Termination w.r.t. Q proof of /tmp/tmp0YtL7M/MYNAT_complete_noand

Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 DependencyGraphProof (⇔)

↳4 AND

  ↳5 QDP

    ↳6 QDPOrderProof (⇔)

    ↳7 QDP

    ↳8 DependencyGraphProof (⇔)

    ↳9 QDP

    ↳10 QDPOrderProof (⇔)

    ↳11 QDP

    ↳12 PisEmptyProof (⇔)

    ↳13 TRUE

  ↳14 QDP

    ↳15 QDPOrderProof (⇔)

    ↳16 QDP

    ↳17 DependencyGraphProof (⇔)

    ↳18 TRUE

  ↳19 QDP

    ↳20 QDPOrderProof (⇔)

    ↳21 QDP

    ↳22 DependencyGraphProof (⇔)

    ↳23 QDP

    ↳24 QDPOrderProof (⇔)

    ↳25 QDP

    ↳26 QDPOrderProof (⇔)

    ↳27 QDP

    ↳28 QDPOrderProof (⇔)

    ↳29 QDP

    ↳30 QDPOrderProof (⇔)

    ↳31 QDP

    ↳32 QDPOrderProof (⇔)

    ↳33 QDP

    ↳34 QDPOrderProof (⇔)

    ↳35 QDP

    ↳36 QDPOrderProof (⇔)

    ↳37 QDP

    ↳38 PisEmptyProof (⇔)

    ↳39 TRUE

(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__ISNATKIND(plus(V1, V2)) → A__U41(a__isNatKind(V1), V2)
A__ISNATKIND(plus(V1, V2)) → 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) = x2
tt = tt
A__ISNATKIND(x1) = x1
plus(x1, x2) = plus(x1, x2)
a__isNatKind(x1) = a__isNatKind
s(x1) = x1
x(x1, x2) = x(x1, x2)
A__U61(x1, x2) = A__U61(x1, x2)
0 = 0
a__U41(x1, x2) = a__U41
a__U51(x1) = a__U51
a__U61(x1, x2) = a__U61
isNatKind(x1) = isNatKind
U41(x1, x2) = U41
U51(x1) = U51
U61(x1, x2) = U61
a__U62(x1) = x1
U62(x1) = U62
a__U42(x1) = x1
U42(x1) = U42

Recursive path order with status [RPO].
Quasi-Precedence:

plus₂ > [U62, U42]
x₂ > [a_{_isNatKind}, a_{_U41}, a_{_U61}, U41, U61] > [tt, 0, a_{_U51}, U51] > [U62, U42]
x₂ > [a_{_isNatKind}, a_{_U41}, a_{_U61}, U41, U61] > isNatKind > [U62, U42]
x₂ > A_{_U61₂} > [U62, U42]

Status:

tt: multiset
plus₂: multiset
a_{_isNatKind}: []
x₂: [1,2]
A_{_U61₂}: [2,1]
0: multiset
a_{_U41}: []
a_{_U51}: multiset
a_{_U61}: []
isNatKind: multiset
U41: []
U51: multiset
U61: []
U62: []
U42: []

The following usable rules [FROCOS05] were oriented:

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__isNatKind(X) → isNatKind(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(tt) → tt
a__U51(X) → U51(X)
a__U61(X1, X2) → U61(X1, X2)
a__U61(tt, V2) → a__U62(a__isNatKind(V2))
a__U62(tt) → tt
a__U62(X) → U62(X)
a__U41(tt, V2) → a__U42(a__isNatKind(V2))
a__U42(tt) → tt
a__U42(X) → U42(X)

(7) Obligation:

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

A__U41(tt, V2) → A__ISNATKIND(V2)
A__ISNATKIND(s(V1)) → 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.

(8) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(9) Obligation:

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

A__ISNATKIND(s(V1)) → 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.

(10) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

s₁ > A_{_ISNATKIND₁}

Status:

A_{_ISNATKIND₁}: multiset
s₁: multiset

The following usable rules [FROCOS05] were oriented: none

(11) 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.

