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__isNatKind(V1), V1, V2)
  , a__U12(X1, X2, X3) -> U12(X1, X2, X3)
  , a__U12(tt(), V1, V2) -> a__U13(a__isNatKind(V2), V1, V2)
  , a__isNatKind(X) -> isNatKind(X)
  , a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1))
  , a__isNatKind(0()) -> tt()
  , a__isNatKind(plus(V1, V2)) -> a__U31(a__isNatKind(V1), V2)
  , a__U13(X1, X2, X3) -> U13(X1, X2, X3)
  , a__U13(tt(), V1, V2) -> a__U14(a__isNatKind(V2), V1, V2)
  , a__U14(X1, X2, X3) -> U14(X1, X2, X3)
  , a__U14(tt(), V1, V2) -> a__U15(a__isNat(V1), V2)
  , a__U15(X1, X2) -> U15(X1, X2)
  , a__U15(tt(), V2) -> a__U16(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__isNatKind(V1), V1, V2)
  , a__U16(X) -> U16(X)
  , a__U16(tt()) -> tt()
  , a__U21(X1, X2) -> U21(X1, X2)
  , a__U21(tt(), V1) -> a__U22(a__isNatKind(V1), V1)
  , a__U22(X1, X2) -> U22(X1, X2)
  , a__U22(tt(), V1) -> a__U23(a__isNat(V1))
  , a__U23(X) -> U23(X)
  , a__U23(tt()) -> tt()
  , a__U31(X1, X2) -> U31(X1, X2)
  , a__U31(tt(), V2) -> a__U32(a__isNatKind(V2))
  , a__U32(X) -> U32(X)
  , a__U32(tt()) -> tt()
  , a__U41(X) -> U41(X)
  , a__U41(tt()) -> tt()
  , a__U51(X1, X2) -> U51(X1, X2)
  , a__U51(tt(), N) -> a__U52(a__isNatKind(N), N)
  , a__U52(X1, X2) -> U52(X1, X2)
  , a__U52(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(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
  , mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3)
  , mark(isNatKind(X)) -> a__isNatKind(X)
  , mark(U13(X1, X2, X3)) -> a__U13(mark(X1), X2, X3)
  , mark(U14(X1, X2, X3)) -> a__U14(mark(X1), X2, X3)
  , mark(U15(X1, X2)) -> a__U15(mark(X1), X2)
  , mark(isNat(X)) -> a__isNat(X)
  , mark(U16(X)) -> a__U16(mark(X))
  , mark(U21(X1, X2)) -> a__U21(mark(X1), X2)
  , mark(U22(X1, X2)) -> a__U22(mark(X1), X2)
  , mark(U23(X)) -> a__U23(mark(X))
  , mark(U31(X1, X2)) -> a__U31(mark(X1), X2)
  , mark(U32(X)) -> a__U32(mark(X))
  , mark(U41(X)) -> a__U41(mark(X))
  , mark(U51(X1, X2)) -> a__U51(mark(X1), X2)
  , mark(U52(X1, X2)) -> a__U52(mark(X1), X2)
  , mark(U61(X1, X2, X3)) -> a__U61(mark(X1), X2, X3)
  , mark(U62(X1, X2, X3)) -> a__U62(mark(X1), X2, X3)
  , mark(U63(X1, X2, X3)) -> a__U63(mark(X1), X2, X3)
  , mark(U64(X1, X2, X3)) -> a__U64(mark(X1), X2, X3)
  , a__U61(X1, X2, X3) -> U61(X1, X2, X3)
  , a__U61(tt(), M, N) -> a__U62(a__isNatKind(M), M, N)
  , a__U62(X1, X2, X3) -> U62(X1, X2, X3)
  , a__U62(tt(), M, N) -> a__U63(a__isNat(N), M, N)
  , a__U63(X1, X2, X3) -> U63(X1, X2, X3)
  , a__U63(tt(), M, N) -> a__U64(a__isNatKind(N), M, N)
  , a__U64(X1, X2, X3) -> U64(X1, X2, X3)
  , a__U64(tt(), M, N) -> s(a__plus(mark(N), mark(M)))
  , a__plus(X1, X2) -> plus(X1, X2)
  , a__plus(N, s(M)) -> a__U61(a__isNat(M), M, N)
  , a__plus(N, 0()) -> a__U51(a__isNat(N), N) }
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__isNatKind(V1), V1, V2), a__isNatKind^#(V1))
  , a__U12^#(X1, X2, X3) -> c_3()
  , a__U12^#(tt(), V1, V2) ->
    c_4(a__U13^#(a__isNatKind(V2), V1, V2), a__isNatKind^#(V2))
  , a__isNatKind^#(X) -> c_5()
  , a__isNatKind^#(s(V1)) ->
    c_6(a__U41^#(a__isNatKind(V1)), a__isNatKind^#(V1))
  , a__isNatKind^#(0()) -> c_7()
  , a__isNatKind^#(plus(V1, V2)) ->
    c_8(a__U31^#(a__isNatKind(V1), V2), a__isNatKind^#(V1))
  , a__U13^#(X1, X2, X3) -> c_9()
  , a__U13^#(tt(), V1, V2) ->
    c_10(a__U14^#(a__isNatKind(V2), V1, V2), a__isNatKind^#(V2))
  , a__U41^#(X) -> c_31()
  , a__U41^#(tt()) -> c_32()
  , a__U31^#(X1, X2) -> c_27()
  , a__U31^#(tt(), V2) ->
    c_28(a__U32^#(a__isNatKind(V2)), a__isNatKind^#(V2))
  , a__U14^#(X1, X2, X3) -> c_11()
  , a__U14^#(tt(), V1, V2) ->
    c_12(a__U15^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__U15^#(X1, X2) -> c_13()
  , a__U15^#(tt(), V2) ->
    c_14(a__U16^#(a__isNat(V2)), a__isNat^#(V2))
  , a__isNat^#(X) -> c_15()
  , a__isNat^#(s(V1)) ->
    c_16(a__U21^#(a__isNatKind(V1), V1), a__isNatKind^#(V1))
  , a__isNat^#(0()) -> c_17()
  , a__isNat^#(plus(V1, V2)) ->
    c_18(a__U11^#(a__isNatKind(V1), V1, V2), a__isNatKind^#(V1))
  , a__U16^#(X) -> c_19()
  , a__U16^#(tt()) -> c_20()
  , a__U21^#(X1, X2) -> c_21()
  , a__U21^#(tt(), V1) ->
    c_22(a__U22^#(a__isNatKind(V1), V1), a__isNatKind^#(V1))
  , a__U22^#(X1, X2) -> c_23()
  , a__U22^#(tt(), V1) ->
    c_24(a__U23^#(a__isNat(V1)), a__isNat^#(V1))
  , a__U23^#(X) -> c_25()
  , a__U23^#(tt()) -> c_26()
  , a__U32^#(X) -> c_29()
  , a__U32^#(tt()) -> c_30()
  , a__U51^#(X1, X2) -> c_33()
  , a__U51^#(tt(), N) ->
    c_34(a__U52^#(a__isNatKind(N), N), a__isNatKind^#(N))
  , a__U52^#(X1, X2) -> c_35()
  , a__U52^#(tt(), N) -> c_36(mark^#(N))
  , mark^#(tt()) -> c_37()
  , mark^#(s(X)) -> c_38(mark^#(X))
