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__isNat(x(V1, V2)) ->
    a__U31(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, X3) -> U31(X1, X2, X3)
  , a__U31(tt(), V1, V2) -> a__U32(a__isNat(V1), V2)
  , a__U32(X1, X2) -> U32(X1, X2)
  , a__U32(tt(), V2) -> a__U33(a__isNat(V2))
  , a__U33(X) -> U33(X)
  , a__U33(tt()) -> tt()
  , a__U41(X1, X2) -> U41(X1, X2)
  , a__U41(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(x(X1, X2)) -> a__x(mark(X1), mark(X2))
  , 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, X3)) -> a__U31(mark(X1), X2, X3)
  , mark(U32(X1, X2)) -> a__U32(mark(X1), X2)
  , mark(U33(X)) -> a__U33(mark(X))
  , mark(U41(X1, X2)) -> a__U41(mark(X1), X2)
  , mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3)
  , mark(U61(X)) -> a__U61(mark(X))
  , mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3)
  , a__U51(X1, X2, X3) -> U51(X1, X2, X3)
  , a__U51(tt(), M, N) -> s(a__plus(mark(N), mark(M)))
  , a__plus(X1, X2) -> plus(X1, X2)
  , a__plus(N, s(M)) ->
    a__U51(a__and(a__and(a__isNat(M), isNatKind(M)),
                  and(isNat(N), isNatKind(N))),
           M,
           N)
  , a__plus(N, 0()) -> a__U41(a__and(a__isNat(N), isNatKind(N)), N)
  , a__U61(X) -> U61(X)
  , a__U61(tt()) -> 0()
  , a__U71(X1, X2, X3) -> U71(X1, X2, X3)
  , a__U71(tt(), M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N))
  , a__x(X1, X2) -> x(X1, X2)
  , a__x(N, s(M)) ->
    a__U71(a__and(a__and(a__isNat(M), isNatKind(M)),
                  and(isNat(N), isNatKind(N))),
           M,
           N)
  , a__x(N, 0()) -> a__U61(a__and(a__isNat(N), isNatKind(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))
  , a__isNatKind(x(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__isNat^#(x(V1, V2)) ->
    c_9(a__U31^#(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2),
        a__and^#(a__isNatKind(V1), isNatKind(V2)),
        a__isNatKind^#(V1))
  , a__U13^#(X) -> c_10()
  , a__U13^#(tt()) -> c_11()
  , a__U21^#(X1, X2) -> c_12()
  , a__U21^#(tt(), V1) ->
    c_13(a__U22^#(a__isNat(V1)), a__isNat^#(V1))
  , a__isNatKind^#(X) -> c_58()
  , a__isNatKind^#(s(V1)) -> c_59(a__isNatKind^#(V1))
  , a__isNatKind^#(0()) -> c_60()
  , a__isNatKind^#(plus(V1, V2)) ->
    c_61(a__and^#(a__isNatKind(V1), isNatKind(V2)), a__isNatKind^#(V1))
  , a__isNatKind^#(x(V1, V2)) ->
    c_62(a__and^#(a__isNatKind(V1), isNatKind(V2)), a__isNatKind^#(V1))
  , a__and^#(X1, X2) -> c_56()
  , a__and^#(tt(), X) -> c_57(mark^#(X))
  , a__U31^#(X1, X2, X3) -> c_16()
  , a__U31^#(tt(), V1, V2) ->
    c_17(a__U32^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__U22^#(X) -> c_14()
  , a__U22^#(tt()) -> c_15()
  , a__U32^#(X1, X2) -> c_18()
  , a__U32^#(tt(), V2) ->
    c_19(a__U33^#(a__isNat(V2)), a__isNat^#(V2))
  , a__U33^#(X) -> c_20()
  , a__U33^#(tt()) -> c_21()
  , a__U41^#(X1, X2) -> c_22()
  , a__U41^#(tt(), N) -> c_23(mark^#(N))
  , mark^#(tt()) -> c_24()
  , mark^#(s(X)) -> c_25(mark^#(X))
  , mark^#(0()) -> c_26()
  , mark^#(plus(X1, X2)) ->
    c_27(a__plus^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(isNatKind(X)) -> c_28(a__isNatKind^#(X))
  , mark^#(x(X1, X2)) ->
    c_29(a__x^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(and(X1, X2)) -> c_30(a__and^#(mark(X1), X2), mark^#(X1))
  , mark^#(isNat(X)) -> c_31(a__isNat^#(X))
  , mark^#(U11(X1, X2, X3)) ->
    c_32(a__U11^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U12(X1, X2)) -> c_33(a__U12^#(mark(X1), X2), mark^#(X1))
  , mark^#(U13(X)) -> c_34(a__U13^#(mark(X)), mark^#(X))
