MAYBE

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

Strict Trs:
  { a__U11(X1, X2, X3) -> U11(X1, X2, X3)
  , a__U11(tt(), V1, V2) -> a__U12(a__isNat(V1), V2)
  , a__U12(X1, X2) -> U12(X1, X2)
  , a__U12(tt(), V2) -> a__U13(a__isNat(V2))
  , a__isNat(X) -> isNat(X)
  , a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1)
  , a__isNat(0()) -> tt()
  , a__isNat(plus(V1, V2)) ->
    a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
  , a__U13(X) -> U13(X)
  , a__U13(tt()) -> 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__U31(X1, X2) -> U31(X1, X2)
  , a__U31(tt(), N) -> mark(N)
  , mark(tt()) -> tt()
  , mark(s(X)) -> s(mark(X))
  , mark(0()) -> 0()
  , mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2))
  , mark(isNatKind(X)) -> a__isNatKind(X)
  , mark(and(X1, X2)) -> a__and(mark(X1), X2)
  , mark(isNat(X)) -> a__isNat(X)
  , mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
  , mark(U12(X1, X2)) -> a__U12(mark(X1), X2)
  , mark(U13(X)) -> a__U13(mark(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(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3)
  , a__U41(X1, X2, X3) -> U41(X1, X2, X3)
  , a__U41(tt(), M, N) -> s(a__plus(mark(N), mark(M)))
  , a__plus(X1, X2) -> plus(X1, X2)
  , a__plus(N, s(M)) ->
    a__U41(a__and(a__and(a__isNat(M), isNatKind(M)),
                  and(isNat(N), isNatKind(N))),
           M,
           N)
  , a__plus(N, 0()) -> a__U31(a__and(a__isNat(N), isNatKind(N)), N)
  , a__and(X1, X2) -> and(X1, X2)
  , a__and(tt(), X) -> mark(X)
  , a__isNatKind(X) -> isNatKind(X)
  , a__isNatKind(s(V1)) -> a__isNatKind(V1)
  , a__isNatKind(0()) -> tt()
  , a__isNatKind(plus(V1, V2)) ->
    a__and(a__isNatKind(V1), isNatKind(V2)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { a__U11^#(X1, X2, X3) -> c_1()
  , a__U11^#(tt(), V1, V2) ->
    c_2(a__U12^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__U12^#(X1, X2) -> c_3()
  , a__U12^#(tt(), V2) -> c_4(a__U13^#(a__isNat(V2)), a__isNat^#(V2))
  , a__isNat^#(X) -> c_5()
  , a__isNat^#(s(V1)) ->
    c_6(a__U21^#(a__isNatKind(V1), V1), a__isNatKind^#(V1))
  , a__isNat^#(0()) -> c_7()
  , a__isNat^#(plus(V1, V2)) ->
    c_8(a__U11^#(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2),
        a__and^#(a__isNatKind(V1), isNatKind(V2)),
        a__isNatKind^#(V1))
  , a__U13^#(X) -> c_9()
  , a__U13^#(tt()) -> c_10()
  , a__U21^#(X1, X2) -> c_11()
  , a__U21^#(tt(), V1) ->
    c_12(a__U22^#(a__isNat(V1)), a__isNat^#(V1))
  , a__isNatKind^#(X) -> c_38()
  , a__isNatKind^#(s(V1)) -> c_39(a__isNatKind^#(V1))
  , a__isNatKind^#(0()) -> c_40()
  , a__isNatKind^#(plus(V1, V2)) ->
    c_41(a__and^#(a__isNatKind(V1), isNatKind(V2)), a__isNatKind^#(V1))
  , a__and^#(X1, X2) -> c_36()
  , a__and^#(tt(), X) -> c_37(mark^#(X))
  , a__U22^#(X) -> c_13()
  , a__U22^#(tt()) -> c_14()
  , a__U31^#(X1, X2) -> c_15()
  , a__U31^#(tt(), N) -> c_16(mark^#(N))
  , mark^#(tt()) -> c_17()
  , mark^#(s(X)) -> c_18(mark^#(X))
  , mark^#(0()) -> c_19()
  , mark^#(plus(X1, X2)) ->
    c_20(a__plus^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(isNatKind(X)) -> c_21(a__isNatKind^#(X))