  , mark^#(0()) -> c_39()
  , mark^#(plus(X1, X2)) ->
    c_40(a__plus^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(U11(X1, X2, X3)) ->
    c_41(a__U11^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U12(X1, X2, X3)) ->
    c_42(a__U12^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(isNatKind(X)) -> c_43(a__isNatKind^#(X))
  , mark^#(U13(X1, X2, X3)) ->
    c_44(a__U13^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U14(X1, X2, X3)) ->
    c_45(a__U14^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U15(X1, X2)) -> c_46(a__U15^#(mark(X1), X2), mark^#(X1))
  , mark^#(isNat(X)) -> c_47(a__isNat^#(X))
  , mark^#(U16(X)) -> c_48(a__U16^#(mark(X)), mark^#(X))
  , mark^#(U21(X1, X2)) -> c_49(a__U21^#(mark(X1), X2), mark^#(X1))
  , mark^#(U22(X1, X2)) -> c_50(a__U22^#(mark(X1), X2), mark^#(X1))
  , mark^#(U23(X)) -> c_51(a__U23^#(mark(X)), mark^#(X))
  , mark^#(U31(X1, X2)) -> c_52(a__U31^#(mark(X1), X2), mark^#(X1))
  , mark^#(U32(X)) -> c_53(a__U32^#(mark(X)), mark^#(X))
  , mark^#(U41(X)) -> c_54(a__U41^#(mark(X)), mark^#(X))
  , mark^#(U51(X1, X2)) -> c_55(a__U51^#(mark(X1), X2), mark^#(X1))
  , mark^#(U52(X1, X2)) -> c_56(a__U52^#(mark(X1), X2), mark^#(X1))
  , mark^#(U61(X1, X2, X3)) ->
    c_57(a__U61^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U62(X1, X2, X3)) ->
    c_58(a__U62^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U63(X1, X2, X3)) ->
    c_59(a__U63^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U64(X1, X2, X3)) ->
    c_60(a__U64^#(mark(X1), X2, X3), mark^#(X1))
  , a__plus^#(X1, X2) -> c_69()
  , a__plus^#(N, s(M)) ->
    c_70(a__U61^#(a__isNat(M), M, N), a__isNat^#(M))
  , a__plus^#(N, 0()) ->
    c_71(a__U51^#(a__isNat(N), N), a__isNat^#(N))
  , a__U61^#(X1, X2, X3) -> c_61()
  , a__U61^#(tt(), M, N) ->
    c_62(a__U62^#(a__isNatKind(M), M, N), a__isNatKind^#(M))
  , a__U62^#(X1, X2, X3) -> c_63()
  , a__U62^#(tt(), M, N) ->
    c_64(a__U63^#(a__isNat(N), M, N), a__isNat^#(N))
  , a__U63^#(X1, X2, X3) -> c_65()
  , a__U63^#(tt(), M, N) ->
    c_66(a__U64^#(a__isNatKind(N), M, N), a__isNatKind^#(N))
  , a__U64^#(X1, X2, X3) -> c_67()
  , a__U64^#(tt(), M, N) ->
    c_68(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__isNatKind(V1), V1, V2), a__isNatKind^#(V1))
  , a__U12^#(X1, X2, X3) -> c_3()
  , a__U12^#(tt(), V1, V2) ->
    c_4(a__U13^#(a__isNatKind(V2), V1, V2), a__isNatKind^#(V2))
  , a__isNatKind^#(X) -> c_5()
  , a__isNatKind^#(s(V1)) ->
    c_6(a__U41^#(a__isNatKind(V1)), a__isNatKind^#(V1))
  , a__isNatKind^#(0()) -> c_7()
  , a__isNatKind^#(plus(V1, V2)) ->
    c_8(a__U31^#(a__isNatKind(V1), V2), a__isNatKind^#(V1))
  , a__U13^#(X1, X2, X3) -> c_9()
  , a__U13^#(tt(), V1, V2) ->
    c_10(a__U14^#(a__isNatKind(V2), V1, V2), a__isNatKind^#(V2))
  , a__U41^#(X) -> c_31()
  , a__U41^#(tt()) -> c_32()
  , a__U31^#(X1, X2) -> c_27()
  , a__U31^#(tt(), V2) ->
    c_28(a__U32^#(a__isNatKind(V2)), a__isNatKind^#(V2))
  , a__U14^#(X1, X2, X3) -> c_11()
  , a__U14^#(tt(), V1, V2) ->
    c_12(a__U15^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__U15^#(X1, X2) -> c_13()
  , a__U15^#(tt(), V2) ->
    c_14(a__U16^#(a__isNat(V2)), a__isNat^#(V2))
  , a__isNat^#(X) -> c_15()
  , a__isNat^#(s(V1)) ->
    c_16(a__U21^#(a__isNatKind(V1), V1), a__isNatKind^#(V1))
  , a__isNat^#(0()) -> c_17()
  , a__isNat^#(plus(V1, V2)) ->
    c_18(a__U11^#(a__isNatKind(V1), V1, V2), a__isNatKind^#(V1))
  , a__U16^#(X) -> c_19()
  , a__U16^#(tt()) -> c_20()
  , a__U21^#(X1, X2) -> c_21()
  , a__U21^#(tt(), V1) ->
    c_22(a__U22^#(a__isNatKind(V1), V1), a__isNatKind^#(V1))
  , a__U22^#(X1, X2) -> c_23()
  , a__U22^#(tt(), V1) ->
    c_24(a__U23^#(a__isNat(V1)), a__isNat^#(V1))
  , a__U23^#(X) -> c_25()
  , a__U23^#(tt()) -> c_26()
  , a__U32^#(X) -> c_29()
  , a__U32^#(tt()) -> c_30()
  , a__U51^#(X1, X2) -> c_33()
  , a__U51^#(tt(), N) ->
    c_34(a__U52^#(a__isNatKind(N), N), a__isNatKind^#(N))
  , a__U52^#(X1, X2) -> c_35()
  , a__U52^#(tt(), N) -> c_36(mark^#(N))
  , mark^#(tt()) -> c_37()
  , mark^#(s(X)) -> c_38(mark^#(X))
  , mark^#(0()) -> c_39()
  , mark^#(plus(X1, X2)) ->
    c_40(a__plus^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(U11(X1, X2, X3)) ->
    c_41(a__U11^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U12(X1, X2, X3)) ->
    c_42(a__U12^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(isNatKind(X)) -> c_43(a__isNatKind^#(X))
  , mark^#(U13(X1, X2, X3)) ->
    c_44(a__U13^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U14(X1, X2, X3)) ->
    c_45(a__U14^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U15(X1, X2)) -> c_46(a__U15^#(mark(X1), X2), mark^#(X1))
  , mark^#(isNat(X)) -> c_47(a__isNat^#(X))