(12) PisEmptyProof (EQUIVALENT transformation)

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

(13) TRUE

(14) 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.

(15) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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(s(V1)) → A__U21(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__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) = a__isNatKind
A__U14(x1, x2, x3) = A__U14(x2, x3)
A__U15(x1, x2) = A__U15(x1, x2)
a__isNat(x1) = x1
A__ISNAT(x1) = x1
plus(x1, x2) = plus(x1, x2)
A__U11(x1, x2, x3) = A__U11(x2, x3)
s(x1) = s(x1)
A__U21(x1, x2) = A__U21(x1, x2)
A__U22(x1, x2) = A__U22(x1, x2)
x(x1, x2) = x(x1, x2)
A__U31(x1, x2, x3) = A__U31(x1, x2, x3)
A__U32(x1, x2, x3) = A__U32(x1, x2, x3)
A__U33(x1, x2, x3) = A__U33(x1, x2, x3)
A__U34(x1, x2, x3) = A__U34(x2, x3)
A__U35(x1, x2) = A__U35(x1, x2)
0 = 0
a__U41(x1, x2) = a__U41
a__U51(x1) = x1
a__U61(x1, x2) = x1
isNatKind(x1) = isNatKind
a__U11(x1, x2, x3) = a__U11(x2, x3)
a__U21(x1, x2) = x2
a__U31(x1, x2, x3) = a__U31(x1, x2, x3)
isNat(x1) = x1
a__U12(x1, x2, x3) = x2
a__U13(x1, x2, x3) = x2
U13(x1, x2, x3) = x2
a__U14(x1, x2, x3) = x2
U14(x1, x2, x3) = x2
a__U15(x1, x2) = x1
U15(x1, x2) = U15
U11(x1, x2, x3) = U11
U12(x1, x2, x3) = x2
U21(x1, x2) = x2
a__U22(x1, x2) = x2
U22(x1, x2) = x2
a__U23(x1) = x1
U23(x1) = x1
U31(x1, x2, x3) = U31
a__U32(x1, x2, x3) = a__U32(x1, x2, x3)
U32(x1, x2, x3) = U32(x1, x2, x3)
a__U33(x1, x2, x3) = a__U33(x1, x2, x3)
U33(x1, x2, x3) = U33(x1, x2, x3)
a__U34(x1, x2, x3) = a__U34
U34(x1, x2, x3) = U34
a__U35(x1, x2) = a__U35
U35(x1, x2) = U35
a__U36(x1) = a__U36
U36(x1) = U36
a__U16(x1) = a__U16
U16(x1) = U16
U41(x1, x2) = U41
U51(x1) = U51
U61(x1, x2) = U61
a__U62(x1) = x1
U62(x1) = U62
a__U42(x1) = x1
U42(x1) = U42

Recursive path order with status [RPO].
Quasi-Precedence:

[A_{_U12₂}, A_{_U13₂}, plus₂, A_{_U11₂}, a_{_U11₂}] > [tt, a_{_isNatKind}, s₁, 0, a_{_U41}, a_{_U16}, U16, U41] > [A_{_U21₂}, A_{_U22₂}] > [U15, U51, U61, U62, U42]
[A_{_U12₂}, A_{_U13₂}, plus₂, A_{_U11₂}, a_{_U11₂}] > [tt, a_{_isNatKind}, s₁, 0, a_{_U41}, a_{_U16}, U16, U41] > isNatKind > [U15, U51, U61, U62, U42]
[A_{_U12₂}, A_{_U13₂}, plus₂, A_{_U11₂}, a_{_U11₂}] > A_{_U14₂} > A_{_U15₂} > [U15, U51, U61, U62, U42]
[A_{_U12₂}, A_{_U13₂}, plus₂, A_{_U11₂}, a_{_U11₂}] > U11 > [U15, U51, U61, U62, U42]
x₂ > A_{_U31₃} > A_{_U32₃} > A_{_U33₃} > [tt, a_{_isNatKind}, s₁, 0, a_{_U41}, a_{_U16}, U16, U41] > [A_{_U21₂}, A_{_U22₂}] > [U15, U51, U61, U62, U42]
x₂ > A_{_U31₃} > A_{_U32₃} > A_{_U33₃} > [tt, a_{_isNatKind}, s₁, 0, a_{_U41}, a_{_U16}, U16, U41] > isNatKind > [U15, U51, U61, U62, U42]
x₂ > A_{_U31₃} > A_{_U32₃} > A_{_U33₃} > [A_{_U34₂}, A_{_U35₂}] > [U15, U51, U61, U62, U42]
x₂ > [a_{_U31₃}, U31, a_{_U32₃}, U32₃, a_{_U33₃}, U33₃] > [a_{_U34}, U34, a_{_U35}, U35, a_{_U36}, U36] > [tt, a_{_isNatKind}, s₁, 0, a_{_U41}, a_{_U16}, U16, U41] > [A_{_U21₂}, A_{_U22₂}] > [U15, U51, U61, U62, U42]
x₂ > [a_{_U31₃}, U31, a_{_U32₃}, U32₃, a_{_U33₃}, U33₃] > [a_{_U34}, U34, a_{_U35}, U35, a_{_U36}, U36] > [tt, a_{_isNatKind}, s₁, 0, a_{_U41}, a_{_U16}, U16, U41] > isNatKind > [U15, U51, U61, U62, U42]

Status:

A_{_U12₂}: [1,2]
tt: multiset
A_{_U13₂}: [1,2]
a_{_isNatKind}: []
A_{_U14₂}: multiset
A_{_U15₂}: multiset
plus₂: [1,2]
A_{_U11₂}: [1,2]
s₁: [1]
A_{_U21₂}: [1,2]
A_{_U22₂}: [1,2]
x₂: [2,1]
A_{_U31₃}: [1,3,2]
A_{_U32₃}: [3,1,2]
A_{_U33₃}: [1,3,2]
A_{_U34₂}: multiset
A_{_U35₂}: multiset
0: multiset
a_{_U41}: []
isNatKind: multiset
a_{_U11₂}: [1,2]
a_{_U31₃}: [3,2,1]
U15: multiset
U11: multiset
U31: []
a_{_U32₃}: [3,2,1]
U32₃: [3,2,1]
a_{_U33₃}: [3,2,1]
U33₃: [3,2,1]
a_{_U34}: multiset
U34: multiset
a_{_U35}: multiset
U35: multiset
a_{_U36}: multiset
U36: multiset
a_{_U16}: []
U16: []
U41: []
U51: []
U61: multiset
U62: multiset
U42: multiset

The following usable rules [FROCOS05] were oriented:

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__isNatKind(X) → isNatKind(X)
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__isNat(X) → isNat(X)
a__U12(tt, V1, V2) → a__U13(a__isNatKind(V2), 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__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__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(tt) → tt
a__U23(X) → U23(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U31(tt, V1, V2) → a__U32(a__isNatKind(V1), V1, V2)
a__U32(X1, X2, X3) → U32(X1, X2, X3)
a__U32(tt, V1, V2) → a__U33(a__isNatKind(V2), V1, V2)
a__U33(X1, X2, X3) → U33(X1, X2, X3)
a__U33(tt, V1, V2) → a__U34(a__isNatKind(V2), V1, V2)
a__U34(X1, X2, X3) → U34(X1, X2, X3)
a__U34(tt, V1, V2) → a__U35(a__isNat(V1), V2)
a__U35(X1, X2) → U35(X1, X2)
a__U35(tt, V2) → a__U36(a__isNat(V2))
a__U36(tt) → tt
a__U36(X) → U36(X)
a__U15(tt, V2) → a__U16(a__isNat(V2))
a__U16(tt) → tt
a__U16(X) → U16(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(tt) → tt
a__U51(X) → U51(X)
a__U61(X1, X2) → U61(X1, X2)
a__U61(tt, V2) → a__U62(a__isNatKind(V2))
a__U62(tt) → tt
a__U62(X) → U62(X)
a__U41(tt, V2) → a__U42(a__isNatKind(V2))
a__U42(tt) → tt
a__U42(X) → U42(X)