  , mark^#(and(X1, X2)) -> c_22(a__and^#(mark(X1), X2), mark^#(X1))
  , mark^#(isNat(X)) -> c_23(a__isNat^#(X))
  , mark^#(U11(X1, X2, X3)) ->
    c_24(a__U11^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U12(X1, X2)) -> c_25(a__U12^#(mark(X1), X2), mark^#(X1))
  , mark^#(U13(X)) -> c_26(a__U13^#(mark(X)), mark^#(X))
  , mark^#(U21(X1, X2)) -> c_27(a__U21^#(mark(X1), X2), mark^#(X1))
  , mark^#(U22(X)) -> c_28(a__U22^#(mark(X)), mark^#(X))
  , mark^#(U31(X1, X2)) -> c_29(a__U31^#(mark(X1), X2), mark^#(X1))
  , mark^#(U41(X1, X2, X3)) ->
    c_30(a__U41^#(mark(X1), X2, X3), mark^#(X1))
  , a__plus^#(X1, X2) -> c_33()
  , a__plus^#(N, s(M)) ->
    c_34(a__U41^#(a__and(a__and(a__isNat(M), isNatKind(M)),
                         and(isNat(N), isNatKind(N))),
                  M,
                  N),
         a__and^#(a__and(a__isNat(M), isNatKind(M)),
                  and(isNat(N), isNatKind(N))),
         a__and^#(a__isNat(M), isNatKind(M)),
         a__isNat^#(M))
  , a__plus^#(N, 0()) ->
    c_35(a__U31^#(a__and(a__isNat(N), isNatKind(N)), N),
         a__and^#(a__isNat(N), isNatKind(N)),
         a__isNat^#(N))
  , a__U41^#(X1, X2, X3) -> c_31()
  , a__U41^#(tt(), M, N) ->
    c_32(a__plus^#(mark(N), mark(M)), mark^#(N), mark^#(M)) }

and mark the set of starting terms.

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

Strict DPs:
  { a__U11^#(X1, X2, X3) -> c_1()
  , a__U11^#(tt(), V1, V2) ->
    c_2(a__U12^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__U12^#(X1, X2) -> c_3()
  , a__U12^#(tt(), V2) -> c_4(a__U13^#(a__isNat(V2)), a__isNat^#(V2))
  , a__isNat^#(X) -> c_5()
  , a__isNat^#(s(V1)) ->
    c_6(a__U21^#(a__isNatKind(V1), V1), a__isNatKind^#(V1))
  , a__isNat^#(0()) -> c_7()
  , a__isNat^#(plus(V1, V2)) ->
    c_8(a__U11^#(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2),
        a__and^#(a__isNatKind(V1), isNatKind(V2)),
        a__isNatKind^#(V1))
  , a__U13^#(X) -> c_9()
  , a__U13^#(tt()) -> c_10()
  , a__U21^#(X1, X2) -> c_11()
  , a__U21^#(tt(), V1) ->
    c_12(a__U22^#(a__isNat(V1)), a__isNat^#(V1))
  , a__isNatKind^#(X) -> c_38()
  , a__isNatKind^#(s(V1)) -> c_39(a__isNatKind^#(V1))
  , a__isNatKind^#(0()) -> c_40()
  , a__isNatKind^#(plus(V1, V2)) ->
    c_41(a__and^#(a__isNatKind(V1), isNatKind(V2)), a__isNatKind^#(V1))
  , a__and^#(X1, X2) -> c_36()
  , a__and^#(tt(), X) -> c_37(mark^#(X))
  , a__U22^#(X) -> c_13()
  , a__U22^#(tt()) -> c_14()
  , a__U31^#(X1, X2) -> c_15()
  , a__U31^#(tt(), N) -> c_16(mark^#(N))
  , mark^#(tt()) -> c_17()
  , mark^#(s(X)) -> c_18(mark^#(X))
  , mark^#(0()) -> c_19()
  , mark^#(plus(X1, X2)) ->
    c_20(a__plus^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(isNatKind(X)) -> c_21(a__isNatKind^#(X))
  , mark^#(and(X1, X2)) -> c_22(a__and^#(mark(X1), X2), mark^#(X1))
  , mark^#(isNat(X)) -> c_23(a__isNat^#(X))
  , mark^#(U11(X1, X2, X3)) ->
    c_24(a__U11^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U12(X1, X2)) -> c_25(a__U12^#(mark(X1), X2), mark^#(X1))
  , mark^#(U13(X)) -> c_26(a__U13^#(mark(X)), mark^#(X))