  , mark^#(U16(X)) -> c_48(a__U16^#(mark(X)), mark^#(X))
  , mark^#(U21(X1, X2)) -> c_49(a__U21^#(mark(X1), X2), mark^#(X1))
  , mark^#(U22(X1, X2)) -> c_50(a__U22^#(mark(X1), X2), mark^#(X1))
  , mark^#(U23(X)) -> c_51(a__U23^#(mark(X)), mark^#(X))
  , mark^#(U31(X1, X2)) -> c_52(a__U31^#(mark(X1), X2), mark^#(X1))
  , mark^#(U32(X)) -> c_53(a__U32^#(mark(X)), mark^#(X))
  , mark^#(U41(X)) -> c_54(a__U41^#(mark(X)), mark^#(X))
  , mark^#(U51(X1, X2)) -> c_55(a__U51^#(mark(X1), X2), mark^#(X1))
  , mark^#(U52(X1, X2)) -> c_56(a__U52^#(mark(X1), X2), mark^#(X1))
  , mark^#(U61(X1, X2, X3)) ->
    c_57(a__U61^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U62(X1, X2, X3)) ->
    c_58(a__U62^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U63(X1, X2, X3)) ->
    c_59(a__U63^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U64(X1, X2, X3)) ->
    c_60(a__U64^#(mark(X1), X2, X3), mark^#(X1))
  , a__plus^#(X1, X2) -> c_69()
  , a__plus^#(N, s(M)) ->
    c_70(a__U61^#(a__isNat(M), M, N), a__isNat^#(M))
  , a__plus^#(N, 0()) ->
    c_71(a__U51^#(a__isNat(N), N), a__isNat^#(N))
  , a__U61^#(X1, X2, X3) -> c_61()
  , a__U61^#(tt(), M, N) ->
    c_62(a__U62^#(a__isNatKind(M), M, N), a__isNatKind^#(M))
  , a__U62^#(X1, X2, X3) -> c_63()
  , a__U62^#(tt(), M, N) ->
    c_64(a__U63^#(a__isNat(N), M, N), a__isNat^#(N))
  , a__U63^#(X1, X2, X3) -> c_65()
  , a__U63^#(tt(), M, N) ->
    c_66(a__U64^#(a__isNatKind(N), M, N), a__isNatKind^#(N))
  , a__U64^#(X1, X2, X3) -> c_67()
  , a__U64^#(tt(), M, N) ->
    c_68(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__isNatKind(V1), V1, V2)
  , a__U12(X1, X2, X3) -> U12(X1, X2, X3)
  , a__U12(tt(), V1, V2) -> a__U13(a__isNatKind(V2), V1, V2)
  , a__isNatKind(X) -> isNatKind(X)
  , a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1))
  , a__isNatKind(0()) -> tt()
  , a__isNatKind(plus(V1, V2)) -> a__U31(a__isNatKind(V1), V2)
  , a__U13(X1, X2, X3) -> U13(X1, X2, X3)
  , a__U13(tt(), V1, V2) -> a__U14(a__isNatKind(V2), V1, V2)
  , a__U14(X1, X2, X3) -> U14(X1, X2, X3)
  , a__U14(tt(), V1, V2) -> a__U15(a__isNat(V1), V2)
  , a__U15(X1, X2) -> U15(X1, X2)
  , a__U15(tt(), V2) -> a__U16(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__isNatKind(V1), V1, V2)
  , a__U16(X) -> U16(X)
  , a__U16(tt()) -> tt()
  , a__U21(X1, X2) -> U21(X1, X2)
  , a__U21(tt(), V1) -> a__U22(a__isNatKind(V1), V1)
  , a__U22(X1, X2) -> U22(X1, X2)
  , a__U22(tt(), V1) -> a__U23(a__isNat(V1))
  , a__U23(X) -> U23(X)
  , a__U23(tt()) -> tt()
  , a__U31(X1, X2) -> U31(X1, X2)
  , a__U31(tt(), V2) -> a__U32(a__isNatKind(V2))
  , a__U32(X) -> U32(X)
  , a__U32(tt()) -> tt()
  , a__U41(X) -> U41(X)
  , a__U41(tt()) -> tt()
  , a__U51(X1, X2) -> U51(X1, X2)
  , a__U51(tt(), N) -> a__U52(a__isNatKind(N), N)
  , a__U52(X1, X2) -> U52(X1, X2)
  , a__U52(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(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
  , mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3)
  , mark(isNatKind(X)) -> a__isNatKind(X)
  , mark(U13(X1, X2, X3)) -> a__U13(mark(X1), X2, X3)
  , mark(U14(X1, X2, X3)) -> a__U14(mark(X1), X2, X3)
  , mark(U15(X1, X2)) -> a__U15(mark(X1), X2)
  , mark(isNat(X)) -> a__isNat(X)
  , mark(U16(X)) -> a__U16(mark(X))
  , mark(U21(X1, X2)) -> a__U21(mark(X1), X2)
  , mark(U22(X1, X2)) -> a__U22(mark(X1), X2)
  , mark(U23(X)) -> a__U23(mark(X))
  , mark(U31(X1, X2)) -> a__U31(mark(X1), X2)
  , mark(U32(X)) -> a__U32(mark(X))
  , mark(U41(X)) -> a__U41(mark(X))
  , mark(U51(X1, X2)) -> a__U51(mark(X1), X2)
  , mark(U52(X1, X2)) -> a__U52(mark(X1), X2)
  , mark(U61(X1, X2, X3)) -> a__U61(mark(X1), X2, X3)
  , mark(U62(X1, X2, X3)) -> a__U62(mark(X1), X2, X3)
  , mark(U63(X1, X2, X3)) -> a__U63(mark(X1), X2, X3)
  , mark(U64(X1, X2, X3)) -> a__U64(mark(X1), X2, X3)
  , a__U61(X1, X2, X3) -> U61(X1, X2, X3)
  , a__U61(tt(), M, N) -> a__U62(a__isNatKind(M), M, N)
  , a__U62(X1, X2, X3) -> U62(X1, X2, X3)
  , a__U62(tt(), M, N) -> a__U63(a__isNat(N), M, N)
  , a__U63(X1, X2, X3) -> U63(X1, X2, X3)
  , a__U63(tt(), M, N) -> a__U64(a__isNatKind(N), M, N)
  , a__U64(X1, X2, X3) -> U64(X1, X2, X3)
  , a__U64(tt(), M, N) -> s(a__plus(mark(N), mark(M)))
  , a__plus(X1, X2) -> plus(X1, X2)
  , a__plus(N, s(M)) -> a__U61(a__isNat(M), M, N)
  , a__plus(N, 0()) -> a__U51(a__isNat(N), N) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of
{1,3,5,7,9,11,12,13,15,17,19,21,23,24,25,27,29,30,31,32,33,35,37,39,61,64,66,68,70}
by applications of
Pre({1,3,5,7,9,11,12,13,15,17,19,21,23,24,25,27,29,30,31,32,33,35,37,39,61,64,66,68,70})
=
{2,4,6,8,10,14,16,18,20,22,26,28,34,36,38,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,62,63,65,67,69,71}.