  , mark^#(U21(X1, X2)) -> c_35(a__U21^#(mark(X1), X2), mark^#(X1))
  , mark^#(U22(X)) -> c_36(a__U22^#(mark(X)), mark^#(X))
  , mark^#(U31(X1, X2, X3)) ->
    c_37(a__U31^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U32(X1, X2)) -> c_38(a__U32^#(mark(X1), X2), mark^#(X1))
  , mark^#(U33(X)) -> c_39(a__U33^#(mark(X)), mark^#(X))
  , mark^#(U41(X1, X2)) -> c_40(a__U41^#(mark(X1), X2), mark^#(X1))
  , mark^#(U51(X1, X2, X3)) ->
    c_41(a__U51^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U61(X)) -> c_42(a__U61^#(mark(X)), mark^#(X))
  , mark^#(U71(X1, X2, X3)) ->
    c_43(a__U71^#(mark(X1), X2, X3), mark^#(X1))
  , a__plus^#(X1, X2) -> c_46()
  , a__plus^#(N, s(M)) ->
    c_47(a__U51^#(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_48(a__U41^#(a__and(a__isNat(N), isNatKind(N)), N),
         a__and^#(a__isNat(N), isNatKind(N)),
         a__isNat^#(N))
  , a__x^#(X1, X2) -> c_53()
  , a__x^#(N, s(M)) ->
    c_54(a__U71^#(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__x^#(N, 0()) ->
    c_55(a__U61^#(a__and(a__isNat(N), isNatKind(N))),
         a__and^#(a__isNat(N), isNatKind(N)),
         a__isNat^#(N))
  , a__U51^#(X1, X2, X3) -> c_44()
  , a__U51^#(tt(), M, N) ->
    c_45(a__plus^#(mark(N), mark(M)), mark^#(N), mark^#(M))
  , a__U61^#(X) -> c_49()
  , a__U61^#(tt()) -> c_50()
  , a__U71^#(X1, X2, X3) -> c_51()
  , a__U71^#(tt(), M, N) ->
    c_52(a__plus^#(a__x(mark(N), mark(M)), mark(N)),
         a__x^#(mark(N), mark(M)),
         mark^#(N),
         mark^#(M),
         mark^#(N)) }

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__isNat^#(x(V1, V2)) ->
    c_9(a__U31^#(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2),
        a__and^#(a__isNatKind(V1), isNatKind(V2)),
        a__isNatKind^#(V1))
  , a__U13^#(X) -> c_10()
  , a__U13^#(tt()) -> c_11()
  , a__U21^#(X1, X2) -> c_12()
  , a__U21^#(tt(), V1) ->
    c_13(a__U22^#(a__isNat(V1)), a__isNat^#(V1))
  , a__isNatKind^#(X) -> c_58()
  , a__isNatKind^#(s(V1)) -> c_59(a__isNatKind^#(V1))
  , a__isNatKind^#(0()) -> c_60()
  , a__isNatKind^#(plus(V1, V2)) ->
    c_61(a__and^#(a__isNatKind(V1), isNatKind(V2)), a__isNatKind^#(V1))
  , a__isNatKind^#(x(V1, V2)) ->
    c_62(a__and^#(a__isNatKind(V1), isNatKind(V2)), a__isNatKind^#(V1))
  , a__and^#(X1, X2) -> c_56()
  , a__and^#(tt(), X) -> c_57(mark^#(X))
  , a__U31^#(X1, X2, X3) -> c_16()
  , a__U31^#(tt(), V1, V2) ->
    c_17(a__U32^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__U22^#(X) -> c_14()
  , a__U22^#(tt()) -> c_15()
  , a__U32^#(X1, X2) -> c_18()
  , a__U32^#(tt(), V2) ->
    c_19(a__U33^#(a__isNat(V2)), a__isNat^#(V2))
  , a__U33^#(X) -> c_20()
  , a__U33^#(tt()) -> c_21()
  , a__U41^#(X1, X2) -> c_22()
  , a__U41^#(tt(), N) -> c_23(mark^#(N))
  , mark^#(tt()) -> c_24()
  , mark^#(s(X)) -> c_25(mark^#(X))
  , mark^#(0()) -> c_26()
  , mark^#(plus(X1, X2)) ->
    c_27(a__plus^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(isNatKind(X)) -> c_28(a__isNatKind^#(X))
  , mark^#(x(X1, X2)) ->
    c_29(a__x^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(and(X1, X2)) -> c_30(a__and^#(mark(X1), X2), mark^#(X1))
  , mark^#(isNat(X)) -> c_31(a__isNat^#(X))
  , mark^#(U11(X1, X2, X3)) ->
    c_32(a__U11^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U12(X1, X2)) -> c_33(a__U12^#(mark(X1), X2), mark^#(X1))
  , mark^#(U13(X)) -> c_34(a__U13^#(mark(X)), mark^#(X))
  , mark^#(U21(X1, X2)) -> c_35(a__U21^#(mark(X1), X2), mark^#(X1))
  , mark^#(U22(X)) -> c_36(a__U22^#(mark(X)), mark^#(X))
