Termination w.r.t. Q proof of /tmp/tmpqibfJu/PEANO_complete

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

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 DependencyGraphProof (⇔)

↳4 QDP

↳5 QDPOrderProof (⇔)

↳6 QDP

↳7 DependencyGraphProof (⇔)

↳8 AND

    ↳9 QDP

        ↳10 QDPOrderProof (⇔)

        ↳11 QDP

        ↳12 PisEmptyProof (⇔)

        ↳13 TRUE

    ↳14 QDP

        ↳15 QDPOrderProof (⇔)

        ↳16 QDP

        ↳17 QDPOrderProof (⇔)

        ↳18 QDP

        ↳19 QDPOrderProof (⇔)

        ↳20 QDP

        ↳21 QDPOrderProof (⇔)

        ↳22 QDP

        ↳23 DependencyGraphProof (⇔)

        ↳24 AND

            ↳25 QDP

                ↳26 QDPOrderProof (⇔)

                ↳27 QDP

                ↳28 PisEmptyProof (⇔)

                ↳29 TRUE

            ↳30 QDP

                ↳31 QDPOrderProof (⇔)

                ↳32 QDP

                ↳33 PisEmptyProof (⇔)

                ↳34 TRUE

            ↳35 QDP

                ↳36 QDPOrderProof (⇔)

                ↳37 QDP

                ↳38 QDPOrderProof (⇔)

                ↳39 QDP

                ↳40 QDPOrderProof (⇔)

                ↳41 QDP

                ↳42 PisEmptyProof (⇔)

                ↳43 TRUE

(0) Obligation:

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

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, N) → mark(N)
a__U41(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__plus(N, 0) → a__U31(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U41(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
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) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2) → U31(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

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

(2) Obligation:

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

A__U11(tt, V1, V2) → A__U12(a__isNat(V1), V2)
A__U11(tt, V1, V2) → A__ISNAT(V1)
A__U12(tt, V2) → A__U13(a__isNat(V2))
A__U12(tt, V2) → A__ISNAT(V2)
A__U21(tt, V1) → A__U22(a__isNat(V1))
A__U21(tt, V1) → A__ISNAT(V1)
A__U31(tt, N) → MARK(N)
A__U41(tt, M, N) → A__PLUS(mark(N), mark(M))
A__U41(tt, M, N) → MARK(N)
A__U41(tt, M, N) → MARK(M)
A__AND(tt, X) → MARK(X)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__ISNAT(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(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__AND(a__isNatKind(V1), isNatKind(V2))
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__PLUS(N, 0) → A__U31(a__and(a__isNat(N), isNatKind(N)), N)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, 0) → A__ISNAT(N)
A__PLUS(N, s(M)) → A__U41(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
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)) → A__U12(mark(X1), X2)
MARK(U12(X1, X2)) → MARK(X1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U13(X)) → A__U13(mark(X))
MARK(U13(X)) → MARK(X)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X)) → A__U22(mark(X))
MARK(U22(X)) → MARK(X)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
MARK(U41(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(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(s(X)) → MARK(X)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, N) → mark(N)
a__U41(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__plus(N, 0) → a__U31(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U41(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
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) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2) → U31(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

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

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.

(4) Obligation:

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

A__U12(tt, V2) → A__ISNAT(V2)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__U11(tt, V1, V2) → A__U12(a__isNat(V1), V2)
A__U11(tt, V1, V2) → A__ISNAT(V1)
A__ISNAT(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__AND(tt, X) → MARK(X)
MARK(U11(X1, X2, X3)) → A__U11(mark(X1), X2, X3)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → A__U12(mark(X1), X2)
MARK(U12(X1, X2)) → MARK(X1)
MARK(isNat(X)) → A__ISNAT(X)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U21(tt, V1) → A__ISNAT(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U13(X)) → MARK(X)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X)) → MARK(X)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
A__U31(tt, N) → MARK(N)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
A__U41(tt, M, N) → A__PLUS(mark(N), mark(M))
A__PLUS(N, 0) → A__U31(a__and(a__isNat(N), isNatKind(N)), N)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, 0) → A__ISNAT(N)
A__PLUS(N, s(M)) → A__U41(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U41(tt, M, N) → MARK(N)
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__PLUS(N, s(M)) → A__ISNAT(M)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(s(X)) → MARK(X)
A__U41(tt, M, N) → MARK(M)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, N) → mark(N)
a__U41(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__plus(N, 0) → a__U31(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U41(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
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) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2) → U31(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