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__isNatKind(V1), V1, V2), a__isNatKind^#(V1))
    , 3: a__U12^#(X1, X2, X3) -> c_3()
    , 4: a__U12^#(tt(), V1, V2) ->
         c_4(a__U13^#(a__isNatKind(V2), V1, V2), a__isNatKind^#(V2))
    , 5: a__isNatKind^#(X) -> c_5()
    , 6: a__isNatKind^#(s(V1)) ->
         c_6(a__U41^#(a__isNatKind(V1)), a__isNatKind^#(V1))
    , 7: a__isNatKind^#(0()) -> c_7()
    , 8: a__isNatKind^#(plus(V1, V2)) ->
         c_8(a__U31^#(a__isNatKind(V1), V2), a__isNatKind^#(V1))
    , 9: a__U13^#(X1, X2, X3) -> c_9()
    , 10: a__U13^#(tt(), V1, V2) ->
          c_10(a__U14^#(a__isNatKind(V2), V1, V2), a__isNatKind^#(V2))
    , 11: a__U41^#(X) -> c_31()
    , 12: a__U41^#(tt()) -> c_32()
    , 13: a__U31^#(X1, X2) -> c_27()
    , 14: a__U31^#(tt(), V2) ->
          c_28(a__U32^#(a__isNatKind(V2)), a__isNatKind^#(V2))
    , 15: a__U14^#(X1, X2, X3) -> c_11()
    , 16: a__U14^#(tt(), V1, V2) ->
          c_12(a__U15^#(a__isNat(V1), V2), a__isNat^#(V1))
    , 17: a__U15^#(X1, X2) -> c_13()
    , 18: a__U15^#(tt(), V2) ->
          c_14(a__U16^#(a__isNat(V2)), a__isNat^#(V2))
    , 19: a__isNat^#(X) -> c_15()
    , 20: a__isNat^#(s(V1)) ->
          c_16(a__U21^#(a__isNatKind(V1), V1), a__isNatKind^#(V1))
    , 21: a__isNat^#(0()) -> c_17()
    , 22: a__isNat^#(plus(V1, V2)) ->
          c_18(a__U11^#(a__isNatKind(V1), V1, V2), a__isNatKind^#(V1))
    , 23: a__U16^#(X) -> c_19()
    , 24: a__U16^#(tt()) -> c_20()
    , 25: a__U21^#(X1, X2) -> c_21()
    , 26: a__U21^#(tt(), V1) ->
          c_22(a__U22^#(a__isNatKind(V1), V1), a__isNatKind^#(V1))
    , 27: a__U22^#(X1, X2) -> c_23()
    , 28: a__U22^#(tt(), V1) ->
          c_24(a__U23^#(a__isNat(V1)), a__isNat^#(V1))
    , 29: a__U23^#(X) -> c_25()
    , 30: a__U23^#(tt()) -> c_26()
    , 31: a__U32^#(X) -> c_29()
    , 32: a__U32^#(tt()) -> c_30()
    , 33: a__U51^#(X1, X2) -> c_33()
    , 34: a__U51^#(tt(), N) ->
          c_34(a__U52^#(a__isNatKind(N), N), a__isNatKind^#(N))
    , 35: a__U52^#(X1, X2) -> c_35()
    , 36: a__U52^#(tt(), N) -> c_36(mark^#(N))
    , 37: mark^#(tt()) -> c_37()
    , 38: mark^#(s(X)) -> c_38(mark^#(X))
    , 39: mark^#(0()) -> c_39()
    , 40: mark^#(plus(X1, X2)) ->
          c_40(a__plus^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
    , 41: mark^#(U11(X1, X2, X3)) ->
          c_41(a__U11^#(mark(X1), X2, X3), mark^#(X1))
    , 42: mark^#(U12(X1, X2, X3)) ->
          c_42(a__U12^#(mark(X1), X2, X3), mark^#(X1))
    , 43: mark^#(isNatKind(X)) -> c_43(a__isNatKind^#(X))
    , 44: mark^#(U13(X1, X2, X3)) ->
          c_44(a__U13^#(mark(X1), X2, X3), mark^#(X1))
    , 45: mark^#(U14(X1, X2, X3)) ->
          c_45(a__U14^#(mark(X1), X2, X3), mark^#(X1))
    , 46: mark^#(U15(X1, X2)) ->
          c_46(a__U15^#(mark(X1), X2), mark^#(X1))
    , 47: mark^#(isNat(X)) -> c_47(a__isNat^#(X))
    , 48: mark^#(U16(X)) -> c_48(a__U16^#(mark(X)), mark^#(X))
    , 49: mark^#(U21(X1, X2)) ->
          c_49(a__U21^#(mark(X1), X2), mark^#(X1))
    , 50: mark^#(U22(X1, X2)) ->
          c_50(a__U22^#(mark(X1), X2), mark^#(X1))
    , 51: mark^#(U23(X)) -> c_51(a__U23^#(mark(X)), mark^#(X))
    , 52: mark^#(U31(X1, X2)) ->
          c_52(a__U31^#(mark(X1), X2), mark^#(X1))
    , 53: mark^#(U32(X)) -> c_53(a__U32^#(mark(X)), mark^#(X))
    , 54: mark^#(U41(X)) -> c_54(a__U41^#(mark(X)), mark^#(X))
    , 55: mark^#(U51(X1, X2)) ->
          c_55(a__U51^#(mark(X1), X2), mark^#(X1))
    , 56: mark^#(U52(X1, X2)) ->
          c_56(a__U52^#(mark(X1), X2), mark^#(X1))
    , 57: mark^#(U61(X1, X2, X3)) ->
          c_57(a__U61^#(mark(X1), X2, X3), mark^#(X1))
    , 58: mark^#(U62(X1, X2, X3)) ->
          c_58(a__U62^#(mark(X1), X2, X3), mark^#(X1))
    , 59: mark^#(U63(X1, X2, X3)) ->
          c_59(a__U63^#(mark(X1), X2, X3), mark^#(X1))
    , 60: mark^#(U64(X1, X2, X3)) ->
          c_60(a__U64^#(mark(X1), X2, X3), mark^#(X1))
    , 61: a__plus^#(X1, X2) -> c_69()
    , 62: a__plus^#(N, s(M)) ->
          c_70(a__U61^#(a__isNat(M), M, N), a__isNat^#(M))
    , 63: a__plus^#(N, 0()) ->
          c_71(a__U51^#(a__isNat(N), N), a__isNat^#(N))
    , 64: a__U61^#(X1, X2, X3) -> c_61()
    , 65: a__U61^#(tt(), M, N) ->
          c_62(a__U62^#(a__isNatKind(M), M, N), a__isNatKind^#(M))
    , 66: a__U62^#(X1, X2, X3) -> c_63()
    , 67: a__U62^#(tt(), M, N) ->
          c_64(a__U63^#(a__isNat(N), M, N), a__isNat^#(N))
    , 68: a__U63^#(X1, X2, X3) -> c_65()
    , 69: a__U63^#(tt(), M, N) ->
          c_66(a__U64^#(a__isNatKind(N), M, N), a__isNatKind^#(N))
    , 70: a__U64^#(X1, X2, X3) -> c_67()
    , 71: a__U64^#(tt(), M, N) ->
          c_68(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__isNatKind(V1), V1, V2), a__isNatKind^#(V1))
  , a__U12^#(tt(), V1, V2) ->
    c_4(a__U13^#(a__isNatKind(V2), V1, V2), a__isNatKind^#(V2))
  , a__isNatKind^#(s(V1)) ->
    c_6(a__U41^#(a__isNatKind(V1)), a__isNatKind^#(V1))
  , a__isNatKind^#(plus(V1, V2)) ->
    c_8(a__U31^#(a__isNatKind(V1), V2), a__isNatKind^#(V1))
