Termination w.r.t. Q proof of /tmp/tmpRKqb9K/PEANO_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 QDPOrderProof (⇔)

        ↳13 QDP

        ↳14 PisEmptyProof (⇔)

        ↳15 TRUE

    ↳16 QDP

        ↳17 QDPOrderProof (⇔)

        ↳18 QDP

        ↳19 QDPOrderProof (⇔)

        ↳20 QDP

        ↳21 DependencyGraphProof (⇔)

        ↳22 TRUE

    ↳23 QDP

        ↳24 QDPOrderProof (⇔)

        ↳25 QDP

        ↳26 QDPOrderProof (⇔)

        ↳27 QDP

        ↳28 QDPOrderProof (⇔)

        ↳29 QDP

        ↳30 DependencyGraphProof (⇔)

        ↳31 AND

            ↳32 QDP

                ↳33 QDPOrderProof (⇔)

                ↳34 QDP

                ↳35 PisEmptyProof (⇔)

                ↳36 TRUE

            ↳37 QDP

                ↳38 QDPOrderProof (⇔)

                ↳39 QDP

                ↳40 QDPOrderProof (⇔)

                ↳41 QDP

                ↳42 QDPOrderProof (⇔)

                ↳43 QDP

                ↳44 QDPOrderProof (⇔)

                ↳45 QDP

                ↳46 QDPOrderProof (⇔)

                ↳47 QDP

                ↳48 QDPOrderProof (⇔)

                ↳49 QDP

                ↳50 QDPOrderProof (⇔)

                ↳51 QDP

                ↳52 QDPOrderProof (⇔)

                ↳53 QDP

                ↳54 QDPOrderProof (⇔)

                ↳55 QDP

                ↳56 PisEmptyProof (⇔)

                ↳57 TRUE

(0) Obligation:

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

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, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
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__isNat(X) → isNat(X)
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) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

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__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, V2) → A__U32(a__isNatKind(V2))
A__U31(tt, V2) → A__ISNATKIND(V2)
A__U51(tt, N) → A__U52(a__isNatKind(N), N)
A__U51(tt, N) → A__ISNATKIND(N)
A__U52(tt, N) → MARK(N)
A__U61(tt, M, N) → A__U62(a__isNatKind(M), M, N)
A__U61(tt, M, N) → A__ISNATKIND(M)
A__U62(tt, M, N) → A__U63(a__isNat(N), M, N)
A__U62(tt, M, N) → A__ISNAT(N)
A__U63(tt, M, N) → A__U64(a__isNatKind(N), M, N)
A__U63(tt, M, N) → A__ISNATKIND(N)
A__U64(tt, M, N) → A__PLUS(mark(N), mark(M))
A__U64(tt, M, N) → MARK(N)
A__U64(tt, M, N) → MARK(M)
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__ISNATKIND(plus(V1, V2)) → A__U31(a__isNatKind(V1), V2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(s(V1)) → A__U41(a__isNatKind(V1))
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__PLUS(N, 0) → A__U51(a__isNat(N), N)
A__PLUS(N, 0) → A__ISNAT(N)
A__PLUS(N, s(M)) → A__U61(a__isNat(M), M, N)
A__PLUS(N, s(M)) → A__ISNAT(M)
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(isNatKind(X)) → A__ISNATKIND(X)
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(isNat(X)) → A__ISNAT(X)
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)) → A__U31(mark(X1), X2)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → A__U32(mark(X))
MARK(U32(X)) → MARK(X)
MARK(U41(X)) → A__U41(mark(X))
MARK(U41(X)) → MARK(X)
MARK(U51(X1, X2)) → A__U51(mark(X1), X2)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U61(X1, X2, X3)) → A__U61(mark(X1), X2, X3)
MARK(U61(X1, X2, X3)) → MARK(X1)
MARK(U62(X1, X2, X3)) → A__U62(mark(X1), X2, X3)
MARK(U62(X1, X2, X3)) → MARK(X1)
MARK(U63(X1, X2, X3)) → A__U63(mark(X1), X2, X3)
MARK(U63(X1, X2, X3)) → MARK(X1)
MARK(U64(X1, X2, X3)) → A__U64(mark(X1), X2, X3)
MARK(U64(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(s(X)) → MARK(X)

The TRS R consists of the following rules:

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, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
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__isNat(X) → isNat(X)
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) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

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 30 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

A__U31(tt, V2) → A__ISNATKIND(V2)
A__ISNATKIND(plus(V1, V2)) → A__U31(a__isNatKind(V1), V2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)

The TRS R consists of the following rules:

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, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
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__isNat(X) → isNat(X)
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) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

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__U31(a__isNatKind(V1), V2)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
A__U31(x0, x1, x2) = A__U31(x0)
A__ISNATKIND(x0, x1) = A__ISNATKIND(x0)

Tags:
A__U31 has argument tags [2,1,1] and root tag 1
A__ISNATKIND has argument tags [2,0] and root tag 1

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(0) = 1
POL(A__ISNATKIND(x₁)) = 1 + x₁
POL(A__U31(x₁, x₂)) = x₁ + x₂
POL(U31(x₁, x₂)) = x₁
POL(U32(x₁)) = x₁
POL(U41(x₁)) = 0
POL(a__U31(x₁, x₂)) = x₁ + x₂
POL(a__U32(x₁)) = 1 + x₁
POL(a__U41(x₁)) = x₁
POL(a__isNatKind(x₁)) = x₁
POL(isNatKind(x₁)) = x₁
POL(plus(x₁, x₂)) = x₁ + x₂
POL(s(x₁)) = x₁
POL(tt) = 1

The following usable rules [FROCOS05] were oriented:

a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__isNatKind(X) → isNatKind(X)
a__U31(tt, V2) → a__U32(a__isNatKind(V2))
a__U31(X1, X2) → U31(X1, X2)
a__U41(tt) → tt
a__U41(X) → U41(X)
a__U32(tt) → tt
a__U32(X) → U32(X)

(7) Obligation:

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

A__U31(tt, V2) → A__ISNATKIND(V2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)

The TRS R consists of the following rules:

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, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
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__isNat(X) → isNat(X)
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) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

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)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)

The TRS R consists of the following rules:

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, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
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__isNat(X) → isNat(X)
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) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

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: SCNP Order with the following components:
Level mapping:
Top level AFS:
A__ISNATKIND(x0, x1) = A__ISNATKIND(x1)

Tags:
A__ISNATKIND has argument tags [0,0] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(A__ISNATKIND(x₁)) = 1
POL(plus(x₁, x₂)) = x₁
POL(s(x₁)) = 1 + x₁

The following usable rules [FROCOS05] were oriented: none

(11) Obligation:

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

A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)

The TRS R consists of the following rules:

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, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
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__isNat(X) → isNat(X)
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) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

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

(12) 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__ISNATKIND(V1)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
A__ISNATKIND(x0, x1) = A__ISNATKIND(x1)

Tags:
A__ISNATKIND has argument tags [1,1] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(A__ISNATKIND(x₁)) = 0
POL(plus(x₁, x₂)) = 1 + x₁

The following usable rules [FROCOS05] were oriented: none

(13) Obligation:

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

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, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
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__isNat(X) → isNat(X)
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) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

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

(14) PisEmptyProof (EQUIVALENT transformation)

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

(15) TRUE

(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__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__U14(tt, V1, V2) → A__ISNAT(V1)