  , mark^#(U21(X1, X2)) -> c_27(a__U21^#(mark(X1), X2), mark^#(X1))
  , mark^#(U22(X)) -> c_28(a__U22^#(mark(X)), mark^#(X))
  , mark^#(U31(X1, X2)) -> c_29(a__U31^#(mark(X1), X2), mark^#(X1))
  , mark^#(U41(X1, X2, X3)) ->
    c_30(a__U41^#(mark(X1), X2, X3), mark^#(X1))
  , a__plus^#(X1, X2) -> c_33()
  , a__plus^#(N, s(M)) ->
    c_34(a__U41^#(a__and(a__and(a__isNat(M), isNatKind(M)),
                         and(isNat(N), isNatKind(N))),
                  M,
                  N),
         a__and^#(a__and(a__isNat(M), isNatKind(M)),
                  and(isNat(N), isNatKind(N))),
         a__and^#(a__isNat(M), isNatKind(M)),
         a__isNat^#(M))
  , a__plus^#(N, 0()) ->
    c_35(a__U31^#(a__and(a__isNat(N), isNatKind(N)), N),
         a__and^#(a__isNat(N), isNatKind(N)),
         a__isNat^#(N))
  , a__U41^#(X1, X2, X3) -> c_31()
  , a__U41^#(tt(), M, N) ->
    c_32(a__plus^#(mark(N), mark(M)), mark^#(N), mark^#(M)) }
Weak Trs:
  { a__U11(X1, X2, X3) -> U11(X1, X2, X3)
  , a__U11(tt(), V1, V2) -> a__U12(a__isNat(V1), V2)
  , a__U12(X1, X2) -> U12(X1, X2)
  , a__U12(tt(), V2) -> a__U13(a__isNat(V2))
  , a__isNat(X) -> isNat(X)
  , a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1)
  , a__isNat(0()) -> tt()
  , a__isNat(plus(V1, V2)) ->
    a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
  , a__U13(X) -> U13(X)
  , a__U13(tt()) -> 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__U31(X1, X2) -> U31(X1, X2)
  , a__U31(tt(), N) -> mark(N)
  , mark(tt()) -> tt()
  , mark(s(X)) -> s(mark(X))
  , mark(0()) -> 0()
  , mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2))
  , mark(isNatKind(X)) -> a__isNatKind(X)
  , mark(and(X1, X2)) -> a__and(mark(X1), X2)
  , mark(isNat(X)) -> a__isNat(X)
  , mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
  , mark(U12(X1, X2)) -> a__U12(mark(X1), X2)
  , mark(U13(X)) -> a__U13(mark(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(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3)
  , a__U41(X1, X2, X3) -> U41(X1, X2, X3)
  , a__U41(tt(), M, N) -> s(a__plus(mark(N), mark(M)))
  , a__plus(X1, X2) -> plus(X1, X2)
  , a__plus(N, s(M)) ->
    a__U41(a__and(a__and(a__isNat(M), isNatKind(M)),
                  and(isNat(N), isNatKind(N))),
           M,
           N)
  , a__plus(N, 0()) -> a__U31(a__and(a__isNat(N), isNatKind(N)), N)
  , a__and(X1, X2) -> and(X1, X2)
  , a__and(tt(), X) -> mark(X)
  , a__isNatKind(X) -> isNatKind(X)
  , a__isNatKind(s(V1)) -> a__isNatKind(V1)
  , a__isNatKind(0()) -> tt()
  , a__isNatKind(plus(V1, V2)) ->
    a__and(a__isNatKind(V1), isNatKind(V2)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

  DPs:
    { 1: a__U11^#(X1, X2, X3) -> c_1()
    , 2: a__U11^#(tt(), V1, V2) ->
         c_2(a__U12^#(a__isNat(V1), V2), a__isNat^#(V1))
    , 3: a__U12^#(X1, X2) -> c_3()
    , 4: a__U12^#(tt(), V2) ->
         c_4(a__U13^#(a__isNat(V2)), a__isNat^#(V2))
    , 5: a__isNat^#(X) -> c_5()
    , 6: a__isNat^#(s(V1)) ->
         c_6(a__U21^#(a__isNatKind(V1), V1), a__isNatKind^#(V1))
    , 7: a__isNat^#(0()) -> c_7()