  , a__U13^#(tt(), V1, V2) ->
    c_10(a__U14^#(a__isNatKind(V2), V1, V2), a__isNatKind^#(V2))
  , a__U31^#(tt(), V2) ->
    c_28(a__U32^#(a__isNatKind(V2)), a__isNatKind^#(V2))
  , a__U14^#(tt(), V1, V2) ->
    c_12(a__U15^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__U15^#(tt(), V2) ->
    c_14(a__U16^#(a__isNat(V2)), a__isNat^#(V2))
  , a__isNat^#(s(V1)) ->
    c_16(a__U21^#(a__isNatKind(V1), V1), a__isNatKind^#(V1))
  , a__isNat^#(plus(V1, V2)) ->
    c_18(a__U11^#(a__isNatKind(V1), V1, V2), a__isNatKind^#(V1))
  , a__U21^#(tt(), V1) ->
    c_22(a__U22^#(a__isNatKind(V1), V1), a__isNatKind^#(V1))
  , a__U22^#(tt(), V1) ->
    c_24(a__U23^#(a__isNat(V1)), a__isNat^#(V1))
  , a__U51^#(tt(), N) ->
    c_34(a__U52^#(a__isNatKind(N), N), a__isNatKind^#(N))
  , a__U52^#(tt(), N) -> c_36(mark^#(N))
  , mark^#(s(X)) -> c_38(mark^#(X))
  , mark^#(plus(X1, X2)) ->
    c_40(a__plus^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(U11(X1, X2, X3)) ->
    c_41(a__U11^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U12(X1, X2, X3)) ->
    c_42(a__U12^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(isNatKind(X)) -> c_43(a__isNatKind^#(X))
  , mark^#(U13(X1, X2, X3)) ->
    c_44(a__U13^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U14(X1, X2, X3)) ->
    c_45(a__U14^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U15(X1, X2)) -> c_46(a__U15^#(mark(X1), X2), mark^#(X1))
  , mark^#(isNat(X)) -> c_47(a__isNat^#(X))
  , mark^#(U16(X)) -> c_48(a__U16^#(mark(X)), mark^#(X))
  , mark^#(U21(X1, X2)) -> c_49(a__U21^#(mark(X1), X2), mark^#(X1))
  , mark^#(U22(X1, X2)) -> c_50(a__U22^#(mark(X1), X2), mark^#(X1))
  , mark^#(U23(X)) -> c_51(a__U23^#(mark(X)), mark^#(X))
  , mark^#(U31(X1, X2)) -> c_52(a__U31^#(mark(X1), X2), mark^#(X1))
  , mark^#(U32(X)) -> c_53(a__U32^#(mark(X)), mark^#(X))
  , mark^#(U41(X)) -> c_54(a__U41^#(mark(X)), mark^#(X))
  , mark^#(U51(X1, X2)) -> c_55(a__U51^#(mark(X1), X2), mark^#(X1))
  , mark^#(U52(X1, X2)) -> c_56(a__U52^#(mark(X1), X2), mark^#(X1))
  , mark^#(U61(X1, X2, X3)) ->
    c_57(a__U61^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U62(X1, X2, X3)) ->
    c_58(a__U62^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U63(X1, X2, X3)) ->
    c_59(a__U63^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U64(X1, X2, X3)) ->
    c_60(a__U64^#(mark(X1), X2, X3), mark^#(X1))
  , a__plus^#(N, s(M)) ->
    c_70(a__U61^#(a__isNat(M), M, N), a__isNat^#(M))
  , a__plus^#(N, 0()) ->
    c_71(a__U51^#(a__isNat(N), N), a__isNat^#(N))
  , a__U61^#(tt(), M, N) ->
    c_62(a__U62^#(a__isNatKind(M), M, N), a__isNatKind^#(M))
  , a__U62^#(tt(), M, N) ->
    c_64(a__U63^#(a__isNat(N), M, N), a__isNat^#(N))
  , a__U63^#(tt(), M, N) ->
    c_66(a__U64^#(a__isNatKind(N), M, N), a__isNatKind^#(N))
  , a__U64^#(tt(), M, N) ->
    c_68(a__plus^#(mark(N), mark(M)), mark^#(N), mark^#(M)) }
Weak DPs:
  { a__U11^#(X1, X2, X3) -> c_1()
  , a__U12^#(X1, X2, X3) -> c_3()
  , a__isNatKind^#(X) -> c_5()
  , a__isNatKind^#(0()) -> c_7()
  , a__U13^#(X1, X2, X3) -> c_9()
  , a__U41^#(X) -> c_31()
  , a__U41^#(tt()) -> c_32()
  , a__U31^#(X1, X2) -> c_27()
  , a__U14^#(X1, X2, X3) -> c_11()
  , a__U15^#(X1, X2) -> c_13()
  , a__isNat^#(X) -> c_15()
  , a__isNat^#(0()) -> c_17()
  , a__U16^#(X) -> c_19()
  , a__U16^#(tt()) -> c_20()
  , a__U21^#(X1, X2) -> c_21()
  , a__U22^#(X1, X2) -> c_23()
  , a__U23^#(X) -> c_25()
  , a__U23^#(tt()) -> c_26()
  , a__U32^#(X) -> c_29()
  , a__U32^#(tt()) -> c_30()
  , a__U51^#(X1, X2) -> c_33()
  , a__U52^#(X1, X2) -> c_35()
  , mark^#(tt()) -> c_37()
  , mark^#(0()) -> c_39()
  , a__plus^#(X1, X2) -> c_69()
  , a__U61^#(X1, X2, X3) -> c_61()
  , a__U62^#(X1, X2, X3) -> c_63()
  , a__U63^#(X1, X2, X3) -> c_65()
  , a__U64^#(X1, X2, X3) -> c_67() }
Weak Trs:
  { a__U11(X1, X2, X3) -> U11(X1, X2, X3)
  , a__U11(tt(), V1, V2) -> a__U12(a__isNatKind(V1), V1, V2)
  , a__U12(X1, X2, X3) -> U12(X1, X2, X3)
  , a__U12(tt(), V1, V2) -> a__U13(a__isNatKind(V2), V1, V2)
  , a__isNatKind(X) -> isNatKind(X)
  , a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1))
  , a__isNatKind(0()) -> tt()
  , a__isNatKind(plus(V1, V2)) -> a__U31(a__isNatKind(V1), V2)
  , a__U13(X1, X2, X3) -> U13(X1, X2, X3)
  , a__U13(tt(), V1, V2) -> a__U14(a__isNatKind(V2), V1, V2)
  , a__U14(X1, X2, X3) -> U14(X1, X2, X3)
  , a__U14(tt(), V1, V2) -> a__U15(a__isNat(V1), V2)
  , a__U15(X1, X2) -> U15(X1, X2)
  , a__U15(tt(), V2) -> a__U16(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__isNatKind(V1), V1, V2)
  , a__U16(X) -> U16(X)
  , a__U16(tt()) -> tt()
  , a__U21(X1, X2) -> U21(X1, X2)
  , a__U21(tt(), V1) -> a__U22(a__isNatKind(V1), V1)
  , a__U22(X1, X2) -> U22(X1, X2)
  , a__U22(tt(), V1) -> a__U23(a__isNat(V1))
  , a__U23(X) -> U23(X)
  , a__U23(tt()) -> tt()
  , a__U31(X1, X2) -> U31(X1, X2)
  , a__U31(tt(), V2) -> a__U32(a__isNatKind(V2))
  , a__U32(X) -> U32(X)
  , a__U32(tt()) -> tt()
  , a__U41(X) -> U41(X)