The TRS R consists of the following rules:

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, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
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__isNat(X) → isNat(X)
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) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

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

(17) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A__U12(tt, V1, V2) → A__U13(a__isNatKind(V2), V1, V2)
A__U13(tt, V1, V2) → A__U14(a__isNatKind(V2), V1, V2)
A__U14(tt, V1, V2) → A__U15(a__isNat(V1), V2)
A__U15(tt, V2) → A__ISNAT(V2)
A__ISNAT(plus(V1, V2)) → A__U11(a__isNatKind(V1), V1, V2)
A__U11(tt, V1, V2) → A__U12(a__isNatKind(V1), V1, V2)
A__U14(tt, V1, V2) → A__ISNAT(V1)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
A__U12(x0, x1, x2, x3) = A__U12(x1, x2, x3)
A__U13(x0, x1, x2, x3) = A__U13(x0, x2)
A__U14(x0, x1, x2, x3) = A__U14(x0, x3)
A__U15(x0, x1, x2) = A__U15(x2)
A__ISNAT(x0, x1) = A__ISNAT(x0)
A__U11(x0, x1, x2, x3) = A__U11(x0)
A__U21(x0, x1, x2) = A__U21(x2)
A__U22(x0, x1, x2) = A__U22(x0)

Tags:
A__U12 has argument tags [28,1,4,3] and root tag 7
A__U13 has argument tags [3,29,3,30] and root tag 4
A__U14 has argument tags [3,31,31,3] and root tag 0
A__U15 has argument tags [13,29,2] and root tag 7
A__ISNAT has argument tags [2,1] and root tag 6
A__U11 has argument tags [2,0,0,0] and root tag 0
A__U21 has argument tags [2,16,2] and root tag 6
A__U22 has argument tags [2,8,0] and root tag 6

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(0) = 1
POL(A__ISNAT(x₁)) = x₁
POL(A__U11(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(A__U12(x₁, x₂, x₃)) = 1
POL(A__U13(x₁, x₂, x₃)) = x₃
POL(A__U14(x₁, x₂, x₃)) = x₂
POL(A__U15(x₁, x₂)) = x₁
POL(A__U21(x₁, x₂)) = x₁
POL(A__U22(x₁, x₂)) = x₂
POL(U11(x₁, x₂, x₃)) = 1 + x₃
POL(U12(x₁, x₂, x₃)) = x₃
POL(U13(x₁, x₂, x₃)) = 0
POL(U14(x₁, x₂, x₃)) = 0
POL(U15(x₁, x₂)) = 0
POL(U16(x₁)) = 0
POL(U21(x₁, x₂)) = 0
POL(U22(x₁, x₂)) = 0
POL(U23(x₁)) = 0
POL(U31(x₁, x₂)) = 1
POL(U32(x₁)) = 0
POL(U41(x₁)) = 0
POL(a__U11(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(a__U12(x₁, x₂, x₃)) = x₃
POL(a__U13(x₁, x₂, x₃)) = 0
POL(a__U14(x₁, x₂, x₃)) = 0
POL(a__U15(x₁, x₂)) = 0
POL(a__U16(x₁)) = 0
POL(a__U21(x₁, x₂)) = x₂
POL(a__U22(x₁, x₂)) = x₂
POL(a__U23(x₁)) = 0
POL(a__U31(x₁, x₂)) = 1 + x₂
POL(a__U32(x₁)) = x₁
POL(a__U41(x₁)) = x₁
POL(a__isNat(x₁)) = 1 + x₁
POL(a__isNatKind(x₁)) = 1 + x₁
POL(isNat(x₁)) = 1
POL(isNatKind(x₁)) = x₁
POL(plus(x₁, x₂)) = 1 + x₁ + x₂
POL(s(x₁)) = x₁
POL(tt) = 0

The following usable rules [FROCOS05] were oriented:

a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
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) → isNat(X)
a__U11(tt, V1, V2) → a__U12(a__isNatKind(V1), V1, V2)
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U12(tt, V1, V2) → a__U13(a__isNatKind(V2), V1, V2)
a__U22(tt, V1) → a__U23(a__isNat(V1))
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__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__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(tt) → tt
a__U23(X) → U23(X)
a__U16(tt) → tt
a__U16(X) → U16(X)
a__U31(tt, V2) → a__U32(a__isNatKind(V2))
a__U31(X1, X2) → U31(X1, X2)
a__U41(tt) → tt
a__U41(X) → U41(X)
a__U32(tt) → tt
a__U32(X) → U32(X)

(18) Obligation:

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

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)

The TRS R consists of the following rules:

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, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
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__isNat(X) → isNat(X)
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) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

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

(19) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U21(tt, V1) → A__U22(a__isNatKind(V1), V1)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
A__ISNAT(x0, x1) = A__ISNAT(x1)
A__U21(x0, x1, x2) = A__U21(x0)
A__U22(x0, x1, x2) = A__U22(x0, x2)

Tags:
A__ISNAT has argument tags [4,2] and root tag 0
A__U21 has argument tags [2,3,1] and root tag 2
A__U22 has argument tags [2,4,2] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(0) = 0
POL(A__ISNAT(x₁)) = 0
POL(A__U21(x₁, x₂)) = x₂
POL(A__U22(x₁, x₂)) = 0
POL(U31(x₁, x₂)) = x₁ + x₂
POL(U32(x₁)) = 1 + x₁
POL(U41(x₁)) = 0
POL(a__U31(x₁, x₂)) = 1 + x₁ + x₂
POL(a__U32(x₁)) = 0
POL(a__U41(x₁)) = 1 + x₁
POL(a__isNatKind(x₁)) = x₁
POL(isNatKind(x₁)) = 1
POL(plus(x₁, x₂)) = 0
POL(s(x₁)) = 1 + x₁
POL(tt) = 1

The following usable rules [FROCOS05] were oriented: none

(20) Obligation:

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

A__U22(tt, V1) → A__ISNAT(V1)

The TRS R consists of the following rules:

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, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
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__isNat(X) → isNat(X)
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) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

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

(21) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(22) TRUE

(23) Obligation:

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

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)) → MARK(X1)
MARK(U32(X)) → MARK(X)
MARK(U41(X)) → MARK(X)
MARK(U51(X1, X2)) → A__U51(mark(X1), X2)
A__U51(tt, N) → A__U52(a__isNatKind(N), N)
A__U52(tt, N) → MARK(N)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U61(X1, X2, X3)) → A__U61(mark(X1), X2, X3)
A__U61(tt, M, N) → A__U62(a__isNatKind(M), M, N)
A__U62(tt, M, N) → A__U63(a__isNat(N), M, N)
A__U63(tt, M, N) → A__U64(a__isNatKind(N), M, N)
A__U64(tt, M, N) → A__PLUS(mark(N), mark(M))
A__PLUS(N, 0) → A__U51(a__isNat(N), N)
A__PLUS(N, s(M)) → A__U61(a__isNat(M), M, N)
A__U64(tt, M, N) → MARK(N)
MARK(U61(X1, X2, X3)) → MARK(X1)
MARK(U62(X1, X2, X3)) → A__U62(mark(X1), X2, X3)
MARK(U62(X1, X2, X3)) → MARK(X1)
MARK(U63(X1, X2, X3)) → A__U63(mark(X1), X2, X3)
MARK(U63(X1, X2, X3)) → MARK(X1)
MARK(U64(X1, X2, X3)) → A__U64(mark(X1), X2, X3)
A__U64(tt, M, N) → MARK(M)
MARK(U64(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(s(X)) → MARK(X)

