Termination w.r.t. Q proof of /tmp/tmpV0wGnr/MYNAT_complete_noand

Termination w.r.t. Q of the given QTRS could not be shown:

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 DependencyGraphProof (⇔)

↳4 AND

  ↳5 QDP

    ↳6 QDPOrderProof (⇔)

    ↳7 QDP

    ↳8 PisEmptyProof (⇔)

    ↳9 TRUE

  ↳10 QDP

    ↳11 QDPOrderProof (⇔)

    ↳12 QDP

    ↳13 PisEmptyProof (⇔)

    ↳14 TRUE

  ↳15 QDP

(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

a__U101(tt, M, N) → a__U102(a__isNatKind(M), M, N)
a__U102(tt, M, N) → a__U103(a__isNat(N), M, N)
a__U103(tt, M, N) → a__U104(a__isNatKind(N), M, N)
a__U104(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__U11(tt, V1, V2) → a__U12(a__isNatKind(V1), V1, V2)
a__U12(tt, V1, V2) → a__U13(a__isNatKind(V2), V1, V2)
a__U13(tt, V1, V2) → a__U14(a__isNatKind(V2), V1, V2)
a__U14(tt, V1, V2) → a__U15(a__isNat(V1), V2)
a__U15(tt, V2) → a__U16(a__isNat(V2))
a__U16(tt) → tt
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNatKind(V1), V1, V2)
a__U32(tt, V1, V2) → a__U33(a__isNatKind(V2), V1, V2)
a__U33(tt, V1, V2) → a__U34(a__isNatKind(V2), V1, V2)
a__U34(tt, V1, V2) → a__U35(a__isNat(V1), V2)
a__U35(tt, V2) → a__U36(a__isNat(V2))
a__U36(tt) → tt
a__U41(tt, V2) → a__U42(a__isNatKind(V2))
a__U42(tt) → tt
a__U51(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatKind(V2))
a__U62(tt) → tt
a__U71(tt, N) → a__U72(a__isNatKind(N), N)
a__U72(tt, N) → mark(N)
a__U81(tt, M, N) → a__U82(a__isNatKind(M), M, N)
a__U82(tt, M, N) → a__U83(a__isNat(N), M, N)
a__U83(tt, M, N) → a__U84(a__isNatKind(N), M, N)
a__U84(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U91(tt, N) → a__U92(a__isNatKind(N))
a__U92(tt) → 0
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNatKind(V1), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__isNatKind(V1), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U41(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U51(a__isNatKind(V1))
a__isNatKind(x(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__plus(N, 0) → a__U71(a__isNat(N), N)
a__plus(N, s(M)) → a__U81(a__isNat(M), M, N)
a__x(N, 0) → a__U91(a__isNat(N), N)
a__x(N, s(M)) → a__U101(a__isNat(M), M, N)
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(isNat(X)) → a__isNat(X)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(x(X1, X2)) → a__x(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(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(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, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2, X3)) → a__U32(mark(X1), X2, X3)
mark(U33(X1, X2, X3)) → a__U33(mark(X1), X2, X3)
mark(U34(X1, X2, X3)) → a__U34(mark(X1), X2, X3)
mark(U35(X1, X2)) → a__U35(mark(X1), X2)
mark(U36(X)) → a__U36(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(U51(X)) → a__U51(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(U72(X1, X2)) → a__U72(mark(X1), X2)
mark(U81(X1, X2, X3)) → a__U81(mark(X1), X2, X3)
mark(U82(X1, X2, X3)) → a__U82(mark(X1), X2, X3)
mark(U83(X1, X2, X3)) → a__U83(mark(X1), X2, X3)
mark(U84(X1, X2, X3)) → a__U84(mark(X1), X2, X3)
mark(U91(X1, X2)) → a__U91(mark(X1), X2)
mark(U92(X)) → a__U92(mark(X))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U101(X1, X2, X3) → U101(X1, X2, X3)
a__U102(X1, X2, X3) → U102(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U103(X1, X2, X3) → U103(X1, X2, X3)
a__isNat(X) → isNat(X)
a__U104(X1, X2, X3) → U104(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__x(X1, X2) → x(X1, X2)
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__U13(X1, X2, X3) → U13(X1, X2, X3)
a__U14(X1, X2, X3) → U14(X1, X2, X3)
a__U15(X1, X2) → U15(X1, X2)
a__U16(X) → U16(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2, X3) → U32(X1, X2, X3)
a__U33(X1, X2, X3) → U33(X1, X2, X3)
a__U34(X1, X2, X3) → U34(X1, X2, X3)
a__U35(X1, X2) → U35(X1, X2)
a__U36(X) → U36(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__U51(X) → U51(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X1, X2) → U72(X1, X2)
a__U81(X1, X2, X3) → U81(X1, X2, X3)
a__U82(X1, X2, X3) → U82(X1, X2, X3)
a__U83(X1, X2, X3) → U83(X1, X2, X3)
a__U84(X1, X2, X3) → U84(X1, X2, X3)
a__U91(X1, X2) → U91(X1, X2)
a__U92(X) → U92(X)