(16) 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__ISNAT(plus(V1, V2)) → A__U11(a__isNatKind(V1), V1, V2)
A__U11(tt, V1, V2) → A__U12(a__isNatKind(V1), V1, V2)
A__U21(tt, V1) → A__U22(a__isNatKind(V1), V1)
A__U34(tt, V1, V2) → A__U35(a__isNat(V1), V2)

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.

(17) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 5 less nodes.

(18) TRUE

(19) 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.

(20) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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)
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__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(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(s(X)) → MARK(X)
A__U104(tt, M, N) → MARK(M)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
A__U102(x1, x2, x3) = A__U102(x1, x2, x3)
tt = tt
A__U103(x1, x2, x3) = A__U103(x1, x2, x3)
a__isNat(x1) = a__isNat
A__U104(x1, x2, x3) = A__U104(x1, x2, x3)
a__isNatKind(x1) = a__isNatKind
A__PLUS(x1, x2) = A__PLUS(x1, x2)
a__x(x1, x2) = a__x(x1, x2)
mark(x1) = x1
0 = 0
A__U71(x1, x2) = A__U71(x1, x2)
A__U72(x1, x2) = A__U72(x1, x2)
MARK(x1) = MARK(x1)
U101(x1, x2, x3) = U101(x1, x2, x3)
A__U101(x1, x2, x3) = A__U101(x1, x2, x3)
U102(x1, x2, x3) = U102(x1, x2, x3)
U103(x1, x2, x3) = U103(x1, x2, x3)
U104(x1, x2, x3) = U104(x1, x2, x3)
A__X(x1, x2) = A__X(x1, x2)
s(x1) = s(x1)
plus(x1, x2) = plus(x1, x2)
A__U81(x1, x2, x3) = A__U81(x2, x3)
A__U82(x1, x2, x3) = A__U82(x2, x3)
A__U83(x1, x2, x3) = A__U83(x2, x3)
A__U84(x1, x2, x3) = A__U84(x2, x3)
x(x1, x2) = x(x1, x2)
U11(x1, x2, x3) = x1
U12(x1, x2, x3) = x1
U13(x1, x2, x3) = x1
U14(x1, x2, x3) = x1
U15(x1, x2) = x1
U16(x1) = x1
U21(x1, x2) = x1
U22(x1, x2) = x1
U23(x1) = x1
U31(x1, x2, x3) = x1
U32(x1, x2, x3) = x1
U33(x1, x2, x3) = x1
U34(x1, x2, x3) = x1
U35(x1, x2) = x1
U36(x1) = x1
U41(x1, x2) = x1
U42(x1) = x1
U51(x1) = x1
U61(x1, x2) = x1
U62(x1) = x1
U71(x1, x2) = U71(x1, x2)
U72(x1, x2) = U72(x1, x2)
U81(x1, x2, x3) = U81(x1, x2, x3)
U82(x1, x2, x3) = U82(x1, x2, x3)
U83(x1, x2, x3) = U83(x1, x2, x3)
U84(x1, x2, x3) = U84(x1, x2, x3)
U91(x1, x2) = x1
U92(x1) = x1
a__U11(x1, x2, x3) = x1
a__U21(x1, x2) = x1
a__U31(x1, x2, x3) = x1
isNat(x1) = isNat
a__U41(x1, x2) = x1
a__U51(x1) = x1
a__U61(x1, x2) = x1
isNatKind(x1) = isNatKind
a__U102(x1, x2, x3) = a__U102(x1, x2, x3)
a__U103(x1, x2, x3) = a__U103(x1, x2, x3)
a__U104(x1, x2, x3) = a__U104(x1, x2, x3)
a__plus(x1, x2) = a__plus(x1, x2)
a__U71(x1, x2) = a__U71(x1, x2)
a__U72(x1, x2) = a__U72(x1, x2)
a__U101(x1, x2, x3) = a__U101(x1, x2, x3)
a__U12(x1, x2, x3) = x1
a__U13(x1, x2, x3) = x1
a__U14(x1, x2, x3) = x1
a__U15(x1, x2) = x1
a__U16(x1) = x1
a__U22(x1, x2) = x1
a__U23(x1) = x1
a__U32(x1, x2, x3) = x1
a__U33(x1, x2, x3) = x1
a__U34(x1, x2, x3) = x1
a__U35(x1, x2) = x1
a__U36(x1) = x1
a__U42(x1) = x1
a__U62(x1) = x1
a__U81(x1, x2, x3) = a__U81(x1, x2, x3)
a__U82(x1, x2, x3) = a__U82(x1, x2, x3)
a__U83(x1, x2, x3) = a__U83(x1, x2, x3)
a__U84(x1, x2, x3) = a__U84(x1, x2, x3)
a__U91(x1, x2) = x1
a__U92(x1) = x1