    , 8: a__isNat^#(plus(V1, V2)) ->
         c_8(a__U11^#(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2),
             a__and^#(a__isNatKind(V1), isNatKind(V2)),
             a__isNatKind^#(V1))
    , 9: a__U13^#(X) -> c_9()
    , 10: a__U13^#(tt()) -> c_10()
    , 11: a__U21^#(X1, X2) -> c_11()
    , 12: a__U21^#(tt(), V1) ->
          c_12(a__U22^#(a__isNat(V1)), a__isNat^#(V1))
    , 13: a__isNatKind^#(X) -> c_38()
    , 14: a__isNatKind^#(s(V1)) -> c_39(a__isNatKind^#(V1))
    , 15: a__isNatKind^#(0()) -> c_40()
    , 16: a__isNatKind^#(plus(V1, V2)) ->
          c_41(a__and^#(a__isNatKind(V1), isNatKind(V2)), a__isNatKind^#(V1))
    , 17: a__and^#(X1, X2) -> c_36()
    , 18: a__and^#(tt(), X) -> c_37(mark^#(X))
    , 19: a__U22^#(X) -> c_13()
    , 20: a__U22^#(tt()) -> c_14()
    , 21: a__U31^#(X1, X2) -> c_15()
    , 22: a__U31^#(tt(), N) -> c_16(mark^#(N))
    , 23: mark^#(tt()) -> c_17()
    , 24: mark^#(s(X)) -> c_18(mark^#(X))
    , 25: mark^#(0()) -> c_19()
    , 26: mark^#(plus(X1, X2)) ->
          c_20(a__plus^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
    , 27: mark^#(isNatKind(X)) -> c_21(a__isNatKind^#(X))
    , 28: mark^#(and(X1, X2)) ->
          c_22(a__and^#(mark(X1), X2), mark^#(X1))
    , 29: mark^#(isNat(X)) -> c_23(a__isNat^#(X))
    , 30: mark^#(U11(X1, X2, X3)) ->
          c_24(a__U11^#(mark(X1), X2, X3), mark^#(X1))
    , 31: mark^#(U12(X1, X2)) ->
          c_25(a__U12^#(mark(X1), X2), mark^#(X1))
    , 32: mark^#(U13(X)) -> c_26(a__U13^#(mark(X)), mark^#(X))
    , 33: mark^#(U21(X1, X2)) ->
          c_27(a__U21^#(mark(X1), X2), mark^#(X1))
    , 34: mark^#(U22(X)) -> c_28(a__U22^#(mark(X)), mark^#(X))
    , 35: mark^#(U31(X1, X2)) ->
          c_29(a__U31^#(mark(X1), X2), mark^#(X1))
    , 36: mark^#(U41(X1, X2, X3)) ->
          c_30(a__U41^#(mark(X1), X2, X3), mark^#(X1))
    , 37: a__plus^#(X1, X2) -> c_33()
    , 38: a__plus^#(N, s(M)) ->
          c_34(a__U41^#(a__and(a__and(a__isNat(M), isNatKind(M)),
                               and(isNat(N), isNatKind(N))),
                        M,
                        N),
               a__and^#(a__and(a__isNat(M), isNatKind(M)),
                        and(isNat(N), isNatKind(N))),
               a__and^#(a__isNat(M), isNatKind(M)),
               a__isNat^#(M))
    , 39: a__plus^#(N, 0()) ->
          c_35(a__U31^#(a__and(a__isNat(N), isNatKind(N)), N),
               a__and^#(a__isNat(N), isNatKind(N)),
               a__isNat^#(N))
    , 40: a__U41^#(X1, X2, X3) -> c_31()
    , 41: a__U41^#(tt(), M, N) ->
          c_32(a__plus^#(mark(N), mark(M)), mark^#(N), mark^#(M)) }

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

Strict DPs:
  { a__U11^#(tt(), V1, V2) ->
    c_2(a__U12^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__U12^#(tt(), V2) -> c_4(a__U13^#(a__isNat(V2)), a__isNat^#(V2))
  , a__isNat^#(s(V1)) ->
    c_6(a__U21^#(a__isNatKind(V1), V1), a__isNatKind^#(V1))
  , a__isNat^#(plus(V1, V2)) ->
    c_8(a__U11^#(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2),
        a__and^#(a__isNatKind(V1), isNatKind(V2)),