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

(5) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A__PLUS(N, 0) → A__U31(a__and(a__isNat(N), isNatKind(N)), N)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, 0) → A__ISNAT(N)
A__U41(tt, M, N) → MARK(N)
MARK(U41(X1, X2, X3)) → MARK(X1)
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__PLUS(N, s(M)) → A__ISNAT(M)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
A__U41(tt, M, N) → MARK(M)

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) = A__U12(x0, x1)
A__ISNAT(x0, x1) = A__ISNAT(x0)
A__U11(x0, x1, x2, x3) = A__U11(x1)
A__AND(x0, x1, x2) = A__AND(x1, x2)
MARK(x0, x1) = MARK(x1)
A__ISNATKIND(x0, x1) = A__ISNATKIND(x0)
A__U21(x0, x1, x2) = A__U21(x0, x1)
A__U31(x0, x1, x2) = A__U31(x0, x1)
A__U41(x0, x1, x2, x3) = A__U41(x0, x2, x3)
A__PLUS(x0, x1, x2) = A__PLUS(x0)

Tags:
A__U12 has argument tags [2,2,17] and root tag 0
A__ISNAT has argument tags [2,24] and root tag 0
A__U11 has argument tags [3,2,25,25] and root tag 0
A__AND has argument tags [4,1,2] and root tag 0
MARK has argument tags [9,2] and root tag 0
A__ISNATKIND has argument tags [2,27] and root tag 0
A__U21 has argument tags [2,0,1] and root tag 0
A__U31 has argument tags [2,0,16] and root tag 0
A__U41 has argument tags [2,7,1,0] and root tag 0
A__PLUS has argument tags [2,0,0] 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__AND(x₁, x₂)) = x₁
POL(A__ISNAT(x₁)) = 0
POL(A__ISNATKIND(x₁)) = 0
POL(A__PLUS(x₁, x₂)) = 1 + x₁ + x₂
POL(A__U11(x₁, x₂, x₃)) = x₁ + x₃
POL(A__U12(x₁, x₂)) = x₁
POL(A__U21(x₁, x₂)) = x₁
POL(A__U31(x₁, x₂)) = x₂
POL(A__U41(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(MARK(x₁)) = 0
POL(U11(x₁, x₂, x₃)) = x₁
POL(U12(x₁, x₂)) = x₁
POL(U13(x₁)) = x₁
POL(U21(x₁, x₂)) = x₁
POL(U22(x₁)) = x₁
POL(U31(x₁, x₂)) = x₁ + x₂
POL(U41(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(a__U11(x₁, x₂, x₃)) = x₁
POL(a__U12(x₁, x₂)) = x₁
POL(a__U13(x₁)) = x₁
POL(a__U21(x₁, x₂)) = x₁
POL(a__U22(x₁)) = x₁
POL(a__U31(x₁, x₂)) = x₁ + x₂
POL(a__U41(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(a__and(x₁, x₂)) = x₁ + x₂
POL(a__isNat(x₁)) = 0
POL(a__isNatKind(x₁)) = 0
POL(a__plus(x₁, x₂)) = 1 + x₁ + x₂
POL(and(x₁, x₂)) = 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:

a__isNatKind(0) → tt
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
a__U31(tt, N) → mark(N)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__plus(N, 0) → a__U31(a__and(a__isNat(N), isNatKind(N)), N)
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__and(tt, X) → mark(X)
mark(isNatKind(X)) → a__isNatKind(X)
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(X) → isNatKind(X)
a__and(X1, X2) → and(X1, X2)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(X) → isNat(X)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__plus(N, s(M)) → a__U41(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U41(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U22(tt) → tt
a__U22(X) → U22(X)
a__U31(X1, X2) → U31(X1, X2)
a__plus(X1, X2) → plus(X1, X2)
a__U13(tt) → tt
a__U13(X) → U13(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)

(6) Obligation:

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

A__U12(tt, V2) → A__ISNAT(V2)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__U11(tt, V1, V2) → A__U12(a__isNat(V1), V2)
A__U11(tt, V1, V2) → A__ISNAT(V1)
A__ISNAT(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__AND(tt, X) → MARK(X)
MARK(U11(X1, X2, X3)) → A__U11(mark(X1), X2, X3)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → A__U12(mark(X1), X2)
MARK(U12(X1, X2)) → MARK(X1)
MARK(isNat(X)) → A__ISNAT(X)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U21(tt, V1) → A__ISNAT(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U13(X)) → MARK(X)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X)) → MARK(X)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
A__U31(tt, N) → MARK(N)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
A__U41(tt, M, N) → A__PLUS(mark(N), mark(M))
A__PLUS(N, s(M)) → A__U41(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(s(X)) → MARK(X)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, N) → mark(N)
a__U41(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__plus(N, 0) → a__U31(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U41(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
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) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2) → U31(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