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A__U101(tt, M, N) → A__U102(a__isNatKind(M), M, N)
A__U101(tt, M, N) → A__ISNATKIND(M)
A__U102(tt, M, N) → A__U103(a__isNat(N), M, N)
A__U102(tt, M, N) → A__ISNAT(N)
A__U103(tt, M, N) → A__U104(a__isNatKind(N), M, N)
A__U103(tt, M, N) → A__ISNATKIND(N)
A__U104(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
A__U104(tt, M, N) → A__X(mark(N), mark(M))
A__U104(tt, M, N) → MARK(N)
A__U104(tt, M, N) → MARK(M)
A__U11(tt, V1, V2) → A__U12(a__isNatKind(V1), V1, V2)
A__U11(tt, V1, V2) → A__ISNATKIND(V1)
A__U12(tt, V1, V2) → A__U13(a__isNatKind(V2), V1, V2)
A__U12(tt, V1, V2) → A__ISNATKIND(V2)
A__U13(tt, V1, V2) → A__U14(a__isNatKind(V2), V1, V2)
A__U13(tt, V1, V2) → A__ISNATKIND(V2)
A__U14(tt, V1, V2) → A__U15(a__isNat(V1), V2)
A__U14(tt, V1, V2) → A__ISNAT(V1)
A__U15(tt, V2) → A__U16(a__isNat(V2))
A__U15(tt, V2) → A__ISNAT(V2)
A__U21(tt, V1) → A__U22(a__isNatKind(V1), V1)
A__U21(tt, V1) → A__ISNATKIND(V1)
A__U22(tt, V1) → A__U23(a__isNat(V1))
A__U22(tt, V1) → A__ISNAT(V1)
A__U31(tt, V1, V2) → A__U32(a__isNatKind(V1), V1, V2)
A__U31(tt, V1, V2) → A__ISNATKIND(V1)
A__U32(tt, V1, V2) → A__U33(a__isNatKind(V2), V1, V2)
A__U32(tt, V1, V2) → A__ISNATKIND(V2)
A__U33(tt, V1, V2) → A__U34(a__isNatKind(V2), V1, V2)
A__U33(tt, V1, V2) → A__ISNATKIND(V2)
A__U34(tt, V1, V2) → A__U35(a__isNat(V1), V2)
A__U34(tt, V1, V2) → A__ISNAT(V1)
A__U35(tt, V2) → A__U36(a__isNat(V2))
A__U35(tt, V2) → A__ISNAT(V2)
A__U41(tt, V2) → A__U42(a__isNatKind(V2))
A__U41(tt, V2) → A__ISNATKIND(V2)
A__U61(tt, V2) → A__U62(a__isNatKind(V2))
A__U61(tt, V2) → A__ISNATKIND(V2)
A__U71(tt, N) → A__U72(a__isNatKind(N), N)
A__U71(tt, N) → A__ISNATKIND(N)
A__U72(tt, N) → MARK(N)
A__U81(tt, M, N) → A__U82(a__isNatKind(M), M, N)
A__U81(tt, M, N) → A__ISNATKIND(M)
A__U82(tt, M, N) → A__U83(a__isNat(N), M, N)
A__U82(tt, M, N) → A__ISNAT(N)
A__U83(tt, M, N) → A__U84(a__isNatKind(N), M, N)
A__U83(tt, M, N) → A__ISNATKIND(N)
A__U84(tt, M, N) → A__PLUS(mark(N), mark(M))
A__U84(tt, M, N) → MARK(N)
A__U84(tt, M, N) → MARK(M)
A__U91(tt, N) → A__U92(a__isNatKind(N))
A__U91(tt, N) → A__ISNATKIND(N)
A__ISNAT(plus(V1, V2)) → A__U11(a__isNatKind(V1), V1, V2)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
A__ISNAT(x(V1, V2)) → A__U31(a__isNatKind(V1), V1, V2)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(plus(V1, V2)) → A__U41(a__isNatKind(V1), V2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(s(V1)) → A__U51(a__isNatKind(V1))
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__ISNATKIND(x(V1, V2)) → A__U61(a__isNatKind(V1), V2)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__PLUS(N, 0) → A__U71(a__isNat(N), N)
A__PLUS(N, 0) → A__ISNAT(N)
A__PLUS(N, s(M)) → A__U81(a__isNat(M), M, N)
A__PLUS(N, s(M)) → A__ISNAT(M)
A__X(N, 0) → A__U91(a__isNat(N), N)
A__X(N, 0) → A__ISNAT(N)
A__X(N, s(M)) → A__U101(a__isNat(M), M, N)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U101(X1, X2, X3)) → A__U101(mark(X1), X2, X3)
MARK(U101(X1, X2, X3)) → MARK(X1)
MARK(U102(X1, X2, X3)) → A__U102(mark(X1), X2, X3)
MARK(U102(X1, X2, X3)) → MARK(X1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(U103(X1, X2, X3)) → A__U103(mark(X1), X2, X3)
MARK(U103(X1, X2, X3)) → MARK(X1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U104(X1, X2, X3)) → A__U104(mark(X1), X2, X3)
MARK(U104(X1, X2, X3)) → MARK(X1)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
MARK(x(X1, X2)) → MARK(X1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U11(X1, X2, X3)) → A__U11(mark(X1), X2, X3)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2, X3)) → A__U12(mark(X1), X2, X3)
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U13(X1, X2, X3)) → A__U13(mark(X1), X2, X3)
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → A__U14(mark(X1), X2, X3)
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U15(X1, X2)) → A__U15(mark(X1), X2)
MARK(U15(X1, X2)) → MARK(X1)
MARK(U16(X)) → A__U16(mark(X))
MARK(U16(X)) → MARK(X)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → A__U22(mark(X1), X2)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U23(X)) → A__U23(mark(X))
MARK(U23(X)) → MARK(X)
MARK(U31(X1, X2, X3)) → A__U31(mark(X1), X2, X3)
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(U32(X1, X2, X3)) → A__U32(mark(X1), X2, X3)
MARK(U32(X1, X2, X3)) → MARK(X1)
MARK(U33(X1, X2, X3)) → A__U33(mark(X1), X2, X3)
MARK(U33(X1, X2, X3)) → MARK(X1)
MARK(U34(X1, X2, X3)) → A__U34(mark(X1), X2, X3)
MARK(U34(X1, X2, X3)) → MARK(X1)
MARK(U35(X1, X2)) → A__U35(mark(X1), X2)
MARK(U35(X1, X2)) → MARK(X1)
MARK(U36(X)) → A__U36(mark(X))
MARK(U36(X)) → MARK(X)
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → A__U42(mark(X))
MARK(U42(X)) → MARK(X)
MARK(U51(X)) → A__U51(mark(X))
MARK(U51(X)) → MARK(X)
MARK(U61(X1, X2)) → A__U61(mark(X1), X2)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U62(X)) → A__U62(mark(X))
MARK(U62(X)) → MARK(X)
MARK(U71(X1, X2)) → A__U71(mark(X1), X2)
MARK(U71(X1, X2)) → MARK(X1)
MARK(U72(X1, X2)) → A__U72(mark(X1), X2)
MARK(U72(X1, X2)) → MARK(X1)
MARK(U81(X1, X2, X3)) → A__U81(mark(X1), X2, X3)
MARK(U81(X1, X2, X3)) → MARK(X1)
MARK(U82(X1, X2, X3)) → A__U82(mark(X1), X2, X3)
MARK(U82(X1, X2, X3)) → MARK(X1)
MARK(U83(X1, X2, X3)) → A__U83(mark(X1), X2, X3)
MARK(U83(X1, X2, X3)) → MARK(X1)
MARK(U84(X1, X2, X3)) → A__U84(mark(X1), X2, X3)
MARK(U84(X1, X2, X3)) → MARK(X1)
MARK(U91(X1, X2)) → A__U91(mark(X1), X2)
MARK(U91(X1, X2)) → MARK(X1)
MARK(U92(X)) → A__U92(mark(X))
MARK(U92(X)) → MARK(X)
MARK(s(X)) → MARK(X)