  , mark^#(U31(X1, X2, X3)) ->
    c_37(a__U31^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U32(X1, X2)) -> c_38(a__U32^#(mark(X1), X2), mark^#(X1))
  , mark^#(U33(X)) -> c_39(a__U33^#(mark(X)), mark^#(X))
  , mark^#(U41(X1, X2)) -> c_40(a__U41^#(mark(X1), X2), mark^#(X1))
  , mark^#(U51(X1, X2, X3)) ->
    c_41(a__U51^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U61(X)) -> c_42(a__U61^#(mark(X)), mark^#(X))
  , mark^#(U71(X1, X2, X3)) ->
    c_43(a__U71^#(mark(X1), X2, X3), mark^#(X1))
  , a__plus^#(X1, X2) -> c_46()
  , a__plus^#(N, s(M)) ->
    c_47(a__U51^#(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_48(a__U41^#(a__and(a__isNat(N), isNatKind(N)), N),
         a__and^#(a__isNat(N), isNatKind(N)),
         a__isNat^#(N))
  , a__x^#(X1, X2) -> c_53()
  , a__x^#(N, s(M)) ->
    c_54(a__U71^#(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__x^#(N, 0()) ->
    c_55(a__U61^#(a__and(a__isNat(N), isNatKind(N))),
         a__and^#(a__isNat(N), isNatKind(N)),
         a__isNat^#(N))
  , a__U51^#(X1, X2, X3) -> c_44()
  , a__U51^#(tt(), M, N) ->
    c_45(a__plus^#(mark(N), mark(M)), mark^#(N), mark^#(M))
  , a__U61^#(X) -> c_49()
  , a__U61^#(tt()) -> c_50()
  , a__U71^#(X1, X2, X3) -> c_51()
  , a__U71^#(tt(), M, N) ->
    c_52(a__plus^#(a__x(mark(N), mark(M)), mark(N)),
         a__x^#(mark(N), mark(M)),
         mark^#(N),
         mark^#(M),
         mark^#(N)) }
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__isNat(x(V1, V2)) ->
    a__U31(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, X3) -> U31(X1, X2, X3)
  , a__U31(tt(), V1, V2) -> a__U32(a__isNat(V1), V2)
  , a__U32(X1, X2) -> U32(X1, X2)
  , a__U32(tt(), V2) -> a__U33(a__isNat(V2))
  , a__U33(X) -> U33(X)
  , a__U33(tt()) -> tt()
  , a__U41(X1, X2) -> U41(X1, X2)
  , a__U41(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(x(X1, X2)) -> a__x(mark(X1), mark(X2))
  , 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, X3)) -> a__U31(mark(X1), X2, X3)
  , mark(U32(X1, X2)) -> a__U32(mark(X1), X2)
  , mark(U33(X)) -> a__U33(mark(X))
  , mark(U41(X1, X2)) -> a__U41(mark(X1), X2)
  , mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3)
  , mark(U61(X)) -> a__U61(mark(X))
  , mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3)
  , a__U51(X1, X2, X3) -> U51(X1, X2, X3)
  , a__U51(tt(), M, N) -> s(a__plus(mark(N), mark(M)))
  , a__plus(X1, X2) -> plus(X1, X2)
  , a__plus(N, s(M)) ->
    a__U51(a__and(a__and(a__isNat(M), isNatKind(M)),
                  and(isNat(N), isNatKind(N))),
           M,
           N)
  , a__plus(N, 0()) -> a__U41(a__and(a__isNat(N), isNatKind(N)), N)
  , a__U61(X) -> U61(X)
  , a__U61(tt()) -> 0()
  , a__U71(X1, X2, X3) -> U71(X1, X2, X3)
  , a__U71(tt(), M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N))
  , a__x(X1, X2) -> x(X1, X2)
  , a__x(N, s(M)) ->
    a__U71(a__and(a__and(a__isNat(M), isNatKind(M)),
                  and(isNat(N), isNatKind(N))),
           M,
           N)
  , a__x(N, 0()) -> a__U61(a__and(a__isNat(N), isNatKind(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))
  , a__isNatKind(x(V1, V2)) ->
    a__and(a__isNatKind(V1), isNatKind(V2)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of
{1,3,5,7,10,11,12,14,16,19,21,23,24,25,27,28,29,31,33,51,54,57,59,60,61}
by applications of
Pre({1,3,5,7,10,11,12,14,16,19,21,23,24,25,27,28,29,31,33,51,54,57,59,60,61})
=
{2,4,6,8,9,13,15,17,18,20,22,26,30,32,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,52,53,55,56,58,62}.