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

(8) Complex Obligation (AND)

(9) Obligation:

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

A__PLUS(N, s(M)) → A__U41(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U41(tt, M, N) → A__PLUS(mark(N), mark(M))

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, N) → mark(N)
a__U41(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__plus(N, 0) → a__U31(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U41(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
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) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2) → U31(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

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

(10) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A__PLUS(N, s(M)) → A__U41(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U41(tt, M, N) → A__PLUS(mark(N), mark(M))

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
A__PLUS(x0, x1, x2) = A__PLUS(x2)
A__U41(x0, x1, x2, x3) = A__U41(x0)

Tags:
A__PLUS has argument tags [5,5,5] and root tag 1
A__U41 has argument tags [6,6,2,2] 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__PLUS(x₁, x₂)) = x₁
POL(A__U41(x₁, x₂, x₃)) = x₂
POL(U11(x₁, x₂, x₃)) = x₁
POL(U12(x₁, x₂)) = 0
POL(U13(x₁)) = x₁
POL(U21(x₁, x₂)) = 0
POL(U22(x₁)) = 0
POL(U31(x₁, x₂)) = x₂
POL(U41(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(a__U11(x₁, x₂, x₃)) = x₁
POL(a__U12(x₁, x₂)) = 0
POL(a__U13(x₁)) = x₁
POL(a__U21(x₁, x₂)) = 0
POL(a__U22(x₁)) = 0
POL(a__U31(x₁, x₂)) = x₂
POL(a__U41(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(a__and(x₁, x₂)) = x₂
POL(a__isNat(x₁)) = 0
POL(a__isNatKind(x₁)) = 0
POL(a__plus(x₁, x₂)) = x₁ + x₂
POL(and(x₁, x₂)) = x₂
POL(isNat(x₁)) = 0
POL(isNatKind(x₁)) = 0
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__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(X) → isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
a__U31(tt, N) → mark(N)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__plus(N, 0) → a__U31(a__and(a__isNat(N), isNatKind(N)), N)
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__and(tt, X) → mark(X)
mark(isNatKind(X)) → a__isNatKind(X)
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__and(X1, X2) → and(X1, X2)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__plus(N, s(M)) → a__U41(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U41(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__isNatKind(0) → tt
a__isNatKind(X) → isNatKind(X)
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U22(tt) → tt
a__U22(X) → U22(X)
a__U31(X1, X2) → U31(X1, X2)
a__plus(X1, X2) → plus(X1, X2)
a__U13(tt) → tt
a__U13(X) → U13(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)

(11) Obligation:

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

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, N) → mark(N)
a__U41(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__plus(N, 0) → a__U31(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U41(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
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) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2) → U31(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

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

(12) PisEmptyProof (EQUIVALENT transformation)

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

(13) TRUE

(14) Obligation:

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

A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__U11(tt, V1, V2) → A__U12(a__isNat(V1), V2)
A__U12(tt, V2) → A__ISNAT(V2)
A__ISNAT(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__AND(tt, X) → MARK(X)
MARK(U11(X1, X2, X3)) → A__U11(mark(X1), X2, X3)
A__U11(tt, V1, V2) → A__ISNAT(V1)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U21(tt, V1) → A__ISNAT(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → A__U12(mark(X1), X2)
MARK(U12(X1, X2)) → MARK(X1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U13(X)) → MARK(X)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X)) → MARK(X)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
A__U31(tt, N) → MARK(N)
MARK(U31(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(s(X)) → MARK(X)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, N) → mark(N)
a__U41(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__plus(N, 0) → a__U31(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U41(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
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) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2) → U31(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

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

(15) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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:
A__ISNAT(x0, x1) = A__ISNAT(x0)
A__U11(x0, x1, x2, x3) = A__U11(x0)
A__U12(x0, x1, x2) = A__U12(x0, x1)
A__AND(x0, x1, x2) = A__AND(x1, x2)
MARK(x0, x1) = MARK(x1)
A__ISNATKIND(x0, x1) = A__ISNATKIND(x0)
A__U21(x0, x1, x2) = A__U21(x0, x1)
A__U31(x0, x1, x2) = A__U31(x0, x2)