The TRS R consists of the following rules:

a__U101(tt, M, N) → a__U102(a__isNatKind(M), M, N)
a__U102(tt, M, N) → a__U103(a__isNat(N), M, N)
a__U103(tt, M, N) → a__U104(a__isNatKind(N), M, N)
a__U104(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__U11(tt, V1, V2) → a__U12(a__isNatKind(V1), V1, V2)
a__U12(tt, V1, V2) → a__U13(a__isNatKind(V2), V1, V2)
a__U13(tt, V1, V2) → a__U14(a__isNatKind(V2), V1, V2)
a__U14(tt, V1, V2) → a__U15(a__isNat(V1), V2)
a__U15(tt, V2) → a__U16(a__isNat(V2))
a__U16(tt) → tt
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNatKind(V1), V1, V2)
a__U32(tt, V1, V2) → a__U33(a__isNatKind(V2), V1, V2)
a__U33(tt, V1, V2) → a__U34(a__isNatKind(V2), V1, V2)
a__U34(tt, V1, V2) → a__U35(a__isNat(V1), V2)
a__U35(tt, V2) → a__U36(a__isNat(V2))
a__U36(tt) → tt
a__U41(tt, V2) → a__U42(a__isNatKind(V2))
a__U42(tt) → tt
a__U51(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatKind(V2))
a__U62(tt) → tt
a__U71(tt, N) → a__U72(a__isNatKind(N), N)
a__U72(tt, N) → mark(N)
a__U81(tt, M, N) → a__U82(a__isNatKind(M), M, N)
a__U82(tt, M, N) → a__U83(a__isNat(N), M, N)
a__U83(tt, M, N) → a__U84(a__isNatKind(N), M, N)
a__U84(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U91(tt, N) → a__U92(a__isNatKind(N))
a__U92(tt) → 0
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNatKind(V1), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__isNatKind(V1), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U41(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U51(a__isNatKind(V1))
a__isNatKind(x(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__plus(N, 0) → a__U71(a__isNat(N), N)
a__plus(N, s(M)) → a__U81(a__isNat(M), M, N)
a__x(N, 0) → a__U91(a__isNat(N), N)
a__x(N, s(M)) → a__U101(a__isNat(M), M, N)
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(isNat(X)) → a__isNat(X)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(x(X1, X2)) → a__x(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(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(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, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2, X3)) → a__U32(mark(X1), X2, X3)
mark(U33(X1, X2, X3)) → a__U33(mark(X1), X2, X3)
mark(U34(X1, X2, X3)) → a__U34(mark(X1), X2, X3)
mark(U35(X1, X2)) → a__U35(mark(X1), X2)
mark(U36(X)) → a__U36(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(U51(X)) → a__U51(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(U72(X1, X2)) → a__U72(mark(X1), X2)
mark(U81(X1, X2, X3)) → a__U81(mark(X1), X2, X3)
mark(U82(X1, X2, X3)) → a__U82(mark(X1), X2, X3)
mark(U83(X1, X2, X3)) → a__U83(mark(X1), X2, X3)
mark(U84(X1, X2, X3)) → a__U84(mark(X1), X2, X3)
mark(U91(X1, X2)) → a__U91(mark(X1), X2)
mark(U92(X)) → a__U92(mark(X))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U101(X1, X2, X3) → U101(X1, X2, X3)
a__U102(X1, X2, X3) → U102(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U103(X1, X2, X3) → U103(X1, X2, X3)
a__isNat(X) → isNat(X)
a__U104(X1, X2, X3) → U104(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__x(X1, X2) → x(X1, X2)
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__U13(X1, X2, X3) → U13(X1, X2, X3)
a__U14(X1, X2, X3) → U14(X1, X2, X3)
a__U15(X1, X2) → U15(X1, X2)
a__U16(X) → U16(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2, X3) → U32(X1, X2, X3)
a__U33(X1, X2, X3) → U33(X1, X2, X3)
a__U34(X1, X2, X3) → U34(X1, X2, X3)
a__U35(X1, X2) → U35(X1, X2)
a__U36(X) → U36(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__U51(X) → U51(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X1, X2) → U72(X1, X2)
a__U81(X1, X2, X3) → U81(X1, X2, X3)
a__U82(X1, X2, X3) → U82(X1, X2, X3)
a__U83(X1, X2, X3) → U83(X1, X2, X3)
a__U84(X1, X2, X3) → U84(X1, X2, X3)
a__U91(X1, X2) → U91(X1, X2)
a__U92(X) → U92(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 54 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A__U41(tt, V2) → A__ISNATKIND(V2)
A__ISNATKIND(plus(V1, V2)) → A__U41(a__isNatKind(V1), V2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__ISNATKIND(x(V1, V2)) → A__U61(a__isNatKind(V1), V2)
A__U61(tt, V2) → A__ISNATKIND(V2)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)

The TRS R consists of the following rules:

a__U101(tt, M, N) → a__U102(a__isNatKind(M), M, N)
a__U102(tt, M, N) → a__U103(a__isNat(N), M, N)
a__U103(tt, M, N) → a__U104(a__isNatKind(N), M, N)
a__U104(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__U11(tt, V1, V2) → a__U12(a__isNatKind(V1), V1, V2)
a__U12(tt, V1, V2) → a__U13(a__isNatKind(V2), V1, V2)
a__U13(tt, V1, V2) → a__U14(a__isNatKind(V2), V1, V2)
a__U14(tt, V1, V2) → a__U15(a__isNat(V1), V2)
a__U15(tt, V2) → a__U16(a__isNat(V2))
a__U16(tt) → tt
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNatKind(V1), V1, V2)
a__U32(tt, V1, V2) → a__U33(a__isNatKind(V2), V1, V2)
a__U33(tt, V1, V2) → a__U34(a__isNatKind(V2), V1, V2)
a__U34(tt, V1, V2) → a__U35(a__isNat(V1), V2)
a__U35(tt, V2) → a__U36(a__isNat(V2))
a__U36(tt) → tt
a__U41(tt, V2) → a__U42(a__isNatKind(V2))
a__U42(tt) → tt
a__U51(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatKind(V2))
a__U62(tt) → tt
a__U71(tt, N) → a__U72(a__isNatKind(N), N)
a__U72(tt, N) → mark(N)
a__U81(tt, M, N) → a__U82(a__isNatKind(M), M, N)
a__U82(tt, M, N) → a__U83(a__isNat(N), M, N)
a__U83(tt, M, N) → a__U84(a__isNatKind(N), M, N)
a__U84(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U91(tt, N) → a__U92(a__isNatKind(N))
a__U92(tt) → 0
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNatKind(V1), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__isNatKind(V1), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U41(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U51(a__isNatKind(V1))
a__isNatKind(x(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__plus(N, 0) → a__U71(a__isNat(N), N)
a__plus(N, s(M)) → a__U81(a__isNat(M), M, N)
a__x(N, 0) → a__U91(a__isNat(N), N)
a__x(N, s(M)) → a__U101(a__isNat(M), M, N)
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(isNat(X)) → a__isNat(X)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(x(X1, X2)) → a__x(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(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(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, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2, X3)) → a__U32(mark(X1), X2, X3)
mark(U33(X1, X2, X3)) → a__U33(mark(X1), X2, X3)
mark(U34(X1, X2, X3)) → a__U34(mark(X1), X2, X3)
mark(U35(X1, X2)) → a__U35(mark(X1), X2)
mark(U36(X)) → a__U36(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(U51(X)) → a__U51(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(U72(X1, X2)) → a__U72(mark(X1), X2)
mark(U81(X1, X2, X3)) → a__U81(mark(X1), X2, X3)
mark(U82(X1, X2, X3)) → a__U82(mark(X1), X2, X3)
mark(U83(X1, X2, X3)) → a__U83(mark(X1), X2, X3)
mark(U84(X1, X2, X3)) → a__U84(mark(X1), X2, X3)
mark(U91(X1, X2)) → a__U91(mark(X1), X2)
mark(U92(X)) → a__U92(mark(X))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U101(X1, X2, X3) → U101(X1, X2, X3)
a__U102(X1, X2, X3) → U102(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U103(X1, X2, X3) → U103(X1, X2, X3)
a__isNat(X) → isNat(X)
a__U104(X1, X2, X3) → U104(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__x(X1, X2) → x(X1, X2)
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__U13(X1, X2, X3) → U13(X1, X2, X3)
a__U14(X1, X2, X3) → U14(X1, X2, X3)
a__U15(X1, X2) → U15(X1, X2)
a__U16(X) → U16(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2, X3) → U32(X1, X2, X3)
a__U33(X1, X2, X3) → U33(X1, X2, X3)
a__U34(X1, X2, X3) → U34(X1, X2, X3)
a__U35(X1, X2) → U35(X1, X2)
a__U36(X) → U36(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__U51(X) → U51(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X1, X2) → U72(X1, X2)
a__U81(X1, X2, X3) → U81(X1, X2, X3)
a__U82(X1, X2, X3) → U82(X1, X2, X3)
a__U83(X1, X2, X3) → U83(X1, X2, X3)
a__U84(X1, X2, X3) → U84(X1, X2, X3)
a__U91(X1, X2) → U91(X1, X2)
a__U92(X) → U92(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A__U41(tt, V2) → A__ISNATKIND(V2)
A__ISNATKIND(plus(V1, V2)) → A__U41(a__isNatKind(V1), V2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__ISNATKIND(x(V1, V2)) → A__U61(a__isNatKind(V1), V2)
A__U61(tt, V2) → A__ISNATKIND(V2)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
A__U41(x1, x2) = A__U41(x1, x2)
tt = tt
A__ISNATKIND(x1) = A__ISNATKIND(x1)
plus(x1, x2) = plus(x1, x2)
a__isNatKind(x1) = x1
s(x1) = s(x1)
x(x1, x2) = x(x1, x2)
A__U61(x1, x2) = A__U61(x1, x2)
a__U42(x1) = x1
a__U62(x1) = x1
U62(x1) = x1
a__U51(x1) = x1
a__U41(x1, x2) = x2
isNatKind(x1) = isNatKind
a__U61(x1, x2) = x2
0 = 0
U42(x1) = x1
U41(x1, x2) = x2
U61(x1, x2) = x2
U51(x1) = x1

Lexicographic Path Order [LPO].
Precedence:

plus₂ > A_{_U41₂} > A_{_ISNATKIND₁} > isNatKind
s₁ > A_{_ISNATKIND₁} > isNatKind
x₂ > A_{_U61₂} > A_{_ISNATKIND₁} > isNatKind
0 > tt > isNatKind

The following usable rules [FROCOS05] were oriented:

a__U42(tt) → tt
a__U62(X) → U62(X)
a__U51(tt) → tt
a__U41(tt, V2) → a__U42(a__isNatKind(V2))
a__isNatKind(X) → isNatKind(X)
a__U61(tt, V2) → a__U62(a__isNatKind(V2))
a__U62(tt) → tt
a__isNatKind(plus(V1, V2)) → a__U41(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__U42(X) → U42(X)
a__isNatKind(x(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__U41(X1, X2) → U41(X1, X2)
a__isNatKind(s(V1)) → a__U51(a__isNatKind(V1))
a__U61(X1, X2) → U61(X1, X2)
a__U51(X) → U51(X)

(7) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

a__U101(tt, M, N) → a__U102(a__isNatKind(M), M, N)
a__U102(tt, M, N) → a__U103(a__isNat(N), M, N)
a__U103(tt, M, N) → a__U104(a__isNatKind(N), M, N)
a__U104(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__U11(tt, V1, V2) → a__U12(a__isNatKind(V1), V1, V2)
a__U12(tt, V1, V2) → a__U13(a__isNatKind(V2), V1, V2)
a__U13(tt, V1, V2) → a__U14(a__isNatKind(V2), V1, V2)
a__U14(tt, V1, V2) → a__U15(a__isNat(V1), V2)
a__U15(tt, V2) → a__U16(a__isNat(V2))
a__U16(tt) → tt
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNatKind(V1), V1, V2)
a__U32(tt, V1, V2) → a__U33(a__isNatKind(V2), V1, V2)
a__U33(tt, V1, V2) → a__U34(a__isNatKind(V2), V1, V2)
a__U34(tt, V1, V2) → a__U35(a__isNat(V1), V2)
a__U35(tt, V2) → a__U36(a__isNat(V2))
a__U36(tt) → tt
a__U41(tt, V2) → a__U42(a__isNatKind(V2))
a__U42(tt) → tt
a__U51(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatKind(V2))
a__U62(tt) → tt
a__U71(tt, N) → a__U72(a__isNatKind(N), N)
a__U72(tt, N) → mark(N)
a__U81(tt, M, N) → a__U82(a__isNatKind(M), M, N)
a__U82(tt, M, N) → a__U83(a__isNat(N), M, N)
a__U83(tt, M, N) → a__U84(a__isNatKind(N), M, N)
a__U84(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U91(tt, N) → a__U92(a__isNatKind(N))
a__U92(tt) → 0
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNatKind(V1), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__isNatKind(V1), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U41(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U51(a__isNatKind(V1))
a__isNatKind(x(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__plus(N, 0) → a__U71(a__isNat(N), N)
a__plus(N, s(M)) → a__U81(a__isNat(M), M, N)
a__x(N, 0) → a__U91(a__isNat(N), N)
a__x(N, s(M)) → a__U101(a__isNat(M), M, N)
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(isNat(X)) → a__isNat(X)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(x(X1, X2)) → a__x(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(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(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, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2, X3)) → a__U32(mark(X1), X2, X3)
mark(U33(X1, X2, X3)) → a__U33(mark(X1), X2, X3)
mark(U34(X1, X2, X3)) → a__U34(mark(X1), X2, X3)
mark(U35(X1, X2)) → a__U35(mark(X1), X2)
mark(U36(X)) → a__U36(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(U51(X)) → a__U51(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(U72(X1, X2)) → a__U72(mark(X1), X2)
mark(U81(X1, X2, X3)) → a__U81(mark(X1), X2, X3)
mark(U82(X1, X2, X3)) → a__U82(mark(X1), X2, X3)
mark(U83(X1, X2, X3)) → a__U83(mark(X1), X2, X3)
mark(U84(X1, X2, X3)) → a__U84(mark(X1), X2, X3)
mark(U91(X1, X2)) → a__U91(mark(X1), X2)
mark(U92(X)) → a__U92(mark(X))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U101(X1, X2, X3) → U101(X1, X2, X3)
a__U102(X1, X2, X3) → U102(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U103(X1, X2, X3) → U103(X1, X2, X3)
a__isNat(X) → isNat(X)
a__U104(X1, X2, X3) → U104(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__x(X1, X2) → x(X1, X2)
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__U13(X1, X2, X3) → U13(X1, X2, X3)
a__U14(X1, X2, X3) → U14(X1, X2, X3)
a__U15(X1, X2) → U15(X1, X2)
a__U16(X) → U16(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2, X3) → U32(X1, X2, X3)
a__U33(X1, X2, X3) → U33(X1, X2, X3)
a__U34(X1, X2, X3) → U34(X1, X2, X3)
a__U35(X1, X2) → U35(X1, X2)
a__U36(X) → U36(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__U51(X) → U51(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X1, X2) → U72(X1, X2)
a__U81(X1, X2, X3) → U81(X1, X2, X3)
a__U82(X1, X2, X3) → U82(X1, X2, X3)
a__U83(X1, X2, X3) → U83(X1, X2, X3)
a__U84(X1, X2, X3) → U84(X1, X2, X3)
a__U91(X1, X2) → U91(X1, X2)
a__U92(X) → U92(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(8) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(9) TRUE