  , a__U41(tt()) -> tt()
  , a__U51(X1, X2) -> U51(X1, X2)
  , a__U51(tt(), N) -> a__U52(a__isNatKind(N), N)
  , a__U52(X1, X2) -> U52(X1, X2)
  , a__U52(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(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
  , mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3)
  , mark(isNatKind(X)) -> a__isNatKind(X)
  , mark(U13(X1, X2, X3)) -> a__U13(mark(X1), X2, X3)
  , mark(U14(X1, X2, X3)) -> a__U14(mark(X1), X2, X3)
  , mark(U15(X1, X2)) -> a__U15(mark(X1), X2)
  , mark(isNat(X)) -> a__isNat(X)
  , mark(U16(X)) -> a__U16(mark(X))
  , mark(U21(X1, X2)) -> a__U21(mark(X1), X2)
  , mark(U22(X1, X2)) -> a__U22(mark(X1), X2)
  , mark(U23(X)) -> a__U23(mark(X))
  , mark(U31(X1, X2)) -> a__U31(mark(X1), X2)
  , mark(U32(X)) -> a__U32(mark(X))
  , mark(U41(X)) -> a__U41(mark(X))
  , mark(U51(X1, X2)) -> a__U51(mark(X1), X2)
  , mark(U52(X1, X2)) -> a__U52(mark(X1), X2)
  , mark(U61(X1, X2, X3)) -> a__U61(mark(X1), X2, X3)
  , mark(U62(X1, X2, X3)) -> a__U62(mark(X1), X2, X3)
  , mark(U63(X1, X2, X3)) -> a__U63(mark(X1), X2, X3)
  , mark(U64(X1, X2, X3)) -> a__U64(mark(X1), X2, X3)
  , a__U61(X1, X2, X3) -> U61(X1, X2, X3)
  , a__U61(tt(), M, N) -> a__U62(a__isNatKind(M), M, N)
  , a__U62(X1, X2, X3) -> U62(X1, X2, X3)
  , a__U62(tt(), M, N) -> a__U63(a__isNat(N), M, N)
  , a__U63(X1, X2, X3) -> U63(X1, X2, X3)
  , a__U63(tt(), M, N) -> a__U64(a__isNatKind(N), M, N)
  , a__U64(X1, X2, X3) -> U64(X1, X2, X3)
  , a__U64(tt(), M, N) -> s(a__plus(mark(N), mark(M)))
  , a__plus(X1, X2) -> plus(X1, X2)
  , a__plus(N, s(M)) -> a__U61(a__isNat(M), M, N)
  , a__plus(N, 0()) -> a__U51(a__isNat(N), N) }
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, X3) -> c_3()
, a__isNatKind^#(X) -> c_5()
, a__isNatKind^#(0()) -> c_7()
, a__U13^#(X1, X2, X3) -> c_9()
, a__U41^#(X) -> c_31()
, a__U41^#(tt()) -> c_32()
, a__U31^#(X1, X2) -> c_27()
, a__U14^#(X1, X2, X3) -> c_11()
, a__U15^#(X1, X2) -> c_13()
, a__isNat^#(X) -> c_15()
, a__isNat^#(0()) -> c_17()
, a__U16^#(X) -> c_19()
, a__U16^#(tt()) -> c_20()
, a__U21^#(X1, X2) -> c_21()
, a__U22^#(X1, X2) -> c_23()
, a__U23^#(X) -> c_25()
, a__U23^#(tt()) -> c_26()
, a__U32^#(X) -> c_29()
, a__U32^#(tt()) -> c_30()
, a__U51^#(X1, X2) -> c_33()
, a__U52^#(X1, X2) -> c_35()
, mark^#(tt()) -> c_37()
, mark^#(0()) -> c_39()
, a__plus^#(X1, X2) -> c_69()
, a__U61^#(X1, X2, X3) -> c_61()
, a__U62^#(X1, X2, X3) -> c_63()
, a__U63^#(X1, X2, X3) -> c_65()
, a__U64^#(X1, X2, X3) -> c_67() }

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__isNatKind(V1), V1, V2), a__isNatKind^#(V1))
  , a__U12^#(tt(), V1, V2) ->
    c_4(a__U13^#(a__isNatKind(V2), V1, V2), a__isNatKind^#(V2))
  , a__isNatKind^#(s(V1)) ->
    c_6(a__U41^#(a__isNatKind(V1)), a__isNatKind^#(V1))
  , a__isNatKind^#(plus(V1, V2)) ->
    c_8(a__U31^#(a__isNatKind(V1), V2), a__isNatKind^#(V1))
  , a__U13^#(tt(), V1, V2) ->
    c_10(a__U14^#(a__isNatKind(V2), V1, V2), a__isNatKind^#(V2))
  , a__U31^#(tt(), V2) ->
    c_28(a__U32^#(a__isNatKind(V2)), a__isNatKind^#(V2))
  , a__U14^#(tt(), V1, V2) ->
    c_12(a__U15^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__U15^#(tt(), V2) ->
    c_14(a__U16^#(a__isNat(V2)), a__isNat^#(V2))
  , a__isNat^#(s(V1)) ->
    c_16(a__U21^#(a__isNatKind(V1), V1), a__isNatKind^#(V1))
  , a__isNat^#(plus(V1, V2)) ->
    c_18(a__U11^#(a__isNatKind(V1), V1, V2), a__isNatKind^#(V1))
  , a__U21^#(tt(), V1) ->
    c_22(a__U22^#(a__isNatKind(V1), V1), a__isNatKind^#(V1))
  , a__U22^#(tt(), V1) ->
    c_24(a__U23^#(a__isNat(V1)), a__isNat^#(V1))
  , a__U51^#(tt(), N) ->
    c_34(a__U52^#(a__isNatKind(N), N), a__isNatKind^#(N))
  , a__U52^#(tt(), N) -> c_36(mark^#(N))
  , mark^#(s(X)) -> c_38(mark^#(X))
  , mark^#(plus(X1, X2)) ->
    c_40(a__plus^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(U11(X1, X2, X3)) ->
    c_41(a__U11^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U12(X1, X2, X3)) ->
    c_42(a__U12^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(isNatKind(X)) -> c_43(a__isNatKind^#(X))
  , mark^#(U13(X1, X2, X3)) ->
    c_44(a__U13^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U14(X1, X2, X3)) ->
    c_45(a__U14^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U15(X1, X2)) -> c_46(a__U15^#(mark(X1), X2), mark^#(X1))
  , mark^#(isNat(X)) -> c_47(a__isNat^#(X))
  , mark^#(U16(X)) -> c_48(a__U16^#(mark(X)), mark^#(X))
  , mark^#(U21(X1, X2)) -> c_49(a__U21^#(mark(X1), X2), mark^#(X1))
  , mark^#(U22(X1, X2)) -> c_50(a__U22^#(mark(X1), X2), mark^#(X1))
  , mark^#(U23(X)) -> c_51(a__U23^#(mark(X)), mark^#(X))
  , mark^#(U31(X1, X2)) -> c_52(a__U31^#(mark(X1), X2), mark^#(X1))
  , mark^#(U32(X)) -> c_53(a__U32^#(mark(X)), mark^#(X))
  , mark^#(U41(X)) -> c_54(a__U41^#(mark(X)), mark^#(X))
  , mark^#(U51(X1, X2)) -> c_55(a__U51^#(mark(X1), X2), mark^#(X1))
  , mark^#(U52(X1, X2)) -> c_56(a__U52^#(mark(X1), X2), mark^#(X1))
  , mark^#(U61(X1, X2, X3)) ->