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__isNat^#(x(V1, V2)) ->
         c_9(a__U31^#(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2),
             a__and^#(a__isNatKind(V1), isNatKind(V2)),
             a__isNatKind^#(V1))
    , 10: a__U13^#(X) -> c_10()
    , 11: a__U13^#(tt()) -> c_11()
    , 12: a__U21^#(X1, X2) -> c_12()
    , 13: a__U21^#(tt(), V1) ->
          c_13(a__U22^#(a__isNat(V1)), a__isNat^#(V1))
    , 14: a__isNatKind^#(X) -> c_58()
    , 15: a__isNatKind^#(s(V1)) -> c_59(a__isNatKind^#(V1))
    , 16: a__isNatKind^#(0()) -> c_60()
    , 17: a__isNatKind^#(plus(V1, V2)) ->
          c_61(a__and^#(a__isNatKind(V1), isNatKind(V2)), a__isNatKind^#(V1))
    , 18: a__isNatKind^#(x(V1, V2)) ->
          c_62(a__and^#(a__isNatKind(V1), isNatKind(V2)), a__isNatKind^#(V1))
    , 19: a__and^#(X1, X2) -> c_56()
    , 20: a__and^#(tt(), X) -> c_57(mark^#(X))
    , 21: a__U31^#(X1, X2, X3) -> c_16()
    , 22: a__U31^#(tt(), V1, V2) ->
          c_17(a__U32^#(a__isNat(V1), V2), a__isNat^#(V1))
    , 23: a__U22^#(X) -> c_14()
    , 24: a__U22^#(tt()) -> c_15()
    , 25: a__U32^#(X1, X2) -> c_18()
    , 26: a__U32^#(tt(), V2) ->
          c_19(a__U33^#(a__isNat(V2)), a__isNat^#(V2))
    , 27: a__U33^#(X) -> c_20()
    , 28: a__U33^#(tt()) -> c_21()
    , 29: a__U41^#(X1, X2) -> c_22()
    , 30: a__U41^#(tt(), N) -> c_23(mark^#(N))
    , 31: mark^#(tt()) -> c_24()
    , 32: mark^#(s(X)) -> c_25(mark^#(X))
    , 33: mark^#(0()) -> c_26()
    , 34: mark^#(plus(X1, X2)) ->
          c_27(a__plus^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
    , 35: mark^#(isNatKind(X)) -> c_28(a__isNatKind^#(X))
    , 36: mark^#(x(X1, X2)) ->
          c_29(a__x^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
    , 37: mark^#(and(X1, X2)) ->
          c_30(a__and^#(mark(X1), X2), mark^#(X1))
    , 38: mark^#(isNat(X)) -> c_31(a__isNat^#(X))
    , 39: mark^#(U11(X1, X2, X3)) ->
          c_32(a__U11^#(mark(X1), X2, X3), mark^#(X1))
    , 40: mark^#(U12(X1, X2)) ->
          c_33(a__U12^#(mark(X1), X2), mark^#(X1))
    , 41: mark^#(U13(X)) -> c_34(a__U13^#(mark(X)), mark^#(X))
    , 42: mark^#(U21(X1, X2)) ->
          c_35(a__U21^#(mark(X1), X2), mark^#(X1))
    , 43: mark^#(U22(X)) -> c_36(a__U22^#(mark(X)), mark^#(X))
    , 44: mark^#(U31(X1, X2, X3)) ->
          c_37(a__U31^#(mark(X1), X2, X3), mark^#(X1))
    , 45: mark^#(U32(X1, X2)) ->
          c_38(a__U32^#(mark(X1), X2), mark^#(X1))
    , 46: mark^#(U33(X)) -> c_39(a__U33^#(mark(X)), mark^#(X))
    , 47: mark^#(U41(X1, X2)) ->
          c_40(a__U41^#(mark(X1), X2), mark^#(X1))
    , 48: mark^#(U51(X1, X2, X3)) ->
          c_41(a__U51^#(mark(X1), X2, X3), mark^#(X1))
    , 49: mark^#(U61(X)) -> c_42(a__U61^#(mark(X)), mark^#(X))
    , 50: mark^#(U71(X1, X2, X3)) ->
          c_43(a__U71^#(mark(X1), X2, X3), mark^#(X1))
    , 51: a__plus^#(X1, X2) -> c_46()
    , 52: a__plus^#(N, s(M)) ->
          c_47(a__U51^#(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))
    , 53: a__plus^#(N, 0()) ->
          c_48(a__U41^#(a__and(a__isNat(N), isNatKind(N)), N),
               a__and^#(a__isNat(N), isNatKind(N)),
               a__isNat^#(N))
    , 54: a__x^#(X1, X2) -> c_53()
    , 55: a__x^#(N, s(M)) ->
          c_54(a__U71^#(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))
    , 56: a__x^#(N, 0()) ->
          c_55(a__U61^#(a__and(a__isNat(N), isNatKind(N))),
               a__and^#(a__isNat(N), isNatKind(N)),
               a__isNat^#(N))
    , 57: a__U51^#(X1, X2, X3) -> c_44()
    , 58: a__U51^#(tt(), M, N) ->
          c_45(a__plus^#(mark(N), mark(M)), mark^#(N), mark^#(M))
    , 59: a__U61^#(X) -> c_49()
    , 60: a__U61^#(tt()) -> c_50()
    , 61: a__U71^#(X1, X2, X3) -> c_51()
    , 62: a__U71^#(tt(), M, N) ->
          c_52(a__plus^#(a__x(mark(N), mark(M)), mark(N)),