The TRS R consists of the following rules:

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, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
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__isNat(X) → isNat(X)
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) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

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(s(X)) → MARK(X)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
MARK(x0, x1) = MARK(x1)
A__U51(x0, x1, x2) = A__U51(x0, x1)
A__U52(x0, x1, x2) = A__U52(x0, x2)
A__U61(x0, x1, x2, x3) = A__U61(x0, x2, x3)
A__U62(x0, x1, x2, x3) = A__U62(x0)
A__U63(x0, x1, x2, x3) = A__U63(x0)
A__U64(x0, x1, x2, x3) = A__U64(x0, x1, x2, x3)
A__PLUS(x0, x1, x2) = A__PLUS(x0)

Tags:
MARK has argument tags [4,8] and root tag 0
A__U51 has argument tags [8,8,18] and root tag 0
A__U52 has argument tags [6,16,8] and root tag 0
A__U61 has argument tags [8,16,4,0] and root tag 0
A__U62 has argument tags [8,0,3,16] and root tag 0
A__U63 has argument tags [8,0,21,1] and root tag 0
A__U64 has argument tags [8,8,3,0] and root tag 0
A__PLUS has argument tags [8,16,13] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(0) = 1
POL(A__PLUS(x₁, x₂)) = x₁ + x₂
POL(A__U51(x₁, x₂)) = x₂
POL(A__U52(x₁, x₂)) = 0
POL(A__U61(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(A__U62(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(A__U63(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(A__U64(x₁, x₂, x₃)) = x₂ + x₃
POL(MARK(x₁)) = 0
POL(U11(x₁, x₂, x₃)) = x₁
POL(U12(x₁, x₂, x₃)) = x₁
POL(U13(x₁, x₂, x₃)) = x₁
POL(U14(x₁, x₂, x₃)) = x₁
POL(U15(x₁, x₂)) = x₁
POL(U16(x₁)) = x₁
POL(U21(x₁, x₂)) = x₁
POL(U22(x₁, x₂)) = x₁
POL(U23(x₁)) = x₁
POL(U31(x₁, x₂)) = x₁
POL(U32(x₁)) = x₁
POL(U41(x₁)) = x₁
POL(U51(x₁, x₂)) = x₁ + x₂
POL(U52(x₁, x₂)) = x₁ + x₂
POL(U61(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U62(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U63(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U64(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(a__U11(x₁, x₂, x₃)) = x₁
POL(a__U12(x₁, x₂, x₃)) = x₁
POL(a__U13(x₁, x₂, x₃)) = x₁
POL(a__U14(x₁, x₂, x₃)) = x₁
POL(a__U15(x₁, x₂)) = x₁
POL(a__U16(x₁)) = x₁
POL(a__U21(x₁, x₂)) = x₁
POL(a__U22(x₁, x₂)) = x₁
POL(a__U23(x₁)) = x₁
POL(a__U31(x₁, x₂)) = x₁
POL(a__U32(x₁)) = x₁
POL(a__U41(x₁)) = x₁
POL(a__U51(x₁, x₂)) = x₁ + x₂
POL(a__U52(x₁, x₂)) = x₁ + x₂
POL(a__U61(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(a__U62(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(a__U63(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(a__U64(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(a__isNat(x₁)) = 1
POL(a__isNatKind(x₁)) = 1
POL(a__plus(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = 1
POL(isNatKind(x₁)) = 1
POL(mark(x₁)) = x₁
POL(plus(x₁, x₂)) = x₁ + x₂
POL(s(x₁)) = 1 + x₁
POL(tt) = 1

The following usable rules [FROCOS05] were oriented:

mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__plus(N, 0) → a__U51(a__isNat(N), N)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
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) → isNat(X)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
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, V2) → a__U32(a__isNatKind(V2))
a__U31(X1, X2) → U31(X1, X2)
a__U32(tt) → tt
a__U32(X) → U32(X)
a__U41(tt) → tt
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)

(25) Obligation:

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

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)) → MARK(X1)
MARK(U32(X)) → MARK(X)
MARK(U41(X)) → MARK(X)
MARK(U51(X1, X2)) → A__U51(mark(X1), X2)
A__U51(tt, N) → A__U52(a__isNatKind(N), N)
A__U52(tt, N) → MARK(N)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U61(X1, X2, X3)) → A__U61(mark(X1), X2, X3)
A__U61(tt, M, N) → A__U62(a__isNatKind(M), M, N)
A__U62(tt, M, N) → A__U63(a__isNat(N), M, N)
A__U63(tt, M, N) → A__U64(a__isNatKind(N), M, N)
A__U64(tt, M, N) → A__PLUS(mark(N), mark(M))
A__PLUS(N, 0) → A__U51(a__isNat(N), N)
A__PLUS(N, s(M)) → A__U61(a__isNat(M), M, N)
A__U64(tt, M, N) → MARK(N)
MARK(U61(X1, X2, X3)) → MARK(X1)
MARK(U62(X1, X2, X3)) → A__U62(mark(X1), X2, X3)
MARK(U62(X1, X2, X3)) → MARK(X1)
MARK(U63(X1, X2, X3)) → A__U63(mark(X1), X2, X3)
MARK(U63(X1, X2, X3)) → MARK(X1)
MARK(U64(X1, X2, X3)) → A__U64(mark(X1), X2, X3)
A__U64(tt, M, N) → MARK(M)
MARK(U64(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)

The TRS R consists of the following rules:

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, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
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__isNat(X) → isNat(X)
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) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

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(U51(X1, X2)) → MARK(X1)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
MARK(x0, x1) = MARK(x1)
A__U51(x0, x1, x2) = A__U51(x0, x1, x2)
A__U52(x0, x1, x2) = A__U52(x2)
A__U61(x0, x1, x2, x3) = A__U61(x0, x2)
A__U62(x0, x1, x2, x3) = A__U62(x0, x3)
A__U63(x0, x1, x2, x3) = A__U63(x1, x2, x3)
A__U64(x0, x1, x2, x3) = A__U64(x0, x2, x3)
A__PLUS(x0, x1, x2) = A__PLUS(x0, x2)