    c_57(a__U61^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U62(X1, X2, X3)) ->
    c_58(a__U62^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U63(X1, X2, X3)) ->
    c_59(a__U63^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U64(X1, X2, X3)) ->
    c_60(a__U64^#(mark(X1), X2, X3), mark^#(X1))
  , a__plus^#(N, s(M)) ->
    c_70(a__U61^#(a__isNat(M), M, N), a__isNat^#(M))
  , a__plus^#(N, 0()) ->
    c_71(a__U51^#(a__isNat(N), N), a__isNat^#(N))
  , a__U61^#(tt(), M, N) ->
    c_62(a__U62^#(a__isNatKind(M), M, N), a__isNatKind^#(M))
  , a__U62^#(tt(), M, N) ->
    c_64(a__U63^#(a__isNat(N), M, N), a__isNat^#(N))
  , a__U63^#(tt(), M, N) ->
    c_66(a__U64^#(a__isNatKind(N), M, N), a__isNatKind^#(N))
  , a__U64^#(tt(), M, N) ->
    c_68(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__isNatKind(V1), V1, V2)
  , a__U12(X1, X2, X3) -> U12(X1, X2, X3)
  , a__U12(tt(), V1, V2) -> a__U13(a__isNatKind(V2), V1, V2)
  , a__isNatKind(X) -> isNatKind(X)
  , a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1))
  , a__isNatKind(0()) -> tt()
  , a__isNatKind(plus(V1, V2)) -> a__U31(a__isNatKind(V1), V2)
  , a__U13(X1, X2, X3) -> U13(X1, X2, X3)
  , a__U13(tt(), V1, V2) -> a__U14(a__isNatKind(V2), V1, V2)
  , a__U14(X1, X2, X3) -> U14(X1, X2, X3)
  , a__U14(tt(), V1, V2) -> a__U15(a__isNat(V1), V2)
  , a__U15(X1, X2) -> U15(X1, X2)
  , a__U15(tt(), V2) -> a__U16(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__isNatKind(V1), V1, V2)
  , a__U16(X) -> U16(X)
  , a__U16(tt()) -> tt()
  , a__U21(X1, X2) -> U21(X1, X2)
  , a__U21(tt(), V1) -> a__U22(a__isNatKind(V1), V1)
  , a__U22(X1, X2) -> U22(X1, X2)
  , a__U22(tt(), V1) -> a__U23(a__isNat(V1))
  , a__U23(X) -> U23(X)
  , a__U23(tt()) -> tt()
  , a__U31(X1, X2) -> U31(X1, X2)
  , a__U31(tt(), V2) -> a__U32(a__isNatKind(V2))
  , a__U32(X) -> U32(X)
  , a__U32(tt()) -> tt()
  , a__U41(X) -> U41(X)
  , a__U41(tt()) -> tt()
  , a__U51(X1, X2) -> U51(X1, X2)
  , a__U51(tt(), N) -> a__U52(a__isNatKind(N), N)
  , a__U52(X1, X2) -> U52(X1, X2)
  , a__U52(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(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
  , mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3)
  , mark(isNatKind(X)) -> a__isNatKind(X)
  , mark(U13(X1, X2, X3)) -> a__U13(mark(X1), X2, X3)
  , mark(U14(X1, X2, X3)) -> a__U14(mark(X1), X2, X3)
  , mark(U15(X1, X2)) -> a__U15(mark(X1), X2)
  , mark(isNat(X)) -> a__isNat(X)
  , mark(U16(X)) -> a__U16(mark(X))
  , mark(U21(X1, X2)) -> a__U21(mark(X1), X2)
  , mark(U22(X1, X2)) -> a__U22(mark(X1), X2)
  , mark(U23(X)) -> a__U23(mark(X))
  , mark(U31(X1, X2)) -> a__U31(mark(X1), X2)
  , mark(U32(X)) -> a__U32(mark(X))
  , mark(U41(X)) -> a__U41(mark(X))
  , mark(U51(X1, X2)) -> a__U51(mark(X1), X2)
  , mark(U52(X1, X2)) -> a__U52(mark(X1), X2)
  , mark(U61(X1, X2, X3)) -> a__U61(mark(X1), X2, X3)
  , mark(U62(X1, X2, X3)) -> a__U62(mark(X1), X2, X3)
  , mark(U63(X1, X2, X3)) -> a__U63(mark(X1), X2, X3)
  , mark(U64(X1, X2, X3)) -> a__U64(mark(X1), X2, X3)
  , a__U61(X1, X2, X3) -> U61(X1, X2, X3)
  , a__U61(tt(), M, N) -> a__U62(a__isNatKind(M), M, N)
  , a__U62(X1, X2, X3) -> U62(X1, X2, X3)
  , a__U62(tt(), M, N) -> a__U63(a__isNat(N), M, N)
  , a__U63(X1, X2, X3) -> U63(X1, X2, X3)
  , a__U63(tt(), M, N) -> a__U64(a__isNatKind(N), M, N)
  , a__U64(X1, X2, X3) -> U64(X1, X2, X3)
  , a__U64(tt(), M, N) -> s(a__plus(mark(N), mark(M)))
  , a__plus(X1, X2) -> plus(X1, X2)
  , a__plus(N, s(M)) -> a__U61(a__isNat(M), M, N)
  , a__plus(N, 0()) -> a__U51(a__isNat(N), N) }
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__isNatKind^#(s(V1)) ->
    c_6(a__U41^#(a__isNatKind(V1)), a__isNatKind^#(V1))
  , a__U31^#(tt(), V2) ->
    c_28(a__U32^#(a__isNatKind(V2)), a__isNatKind^#(V2))
  , a__U15^#(tt(), V2) ->
    c_14(a__U16^#(a__isNat(V2)), a__isNat^#(V2))
  , a__U22^#(tt(), V1) ->
    c_24(a__U23^#(a__isNat(V1)), a__isNat^#(V1))
  , mark^#(U16(X)) -> c_48(a__U16^#(mark(X)), mark^#(X))
  , mark^#(U23(X)) -> c_51(a__U23^#(mark(X)), mark^#(X))
  , mark^#(U32(X)) -> c_53(a__U32^#(mark(X)), mark^#(X))
  , mark^#(U41(X)) -> c_54(a__U41^#(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__isNatKind(V1), V1, V2), a__isNatKind^#(V1))
  , a__U12^#(tt(), V1, V2) ->
    c_2(a__U13^#(a__isNatKind(V2), V1, V2), a__isNatKind^#(V2))
  , a__isNatKind^#(s(V1)) -> c_3(a__isNatKind^#(V1))
  , a__isNatKind^#(plus(V1, V2)) ->
    c_4(a__U31^#(a__isNatKind(V1), V2), a__isNatKind^#(V1))
  , a__U13^#(tt(), V1, V2) ->
    c_5(a__U14^#(a__isNatKind(V2), V1, V2), a__isNatKind^#(V2))
  , a__U31^#(tt(), V2) -> c_6(a__isNatKind^#(V2))
  , a__U14^#(tt(), V1, V2) ->
    c_7(a__U15^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__U15^#(tt(), V2) -> c_8(a__isNat^#(V2))
  , a__isNat^#(s(V1)) ->
    c_9(a__U21^#(a__isNatKind(V1), V1), a__isNatKind^#(V1))
  , a__isNat^#(plus(V1, V2)) ->
    c_10(a__U11^#(a__isNatKind(V1), V1, V2), a__isNatKind^#(V1))
  , a__U21^#(tt(), V1) ->
    c_11(a__U22^#(a__isNatKind(V1), V1), a__isNatKind^#(V1))