               a__x^#(mark(N), mark(M)),
               mark^#(N),
               mark^#(M),
               mark^#(N)) }

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__isNat^#(x(V1, V2)) ->
    c_9(a__U31^#(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2),
        a__and^#(a__isNatKind(V1), isNatKind(V2)),
        a__isNatKind^#(V1))
  , a__U21^#(tt(), V1) ->
    c_13(a__U22^#(a__isNat(V1)), a__isNat^#(V1))
  , a__isNatKind^#(s(V1)) -> c_59(a__isNatKind^#(V1))
  , a__isNatKind^#(plus(V1, V2)) ->
    c_61(a__and^#(a__isNatKind(V1), isNatKind(V2)), a__isNatKind^#(V1))
  , a__isNatKind^#(x(V1, V2)) ->
    c_62(a__and^#(a__isNatKind(V1), isNatKind(V2)), a__isNatKind^#(V1))
  , a__and^#(tt(), X) -> c_57(mark^#(X))
  , a__U31^#(tt(), V1, V2) ->
    c_17(a__U32^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__U32^#(tt(), V2) ->
    c_19(a__U33^#(a__isNat(V2)), a__isNat^#(V2))
  , a__U41^#(tt(), N) -> c_23(mark^#(N))
  , mark^#(s(X)) -> c_25(mark^#(X))
  , mark^#(plus(X1, X2)) ->
    c_27(a__plus^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(isNatKind(X)) -> c_28(a__isNatKind^#(X))
  , mark^#(x(X1, X2)) ->
    c_29(a__x^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(and(X1, X2)) -> c_30(a__and^#(mark(X1), X2), mark^#(X1))
  , mark^#(isNat(X)) -> c_31(a__isNat^#(X))
  , mark^#(U11(X1, X2, X3)) ->
    c_32(a__U11^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U12(X1, X2)) -> c_33(a__U12^#(mark(X1), X2), mark^#(X1))
  , mark^#(U13(X)) -> c_34(a__U13^#(mark(X)), mark^#(X))
  , mark^#(U21(X1, X2)) -> c_35(a__U21^#(mark(X1), X2), mark^#(X1))
  , mark^#(U22(X)) -> c_36(a__U22^#(mark(X)), mark^#(X))
  , mark^#(U31(X1, X2, X3)) ->
    c_37(a__U31^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U32(X1, X2)) -> c_38(a__U32^#(mark(X1), X2), mark^#(X1))
  , mark^#(U33(X)) -> c_39(a__U33^#(mark(X)), mark^#(X))
  , mark^#(U41(X1, X2)) -> c_40(a__U41^#(mark(X1), X2), mark^#(X1))
  , mark^#(U51(X1, X2, X3)) ->
    c_41(a__U51^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U61(X)) -> c_42(a__U61^#(mark(X)), mark^#(X))
  , mark^#(U71(X1, X2, X3)) ->
    c_43(a__U71^#(mark(X1), X2, X3), mark^#(X1))
  , a__plus^#(N, s(M)) ->
    c_47(a__U51^#(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_48(a__U41^#(a__and(a__isNat(N), isNatKind(N)), N),
         a__and^#(a__isNat(N), isNatKind(N)),
         a__isNat^#(N))
  , a__x^#(N, s(M)) ->
    c_54(a__U71^#(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__x^#(N, 0()) ->
    c_55(a__U61^#(a__and(a__isNat(N), isNatKind(N))),
         a__and^#(a__isNat(N), isNatKind(N)),
         a__isNat^#(N))
  , a__U51^#(tt(), M, N) ->
    c_45(a__plus^#(mark(N), mark(M)), mark^#(N), mark^#(M))
  , a__U71^#(tt(), M, N) ->
    c_52(a__plus^#(a__x(mark(N), mark(M)), mark(N)),
         a__x^#(mark(N), mark(M)),
         mark^#(N),
         mark^#(M),
         mark^#(N)) }
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_10()
  , a__U13^#(tt()) -> c_11()
  , a__U21^#(X1, X2) -> c_12()
  , a__isNatKind^#(X) -> c_58()
  , a__isNatKind^#(0()) -> c_60()
  , a__and^#(X1, X2) -> c_56()
  , a__U31^#(X1, X2, X3) -> c_16()
  , a__U22^#(X) -> c_14()
  , a__U22^#(tt()) -> c_15()
  , a__U32^#(X1, X2) -> c_18()
  , a__U33^#(X) -> c_20()
  , a__U33^#(tt()) -> c_21()
  , a__U41^#(X1, X2) -> c_22()
  , mark^#(tt()) -> c_24()
  , mark^#(0()) -> c_26()
  , a__plus^#(X1, X2) -> c_46()
  , a__x^#(X1, X2) -> c_53()
  , a__U51^#(X1, X2, X3) -> c_44()
  , a__U61^#(X) -> c_49()
  , a__U61^#(tt()) -> c_50()
  , a__U71^#(X1, X2, X3) -> c_51() }
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__isNat(x(V1, V2)) ->
    a__U31(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, X3) -> U31(X1, X2, X3)
  , a__U31(tt(), V1, V2) -> a__U32(a__isNat(V1), V2)
  , a__U32(X1, X2) -> U32(X1, X2)