Recursive path order with status [RPO].
Quasi-Precedence:

[A_{_U102₃}, A_{_U103₃}, A_{_U104₃}, a_{_x₂}, U101₃, A_{_U101₃}, U102₃, U103₃, U104₃, A_{_X₂}, x₂, a_{_U102₃}, a_{_U103₃}, a_{_U104₃}, a_{_U101₃}] > [plus₂, U71₂, U81₃, U82₃, U83₃, U84₃, a_{_plus₂}, a_{_U71₂}, a_{_U81₃}, a_{_U82₃}, a_{_U83₃}, a_{_U84₃}] > [A_{_PLUS₂}, A_{_U81₂}, A_{_U82₂}, A_{_U83₂}, A_{_U84₂}] > A_{_U71₂} > [A_{_U72₂}, MARK₁, s₁] > [tt, a_{_isNat}, a_{_isNatKind}, 0, isNat, isNatKind]
[A_{_U102₃}, A_{_U103₃}, A_{_U104₃}, a_{_x₂}, U101₃, A_{_U101₃}, U102₃, U103₃, U104₃, A_{_X₂}, x₂, a_{_U102₃}, a_{_U103₃}, a_{_U104₃}, a_{_U101₃}] > [plus₂, U71₂, U81₃, U82₃, U83₃, U84₃, a_{_plus₂}, a_{_U71₂}, a_{_U81₃}, a_{_U82₃}, a_{_U83₃}, a_{_U84₃}] > [U72₂, a_{_U72₂}] > [A_{_U72₂}, MARK₁, s₁] > [tt, a_{_isNat}, a_{_isNatKind}, 0, isNat, isNatKind]

Status:

A_{_U102₃}: [3,2,1]
tt: multiset
A_{_U103₃}: [3,2,1]
a_{_isNat}: multiset
A_{_U104₃}: [3,2,1]
a_{_isNatKind}: multiset
A_{_PLUS₂}: multiset
a_{_x₂}: [1,2]
0: multiset
A_{_U71₂}: multiset
A_{_U72₂}: multiset
MARK₁: multiset
U101₃: [3,2,1]
A_{_U101₃}: [3,2,1]
U102₃: [3,2,1]
U103₃: [3,2,1]
U104₃: [3,2,1]
A_{_X₂}: [1,2]
s₁: multiset
plus₂: multiset
A_{_U81₂}: multiset
A_{_U82₂}: multiset
A_{_U83₂}: multiset
A_{_U84₂}: multiset
x₂: [1,2]
U71₂: multiset
U72₂: multiset
U81₃: multiset
U82₃: multiset
U83₃: multiset
U84₃: multiset
isNat: multiset
isNatKind: multiset
a_{_U102₃}: [3,2,1]
a_{_U103₃}: [3,2,1]
a_{_U104₃}: [3,2,1]
a_{_plus₂}: multiset
a_{_U71₂}: multiset
a_{_U72₂}: multiset
a_{_U101₃}: [3,2,1]
a_{_U81₃}: multiset
a_{_U82₃}: multiset
a_{_U83₃}: multiset
a_{_U84₃}: multiset

The following usable rules [FROCOS05] were oriented:

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

(21) 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__U101(tt, M, N) → A__U102(a__isNatKind(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))
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(U91(X1, X2)) → MARK(X1)
MARK(U92(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.