(10) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A__U12(tt, V1, V2) → A__U13(a__isNatKind(V2), V1, V2)
A__U13(tt, V1, V2) → A__U14(a__isNatKind(V2), V1, V2)
A__U14(tt, V1, V2) → A__U15(a__isNat(V1), V2)
A__U15(tt, V2) → A__ISNAT(V2)
A__ISNAT(plus(V1, V2)) → A__U11(a__isNatKind(V1), V1, V2)
A__U11(tt, V1, V2) → A__U12(a__isNatKind(V1), V1, V2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U21(tt, V1) → A__U22(a__isNatKind(V1), V1)
A__U22(tt, V1) → A__ISNAT(V1)
A__ISNAT(x(V1, V2)) → A__U31(a__isNatKind(V1), V1, V2)
A__U31(tt, V1, V2) → A__U32(a__isNatKind(V1), V1, V2)
A__U32(tt, V1, V2) → A__U33(a__isNatKind(V2), V1, V2)
A__U33(tt, V1, V2) → A__U34(a__isNatKind(V2), V1, V2)
A__U34(tt, V1, V2) → A__U35(a__isNat(V1), V2)
A__U35(tt, V2) → A__ISNAT(V2)
A__U34(tt, V1, V2) → A__ISNAT(V1)
A__U14(tt, V1, V2) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__U101(tt, M, N) → a__U102(a__isNatKind(M), M, N)
a__U102(tt, M, N) → a__U103(a__isNat(N), M, N)
a__U103(tt, M, N) → a__U104(a__isNatKind(N), M, N)
a__U104(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__U11(tt, V1, V2) → a__U12(a__isNatKind(V1), V1, V2)
a__U12(tt, V1, V2) → a__U13(a__isNatKind(V2), V1, V2)
a__U13(tt, V1, V2) → a__U14(a__isNatKind(V2), V1, V2)
a__U14(tt, V1, V2) → a__U15(a__isNat(V1), V2)
a__U15(tt, V2) → a__U16(a__isNat(V2))
a__U16(tt) → tt
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNatKind(V1), V1, V2)
a__U32(tt, V1, V2) → a__U33(a__isNatKind(V2), V1, V2)
a__U33(tt, V1, V2) → a__U34(a__isNatKind(V2), V1, V2)
a__U34(tt, V1, V2) → a__U35(a__isNat(V1), V2)
a__U35(tt, V2) → a__U36(a__isNat(V2))
a__U36(tt) → tt
a__U41(tt, V2) → a__U42(a__isNatKind(V2))
a__U42(tt) → tt
a__U51(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatKind(V2))
a__U62(tt) → tt
a__U71(tt, N) → a__U72(a__isNatKind(N), N)
a__U72(tt, N) → mark(N)
a__U81(tt, M, N) → a__U82(a__isNatKind(M), M, N)
a__U82(tt, M, N) → a__U83(a__isNat(N), M, N)
a__U83(tt, M, N) → a__U84(a__isNatKind(N), M, N)
a__U84(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U91(tt, N) → a__U92(a__isNatKind(N))
a__U92(tt) → 0
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNatKind(V1), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__isNatKind(V1), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U41(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U51(a__isNatKind(V1))
a__isNatKind(x(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__plus(N, 0) → a__U71(a__isNat(N), N)
a__plus(N, s(M)) → a__U81(a__isNat(M), M, N)
a__x(N, 0) → a__U91(a__isNat(N), N)
a__x(N, s(M)) → a__U101(a__isNat(M), M, N)
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(isNat(X)) → a__isNat(X)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(x(X1, X2)) → a__x(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(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(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, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2, X3)) → a__U32(mark(X1), X2, X3)
mark(U33(X1, X2, X3)) → a__U33(mark(X1), X2, X3)
mark(U34(X1, X2, X3)) → a__U34(mark(X1), X2, X3)
mark(U35(X1, X2)) → a__U35(mark(X1), X2)
mark(U36(X)) → a__U36(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(U51(X)) → a__U51(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(U72(X1, X2)) → a__U72(mark(X1), X2)
mark(U81(X1, X2, X3)) → a__U81(mark(X1), X2, X3)
mark(U82(X1, X2, X3)) → a__U82(mark(X1), X2, X3)
mark(U83(X1, X2, X3)) → a__U83(mark(X1), X2, X3)
mark(U84(X1, X2, X3)) → a__U84(mark(X1), X2, X3)
mark(U91(X1, X2)) → a__U91(mark(X1), X2)
mark(U92(X)) → a__U92(mark(X))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U101(X1, X2, X3) → U101(X1, X2, X3)
a__U102(X1, X2, X3) → U102(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U103(X1, X2, X3) → U103(X1, X2, X3)
a__isNat(X) → isNat(X)
a__U104(X1, X2, X3) → U104(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__x(X1, X2) → x(X1, X2)
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__U13(X1, X2, X3) → U13(X1, X2, X3)
a__U14(X1, X2, X3) → U14(X1, X2, X3)
a__U15(X1, X2) → U15(X1, X2)
a__U16(X) → U16(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2, X3) → U32(X1, X2, X3)
a__U33(X1, X2, X3) → U33(X1, X2, X3)
a__U34(X1, X2, X3) → U34(X1, X2, X3)
a__U35(X1, X2) → U35(X1, X2)
a__U36(X) → U36(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__U51(X) → U51(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X1, X2) → U72(X1, X2)
a__U81(X1, X2, X3) → U81(X1, X2, X3)
a__U82(X1, X2, X3) → U82(X1, X2, X3)
a__U83(X1, X2, X3) → U83(X1, X2, X3)
a__U84(X1, X2, X3) → U84(X1, X2, X3)
a__U91(X1, X2) → U91(X1, X2)
a__U92(X) → U92(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A__U12(tt, V1, V2) → A__U13(a__isNatKind(V2), V1, V2)
A__U13(tt, V1, V2) → A__U14(a__isNatKind(V2), V1, V2)
A__U14(tt, V1, V2) → A__U15(a__isNat(V1), V2)
A__U15(tt, V2) → A__ISNAT(V2)
A__ISNAT(plus(V1, V2)) → A__U11(a__isNatKind(V1), V1, V2)
A__U11(tt, V1, V2) → A__U12(a__isNatKind(V1), V1, V2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U21(tt, V1) → A__U22(a__isNatKind(V1), V1)
A__U22(tt, V1) → A__ISNAT(V1)
A__ISNAT(x(V1, V2)) → A__U31(a__isNatKind(V1), V1, V2)
A__U31(tt, V1, V2) → A__U32(a__isNatKind(V1), V1, V2)
A__U32(tt, V1, V2) → A__U33(a__isNatKind(V2), V1, V2)
A__U33(tt, V1, V2) → A__U34(a__isNatKind(V2), V1, V2)
A__U34(tt, V1, V2) → A__U35(a__isNat(V1), V2)
A__U35(tt, V2) → A__ISNAT(V2)
A__U34(tt, V1, V2) → A__ISNAT(V1)
A__U14(tt, V1, V2) → A__ISNAT(V1)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
A__U12(x1, x2, x3) = A__U12(x2, x3)
tt = tt
A__U13(x1, x2, x3) = A__U13(x2, x3)
a__isNatKind(x1) = x1
A__U14(x1, x2, x3) = A__U14(x2, x3)
A__U15(x1, x2) = A__U15(x2)
a__isNat(x1) = a__isNat(x1)
A__ISNAT(x1) = A__ISNAT(x1)
plus(x1, x2) = plus(x1, x2)
A__U11(x1, x2, x3) = A__U11(x1, x2, x3)
s(x1) = s(x1)
A__U21(x1, x2) = A__U21(x2)
A__U22(x1, x2) = A__U22(x2)
x(x1, x2) = x(x1, x2)
A__U31(x1, x2, x3) = A__U31(x2, x3)
A__U32(x1, x2, x3) = A__U32(x2, x3)
A__U33(x1, x2, x3) = A__U33(x2, x3)
A__U34(x1, x2, x3) = A__U34(x2, x3)
A__U35(x1, x2) = A__U35(x2)
a__U42(x1) = a__U42
a__U51(x1) = a__U51
a__U36(x1) = x1
a__U41(x1, x2) = a__U41(x1, x2)
a__U34(x1, x2, x3) = x2
a__U35(x1, x2) = a__U35(x1, x2)
a__U32(x1, x2, x3) = a__U32(x1, x2, x3)
a__U33(x1, x2, x3) = x1
a__U61(x1, x2) = a__U61(x1, x2)
a__U62(x1) = a__U62(x1)
a__U12(x1, x2, x3) = a__U12(x1, x2)
a__U13(x1, x2, x3) = x3
a__U11(x1, x2, x3) = x2
a__U31(x1, x2, x3) = x2
a__U23(x1) = x1
a__U22(x1, x2) = a__U22
a__U21(x1, x2) = x2
a__U16(x1) = a__U16
a__U15(x1, x2) = a__U15
a__U14(x1, x2, x3) = x3
0 = 0
isNatKind(x1) = x1
isNat(x1) = isNat
U12(x1, x2, x3) = U12(x1, x2, x3)
U13(x1, x2, x3) = x1
U11(x1, x2, x3) = U11(x1, x2, x3)
U62(x1) = U62
U15(x1, x2) = U15(x1, x2)
U14(x1, x2, x3) = U14(x1, x2, x3)
U21(x1, x2) = U21(x1, x2)
U16(x1) = U16(x1)
U23(x1) = U23(x1)
U22(x1, x2) = U22(x1, x2)
U32(x1, x2, x3) = U32(x1, x2, x3)
U31(x1, x2, x3) = U31(x1, x2, x3)
U34(x1, x2, x3) = U34(x1, x2, x3)
U33(x1, x2, x3) = U33(x1, x2, x3)
U36(x1) = U36
U35(x1, x2) = U35(x1, x2)
U42(x1) = U42
U41(x1, x2) = U41(x1, x2)
U61(x1, x2) = U61(x1, x2)
U51(x1) = U51