  , a__U32(tt(), V2) -> a__U33(a__isNat(V2))
  , a__U33(X) -> U33(X)
  , a__U33(tt()) -> tt()
  , a__U41(X1, X2) -> U41(X1, X2)
  , a__U41(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(x(X1, X2)) -> a__x(mark(X1), mark(X2))
  , 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, X3)) -> a__U31(mark(X1), X2, X3)
  , mark(U32(X1, X2)) -> a__U32(mark(X1), X2)
  , mark(U33(X)) -> a__U33(mark(X))
  , mark(U41(X1, X2)) -> a__U41(mark(X1), X2)
  , mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3)
  , mark(U61(X)) -> a__U61(mark(X))
  , mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3)
  , a__U51(X1, X2, X3) -> U51(X1, X2, X3)
  , a__U51(tt(), M, N) -> s(a__plus(mark(N), mark(M)))
  , a__plus(X1, X2) -> plus(X1, X2)
  , a__plus(N, s(M)) ->
    a__U51(a__and(a__and(a__isNat(M), isNatKind(M)),
                  and(isNat(N), isNatKind(N))),
           M,
           N)
  , a__plus(N, 0()) -> a__U41(a__and(a__isNat(N), isNatKind(N)), N)
  , a__U61(X) -> U61(X)
  , a__U61(tt()) -> 0()
  , a__U71(X1, X2, X3) -> U71(X1, X2, X3)
  , a__U71(tt(), M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N))
  , a__x(X1, X2) -> x(X1, X2)
  , a__x(N, s(M)) ->
    a__U71(a__and(a__and(a__isNat(M), isNatKind(M)),
                  and(isNat(N), isNatKind(N))),
           M,
           N)
  , a__x(N, 0()) -> a__U61(a__and(a__isNat(N), isNatKind(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))
  , a__isNatKind(x(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_10()
, a__U13^#(tt()) -> c_11()
, a__U21^#(X1, X2) -> c_12()
, a__isNatKind^#(X) -> c_58()
, a__isNatKind^#(0()) -> c_60()
, a__and^#(X1, X2) -> c_56()
, a__U31^#(X1, X2, X3) -> c_16()
, a__U22^#(X) -> c_14()
, a__U22^#(tt()) -> c_15()
, a__U32^#(X1, X2) -> c_18()
, a__U33^#(X) -> c_20()
, a__U33^#(tt()) -> c_21()
, a__U41^#(X1, X2) -> c_22()
, mark^#(tt()) -> c_24()
, mark^#(0()) -> c_26()
, a__plus^#(X1, X2) -> c_46()
, a__x^#(X1, X2) -> c_53()
, a__U51^#(X1, X2, X3) -> c_44()
, a__U61^#(X) -> c_49()
, a__U61^#(tt()) -> c_50()
, a__U71^#(X1, X2, X3) -> c_51() }

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__isNat^#(x(V1, V2)) ->
    c_9(a__U31^#(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2),
        a__and^#(a__isNatKind(V1), isNatKind(V2)),
        a__isNatKind^#(V1))
  , a__U21^#(tt(), V1) ->
    c_13(a__U22^#(a__isNat(V1)), a__isNat^#(V1))
  , a__isNatKind^#(s(V1)) -> c_59(a__isNatKind^#(V1))
  , a__isNatKind^#(plus(V1, V2)) ->
    c_61(a__and^#(a__isNatKind(V1), isNatKind(V2)), a__isNatKind^#(V1))
  , a__isNatKind^#(x(V1, V2)) ->
    c_62(a__and^#(a__isNatKind(V1), isNatKind(V2)), a__isNatKind^#(V1))
  , a__and^#(tt(), X) -> c_57(mark^#(X))
  , a__U31^#(tt(), V1, V2) ->
    c_17(a__U32^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__U32^#(tt(), V2) ->
    c_19(a__U33^#(a__isNat(V2)), a__isNat^#(V2))
  , a__U41^#(tt(), N) -> c_23(mark^#(N))
  , mark^#(s(X)) -> c_25(mark^#(X))
  , mark^#(plus(X1, X2)) ->
    c_27(a__plus^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(isNatKind(X)) -> c_28(a__isNatKind^#(X))
  , mark^#(x(X1, X2)) ->
    c_29(a__x^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(and(X1, X2)) -> c_30(a__and^#(mark(X1), X2), mark^#(X1))
  , mark^#(isNat(X)) -> c_31(a__isNat^#(X))
  , mark^#(U11(X1, X2, X3)) ->
    c_32(a__U11^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U12(X1, X2)) -> c_33(a__U12^#(mark(X1), X2), mark^#(X1))
  , mark^#(U13(X)) -> c_34(a__U13^#(mark(X)), mark^#(X))
  , mark^#(U21(X1, X2)) -> c_35(a__U21^#(mark(X1), X2), mark^#(X1))
  , mark^#(U22(X)) -> c_36(a__U22^#(mark(X)), mark^#(X))
  , mark^#(U31(X1, X2, X3)) ->
    c_37(a__U31^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U32(X1, X2)) -> c_38(a__U32^#(mark(X1), X2), mark^#(X1))
  , mark^#(U33(X)) -> c_39(a__U33^#(mark(X)), mark^#(X))