(22) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 7 less nodes.

(23) Obligation:

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

MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U11(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(U91(X1, X2)) → MARK(X1)
MARK(U92(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.

(24) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U11(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(U32(X1, X2, X3)) → MARK(X1)
MARK(U33(X1, X2, X3)) → MARK(X1)
MARK(U34(X1, X2, X3)) → MARK(X1)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U91(X1, X2)) → MARK(X1)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1) = MARK(x1)
U12(x1, x2, x3) = U12(x1, x2)
U11(x1, x2, x3) = U11(x1, x2, x3)
U13(x1, x2, x3) = x1
U14(x1, x2, x3) = U14(x1, x2, x3)
U15(x1, x2) = U15(x1, x2)
U16(x1) = U16(x1)
U21(x1, x2) = U21(x1, x2)
U22(x1, x2) = U22(x1, x2)
U23(x1) = U23(x1)
U31(x1, x2, x3) = x1
U32(x1, x2, x3) = U32(x1, x2, x3)
U33(x1, x2, x3) = U33(x1, x2, x3)
U34(x1, x2, x3) = U34(x1, x2, x3)
U35(x1, x2) = x1
U36(x1) = x1
U41(x1, x2) = U41(x1, x2)
U42(x1) = x1
U51(x1) = x1
U61(x1, x2) = U61(x1, x2)
U62(x1) = x1
U91(x1, x2) = U91(x1, x2)
U92(x1) = x1

Recursive path order with status [RPO].
Quasi-Precedence:

U14₃ > [MARK₁, U12₂, U16₁, U23₁, U41₂]
U21₂ > [MARK₁, U12₂, U16₁, U23₁, U41₂]
U22₂ > [MARK₁, U12₂, U16₁, U23₁, U41₂]
U34₃ > [MARK₁, U12₂, U16₁, U23₁, U41₂]

Status:

MARK₁: multiset
U12₂: multiset
U11₃: multiset
U14₃: multiset
U15₂: multiset
U16₁: multiset
U21₂: multiset
U22₂: [2,1]
U23₁: multiset
U32₃: multiset
U33₃: multiset
U34₃: multiset
U41₂: multiset
U61₂: multiset
U91₂: multiset

The following usable rules [FROCOS05] were oriented: none

(25) Obligation:

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

MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(U35(X1, X2)) → MARK(X1)
MARK(U36(X)) → MARK(X)
MARK(U42(X)) → MARK(X)
MARK(U51(X)) → MARK(X)
MARK(U62(X)) → MARK(X)
MARK(U92(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.

(26) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(U35(X1, X2)) → MARK(X1)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1) = MARK(x1)
U13(x1, x2, x3) = U13(x1, x2, x3)
U31(x1, x2, x3) = U31(x1, x2, x3)
U35(x1, x2) = U35(x1, x2)
U36(x1) = x1
U42(x1) = x1
U51(x1) = x1
U62(x1) = x1
U92(x1) = x1

Recursive path order with status [RPO].
Quasi-Precedence:

U13₃ > [MARK₁, U31₃]
U35₂ > [MARK₁, U31₃]

Status:

MARK₁: multiset
U13₃: multiset
U31₃: multiset
U35₂: multiset

The following usable rules [FROCOS05] were oriented: none

(27) Obligation:

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

MARK(U36(X)) → MARK(X)
MARK(U42(X)) → MARK(X)
MARK(U51(X)) → MARK(X)
MARK(U62(X)) → MARK(X)
MARK(U92(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.