Lexicographic Path Order [LPO].
Precedence:

a_{_isNat₁} > tt > a_{_U62₁} > U62 > U34₃
a_{_isNat₁} > tt > a_{_U22} > U34₃
plus₂ > A_{_U11₃} > A_{_U12₂} > A_{_U13₂} > A_{_U14₂} > A_{_U15₁} > A_{_ISNAT₁} > U34₃
plus₂ > a_{_U41₂} > a_{_U42} > tt > a_{_U62₁} > U62 > U34₃
plus₂ > a_{_U41₂} > a_{_U42} > tt > a_{_U22} > U34₃
plus₂ > a_{_U41₂} > a_{_U42} > U42 > U34₃
plus₂ > a_{_U41₂} > U41₂ > U34₃
s₁ > A_{_U21₁} > A_{_U22₁} > A_{_ISNAT₁} > U34₃
s₁ > a_{_U51} > tt > a_{_U62₁} > U62 > U34₃
s₁ > a_{_U51} > tt > a_{_U22} > U34₃
s₁ > a_{_U51} > U51 > U34₃
x₂ > A_{_U31₂} > A_{_U32₂} > A_{_U33₂} > A_{_U34₂} > A_{_U35₁} > A_{_ISNAT₁} > U34₃
x₂ > a_{_U61₂} > a_{_U62₁} > U62 > U34₃
x₂ > a_{_U61₂} > U61₂ > U34₃
a_{_U35₂} > U34₃
a_{_U32₃} > U32₃ > U34₃
a_{_U12₂} > U34₃
a_{_U16} > tt > a_{_U62₁} > U62 > U34₃
a_{_U16} > tt > a_{_U22} > U34₃
a_{_U16} > U16₁ > U34₃
a_{_U15} > U34₃
0 > tt > a_{_U62₁} > U62 > U34₃
0 > tt > a_{_U22} > U34₃
isNat > U34₃
U12₃ > U34₃
U11₃ > U34₃
U15₂ > U34₃
U14₃ > U34₃
U21₂ > U34₃
U23₁ > U34₃
U22₂ > U34₃
U31₃ > U34₃
U33₃ > U34₃
U36 > U34₃
U35₂ > U34₃