  , mark^#(U41(X1, X2)) -> c_40(a__U41^#(mark(X1), X2), mark^#(X1))
  , mark^#(U51(X1, X2, X3)) ->
    c_41(a__U51^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U61(X)) -> c_42(a__U61^#(mark(X)), mark^#(X))
  , mark^#(U71(X1, X2, X3)) ->
    c_43(a__U71^#(mark(X1), X2, X3), mark^#(X1))
  , a__plus^#(N, s(M)) ->
    c_47(a__U51^#(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_48(a__U41^#(a__and(a__isNat(N), isNatKind(N)), N),
         a__and^#(a__isNat(N), isNatKind(N)),
         a__isNat^#(N))
  , a__x^#(N, s(M)) ->
    c_54(a__U71^#(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__x^#(N, 0()) ->
    c_55(a__U61^#(a__and(a__isNat(N), isNatKind(N))),
         a__and^#(a__isNat(N), isNatKind(N)),
         a__isNat^#(N))
  , a__U51^#(tt(), M, N) ->
    c_45(a__plus^#(mark(N), mark(M)), mark^#(N), mark^#(M))
  , a__U71^#(tt(), M, N) ->
    c_52(a__plus^#(a__x(mark(N), mark(M)), mark(N)),
         a__x^#(mark(N), mark(M)),
         mark^#(N),
         mark^#(M),
         mark^#(N)) }
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__isNat(x(V1, V2)) ->
    a__U31(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, X3) -> U31(X1, X2, X3)
  , a__U31(tt(), V1, V2) -> a__U32(a__isNat(V1), V2)
  , a__U32(X1, X2) -> U32(X1, X2)
  , a__U32(tt(), V2) -> a__U33(a__isNat(V2))
  , a__U33(X) -> U33(X)
  , a__U33(tt()) -> tt()
  , a__U41(X1, X2) -> U41(X1, X2)
  , a__U41(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(x(X1, X2)) -> a__x(mark(X1), mark(X2))
  , 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, X3)) -> a__U31(mark(X1), X2, X3)
  , mark(U32(X1, X2)) -> a__U32(mark(X1), X2)
  , mark(U33(X)) -> a__U33(mark(X))
  , mark(U41(X1, X2)) -> a__U41(mark(X1), X2)
  , mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3)
  , mark(U61(X)) -> a__U61(mark(X))
  , mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3)
  , a__U51(X1, X2, X3) -> U51(X1, X2, X3)
  , a__U51(tt(), M, N) -> s(a__plus(mark(N), mark(M)))
  , a__plus(X1, X2) -> plus(X1, X2)
  , a__plus(N, s(M)) ->
    a__U51(a__and(a__and(a__isNat(M), isNatKind(M)),
                  and(isNat(N), isNatKind(N))),
           M,
           N)
  , a__plus(N, 0()) -> a__U41(a__and(a__isNat(N), isNatKind(N)), N)
  , a__U61(X) -> U61(X)
  , a__U61(tt()) -> 0()
  , a__U71(X1, X2, X3) -> U71(X1, X2, X3)
  , a__U71(tt(), M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N))
  , a__x(X1, X2) -> x(X1, X2)
  , a__x(N, s(M)) ->
    a__U71(a__and(a__and(a__isNat(M), isNatKind(M)),
                  and(isNat(N), isNatKind(N))),
           M,
           N)
  , a__x(N, 0()) -> a__U61(a__and(a__isNat(N), isNatKind(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))
  , a__isNatKind(x(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_13(a__U22^#(a__isNat(V1)), a__isNat^#(V1))
  , a__U32^#(tt(), V2) ->
    c_19(a__U33^#(a__isNat(V2)), a__isNat^#(V2))
  , mark^#(U13(X)) -> c_34(a__U13^#(mark(X)), mark^#(X))
  , mark^#(U22(X)) -> c_36(a__U22^#(mark(X)), mark^#(X))
  , mark^#(U33(X)) -> c_39(a__U33^#(mark(X)), mark^#(X))
  , mark^#(U61(X)) -> c_42(a__U61^#(mark(X)), mark^#(X))
  , a__x^#(N, 0()) ->
    c_55(a__U61^#(a__and(a__isNat(N), isNatKind(N))),
         a__and^#(a__isNat(N), isNatKind(N)),
         a__isNat^#(N)) }

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__isNat^#(x(V1, V2)) ->
    c_5(a__U31^#(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2),
        a__and^#(a__isNatKind(V1), isNatKind(V2)),
        a__isNatKind^#(V1))
  , a__U21^#(tt(), V1) -> c_6(a__isNat^#(V1))
  , a__isNatKind^#(s(V1)) -> c_7(a__isNatKind^#(V1))
  , a__isNatKind^#(plus(V1, V2)) ->
    c_8(a__and^#(a__isNatKind(V1), isNatKind(V2)), a__isNatKind^#(V1))
  , a__isNatKind^#(x(V1, V2)) ->
    c_9(a__and^#(a__isNatKind(V1), isNatKind(V2)), a__isNatKind^#(V1))