(28) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(U36(X)) → MARK(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1) = MARK(x1)
U36(x1) = U36(x1)
U42(x1) = x1
U51(x1) = x1
U62(x1) = x1
U92(x1) = x1

Recursive path order with status [RPO].
Quasi-Precedence:

U36₁ > MARK₁

Status:

MARK₁: multiset
U36₁: multiset

The following usable rules [FROCOS05] were oriented: none

(29) Obligation:

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

MARK(U42(X)) → MARK(X)
MARK(U51(X)) → MARK(X)
MARK(U62(X)) → MARK(X)
MARK(U92(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.

(30) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(U51(X)) → MARK(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1) = MARK(x1)
U42(x1) = x1
U51(x1) = U51(x1)
U62(x1) = x1
U92(x1) = x1

Recursive path order with status [RPO].
Quasi-Precedence:

[MARK₁, U51₁]

Status:

MARK₁: multiset
U51₁: multiset

The following usable rules [FROCOS05] were oriented: none

(31) Obligation:

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

MARK(U42(X)) → MARK(X)
MARK(U62(X)) → MARK(X)
MARK(U92(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.

(32) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(U42(X)) → MARK(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1) = MARK(x1)
U42(x1) = U42(x1)
U62(x1) = x1
U92(x1) = x1

Recursive path order with status [RPO].
Quasi-Precedence:

[MARK₁, U42₁]

Status:

MARK₁: multiset
U42₁: multiset

The following usable rules [FROCOS05] were oriented: none

(33) Obligation:

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

MARK(U62(X)) → MARK(X)
MARK(U92(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.

(34) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(U62(X)) → MARK(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1) = MARK(x1)
U62(x1) = U62(x1)
U92(x1) = x1

Recursive path order with status [RPO].
Quasi-Precedence:

U62₁ > MARK₁

Status:

MARK₁: multiset
U62₁: multiset

The following usable rules [FROCOS05] were oriented: none

(35) Obligation:

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

MARK(U92(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.

(36) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(U92(X)) → MARK(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

U92₁ > MARK₁

Status:

MARK₁: multiset
U92₁: multiset

The following usable rules [FROCOS05] were oriented: none

(37) 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.

(38) PisEmptyProof (EQUIVALENT transformation)

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

(0) Obligation:

(1) DependencyPairsProof (EQUIVALENT transformation)

(2) Obligation:

(3) DependencyGraphProof (EQUIVALENT transformation)

(4) Complex Obligation (AND)

(5) Obligation:

(6) QDPOrderProof (EQUIVALENT transformation)

(7) Obligation:

(8) DependencyGraphProof (EQUIVALENT transformation)

(9) Obligation:

(10) QDPOrderProof (EQUIVALENT transformation)

(11) Obligation:

(12) PisEmptyProof (EQUIVALENT transformation)

(13) TRUE

(14) Obligation:

(15) QDPOrderProof (EQUIVALENT transformation)

(16) Obligation:

(17) DependencyGraphProof (EQUIVALENT transformation)

(18) TRUE

(19) Obligation:

(20) QDPOrderProof (EQUIVALENT transformation)

(21) Obligation:

(22) DependencyGraphProof (EQUIVALENT transformation)

(23) Obligation:

(24) QDPOrderProof (EQUIVALENT transformation)

(25) Obligation:

(26) QDPOrderProof (EQUIVALENT transformation)

(27) Obligation:

(28) QDPOrderProof (EQUIVALENT transformation)

(29) Obligation:

(30) QDPOrderProof (EQUIVALENT transformation)

(31) Obligation:

(32) QDPOrderProof (EQUIVALENT transformation)

(33) Obligation:

(34) QDPOrderProof (EQUIVALENT transformation)

(35) Obligation:

(36) QDPOrderProof (EQUIVALENT transformation)

(37) Obligation:

(38) PisEmptyProof (EQUIVALENT transformation)

(39) TRUE