)) -> c_19(mark^#(X1))
  , mark^#(s(X)) -> c_20(mark^#(X))
  , mark^#(length(X)) -> c_21(a__length^#(mark(X)), mark^#(X))
  , mark^#(isNatIListKind(X)) -> c_22(a__isNatIListKind^#(X))
  , mark^#(and(X1, X2)) -> c_23(a__and^#(mark(X1), X2), mark^#(X1))
  , mark^#(isNat(X)) -> c_24(a__isNat^#(X))
  , mark^#(isNatKind(X)) -> c_25(a__isNatKind^#(X))
  , mark^#(U11(X1, X2)) -> c_26(a__U11^#(mark(X1), X2), mark^#(X1))
  , mark^#(U12(X)) -> c_27(mark^#(X))
  , mark^#(isNatList(X)) -> c_28(a__isNatList^#(X))
  , mark^#(U21(X1, X2)) -> c_29(a__U21^#(mark(X1), X2), mark^#(X1))
  , mark^#(U22(X)) -> c_30(mark^#(X))
  , mark^#(U31(X1, X2)) -> c_31(a__U31^#(mark(X1), X2), mark^#(X1))
  , mark^#(U32(X)) -> c_32(mark^#(X))
  , mark^#(U41(X1, X2, X3)) ->
    c_33(a__U41^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U42(X1, X2)) -> c_34(a__U42^#(mark(X1), X2), mark^#(X1))
  , mark^#(U43(X)) -> c_35(mark^#(X))
  , mark^#(isNatIList(X)) -> c_36(a__isNatIList^#(X))
  , mark^#(U51(X1, X2, X3)) ->
    c_37(a__U51^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U52(X1, X2)) -> c_38(a__U52^#(mark(X1), X2), mark^#(X1))
  , mark^#(U53(X)) -> c_39(mark^#(X))
  , mark^#(U61(X1, X2)) -> c_40(a__U61^#(mark(X1), X2), mark^#(X1)) }
Weak Trs:
  { a__zeros() -> cons(0(), zeros())
  , a__zeros() -> zeros()
  , a__U11(X1, X2) -> U11(X1, X2)
  , a__U11(tt(), V1) -> a__U12(a__isNatList(V1))
  , a__U12(X) -> U12(X)
  , a__U12(tt()) -> tt()
  , a__isNatList(X) -> isNatList(X)
  , a__isNatList(cons(V1, V2)) ->
    a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
  , a__isNatList(nil()) -> tt()
  , a__U21(X1, X2) -> U21(X1, X2)
  , a__U21(tt(), V1) -> a__U22(a__isNat(V1))
  , a__U22(X) -> U22(X)
  , a__U22(tt()) -> tt()
  , a__isNat(X) -> isNat(X)
  , a__isNat(0()) -> tt()
  , a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1)
  , a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1)
  , a__U31(X1, X2) -> U31(X1, X2)
  , a__U31(tt(), V) -> a__U32(a__isNatList(V))
  , a__U32(X) -> U32(X)
  , a__U32(tt()) -> tt()
  , a__U41(X1, X2, X3) -> U41(X1, X2, X3)
  , a__U41(tt(), V1, V2) -> a__U42(a__isNat(V1), V2)
  , a__U42(X1, X2) -> U42(X1, X2)
  , a__U42(tt(), V2) -> a__U43(a__isNatIList(V2))
  , a__U43(X) -> U43(X)
  , a__U43(tt()) -> tt()
  , a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V)
  , a__isNatIList(X) -> isNatIList(X)
  , a__isNatIList(cons(V1, V2)) ->
    a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
  , a__isNatIList(zeros()) -> tt()
  , a__U51(X1, X2, X3) -> U51(X1, X2, X3)
  , a__U51(tt(), V1, V2) -> a__U52(a__isNat(V1), V2)
  , a__U52(X1, X2) -> U52(X1, X2)
  , a__U52(tt(), V2) -> a__U53(a__isNatList(V2))
  , a__U53(X) -> U53(X)
  , a__U53(tt()) -> tt()
  , a__U61(X1, X2) -> U61(X1, X2)
  , a__U61(tt(), L) -> s(a__length(mark(L)))
  , a__length(X) -> length(X)
  , a__length(cons(N, L)) ->
    a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)),
                  and(isNat(N), isNatKind(N))),
           L)
  , 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(isNatIListKind(X)) -> a__isNatIListKind(X)
  , mark(nil()) -> nil()
  , mark(and(X1, X2)) -> a__and(mark(X1), X2)
  , mark(isNat(X)) -> a__isNat(X)
  , mark(isNatKind(X)) -> a__isNatKind(X)
  , mark(U11(X1, X2)) -> a__U11(mark(X1), X2)
  , mark(U12(X)) -> a__U12(mark(X))
  , mark(isNatList(X)) -> a__isNatList(X)
  , mark(U21(X1, X2)) -> a__U21(mark(X1), X2)
  , mark(U22(X)) -> a__U22(mark(X))
  , mark(U31(X1, X2)) -> a__U31(mark(X1), X2)
  , mark(U32(X)) -> a__U32(mark(X))
  , mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3)
  , mark(U42(X1, X2)) -> a__U42(mark(X1), X2)
  , mark(U43(X)) -> a__U43(mark(X))
  , mark(isNatIList(X)) -> a__isNatIList(X)
  , mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3)
  , mark(U52(X1, X2)) -> a__U52(mark(X1), X2)
  , mark(U53(X)) -> a__U53(mark(X))
  , mark(U61(X1, X2)) -> a__U61(mark(X1), X2)
  , a__and(X1, X2) -> and(X1, X2)
  , a__and(tt(), X) -> mark(X)
  , a__isNatIListKind(X) -> isNatIListKind(X)
  , a__isNatIListKind(cons(V1, V2)) ->
    a__and(a__isNatKind(V1), isNatIListKind(V2))
  , a__isNatIListKind(zeros()) -> tt()
  , a__isNatIListKind(nil()) -> tt()
  , a__isNatKind(X) -> isNatKind(X)
  , a__isNatKind(0()) -> tt()
  , a__isNatKind(s(V1)) -> a__isNatKind(V1)
  , a__isNatKind(length(V1)) -> a__isNatIListKind(V1) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..