  , a__U22^#(tt(), V1) -> c_12(a__isNat^#(V1))
  , a__U51^#(tt(), N) ->
    c_13(a__U52^#(a__isNatKind(N), N), a__isNatKind^#(N))
  , a__U52^#(tt(), N) -> c_14(mark^#(N))
  , mark^#(s(X)) -> c_15(mark^#(X))
  , mark^#(plus(X1, X2)) ->
    c_16(a__plus^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(U11(X1, X2, X3)) ->
    c_17(a__U11^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U12(X1, X2, X3)) ->
    c_18(a__U12^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(isNatKind(X)) -> c_19(a__isNatKind^#(X))
  , mark^#(U13(X1, X2, X3)) ->
    c_20(a__U13^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U14(X1, X2, X3)) ->
    c_21(a__U14^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U15(X1, X2)) -> c_22(a__U15^#(mark(X1), X2), mark^#(X1))
  , mark^#(isNat(X)) -> c_23(a__isNat^#(X))
  , mark^#(U16(X)) -> c_24(mark^#(X))
  , mark^#(U21(X1, X2)) -> c_25(a__U21^#(mark(X1), X2), mark^#(X1))
  , mark^#(U22(X1, X2)) -> c_26(a__U22^#(mark(X1), X2), mark^#(X1))
  , mark^#(U23(X)) -> c_27(mark^#(X))
  , mark^#(U31(X1, X2)) -> c_28(a__U31^#(mark(X1), X2), mark^#(X1))
  , mark^#(U32(X)) -> c_29(mark^#(X))
  , mark^#(U41(X)) -> c_30(mark^#(X))
  , mark^#(U51(X1, X2)) -> c_31(a__U51^#(mark(X1), X2), mark^#(X1))
  , mark^#(U52(X1, X2)) -> c_32(a__U52^#(mark(X1), X2), mark^#(X1))
  , mark^#(U61(X1, X2, X3)) ->
    c_33(a__U61^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U62(X1, X2, X3)) ->
    c_34(a__U62^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U63(X1, X2, X3)) ->
    c_35(a__U63^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U64(X1, X2, X3)) ->
    c_36(a__U64^#(mark(X1), X2, X3), mark^#(X1))
  , a__plus^#(N, s(M)) ->
    c_37(a__U61^#(a__isNat(M), M, N), a__isNat^#(M))
  , a__plus^#(N, 0()) ->
    c_38(a__U51^#(a__isNat(N), N), a__isNat^#(N))
  , a__U61^#(tt(), M, N) ->
    c_39(a__U62^#(a__isNatKind(M), M, N), a__isNatKind^#(M))
  , a__U62^#(tt(), M, N) ->
    c_40(a__U63^#(a__isNat(N), M, N), a__isNat^#(N))
  , a__U63^#(tt(), M, N) ->
    c_41(a__U64^#(a__isNatKind(N), M, N), a__isNatKind^#(N))
  , a__U64^#(tt(), M, N) ->
    c_42(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__isNatKind(V1), V1, V2)
  , a__U12(X1, X2, X3) -> U12(X1, X2, X3)
  , a__U12(tt(), V1, V2) -> a__U13(a__isNatKind(V2), V1, V2)
  , a__isNatKind(X) -> isNatKind(X)
  , a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1))
  , a__isNatKind(0()) -> tt()
  , a__isNatKind(plus(V1, V2)) -> a__U31(a__isNatKind(V1), V2)
  , a__U13(X1, X2, X3) -> U13(X1, X2, X3)
  , a__U13(tt(), V1, V2) -> a__U14(a__isNatKind(V2), V1, V2)
  , a__U14(X1, X2, X3) -> U14(X1, X2, X3)
  , a__U14(tt(), V1, V2) -> a__U15(a__isNat(V1), V2)
  , a__U15(X1, X2) -> U15(X1, X2)
  , a__U15(tt(), V2) -> a__U16(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__isNatKind(V1), V1, V2)
  , a__U16(X) -> U16(X)
  , a__U16(tt()) -> tt()
  , a__U21(X1, X2) -> U21(X1, X2)
  , a__U21(tt(), V1) -> a__U22(a__isNatKind(V1), V1)
  , a__U22(X1, X2) -> U22(X1, X2)
  , a__U22(tt(), V1) -> a__U23(a__isNat(V1))
  , a__U23(X) -> U23(X)
  , a__U23(tt()) -> tt()
  , a__U31(X1, X2) -> U31(X1, X2)
  , a__U31(tt(), V2) -> a__U32(a__isNatKind(V2))
  , a__U32(X) -> U32(X)
  , a__U32(tt()) -> tt()
  , a__U41(X) -> U41(X)
  , a__U41(tt()) -> tt()
  , a__U51(X1, X2) -> U51(X1, X2)
  , a__U51(tt(), N) -> a__U52(a__isNatKind(N), N)
  , a__U52(X1, X2) -> U52(X1, X2)
  , a__U52(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(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
  , mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3)
  , mark(isNatKind(X)) -> a__isNatKind(X)
  , mark(U13(X1, X2, X3)) -> a__U13(mark(X1), X2, X3)
  , mark(U14(X1, X2, X3)) -> a__U14(mark(X1), X2, X3)
  , mark(U15(X1, X2)) -> a__U15(mark(X1), X2)
  , mark(isNat(X)) -> a__isNat(X)
  , mark(U16(X)) -> a__U16(mark(X))
  , mark(U21(X1, X2)) -> a__U21(mark(X1), X2)
  , mark(U22(X1, X2)) -> a__U22(mark(X1), X2)
  , mark(U23(X)) -> a__U23(mark(X))
  , mark(U31(X1, X2)) -> a__U31(mark(X1), X2)
  , mark(U32(X)) -> a__U32(mark(X))
  , mark(U41(X)) -> a__U41(mark(X))
  , mark(U51(X1, X2)) -> a__U51(mark(X1), X2)
  , mark(U52(X1, X2)) -> a__U52(mark(X1), X2)
  , mark(U61(X1, X2, X3)) -> a__U61(mark(X1), X2, X3)
  , mark(U62(X1, X2, X3)) -> a__U62(mark(X1), X2, X3)
  , mark(U63(X1, X2, X3)) -> a__U63(mark(X1), X2, X3)
  , mark(U64(X1, X2, X3)) -> a__U64(mark(X1), X2, X3)
  , a__U61(X1, X2, X3) -> U61(X1, X2, X3)
  , a__U61(tt(), M, N) -> a__U62(a__isNatKind(M), M, N)
  , a__U62(X1, X2, X3) -> U62(X1, X2, X3)
  , a__U62(tt(), M, N) -> a__U63(a__isNat(N), M, N)
  , a__U63(X1, X2, X3) -> U63(X1, X2, X3)
  , a__U63(tt(), M, N) -> a__U64(a__isNatKind(N), M, N)
  , a__U64(X1, X2, X3) -> U64(X1, X2, X3)
  , a__U64(tt(), M, N) -> s(a__plus(mark(N), mark(M)))
  , a__plus(X1, X2) -> plus(X1, X2)
  , a__plus(N, s(M)) -> a__U61(a__isNat(M), M, N)
  , a__plus(N, 0()) -> a__U51(a__isNat(N), N) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..