  , a__and^#(tt(), X) -> c_10(mark^#(X))
  , a__U31^#(tt(), V1, V2) ->
    c_11(a__U32^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__U32^#(tt(), V2) -> c_12(a__isNat^#(V2))
  , a__U41^#(tt(), N) -> c_13(mark^#(N))
  , mark^#(s(X)) -> c_14(mark^#(X))
  , mark^#(plus(X1, X2)) ->
    c_15(a__plus^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(isNatKind(X)) -> c_16(a__isNatKind^#(X))
  , mark^#(x(X1, X2)) ->
    c_17(a__x^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(and(X1, X2)) -> c_18(a__and^#(mark(X1), X2), mark^#(X1))
  , mark^#(isNat(X)) -> c_19(a__isNat^#(X))
  , mark^#(U11(X1, X2, X3)) ->
    c_20(a__U11^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U12(X1, X2)) -> c_21(a__U12^#(mark(X1), X2), mark^#(X1))
  , mark^#(U13(X)) -> c_22(mark^#(X))
  , mark^#(U21(X1, X2)) -> c_23(a__U21^#(mark(X1), X2), mark^#(X1))
  , mark^#(U22(X)) -> c_24(mark^#(X))
  , mark^#(U31(X1, X2, X3)) ->
    c_25(a__U31^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U32(X1, X2)) -> c_26(a__U32^#(mark(X1), X2), mark^#(X1))
  , mark^#(U33(X)) -> c_27(mark^#(X))
  , mark^#(U41(X1, X2)) -> c_28(a__U41^#(mark(X1), X2), mark^#(X1))
  , mark^#(U51(X1, X2, X3)) ->
    c_29(a__U51^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U61(X)) -> c_30(mark^#(X))
  , mark^#(U71(X1, X2, X3)) ->
    c_31(a__U71^#(mark(X1), X2, X3), mark^#(X1))
  , a__plus^#(N, s(M)) ->
    c_32(a__U51^#(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_33(a__U41^#(a__and(a__isNat(N), isNatKind(N)), N),
         a__and^#(a__isNat(N), isNatKind(N)),
         a__isNat^#(N))
  , a__x^#(N, s(M)) ->
    c_34(a__U71^#(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__x^#(N, 0()) ->
    c_35(a__and^#(a__isNat(N), isNatKind(N)), a__isNat^#(N))
  , a__U51^#(tt(), M, N) ->
    c_36(a__plus^#(mark(N), mark(M)), mark^#(N), mark^#(M))
  , a__U71^#(tt(), M, N) ->
    c_37(a__plus^#(a__x(mark(N), mark(M)), mark(N)),
         a__x^#(mark(N), mark(M)),
         mark^#(N),
         mark^#(M),
         mark^#(N)) }
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__isNat(x(V1, V2)) ->
    a__U31(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, X3) -> U31(X1, X2, X3)
  , a__U31(tt(), V1, V2) -> a__U32(a__isNat(V1), V2)
  , a__U32(X1, X2) -> U32(X1, X2)
  , a__U32(tt(), V2) -> a__U33(a__isNat(V2))
  , a__U33(X) -> U33(X)
  , a__U33(tt()) -> tt()
  , a__U41(X1, X2) -> U41(X1, X2)
  , a__U41(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(x(X1, X2)) -> a__x(mark(X1), mark(X2))
  , 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, X3)) -> a__U31(mark(X1), X2, X3)
  , mark(U32(X1, X2)) -> a__U32(mark(X1), X2)
  , mark(U33(X)) -> a__U33(mark(X))
  , mark(U41(X1, X2)) -> a__U41(mark(X1), X2)
  , mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3)
  , mark(U61(X)) -> a__U61(mark(X))
  , mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3)
  , a__U51(X1, X2, X3) -> U51(X1, X2, X3)
  , a__U51(tt(), M, N) -> s(a__plus(mark(N), mark(M)))
  , a__plus(X1, X2) -> plus(X1, X2)
  , a__plus(N, s(M)) ->
    a__U51(a__and(a__and(a__isNat(M), isNatKind(M)),
                  and(isNat(N), isNatKind(N))),
           M,
           N)
  , a__plus(N, 0()) -> a__U41(a__and(a__isNat(N), isNatKind(N)), N)
  , a__U61(X) -> U61(X)
  , a__U61(tt()) -> 0()
  , a__U71(X1, X2, X3) -> U71(X1, X2, X3)
  , a__U71(tt(), M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N))
  , a__x(X1, X2) -> x(X1, X2)
  , a__x(N, s(M)) ->
    a__U71(a__and(a__and(a__isNat(M), isNatKind(M)),
                  and(isNat(N), isNatKind(N))),
           M,
           N)
  , a__x(N, 0()) -> a__U61(a__and(a__isNat(N), isNatKind(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))
  , a__isNatKind(x(V1, V2)) ->
    a__and(a__isNatKind(V1), isNatKind(V2)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..