Tags:
MARK has argument tags [4,0] and root tag 0
A__U51 has argument tags [0,1,0] and root tag 0
A__U52 has argument tags [3,15,0] and root tag 0
A__U61 has argument tags [0,0,0,1] and root tag 0
A__U62 has argument tags [0,1,16,0] and root tag 0
A__U63 has argument tags [12,0,0,0] and root tag 0
A__U64 has argument tags [0,28,0,0] and root tag 0
A__PLUS has argument tags [0,0,0] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(0) = 1
POL(A__PLUS(x₁, x₂)) = x₁
POL(A__U51(x₁, x₂)) = 1 + x₁
POL(A__U52(x₁, x₂)) = x₁
POL(A__U61(x₁, x₂, x₃)) = x₃
POL(A__U62(x₁, x₂, x₃)) = x₂
POL(A__U63(x₁, x₂, x₃)) = x₃
POL(A__U64(x₁, x₂, x₃)) = x₂
POL(MARK(x₁)) = 0
POL(U11(x₁, x₂, x₃)) = x₁
POL(U12(x₁, x₂, x₃)) = x₁
POL(U13(x₁, x₂, x₃)) = x₁
POL(U14(x₁, x₂, x₃)) = x₁
POL(U15(x₁, x₂)) = x₁
POL(U16(x₁)) = x₁
POL(U21(x₁, x₂)) = x₁
POL(U22(x₁, x₂)) = x₁
POL(U23(x₁)) = x₁
POL(U31(x₁, x₂)) = x₁
POL(U32(x₁)) = x₁
POL(U41(x₁)) = x₁
POL(U51(x₁, x₂)) = 1 + x₁ + x₂
POL(U52(x₁, x₂)) = x₁ + x₂
POL(U61(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U62(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U63(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U64(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(a__U11(x₁, x₂, x₃)) = x₁
POL(a__U12(x₁, x₂, x₃)) = x₁
POL(a__U13(x₁, x₂, x₃)) = x₁
POL(a__U14(x₁, x₂, x₃)) = x₁
POL(a__U15(x₁, x₂)) = x₁
POL(a__U16(x₁)) = x₁
POL(a__U21(x₁, x₂)) = x₁
POL(a__U22(x₁, x₂)) = x₁
POL(a__U23(x₁)) = x₁
POL(a__U31(x₁, x₂)) = x₁
POL(a__U32(x₁)) = x₁
POL(a__U41(x₁)) = x₁
POL(a__U51(x₁, x₂)) = 1 + x₁ + x₂
POL(a__U52(x₁, x₂)) = x₁ + x₂
POL(a__U61(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(a__U62(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(a__U63(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(a__U64(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(a__isNat(x₁)) = 0
POL(a__isNatKind(x₁)) = 0
POL(a__plus(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(mark(x₁)) = x₁
POL(plus(x₁, x₂)) = x₁ + x₂
POL(s(x₁)) = x₁
POL(tt) = 0

The following usable rules [FROCOS05] were oriented:

mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__plus(N, 0) → a__U51(a__isNat(N), N)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
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) → isNat(X)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
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, V2) → a__U32(a__isNatKind(V2))
a__U31(X1, X2) → U31(X1, X2)
a__U32(tt) → tt
a__U32(X) → U32(X)
a__U41(tt) → tt
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)

(27) Obligation:

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

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)) → MARK(X1)
MARK(U32(X)) → MARK(X)
MARK(U41(X)) → MARK(X)
MARK(U51(X1, X2)) → A__U51(mark(X1), X2)
A__U51(tt, N) → A__U52(a__isNatKind(N), N)
A__U52(tt, N) → MARK(N)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U61(X1, X2, X3)) → A__U61(mark(X1), X2, X3)
A__U61(tt, M, N) → A__U62(a__isNatKind(M), M, N)
A__U62(tt, M, N) → A__U63(a__isNat(N), M, N)
A__U63(tt, M, N) → A__U64(a__isNatKind(N), M, N)
A__U64(tt, M, N) → A__PLUS(mark(N), mark(M))
A__PLUS(N, 0) → A__U51(a__isNat(N), N)
A__PLUS(N, s(M)) → A__U61(a__isNat(M), M, N)
A__U64(tt, M, N) → MARK(N)
MARK(U61(X1, X2, X3)) → MARK(X1)
MARK(U62(X1, X2, X3)) → A__U62(mark(X1), X2, X3)
MARK(U62(X1, X2, X3)) → MARK(X1)
MARK(U63(X1, X2, X3)) → A__U63(mark(X1), X2, X3)
MARK(U63(X1, X2, X3)) → MARK(X1)
MARK(U64(X1, X2, X3)) → A__U64(mark(X1), X2, X3)
A__U64(tt, M, N) → MARK(M)
MARK(U64(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)

The TRS R consists of the following rules:

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, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
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__isNat(X) → isNat(X)
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) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

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(U61(X1, X2, X3)) → A__U61(mark(X1), X2, X3)
A__PLUS(N, 0) → A__U51(a__isNat(N), N)
A__U64(tt, M, N) → MARK(N)
MARK(U61(X1, X2, X3)) → MARK(X1)
MARK(U62(X1, X2, X3)) → A__U62(mark(X1), X2, X3)
MARK(U62(X1, X2, X3)) → MARK(X1)
MARK(U63(X1, X2, X3)) → A__U63(mark(X1), X2, X3)
MARK(U63(X1, X2, X3)) → MARK(X1)
MARK(U64(X1, X2, X3)) → A__U64(mark(X1), X2, X3)
A__U64(tt, M, N) → MARK(M)
MARK(U64(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)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
MARK(x0, x1) = MARK(x0, x1)
A__U51(x0, x1, x2) = A__U51(x0, x2)
A__U52(x0, x1, x2) = A__U52(x0, x1, x2)
A__U61(x0, x1, x2, x3) = A__U61(x1, x2, x3)
A__U62(x0, x1, x2, x3) = A__U62(x0, x1, x2)
A__U63(x0, x1, x2, x3) = A__U63(x0, x1, x3)
A__U64(x0, x1, x2, x3) = A__U64(x1, x2, x3)
A__PLUS(x0, x1, x2) = A__PLUS(x1, x2)

Tags:
MARK has argument tags [8,0] and root tag 0
A__U51 has argument tags [0,7,0] and root tag 0
A__U52 has argument tags [8,0,0] and root tag 0
A__U61 has argument tags [19,3,18,0] and root tag 4
A__U62 has argument tags [0,17,18,8] and root tag 4
A__U63 has argument tags [18,9,2,0] and root tag 4
A__U64 has argument tags [9,4,18,0] and root tag 4
A__PLUS has argument tags [19,0,18] and root tag 4

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(0) = 1
POL(A__PLUS(x₁, x₂)) = 0
POL(A__U51(x₁, x₂)) = 1
POL(A__U52(x₁, x₂)) = 0
POL(A__U61(x₁, x₂, x₃)) = 1 + x₃
POL(A__U62(x₁, x₂, x₃)) = x₃
POL(A__U63(x₁, x₂, x₃)) = x₁ + x₂
POL(A__U64(x₁, x₂, x₃)) = 1 + x₃
POL(MARK(x₁)) = 0
POL(U11(x₁, x₂, x₃)) = x₁
POL(U12(x₁, x₂, x₃)) = x₁
POL(U13(x₁, x₂, x₃)) = x₁
POL(U14(x₁, x₂, x₃)) = x₁
POL(U15(x₁, x₂)) = x₁
POL(U16(x₁)) = x₁
POL(U21(x₁, x₂)) = x₁
POL(U22(x₁, x₂)) = x₁
POL(U23(x₁)) = x₁
POL(U31(x₁, x₂)) = x₁
POL(U32(x₁)) = x₁
POL(U41(x₁)) = x₁
POL(U51(x₁, x₂)) = 1 + x₂
POL(U52(x₁, x₂)) = x₁ + x₂
POL(U61(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U62(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U63(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U64(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(a__U11(x₁, x₂, x₃)) = x₁
POL(a__U12(x₁, x₂, x₃)) = x₁
POL(a__U13(x₁, x₂, x₃)) = x₁
POL(a__U14(x₁, x₂, x₃)) = x₁
POL(a__U15(x₁, x₂)) = x₁
POL(a__U16(x₁)) = x₁
POL(a__U21(x₁, x₂)) = x₁
POL(a__U22(x₁, x₂)) = x₁
POL(a__U23(x₁)) = x₁
POL(a__U31(x₁, x₂)) = x₁
POL(a__U32(x₁)) = x₁
POL(a__U41(x₁)) = x₁
POL(a__U51(x₁, x₂)) = 1 + x₂
POL(a__U52(x₁, x₂)) = x₁ + x₂
POL(a__U61(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(a__U62(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(a__U63(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(a__U64(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(a__isNat(x₁)) = 0
POL(a__isNatKind(x₁)) = 0
POL(a__plus(x₁, x₂)) = 1 + x₁ + x₂
POL(isNat(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(mark(x₁)) = x₁
POL(plus(x₁, x₂)) = 1 + x₁ + x₂
POL(s(x₁)) = x₁
POL(tt) = 0