        a__isNatKind^#(V1))
  , a__U21^#(tt(), V1) ->
    c_12(a__U22^#(a__isNat(V1)), a__isNat^#(V1))
  , a__isNatKind^#(s(V1)) -> c_39(a__isNatKind^#(V1))
  , a__isNatKind^#(plus(V1, V2)) ->
    c_41(a__and^#(a__isNatKind(V1), isNatKind(V2)), a__isNatKind^#(V1))
  , a__and^#(tt(), X) -> c_37(mark^#(X))
  , a__U31^#(tt(), N) -> c_16(mark^#(N))
  , mark^#(s(X)) -> c_18(mark^#(X))
  , mark^#(plus(X1, X2)) ->
    c_20(a__plus^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(isNatKind(X)) -> c_21(a__isNatKind^#(X))
  , mark^#(and(X1, X2)) -> c_22(a__and^#(mark(X1), X2), mark^#(X1))
  , mark^#(isNat(X)) -> c_23(a__isNat^#(X))
  , mark^#(U11(X1, X2, X3)) ->
    c_24(a__U11^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U12(X1, X2)) -> c_25(a__U12^#(mark(X1), X2), mark^#(X1))
  , mark^#(U13(X)) -> c_26(a__U13^#(mark(X)), mark^#(X))
  , mark^#(U21(X1, X2)) -> c_27(a__U21^#(mark(X1), X2), mark^#(X1))
  , mark^#(U22(X)) -> c_28(a__U22^#(mark(X)), mark^#(X))
  , mark^#(U31(X1, X2)) -> c_29(a__U31^#(mark(X1), X2), mark^#(X1))
  , mark^#(U41(X1, X2, X3)) ->
    c_30(a__U41^#(mark(X1), X2, X3), mark^#(X1))
  , a__plus^#(N, s(M)) ->
    c_34(a__U41^#(a__and(a__and(a__isNat(M), isNatKind(M)),
                         and(isNat(N), isNatKind(N))),
                  M,
                  N),
         a__and^#(a__and(a__isNat(M), isNatKind(M)),
                  and(isNat(N), isNatKind(N))),
         a__and^#(a__isNat(M), isNatKind(M)),
         a__isNat^#(M))
  , a__plus^#(N, 0()) ->
    c_35(a__U31^#(a__and(a__isNat(N), isNatKind(N)), N),
         a__and^#(a__isNat(N), isNatKind(N)),
         a__isNat^#(N))
  , a__U41^#(tt(), M, N) ->
    c_32(a__plus^#(mark(N), mark(M)), mark^#(N), mark^#(M)) }
Weak DPs:
  { a__U11^#(X1, X2, X3) -> c_1()
  , a__U12^#(X1, X2) -> c_3()
  , a__isNat^#(X) -> c_5()
  , a__isNat^#(0()) -> c_7()
  , a__U13^#(X) -> c_9()
  , a__U13^#(tt()) -> c_10()
  , a__U21^#(X1, X2) -> c_11()
  , a__isNatKind^#(X) -> c_38()
  , a__isNatKind^#(0()) -> c_40()
  , a__and^#(X1, X2) -> c_36()
  , a__U22^#(X) -> c_13()
  , a__U22^#(tt()) -> c_14()
  , a__U31^#(X1, X2) -> c_15()
  , mark^#(tt()) -> c_17()
  , mark^#(0()) -> c_19()
  , a__plus^#(X1, X2) -> c_33()
  , a__U41^#(X1, X2, X3) -> c_31() }
Weak Trs:
  { a__U11(X1, X2, X3) -> U11(X1, X2, X3)
  , a__U11(tt(), V1, V2) -> a__U12(a__isNat(V1), V2)
  , a__U12(X1, X2) -> U12(X1, X2)
  , a__U12(tt(), V2) -> a__U13(a__isNat(V2))
  , a__isNat(X) -> isNat(X)
  , a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1)
  , a__isNat(0()) -> tt()
  , a__isNat(plus(V1, V2)) ->
    a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
  , a__U13(X) -> U13(X)
  , a__U13(tt()) -> 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__U31(X1, X2) -> U31(X1, X2)
  , a__U31(tt(), N) -> mark(N)
  , mark(tt()) -> tt()
  , mark(s(X)) -> s(mark(X))
  , mark(0()) -> 0()
  , mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2))
  , mark(isNatKind(X)) -> a__isNatKind(X)
  , mark(and(X1, X2)) -> a__and(mark(X1), X2)
  , mark(isNat(X)) -> a__isNat(X)
  , mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