The following usable rules [FROCOS05] were oriented:

a__U42(tt) → tt
a__U51(tt) → tt
a__U41(tt, V2) → a__U42(a__isNatKind(V2))
a__U61(tt, V2) → a__U62(a__isNatKind(V2))
a__U62(tt) → tt
a__isNatKind(plus(V1, V2)) → a__U41(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(x(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U51(a__isNatKind(V1))
a__isNatKind(X) → isNatKind(X)
a__U62(X) → U62(X)
a__U42(X) → U42(X)
a__U41(X1, X2) → U41(X1, X2)
a__U61(X1, X2) → U61(X1, X2)
a__U51(X) → U51(X)

(12) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

a__U101(tt, M, N) → a__U102(a__isNatKind(M), M, N)
a__U102(tt, M, N) → a__U103(a__isNat(N), M, N)
a__U103(tt, M, N) → a__U104(a__isNatKind(N), M, N)
a__U104(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__U11(tt, V1, V2) → a__U12(a__isNatKind(V1), V1, V2)
a__U12(tt, V1, V2) → a__U13(a__isNatKind(V2), V1, V2)
a__U13(tt, V1, V2) → a__U14(a__isNatKind(V2), V1, V2)
a__U14(tt, V1, V2) → a__U15(a__isNat(V1), V2)
a__U15(tt, V2) → a__U16(a__isNat(V2))
a__U16(tt) → tt
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNatKind(V1), V1, V2)
a__U32(tt, V1, V2) → a__U33(a__isNatKind(V2), V1, V2)
a__U33(tt, V1, V2) → a__U34(a__isNatKind(V2), V1, V2)
a__U34(tt, V1, V2) → a__U35(a__isNat(V1), V2)
a__U35(tt, V2) → a__U36(a__isNat(V2))
a__U36(tt) → tt
a__U41(tt, V2) → a__U42(a__isNatKind(V2))
a__U42(tt) → tt
a__U51(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatKind(V2))
a__U62(tt) → tt
a__U71(tt, N) → a__U72(a__isNatKind(N), N)
a__U72(tt, N) → mark(N)
a__U81(tt, M, N) → a__U82(a__isNatKind(M), M, N)
a__U82(tt, M, N) → a__U83(a__isNat(N), M, N)
a__U83(tt, M, N) → a__U84(a__isNatKind(N), M, N)
a__U84(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U91(tt, N) → a__U92(a__isNatKind(N))
a__U92(tt) → 0
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNatKind(V1), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__isNatKind(V1), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U41(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U51(a__isNatKind(V1))
a__isNatKind(x(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__plus(N, 0) → a__U71(a__isNat(N), N)
a__plus(N, s(M)) → a__U81(a__isNat(M), M, N)
a__x(N, 0) → a__U91(a__isNat(N), N)
a__x(N, s(M)) → a__U101(a__isNat(M), M, N)
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(isNat(X)) → a__isNat(X)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(x(X1, X2)) → a__x(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(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(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, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2, X3)) → a__U32(mark(X1), X2, X3)
mark(U33(X1, X2, X3)) → a__U33(mark(X1), X2, X3)
mark(U34(X1, X2, X3)) → a__U34(mark(X1), X2, X3)
mark(U35(X1, X2)) → a__U35(mark(X1), X2)
mark(U36(X)) → a__U36(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(U51(X)) → a__U51(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(U72(X1, X2)) → a__U72(mark(X1), X2)
mark(U81(X1, X2, X3)) → a__U81(mark(X1), X2, X3)
mark(U82(X1, X2, X3)) → a__U82(mark(X1), X2, X3)
mark(U83(X1, X2, X3)) → a__U83(mark(X1), X2, X3)
mark(U84(X1, X2, X3)) → a__U84(mark(X1), X2, X3)
mark(U91(X1, X2)) → a__U91(mark(X1), X2)
mark(U92(X)) → a__U92(mark(X))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U101(X1, X2, X3) → U101(X1, X2, X3)
a__U102(X1, X2, X3) → U102(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U103(X1, X2, X3) → U103(X1, X2, X3)
a__isNat(X) → isNat(X)
a__U104(X1, X2, X3) → U104(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__x(X1, X2) → x(X1, X2)
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__U13(X1, X2, X3) → U13(X1, X2, X3)
a__U14(X1, X2, X3) → U14(X1, X2, X3)
a__U15(X1, X2) → U15(X1, X2)
a__U16(X) → U16(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2, X3) → U32(X1, X2, X3)
a__U33(X1, X2, X3) → U33(X1, X2, X3)
a__U34(X1, X2, X3) → U34(X1, X2, X3)
a__U35(X1, X2) → U35(X1, X2)
a__U36(X) → U36(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__U51(X) → U51(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X1, X2) → U72(X1, X2)
a__U81(X1, X2, X3) → U81(X1, X2, X3)
a__U82(X1, X2, X3) → U82(X1, X2, X3)
a__U83(X1, X2, X3) → U83(X1, X2, X3)
a__U84(X1, X2, X3) → U84(X1, X2, X3)
a__U91(X1, X2) → U91(X1, X2)
a__U92(X) → U92(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(13) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(14) TRUE