The following usable rules [FROCOS05] were oriented:

mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__plus(N, 0) → a__U51(a__isNat(N), N)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
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) → isNat(X)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
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, V2) → a__U32(a__isNatKind(V2))
a__U31(X1, X2) → U31(X1, X2)
a__U32(tt) → tt
a__U32(X) → U32(X)
a__U41(tt) → tt
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)

(29) Obligation:

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

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)) → MARK(X1)
MARK(U32(X)) → MARK(X)
MARK(U41(X)) → MARK(X)
MARK(U51(X1, X2)) → A__U51(mark(X1), X2)
A__U51(tt, N) → A__U52(a__isNatKind(N), N)
A__U52(tt, N) → MARK(N)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U52(X1, X2)) → MARK(X1)
A__U61(tt, M, N) → A__U62(a__isNatKind(M), M, N)
A__U62(tt, M, N) → A__U63(a__isNat(N), M, N)
A__U63(tt, M, N) → A__U64(a__isNatKind(N), M, N)
A__U64(tt, M, N) → A__PLUS(mark(N), mark(M))
A__PLUS(N, s(M)) → A__U61(a__isNat(M), M, N)

The TRS R consists of the following rules:

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, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
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__isNat(X) → isNat(X)
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) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

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

(30) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.

(31) Complex Obligation (AND)

(32) Obligation:

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

A__U62(tt, M, N) → A__U63(a__isNat(N), M, N)
A__U63(tt, M, N) → A__U64(a__isNatKind(N), M, N)
A__U64(tt, M, N) → A__PLUS(mark(N), mark(M))
A__PLUS(N, s(M)) → A__U61(a__isNat(M), M, N)
A__U61(tt, M, N) → A__U62(a__isNatKind(M), M, N)

The TRS R consists of the following rules:

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, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
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__isNat(X) → isNat(X)
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) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

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

(33) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A__U62(tt, M, N) → A__U63(a__isNat(N), M, N)
A__U63(tt, M, N) → A__U64(a__isNatKind(N), M, N)
A__U64(tt, M, N) → A__PLUS(mark(N), mark(M))
A__PLUS(N, s(M)) → A__U61(a__isNat(M), M, N)
A__U61(tt, M, N) → A__U62(a__isNatKind(M), M, N)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
A__U62(x0, x1, x2, x3) = A__U62(x0, x2)
A__U63(x0, x1, x2, x3) = A__U63(x0, x1, x2)
A__U64(x0, x1, x2, x3) = A__U64(x2)
A__PLUS(x0, x1, x2) = A__PLUS(x0)
A__U61(x0, x1, x2, x3) = A__U61(x1, x2)

Tags:
A__U62 has argument tags [1,0,0,0] and root tag 4
A__U63 has argument tags [1,1,0,17] and root tag 3
A__U64 has argument tags [2,30,0,2] and root tag 2
A__PLUS has argument tags [0,3,1] and root tag 1
A__U61 has argument tags [0,1,0,0] and root tag 6

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(0) = 0
POL(A__PLUS(x₁, x₂)) = x₂
POL(A__U61(x₁, x₂, x₃)) = x₃
POL(A__U62(x₁, x₂, x₃)) = 0
POL(A__U63(x₁, x₂, x₃)) = 0
POL(A__U64(x₁, x₂, x₃)) = x₁
POL(U11(x₁, x₂, x₃)) = 0
POL(U12(x₁, x₂, x₃)) = 0
POL(U13(x₁, x₂, x₃)) = 0
POL(U14(x₁, x₂, x₃)) = 0
POL(U15(x₁, x₂)) = x₁
POL(U16(x₁)) = x₁
POL(U21(x₁, x₂)) = 0
POL(U22(x₁, x₂)) = 0
POL(U23(x₁)) = 0
POL(U31(x₁, x₂)) = 1
POL(U32(x₁)) = 0
POL(U41(x₁)) = 0
POL(U51(x₁, x₂)) = x₂
POL(U52(x₁, x₂)) = x₂
POL(U61(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(U62(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(U63(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(U64(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(a__U11(x₁, x₂, x₃)) = 0
POL(a__U12(x₁, x₂, x₃)) = 0
POL(a__U13(x₁, x₂, x₃)) = 0
POL(a__U14(x₁, x₂, x₃)) = 0
POL(a__U15(x₁, x₂)) = x₁
POL(a__U16(x₁)) = x₁
POL(a__U21(x₁, x₂)) = 0
POL(a__U22(x₁, x₂)) = 0
POL(a__U23(x₁)) = 0
POL(a__U31(x₁, x₂)) = 1
POL(a__U32(x₁)) = 0
POL(a__U41(x₁)) = 0
POL(a__U51(x₁, x₂)) = x₂
POL(a__U52(x₁, x₂)) = x₂
POL(a__U61(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(a__U62(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(a__U63(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(a__U64(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(a__isNat(x₁)) = 0
POL(a__isNatKind(x₁)) = 1
POL(a__plus(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = 0
POL(isNatKind(x₁)) = 1
POL(mark(x₁)) = x₁
POL(plus(x₁, x₂)) = x₁ + x₂
POL(s(x₁)) = 1 + x₁
POL(tt) = 0

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) → isNat(X)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__isNatKind(X) → isNatKind(X)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__plus(N, 0) → a__U51(a__isNat(N), N)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
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, V2) → a__U32(a__isNatKind(V2))
a__U31(X1, X2) → U31(X1, X2)
a__U32(tt) → tt
a__U32(X) → U32(X)
a__U41(tt) → tt
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)

(34) Obligation:

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

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, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
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__isNat(X) → isNat(X)
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) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

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

(35) PisEmptyProof (EQUIVALENT transformation)

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

(36) TRUE

(37) 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)) → MARK(X1)
MARK(U32(X)) → MARK(X)
MARK(U41(X)) → MARK(X)
MARK(U51(X1, X2)) → A__U51(mark(X1), X2)
A__U51(tt, N) → A__U52(a__isNatKind(N), N)
A__U52(tt, N) → MARK(N)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U52(X1, X2)) → MARK(X1)

The TRS R consists of the following rules:

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, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
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__isNat(X) → isNat(X)
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) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

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

(38) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U51(X1, X2)) → A__U51(mark(X1), X2)
A__U51(tt, N) → A__U52(a__isNatKind(N), N)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
MARK(x0, x1) = MARK(x1)
A__U51(x0, x1, x2) = A__U51(x0, x2)
A__U52(x0, x1, x2) = A__U52(x2)