  , mark(U12(X1, X2)) -> a__U12(mark(X1), X2)
  , mark(U13(X)) -> a__U13(mark(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(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3)
  , a__U41(X1, X2, X3) -> U41(X1, X2, X3)
  , a__U41(tt(), M, N) -> s(a__plus(mark(N), mark(M)))
  , a__plus(X1, X2) -> plus(X1, X2)
  , a__plus(N, s(M)) ->
    a__U41(a__and(a__and(a__isNat(M), isNatKind(M)),
                  and(isNat(N), isNatKind(N))),
           M,
           N)
  , a__plus(N, 0()) -> a__U31(a__and(a__isNat(N), isNatKind(N)), N)
  , a__and(X1, X2) -> and(X1, X2)
  , a__and(tt(), X) -> mark(X)
  , a__isNatKind(X) -> isNatKind(X)
  , a__isNatKind(s(V1)) -> a__isNatKind(V1)
  , a__isNatKind(0()) -> tt()
  , a__isNatKind(plus(V1, V2)) ->
    a__and(a__isNatKind(V1), isNatKind(V2)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.

{ a__U11^#(X1, X2, X3) -> c_1()
, a__U12^#(X1, X2) -> c_3()
, a__isNat^#(X) -> c_5()
, a__isNat^#(0()) -> c_7()
, a__U13^#(X) -> c_9()
, a__U13^#(tt()) -> c_10()
, a__U21^#(X1, X2) -> c_11()
, a__isNatKind^#(X) -> c_38()
, a__isNatKind^#(0()) -> c_40()
, a__and^#(X1, X2) -> c_36()
, a__U22^#(X) -> c_13()
, a__U22^#(tt()) -> c_14()
, a__U31^#(X1, X2) -> c_15()
, mark^#(tt()) -> c_17()
, mark^#(0()) -> c_19()
, a__plus^#(X1, X2) -> c_33()
, a__U41^#(X1, X2, X3) -> c_31() }

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

Strict DPs:
  { a__U11^#(tt(), V1, V2) ->
    c_2(a__U12^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__U12^#(tt(), V2) -> c_4(a__U13^#(a__isNat(V2)), a__isNat^#(V2))
  , a__isNat^#(s(V1)) ->
    c_6(a__U21^#(a__isNatKind(V1), V1), a__isNatKind^#(V1))
  , a__isNat^#(plus(V1, V2)) ->
    c_8(a__U11^#(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2),
        a__and^#(a__isNatKind(V1), isNatKind(V2)),
        a__isNatKind^#(V1))
  , a__U21^#(tt(), V1) ->
    c_12(a__U22^#(a__isNat(V1)), a__isNat^#(V1))
  , a__isNatKind^#(s(V1)) -> c_39(a__isNatKind^#(V1))
  , a__isNatKind^#(plus(V1, V2)) ->
    c_41(a__and^#(a__isNatKind(V1), isNatKind(V2)), a__isNatKind^#(V1))
  , a__and^#(tt(), X) -> c_37(mark^#(X))
  , a__U31^#(tt(), N) -> c_16(mark^#(N))
  , mark^#(s(X)) -> c_18(mark^#(X))
  , mark^#(plus(X1, X2)) ->
    c_20(a__plus^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(isNatKind(X)) -> c_21(a__isNatKind^#(X))
  , mark^#(and(X1, X2)) -> c_22(a__and^#(mark(X1), X2), mark^#(X1))
  , mark^#(isNat(X)) -> c_23(a__isNat^#(X))
  , mark^#(U11(X1, X2, X3)) ->
    c_24(a__U11^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U12(X1, X2)) -> c_25(a__U12^#(mark(X1), X2), mark^#(X1))
  , mark^#(U13(X)) -> c_26(a__U13^#(mark(X)), mark^#(X))
  , mark^#(U21(X1, X2)) -> c_27(a__U21^#(mark(X1), X2), mark^#(X1))
  , mark^#(U22(X)) -> c_28(a__U22^#(mark(X)), mark^#(X))
  , mark^#(U31(X1, X2)) -> c_29(a__U31^#(mark(X1), X2), mark^#(X1))
  , mark^#(U41(X1, X2, X3)) ->
    c_30(a__U41^#(mark(X1), X2, X3), mark^#(X1))
  , a__plus^#(N, s(M)) ->
    c_34(a__U41^#(a__and(a__and(a__isNat(M), isNatKind(M)),
                         and(isNat(N), isNatKind(N))),
                  M,
                  N),