Tags:
A__ISNAT has argument tags [2,28] and root tag 0
A__U11 has argument tags [2,0,3,15] and root tag 0
A__U12 has argument tags [2,0,0] and root tag 0
A__AND has argument tags [0,2,2] and root tag 0
MARK has argument tags [1,2] and root tag 0
A__ISNATKIND has argument tags [2,2] and root tag 0
A__U21 has argument tags [2,0,0] and root tag 0
A__U31 has argument tags [2,17,0] 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__AND(x₁, x₂)) = 1
POL(A__ISNAT(x₁)) = 0
POL(A__ISNATKIND(x₁)) = 0
POL(A__U11(x₁, x₂, x₃)) = 0
POL(A__U12(x₁, x₂)) = 0
POL(A__U21(x₁, x₂)) = x₁
POL(A__U31(x₁, x₂)) = x₁ + x₂
POL(MARK(x₁)) = 1
POL(U11(x₁, x₂, x₃)) = x₁
POL(U12(x₁, x₂)) = x₁
POL(U13(x₁)) = x₁
POL(U21(x₁, x₂)) = x₁
POL(U22(x₁)) = x₁
POL(U31(x₁, x₂)) = x₁ + x₂
POL(U41(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(a__U11(x₁, x₂, x₃)) = x₁
POL(a__U12(x₁, x₂)) = x₁
POL(a__U13(x₁)) = x₁
POL(a__U21(x₁, x₂)) = x₁
POL(a__U22(x₁)) = x₁
POL(a__U31(x₁, x₂)) = x₁ + x₂
POL(a__U41(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(a__and(x₁, x₂)) = x₁ + x₂
POL(a__isNat(x₁)) = 0
POL(a__isNatKind(x₁)) = 0
POL(a__plus(x₁, x₂)) = x₁ + x₂
POL(and(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)) = 1 + x₁
POL(tt) = 0

The following usable rules [FROCOS05] were oriented:

a__isNatKind(0) → tt
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
a__U31(tt, N) → mark(N)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__plus(N, 0) → a__U31(a__and(a__isNat(N), isNatKind(N)), N)
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__and(tt, X) → mark(X)
mark(isNatKind(X)) → a__isNatKind(X)
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(X) → isNatKind(X)
a__and(X1, X2) → and(X1, X2)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(X) → isNat(X)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__plus(N, s(M)) → a__U41(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U41(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U22(tt) → tt
a__U22(X) → U22(X)
a__U31(X1, X2) → U31(X1, X2)
a__plus(X1, X2) → plus(X1, X2)
a__U13(tt) → tt
a__U13(X) → U13(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)

(16) Obligation:

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

A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__U11(tt, V1, V2) → A__U12(a__isNat(V1), V2)
A__U12(tt, V2) → A__ISNAT(V2)
A__ISNAT(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__AND(tt, X) → MARK(X)
MARK(U11(X1, X2, X3)) → A__U11(mark(X1), X2, X3)
A__U11(tt, V1, V2) → A__ISNAT(V1)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U21(tt, V1) → A__ISNAT(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → A__U12(mark(X1), X2)
MARK(U12(X1, X2)) → MARK(X1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U13(X)) → MARK(X)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X)) → MARK(X)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
A__U31(tt, N) → MARK(N)
MARK(U31(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(isNatKind(X)) → A__ISNATKIND(X)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, N) → mark(N)
a__U41(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__plus(N, 0) → a__U31(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U41(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
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) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2) → U31(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

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.

MARK(U31(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:
A__ISNAT(x0, x1) = A__ISNAT(x0)
A__U11(x0, x1, x2, x3) = A__U11(x1)
A__U12(x0, x1, x2) = A__U12(x0, x1)
A__AND(x0, x1, x2) = A__AND(x0, x1, x2)
MARK(x0, x1) = MARK(x1)
A__ISNATKIND(x0, x1) = A__ISNATKIND(x0)
A__U21(x0, x1, x2) = A__U21(x0, x1)
A__U31(x0, x1, x2) = A__U31(x0, x2)