Tags:
MARK has argument tags [2,0] and root tag 0
A__U51 has argument tags [4,3,0] and root tag 2
A__U52 has argument tags [2,2,0] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(0) = 0
POL(A__U51(x₁, x₂)) = 0
POL(A__U52(x₁, x₂)) = 0
POL(MARK(x₁)) = 0
POL(U11(x₁, x₂, x₃)) = x₁ + x₂
POL(U12(x₁, x₂, x₃)) = x₁ + x₃
POL(U13(x₁, x₂, x₃)) = x₁ + x₃
POL(U14(x₁, x₂, x₃)) = 1 + x₁
POL(U15(x₁, x₂)) = x₁ + x₂
POL(U16(x₁)) = x₁
POL(U21(x₁, x₂)) = x₁ + x₂
POL(U22(x₁, x₂)) = x₁ + x₂
POL(U23(x₁)) = x₁
POL(U31(x₁, x₂)) = x₁
POL(U32(x₁)) = x₁
POL(U41(x₁)) = x₁
POL(U51(x₁, x₂)) = 1 + x₁ + x₂
POL(U52(x₁, x₂)) = x₁ + x₂
POL(U61(x₁, x₂, x₃)) = x₃
POL(U62(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U63(x₁, x₂, x₃)) = x₁ + x₃
POL(U64(x₁, x₂, x₃)) = 1 + x₁ + x₃
POL(a__U11(x₁, x₂, x₃)) = 0
POL(a__U12(x₁, x₂, x₃)) = 1 + x₁
POL(a__U13(x₁, x₂, x₃)) = x₂
POL(a__U14(x₁, x₂, x₃)) = x₂ + x₃
POL(a__U15(x₁, x₂)) = 0
POL(a__U16(x₁)) = x₁
POL(a__U21(x₁, x₂)) = 0
POL(a__U22(x₁, x₂)) = 0
POL(a__U23(x₁)) = 1
POL(a__U31(x₁, x₂)) = 0
POL(a__U32(x₁)) = 0
POL(a__U41(x₁)) = 1
POL(a__U51(x₁, x₂)) = x₁
POL(a__U52(x₁, x₂)) = x₂
POL(a__U61(x₁, x₂, x₃)) = x₂
POL(a__U62(x₁, x₂, x₃)) = x₃
POL(a__U63(x₁, x₂, x₃)) = x₂ + x₃
POL(a__U64(x₁, x₂, x₃)) = x₂
POL(a__isNat(x₁)) = x₁
POL(a__isNatKind(x₁)) = 1
POL(a__plus(x₁, x₂)) = 0
POL(isNat(x₁)) = x₁
POL(isNatKind(x₁)) = 0
POL(mark(x₁)) = 0
POL(plus(x₁, x₂)) = x₁ + x₂
POL(s(x₁)) = 1
POL(tt) = 0

The following usable rules [FROCOS05] were oriented: none

(39) 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(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)) → MARK(X1)
MARK(U32(X)) → MARK(X)
MARK(U41(X)) → MARK(X)
A__U52(tt, N) → MARK(N)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U52(X1, X2)) → MARK(X1)

The TRS R consists of the following rules:

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, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
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__isNat(X) → isNat(X)
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) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

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

(40) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(U22(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U41(X)) → MARK(X)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
MARK(x0, x1) = MARK(x1)
A__U52(x0, x1, x2) = A__U52(x0, x2)

Tags:
MARK has argument tags [1,6] and root tag 1
A__U52 has argument tags [6,2,6] and root tag 1

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(0) = 0
POL(A__U52(x₁, x₂)) = 0
POL(MARK(x₁)) = 0
POL(U11(x₁, x₂, x₃)) = x₁ + x₂
POL(U12(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U13(x₁, x₂, x₃)) = x₁
POL(U14(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(U15(x₁, x₂)) = x₁ + x₂
POL(U16(x₁)) = x₁
POL(U21(x₁, x₂)) = x₁ + x₂
POL(U22(x₁, x₂)) = 1 + x₁ + x₂
POL(U23(x₁)) = x₁
POL(U31(x₁, x₂)) = 1 + x₁
POL(U32(x₁)) = x₁
POL(U41(x₁)) = 1 + x₁
POL(U51(x₁, x₂)) = 1
POL(U52(x₁, x₂)) = x₁ + x₂
POL(U61(x₁, x₂, x₃)) = 1 + x₁ + x₃
POL(U62(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U63(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U64(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(a__U11(x₁, x₂, x₃)) = 1 + x₃
POL(a__U12(x₁, x₂, x₃)) = 1 + x₁
POL(a__U13(x₁, x₂, x₃)) = 1 + x₁ + x₂
POL(a__U14(x₁, x₂, x₃)) = x₁
POL(a__U15(x₁, x₂)) = 0
POL(a__U16(x₁)) = 1 + x₁
POL(a__U21(x₁, x₂)) = 1 + x₁
POL(a__U22(x₁, x₂)) = x₁
POL(a__U23(x₁)) = 1 + x₁
POL(a__U31(x₁, x₂)) = x₁
POL(a__U32(x₁)) = 0
POL(a__U41(x₁)) = 0
POL(a__U51(x₁, x₂)) = 1
POL(a__U52(x₁, x₂)) = 1 + x₁
POL(a__U61(x₁, x₂, x₃)) = x₁ + x₂
POL(a__U62(x₁, x₂, x₃)) = 0
POL(a__U63(x₁, x₂, x₃)) = 0
POL(a__U64(x₁, x₂, x₃)) = 0
POL(a__isNat(x₁)) = 1 + x₁
POL(a__isNatKind(x₁)) = 0
POL(a__plus(x₁, x₂)) = 0
POL(isNat(x₁)) = 1
POL(isNatKind(x₁)) = 1
POL(mark(x₁)) = 1 + x₁
POL(plus(x₁, x₂)) = 1
POL(s(x₁)) = 0
POL(tt) = 0

The following usable rules [FROCOS05] were oriented: none

(41) 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(U15(X1, X2)) → MARK(X1)
MARK(U16(X)) → MARK(X)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U23(X)) → MARK(X)
MARK(U32(X)) → MARK(X)
A__U52(tt, N) → MARK(N)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U52(X1, X2)) → MARK(X1)

The TRS R consists of the following rules:

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, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
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__isNat(X) → isNat(X)
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) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

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

(42) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(U11(X1, X2, X3)) → MARK(X1)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
MARK(x0, x1) = MARK(x1)
A__U52(x0, x1, x2) = A__U52(x0, x2)