         a__and^#(a__and(a__isNat(M), isNatKind(M)),
                  and(isNat(N), isNatKind(N))),
         a__and^#(a__isNat(M), isNatKind(M)),
         a__isNat^#(M))
  , a__plus^#(N, 0()) ->
    c_35(a__U31^#(a__and(a__isNat(N), isNatKind(N)), N),
         a__and^#(a__isNat(N), isNatKind(N)),
         a__isNat^#(N))
  , a__U41^#(tt(), M, N) ->
    c_32(a__plus^#(mark(N), mark(M)), mark^#(N), mark^#(M)) }
Weak Trs:
  { a__U11(X1, X2, X3) -> U11(X1, X2, X3)
  , a__U11(tt(), V1, V2) -> a__U12(a__isNat(V1), V2)
  , a__U12(X1, X2) -> U12(X1, X2)
  , a__U12(tt(), V2) -> a__U13(a__isNat(V2))
  , a__isNat(X) -> isNat(X)
  , a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1)
  , a__isNat(0()) -> tt()
  , a__isNat(plus(V1, V2)) ->
    a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
  , a__U13(X) -> U13(X)
  , a__U13(tt()) -> 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__U31(X1, X2) -> U31(X1, X2)
  , a__U31(tt(), N) -> mark(N)
  , mark(tt()) -> tt()
  , mark(s(X)) -> s(mark(X))
  , mark(0()) -> 0()
  , mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2))
  , mark(isNatKind(X)) -> a__isNatKind(X)
  , mark(and(X1, X2)) -> a__and(mark(X1), X2)
  , mark(isNat(X)) -> a__isNat(X)
  , mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
  , mark(U12(X1, X2)) -> a__U12(mark(X1), X2)
  , mark(U13(X)) -> a__U13(mark(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(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3)
  , a__U41(X1, X2, X3) -> U41(X1, X2, X3)
  , a__U41(tt(), M, N) -> s(a__plus(mark(N), mark(M)))
  , a__plus(X1, X2) -> plus(X1, X2)
  , a__plus(N, s(M)) ->
    a__U41(a__and(a__and(a__isNat(M), isNatKind(M)),
                  and(isNat(N), isNatKind(N))),
           M,
           N)
  , a__plus(N, 0()) -> a__U31(a__and(a__isNat(N), isNatKind(N)), N)
  , a__and(X1, X2) -> and(X1, X2)
  , a__and(tt(), X) -> mark(X)
  , a__isNatKind(X) -> isNatKind(X)
  , a__isNatKind(s(V1)) -> a__isNatKind(V1)
  , a__isNatKind(0()) -> tt()
  , a__isNatKind(plus(V1, V2)) ->
    a__and(a__isNatKind(V1), isNatKind(V2)) }
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__U12^#(tt(), V2) -> c_4(a__U13^#(a__isNat(V2)), a__isNat^#(V2))
  , a__U21^#(tt(), V1) ->
    c_12(a__U22^#(a__isNat(V1)), a__isNat^#(V1))
  , mark^#(U13(X)) -> c_26(a__U13^#(mark(X)), mark^#(X))
  , mark^#(U22(X)) -> c_28(a__U22^#(mark(X)), mark^#(X)) }

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

Strict DPs:
  { a__U11^#(tt(), V1, V2) ->
    c_1(a__U12^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__U12^#(tt(), V2) -> c_2(a__isNat^#(V2))
  , a__isNat^#(s(V1)) ->
    c_3(a__U21^#(a__isNatKind(V1), V1), a__isNatKind^#(V1))
  , a__isNat^#(plus(V1, V2)) ->
    c_4(a__U11^#(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2),
        a__and^#(a__isNatKind(V1), isNatKind(V2)),
        a__isNatKind^#(V1))
  , a__U21^#(tt(), V1) -> c_5(a__isNat^#(V1))
  , a__isNatKind^#(s(V1)) -> c_6(a__isNatKind^#(V1))
  , a__isNatKind^#(plus(V1, V2)) ->
    c_7(a__and^#(a__isNatKind(V1), isNatKind(V2)), a__isNatKind^#(V1))
  , a__and^#(tt(), X) -> c_8(mark^#(X))