(15) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A__U102(tt, M, N) → A__U103(a__isNat(N), M, N)
A__U103(tt, M, N) → A__U104(a__isNatKind(N), M, N)
A__U104(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
A__PLUS(N, 0) → A__U71(a__isNat(N), N)
A__U71(tt, N) → A__U72(a__isNatKind(N), N)
A__U72(tt, N) → MARK(N)
MARK(U101(X1, X2, X3)) → A__U101(mark(X1), X2, X3)
A__U101(tt, M, N) → A__U102(a__isNatKind(M), M, N)
MARK(U101(X1, X2, X3)) → MARK(X1)
MARK(U102(X1, X2, X3)) → A__U102(mark(X1), X2, X3)
MARK(U102(X1, X2, X3)) → MARK(X1)
MARK(U103(X1, X2, X3)) → A__U103(mark(X1), X2, X3)
MARK(U103(X1, X2, X3)) → MARK(X1)
MARK(U104(X1, X2, X3)) → A__U104(mark(X1), X2, X3)
A__U104(tt, M, N) → A__X(mark(N), mark(M))
A__X(N, s(M)) → A__U101(a__isNat(M), M, N)
A__U104(tt, M, N) → MARK(N)
MARK(U104(X1, X2, X3)) → MARK(X1)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(N, s(M)) → A__U81(a__isNat(M), M, N)
A__U81(tt, M, N) → A__U82(a__isNatKind(M), M, N)
A__U82(tt, M, N) → A__U83(a__isNat(N), M, N)
A__U83(tt, M, N) → A__U84(a__isNatKind(N), M, N)
A__U84(tt, M, N) → A__PLUS(mark(N), mark(M))
A__U84(tt, M, N) → MARK(N)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
MARK(x(X1, X2)) → MARK(X1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U15(X1, X2)) → MARK(X1)
MARK(U16(X)) → MARK(X)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U23(X)) → MARK(X)
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(U32(X1, X2, X3)) → MARK(X1)
MARK(U33(X1, X2, X3)) → MARK(X1)
MARK(U34(X1, X2, X3)) → MARK(X1)
MARK(U35(X1, X2)) → MARK(X1)
MARK(U36(X)) → MARK(X)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
MARK(U51(X)) → MARK(X)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U62(X)) → MARK(X)
MARK(U71(X1, X2)) → A__U71(mark(X1), X2)
MARK(U71(X1, X2)) → MARK(X1)
MARK(U72(X1, X2)) → A__U72(mark(X1), X2)
MARK(U72(X1, X2)) → MARK(X1)
MARK(U81(X1, X2, X3)) → A__U81(mark(X1), X2, X3)
MARK(U81(X1, X2, X3)) → MARK(X1)
MARK(U82(X1, X2, X3)) → A__U82(mark(X1), X2, X3)
MARK(U82(X1, X2, X3)) → MARK(X1)
MARK(U83(X1, X2, X3)) → A__U83(mark(X1), X2, X3)
MARK(U83(X1, X2, X3)) → MARK(X1)
MARK(U84(X1, X2, X3)) → A__U84(mark(X1), X2, X3)
A__U84(tt, M, N) → MARK(M)
MARK(U84(X1, X2, X3)) → MARK(X1)
MARK(U91(X1, X2)) → MARK(X1)
MARK(U92(X)) → MARK(X)
MARK(s(X)) → MARK(X)
A__U104(tt, M, N) → MARK(M)

The TRS R consists of the following rules:

a__U101(tt, M, N) → a__U102(a__isNatKind(M), M, N)
a__U102(tt, M, N) → a__U103(a__isNat(N), M, N)
a__U103(tt, M, N) → a__U104(a__isNatKind(N), M, N)
a__U104(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__U11(tt, V1, V2) → a__U12(a__isNatKind(V1), V1, V2)
a__U12(tt, V1, V2) → a__U13(a__isNatKind(V2), V1, V2)
a__U13(tt, V1, V2) → a__U14(a__isNatKind(V2), V1, V2)
a__U14(tt, V1, V2) → a__U15(a__isNat(V1), V2)
a__U15(tt, V2) → a__U16(a__isNat(V2))
a__U16(tt) → tt
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNatKind(V1), V1, V2)
a__U32(tt, V1, V2) → a__U33(a__isNatKind(V2), V1, V2)
a__U33(tt, V1, V2) → a__U34(a__isNatKind(V2), V1, V2)
a__U34(tt, V1, V2) → a__U35(a__isNat(V1), V2)
a__U35(tt, V2) → a__U36(a__isNat(V2))
a__U36(tt) → tt
a__U41(tt, V2) → a__U42(a__isNatKind(V2))
a__U42(tt) → tt
a__U51(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatKind(V2))
a__U62(tt) → tt
a__U71(tt, N) → a__U72(a__isNatKind(N), N)
a__U72(tt, N) → mark(N)
a__U81(tt, M, N) → a__U82(a__isNatKind(M), M, N)
a__U82(tt, M, N) → a__U83(a__isNat(N), M, N)
a__U83(tt, M, N) → a__U84(a__isNatKind(N), M, N)
a__U84(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U91(tt, N) → a__U92(a__isNatKind(N))
a__U92(tt) → 0
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNatKind(V1), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__isNatKind(V1), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U41(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U51(a__isNatKind(V1))
a__isNatKind(x(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__plus(N, 0) → a__U71(a__isNat(N), N)
a__plus(N, s(M)) → a__U81(a__isNat(M), M, N)
a__x(N, 0) → a__U91(a__isNat(N), N)
a__x(N, s(M)) → a__U101(a__isNat(M), M, N)
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(isNat(X)) → a__isNat(X)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(x(X1, X2)) → a__x(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(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(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, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2, X3)) → a__U32(mark(X1), X2, X3)
mark(U33(X1, X2, X3)) → a__U33(mark(X1), X2, X3)
mark(U34(X1, X2, X3)) → a__U34(mark(X1), X2, X3)
mark(U35(X1, X2)) → a__U35(mark(X1), X2)
mark(U36(X)) → a__U36(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(U51(X)) → a__U51(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(U72(X1, X2)) → a__U72(mark(X1), X2)
mark(U81(X1, X2, X3)) → a__U81(mark(X1), X2, X3)
mark(U82(X1, X2, X3)) → a__U82(mark(X1), X2, X3)
mark(U83(X1, X2, X3)) → a__U83(mark(X1), X2, X3)
mark(U84(X1, X2, X3)) → a__U84(mark(X1), X2, X3)
mark(U91(X1, X2)) → a__U91(mark(X1), X2)
mark(U92(X)) → a__U92(mark(X))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U101(X1, X2, X3) → U101(X1, X2, X3)
a__U102(X1, X2, X3) → U102(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U103(X1, X2, X3) → U103(X1, X2, X3)
a__isNat(X) → isNat(X)
a__U104(X1, X2, X3) → U104(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__x(X1, X2) → x(X1, X2)
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__U13(X1, X2, X3) → U13(X1, X2, X3)
a__U14(X1, X2, X3) → U14(X1, X2, X3)
a__U15(X1, X2) → U15(X1, X2)
a__U16(X) → U16(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2, X3) → U32(X1, X2, X3)
a__U33(X1, X2, X3) → U33(X1, X2, X3)
a__U34(X1, X2, X3) → U34(X1, X2, X3)
a__U35(X1, X2) → U35(X1, X2)
a__U36(X) → U36(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__U51(X) → U51(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X1, X2) → U72(X1, X2)
a__U81(X1, X2, X3) → U81(X1, X2, X3)
a__U82(X1, X2, X3) → U82(X1, X2, X3)
a__U83(X1, X2, X3) → U83(X1, X2, X3)
a__U84(X1, X2, X3) → U84(X1, X2, X3)
a__U91(X1, X2) → U91(X1, X2)
a__U92(X) → U92(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.