Tags:
MARK has argument tags [7,5] and root tag 0
A__U52 has argument tags [2,7,5] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(0) = 1
POL(A__U52(x₁, x₂)) = 0
POL(MARK(x₁)) = 1
POL(U11(x₁, x₂, x₃)) = 1 + x₁ + x₃
POL(U12(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U13(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U14(x₁, x₂, x₃)) = 1 + x₁
POL(U15(x₁, x₂)) = x₁ + x₂
POL(U16(x₁)) = x₁
POL(U21(x₁, x₂)) = x₁ + x₂
POL(U22(x₁, x₂)) = 1 + x₂
POL(U23(x₁)) = x₁
POL(U31(x₁, x₂)) = 1 + x₁
POL(U32(x₁)) = x₁
POL(U41(x₁)) = 0
POL(U51(x₁, x₂)) = 1 + x₂
POL(U52(x₁, x₂)) = x₁ + x₂
POL(U61(x₁, x₂, x₃)) = x₁ + x₂
POL(U62(x₁, x₂, x₃)) = 1 + x₁ + x₂
POL(U63(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(U64(x₁, x₂, x₃)) = x₂
POL(a__U11(x₁, x₂, x₃)) = 0
POL(a__U12(x₁, x₂, x₃)) = 0
POL(a__U13(x₁, x₂, x₃)) = 1 + x₁
POL(a__U14(x₁, x₂, x₃)) = 0
POL(a__U15(x₁, x₂)) = 0
POL(a__U16(x₁)) = 1 + x₁
POL(a__U21(x₁, x₂)) = 0
POL(a__U22(x₁, x₂)) = 1 + x₁
POL(a__U23(x₁)) = 0
POL(a__U31(x₁, x₂)) = 1 + x₂
POL(a__U32(x₁)) = 0
POL(a__U41(x₁)) = 1 + x₁
POL(a__U51(x₁, x₂)) = x₂
POL(a__U52(x₁, x₂)) = 1 + x₁ + x₂
POL(a__U61(x₁, x₂, x₃)) = 0
POL(a__U62(x₁, x₂, x₃)) = 0
POL(a__U63(x₁, x₂, x₃)) = 1 + x₂
POL(a__U64(x₁, x₂, x₃)) = 1 + x₁ + x₃
POL(a__isNat(x₁)) = 1
POL(a__isNatKind(x₁)) = 1
POL(a__plus(x₁, x₂)) = 1 + x₂
POL(isNat(x₁)) = x₁
POL(isNatKind(x₁)) = 1 + x₁
POL(mark(x₁)) = 1
POL(plus(x₁, x₂)) = 0
POL(s(x₁)) = 0
POL(tt) = 0

The following usable rules [FROCOS05] were oriented: none

(43) Obligation:

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

MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U15(X1, X2)) → MARK(X1)
MARK(U16(X)) → MARK(X)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U23(X)) → MARK(X)
MARK(U32(X)) → MARK(X)
A__U52(tt, N) → MARK(N)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U52(X1, X2)) → MARK(X1)

The TRS R consists of the following rules:

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, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
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__isNat(X) → isNat(X)
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) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

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

(44) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(U32(X)) → MARK(X)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
MARK(x0, x1) = MARK(x0)
A__U52(x0, x1, x2) = A__U52(x0)

Tags:
MARK has argument tags [0,4] and root tag 0
A__U52 has argument tags [0,4,6] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(0) = 0
POL(A__U52(x₁, x₂)) = x₂
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U12(x₁, x₂, x₃)) = x₁
POL(U13(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U14(x₁, x₂, x₃)) = 1 + x₂
POL(U15(x₁, x₂)) = x₁ + x₂
POL(U16(x₁)) = x₁
POL(U21(x₁, x₂)) = x₁ + x₂
POL(U22(x₁, x₂)) = x₂
POL(U23(x₁)) = x₁
POL(U31(x₁, x₂)) = 1 + x₂
POL(U32(x₁)) = 1 + x₁
POL(U41(x₁)) = 1 + x₁
POL(U51(x₁, x₂)) = 1 + x₁ + x₂
POL(U52(x₁, x₂)) = x₁ + x₂
POL(U61(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U62(x₁, x₂, x₃)) = x₂ + x₃
POL(U63(x₁, x₂, x₃)) = 1 + x₂
POL(U64(x₁, x₂, x₃)) = 1 + x₃
POL(a__U11(x₁, x₂, x₃)) = x₂
POL(a__U12(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(a__U13(x₁, x₂, x₃)) = x₂
POL(a__U14(x₁, x₂, x₃)) = x₃
POL(a__U15(x₁, x₂)) = 0
POL(a__U16(x₁)) = 0
POL(a__U21(x₁, x₂)) = 0
POL(a__U22(x₁, x₂)) = x₁ + x₂
POL(a__U23(x₁)) = 1
POL(a__U31(x₁, x₂)) = 0
POL(a__U32(x₁)) = 1
POL(a__U41(x₁)) = 0
POL(a__U51(x₁, x₂)) = 0
POL(a__U52(x₁, x₂)) = x₂
POL(a__U61(x₁, x₂, x₃)) = 1
POL(a__U62(x₁, x₂, x₃)) = 1
POL(a__U63(x₁, x₂, x₃)) = x₃
POL(a__U64(x₁, x₂, x₃)) = 1 + x₁
POL(a__isNat(x₁)) = 0
POL(a__isNatKind(x₁)) = 0
POL(a__plus(x₁, x₂)) = 0
POL(isNat(x₁)) = 1
POL(isNatKind(x₁)) = 1 + x₁
POL(mark(x₁)) = 0
POL(plus(x₁, x₂)) = x₂
POL(s(x₁)) = 0
POL(tt) = 0

The following usable rules [FROCOS05] were oriented: none

(45) Obligation:

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

MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U15(X1, X2)) → MARK(X1)
MARK(U16(X)) → MARK(X)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U23(X)) → MARK(X)
A__U52(tt, N) → MARK(N)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U52(X1, X2)) → MARK(X1)

The TRS R consists of the following rules:

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, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
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__isNat(X) → isNat(X)
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) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

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

(46) 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)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
MARK(x0, x1) = MARK(x1)
A__U52(x0, x1, x2) = A__U52(x0, x2)

Tags:
MARK has argument tags [5,4] and root tag 1
A__U52 has argument tags [4,0,3] and root tag 1

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(0) = 0
POL(A__U52(x₁, x₂)) = x₂
POL(MARK(x₁)) = 0
POL(U11(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(U12(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U13(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U14(x₁, x₂, x₃)) = 1 + x₁ + x₂
POL(U15(x₁, x₂)) = x₁ + x₂
POL(U16(x₁)) = x₁
POL(U21(x₁, x₂)) = x₁ + x₂
POL(U22(x₁, x₂)) = x₁ + x₂
POL(U23(x₁)) = x₁
POL(U31(x₁, x₂)) = 1
POL(U32(x₁)) = x₁
POL(U41(x₁)) = 1 + x₁
POL(U51(x₁, x₂)) = 1 + x₁
POL(U52(x₁, x₂)) = x₁ + x₂
POL(U61(x₁, x₂, x₃)) = 1 + x₁ + x₃
POL(U62(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U63(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U64(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(a__U11(x₁, x₂, x₃)) = 0
POL(a__U12(x₁, x₂, x₃)) = 0
POL(a__U13(x₁, x₂, x₃)) = 1
POL(a__U14(x₁, x₂, x₃)) = x₂
POL(a__U15(x₁, x₂)) = 1 + x₁
POL(a__U16(x₁)) = 1
POL(a__U21(x₁, x₂)) = 1
POL(a__U22(x₁, x₂)) = 0
POL(a__U23(x₁)) = 1 + x₁
POL(a__U31(x₁, x₂)) = 0
POL(a__U32(x₁)) = 0
POL(a__U41(x₁)) = 0
POL(a__U51(x₁, x₂)) = 1
POL(a__U52(x₁, x₂)) = 1
POL(a__U61(x₁, x₂, x₃)) = 0
POL(a__U62(x₁, x₂, x₃)) = 0
POL(a__U63(x₁, x₂, x₃)) = 0
POL(a__U64(x₁, x₂, x₃)) = 0
POL(a__isNat(x₁)) = x₁
POL(a__isNatKind(x₁)) = 0
POL(a__plus(x₁, x₂)) = x₁
POL(isNat(x₁)) = 1 + x₁
POL(isNatKind(x₁)) = 1 + x₁
POL(mark(x₁)) = 0
POL(plus(x₁, x₂)) = 0
POL(s(x₁)) = 0
POL(tt) = 0