  , a__U31^#(tt(), N) -> c_9(mark^#(N))
  , mark^#(s(X)) -> c_10(mark^#(X))
  , mark^#(plus(X1, X2)) ->
    c_11(a__plus^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(isNatKind(X)) -> c_12(a__isNatKind^#(X))
  , mark^#(and(X1, X2)) -> c_13(a__and^#(mark(X1), X2), mark^#(X1))
  , mark^#(isNat(X)) -> c_14(a__isNat^#(X))
  , mark^#(U11(X1, X2, X3)) ->
    c_15(a__U11^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U12(X1, X2)) -> c_16(a__U12^#(mark(X1), X2), mark^#(X1))
  , mark^#(U13(X)) -> c_17(mark^#(X))
  , mark^#(U21(X1, X2)) -> c_18(a__U21^#(mark(X1), X2), mark^#(X1))
  , mark^#(U22(X)) -> c_19(mark^#(X))
  , mark^#(U31(X1, X2)) -> c_20(a__U31^#(mark(X1), X2), mark^#(X1))
  , mark^#(U41(X1, X2, X3)) ->
    c_21(a__U41^#(mark(X1), X2, X3), mark^#(X1))
  , a__plus^#(N, s(M)) ->
    c_22(a__U41^#(a__and(a__and(a__isNat(M), isNatKind(M)),
                         and(isNat(N), isNatKind(N))),
                  M,
                  N),
         a__and^#(a__and(a__isNat(M), isNatKind(M)),
                  and(isNat(N), isNatKind(N))),
         a__and^#(a__isNat(M), isNatKind(M)),
         a__isNat^#(M))
  , a__plus^#(N, 0()) ->
    c_23(a__U31^#(a__and(a__isNat(N), isNatKind(N)), N),
         a__and^#(a__isNat(N), isNatKind(N)),
         a__isNat^#(N))
  , a__U41^#(tt(), M, N) ->
    c_24(a__plus^#(mark(N), mark(M)), mark^#(N), mark^#(M)) }
Weak Trs:
  { a__U11(X1, X2, X3) -> U11(X1, X2, X3)
  , a__U11(tt(), V1, V2) -> a__U12(a__isNat(V1), V2)
  , a__U12(X1, X2) -> U12(X1, X2)
  , a__U12(tt(), V2) -> a__U13(a__isNat(V2))
  , a__isNat(X) -> isNat(X)
  , a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1)
  , a__isNat(0()) -> tt()
  , a__isNat(plus(V1, V2)) ->
    a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
  , a__U13(X) -> U13(X)
  , a__U13(tt()) -> 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__U31(X1, X2) -> U31(X1, X2)
  , a__U31(tt(), N) -> mark(N)
  , mark(tt()) -> tt()
  , mark(s(X)) -> s(mark(X))
  , mark(0()) -> 0()
  , mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2))
  , mark(isNatKind(X)) -> a__isNatKind(X)
  , mark(and(X1, X2)) -> a__and(mark(X1), X2)
  , mark(isNat(X)) -> a__isNat(X)
  , mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
  , mark(U12(X1, X2)) -> a__U12(mark(X1), X2)
  , mark(U13(X)) -> a__U13(mark(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(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3)
  , a__U41(X1, X2, X3) -> U41(X1, X2, X3)
  , a__U41(tt(), M, N) -> s(a__plus(mark(N), mark(M)))
  , a__plus(X1, X2) -> plus(X1, X2)
  , a__plus(N, s(M)) ->
    a__U41(a__and(a__and(a__isNat(M), isNatKind(M)),
                  and(isNat(N), isNatKind(N))),
           M,
           N)
  , a__plus(N, 0()) -> a__U31(a__and(a__isNat(N), isNatKind(N)), N)
  , a__and(X1, X2) -> and(X1, X2)
  , a__and(tt(), X) -> mark(X)
  , a__isNatKind(X) -> isNatKind(X)
  , a__isNatKind(s(V1)) -> a__isNatKind(V1)
  , a__isNatKind(0()) -> tt()
  , a__isNatKind(plus(V1, V2)) ->
    a__and(a__isNatKind(V1), isNatKind(V2)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..