The following usable rules [FROCOS05] were oriented: none

(47) Obligation:

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

MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U15(X1, X2)) → MARK(X1)
MARK(U16(X)) → MARK(X)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U23(X)) → MARK(X)
A__U52(tt, N) → MARK(N)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U52(X1, X2)) → MARK(X1)

The TRS R consists of the following rules:

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, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
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__isNat(X) → isNat(X)
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) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

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

(48) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(U16(X)) → MARK(X)
MARK(U23(X)) → MARK(X)
A__U52(tt, N) → MARK(N)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U52(X1, X2)) → MARK(X1)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
MARK(x0, x1) = MARK(x1)
A__U52(x0, x1, x2) = A__U52(x0)

Tags:
MARK has argument tags [4,5] and root tag 1
A__U52 has argument tags [2,2,3] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(0) = 1
POL(A__U52(x₁, x₂)) = 1 + x₂
POL(MARK(x₁)) = 1
POL(U11(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U12(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U13(x₁, x₂, x₃)) = x₁ + x₂
POL(U14(x₁, x₂, x₃)) = x₁ + x₃
POL(U15(x₁, x₂)) = x₁ + x₂
POL(U16(x₁)) = 1 + x₁
POL(U21(x₁, x₂)) = x₁ + x₂
POL(U22(x₁, x₂)) = 0
POL(U23(x₁)) = 1 + x₁
POL(U31(x₁, x₂)) = x₁ + x₂
POL(U32(x₁)) = 1
POL(U41(x₁)) = 1 + x₁
POL(U51(x₁, x₂)) = x₂
POL(U52(x₁, x₂)) = 1 + x₁ + x₂
POL(U61(x₁, x₂, x₃)) = 1 + x₃
POL(U62(x₁, x₂, x₃)) = x₁ + x₃
POL(U63(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(U64(x₁, x₂, x₃)) = x₁ + x₃
POL(a__U11(x₁, x₂, x₃)) = 0
POL(a__U12(x₁, x₂, x₃)) = 0
POL(a__U13(x₁, x₂, x₃)) = 0
POL(a__U14(x₁, x₂, x₃)) = x₂
POL(a__U15(x₁, x₂)) = 1 + x₁ + x₂
POL(a__U16(x₁)) = 0
POL(a__U21(x₁, x₂)) = 0
POL(a__U22(x₁, x₂)) = 1
POL(a__U23(x₁)) = 0
POL(a__U31(x₁, x₂)) = 1
POL(a__U32(x₁)) = x₁
POL(a__U41(x₁)) = 0
POL(a__U51(x₁, x₂)) = 1 + x₁
POL(a__U52(x₁, x₂)) = 0
POL(a__U61(x₁, x₂, x₃)) = 0
POL(a__U62(x₁, x₂, x₃)) = x₃
POL(a__U63(x₁, x₂, x₃)) = x₂
POL(a__U64(x₁, x₂, x₃)) = 1 + x₁ + x₂
POL(a__isNat(x₁)) = 0
POL(a__isNatKind(x₁)) = 0
POL(a__plus(x₁, x₂)) = 1 + x₂
POL(isNat(x₁)) = 1 + x₁
POL(isNatKind(x₁)) = 1 + x₁
POL(mark(x₁)) = 0
POL(plus(x₁, x₂)) = 1
POL(s(x₁)) = x₁
POL(tt) = 1

The following usable rules [FROCOS05] were oriented: none

(49) Obligation:

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

MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U15(X1, X2)) → MARK(X1)
MARK(U21(X1, X2)) → MARK(X1)

The TRS R consists of the following rules:

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, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
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__isNat(X) → isNat(X)
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) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

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

(50) 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)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
MARK(x0, x1) = MARK(x1)

Tags:
MARK has argument tags [1,1] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(MARK(x₁)) = 1
POL(U13(x₁, x₂, x₃)) = 1 + x₁
POL(U15(x₁, x₂)) = x₁
POL(U21(x₁, x₂)) = x₁

The following usable rules [FROCOS05] were oriented: none

(51) Obligation:

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

MARK(U15(X1, X2)) → MARK(X1)
MARK(U21(X1, X2)) → MARK(X1)

The TRS R consists of the following rules:

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, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
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__isNat(X) → isNat(X)
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) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

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

(52) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(U15(X1, X2)) → MARK(X1)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
MARK(x0, x1) = MARK(x1)

Tags:
MARK has argument tags [0,0] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(MARK(x₁)) = 1
POL(U15(x₁, x₂)) = 1 + x₁
POL(U21(x₁, x₂)) = x₁

The following usable rules [FROCOS05] were oriented: none

(53) Obligation:

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

MARK(U21(X1, X2)) → MARK(X1)

The TRS R consists of the following rules:

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, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
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__isNat(X) → isNat(X)
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) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

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

(54) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(U21(X1, X2)) → MARK(X1)

POL(MARK(x₁)) = 0
POL(U21(x₁, x₂)) = 1 + x₁

The following usable rules [FROCOS05] were oriented: none

(55) Obligation:

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

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, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
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__isNat(X) → isNat(X)
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) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

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

(56) 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) QDPOrderProof (EQUIVALENT transformation)

(13) Obligation:

(14) PisEmptyProof (EQUIVALENT transformation)

(15) TRUE

(16) Obligation:

(17) QDPOrderProof (EQUIVALENT transformation)

(18) Obligation:

(19) QDPOrderProof (EQUIVALENT transformation)

(20) Obligation:

(21) DependencyGraphProof (EQUIVALENT transformation)

(22) TRUE

(23) Obligation:

(24) QDPOrderProof (EQUIVALENT transformation)

(25) Obligation:

(26) QDPOrderProof (EQUIVALENT transformation)

(27) Obligation:

(28) QDPOrderProof (EQUIVALENT transformation)

(29) Obligation:

(30) DependencyGraphProof (EQUIVALENT transformation)

(31) Complex Obligation (AND)

(32) Obligation:

(33) QDPOrderProof (EQUIVALENT transformation)

(34) Obligation:

(35) PisEmptyProof (EQUIVALENT transformation)

(36) TRUE

(37) Obligation:

(38) QDPOrderProof (EQUIVALENT transformation)

(39) Obligation:

(40) QDPOrderProof (EQUIVALENT transformation)

(41) Obligation:

(42) QDPOrderProof (EQUIVALENT transformation)

(43) Obligation:

(44) QDPOrderProof (EQUIVALENT transformation)

(45) Obligation:

(46) QDPOrderProof (EQUIVALENT transformation)

(47) Obligation:

(48) QDPOrderProof (EQUIVALENT transformation)

(49) Obligation:

(50) QDPOrderProof (EQUIVALENT transformation)

(51) Obligation:

(52) QDPOrderProof (EQUIVALENT transformation)

(53) Obligation:

(54) QDPOrderProof (EQUIVALENT transformation)

(55) Obligation:

(56) PisEmptyProof (EQUIVALENT transformation)

(57) TRUE