Tags:
A__ISNAT has argument tags [1,8] and root tag 0
A__U11 has argument tags [18,1,19,28] and root tag 0
A__U12 has argument tags [0,1,23] and root tag 0
A__AND has argument tags [1,0,0] and root tag 0
MARK has argument tags [16,1] and root tag 0
A__ISNATKIND has argument tags [1,0] and root tag 0
A__U21 has argument tags [1,1,0] and root tag 0
A__U31 has argument tags [1,2,1] 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__AND(x₁, x₂)) = x₁ + x₂
POL(A__ISNAT(x₁)) = 0
POL(A__ISNATKIND(x₁)) = 0
POL(A__U11(x₁, x₂, x₃)) = x₁
POL(A__U12(x₁, x₂)) = 0
POL(A__U21(x₁, x₂)) = 0
POL(A__U31(x₁, x₂)) = 1
POL(MARK(x₁)) = 0
POL(U11(x₁, x₂, x₃)) = x₁
POL(U12(x₁, x₂)) = x₁
POL(U13(x₁)) = x₁
POL(U21(x₁, x₂)) = x₁
POL(U22(x₁)) = x₁
POL(U31(x₁, x₂)) = 1 + x₁ + x₂
POL(U41(x₁, x₂, x₃)) = 0
POL(a__U11(x₁, x₂, x₃)) = x₁
POL(a__U12(x₁, x₂)) = x₁
POL(a__U13(x₁)) = x₁
POL(a__U21(x₁, x₂)) = x₁
POL(a__U22(x₁)) = x₁
POL(a__U31(x₁, x₂)) = 1 + x₁ + x₂
POL(a__U41(x₁, x₂, x₃)) = 0
POL(a__and(x₁, x₂)) = x₁ + x₂
POL(a__isNat(x₁)) = 0
POL(a__isNatKind(x₁)) = 0
POL(a__plus(x₁, x₂)) = x₁ + x₂
POL(and(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)) = 0
POL(tt) = 0

The following usable rules [FROCOS05] were oriented:

a__isNatKind(0) → tt
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
a__U31(tt, N) → mark(N)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__plus(N, 0) → a__U31(a__and(a__isNat(N), isNatKind(N)), N)
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__and(tt, X) → mark(X)
mark(isNatKind(X)) → a__isNatKind(X)
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(X) → isNatKind(X)
a__and(X1, X2) → and(X1, X2)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(X) → isNat(X)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__plus(N, s(M)) → a__U41(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U41(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U22(tt) → tt
a__U22(X) → U22(X)
a__U31(X1, X2) → U31(X1, X2)
a__plus(X1, X2) → plus(X1, X2)
a__U13(tt) → tt
a__U13(X) → U13(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)

(18) Obligation:

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

A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__U11(tt, V1, V2) → A__U12(a__isNat(V1), V2)
A__U12(tt, V2) → A__ISNAT(V2)
A__ISNAT(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__AND(tt, X) → MARK(X)
MARK(U11(X1, X2, X3)) → A__U11(mark(X1), X2, X3)
A__U11(tt, V1, V2) → A__ISNAT(V1)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U21(tt, V1) → A__ISNAT(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → A__U12(mark(X1), X2)
MARK(U12(X1, X2)) → MARK(X1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U13(X)) → MARK(X)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X)) → MARK(X)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
A__U31(tt, N) → MARK(N)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(isNatKind(X)) → A__ISNATKIND(X)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, N) → mark(N)
a__U41(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__plus(N, 0) → a__U31(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U41(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
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) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2) → U31(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

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.

MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
A__U31(tt, N) → MARK(N)

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(x0)
A__U11(x0, x1, x2, x3) = A__U11(x0)
A__U12(x0, x1, x2) = A__U12(x0, x1)
A__AND(x0, x1, x2) = A__AND(x0)
MARK(x0, x1) = MARK(x1)
A__ISNATKIND(x0, x1) = A__ISNATKIND(x0)
A__U21(x0, x1, x2) = A__U21(x1)
A__U31(x0, x1, x2) = A__U31(x0, x2)

Tags:
A__ISNAT has argument tags [0,0] and root tag 4
A__U11 has argument tags [0,0,28,24] and root tag 4
A__U12 has argument tags [0,0,0] and root tag 4
A__AND has argument tags [0,1,31] and root tag 4
MARK has argument tags [4,0] and root tag 4
A__ISNATKIND has argument tags [0,28] and root tag 4
A__U21 has argument tags [0,0,22] and root tag 4
A__U31 has argument tags [0,27,16] 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__AND(x₁, x₂)) = x₂
POL(A__ISNAT(x₁)) = 0
POL(A__ISNATKIND(x₁)) = 0
POL(A__U11(x₁, x₂, x₃)) = 0
POL(A__U12(x₁, x₂)) = 0
POL(A__U21(x₁, x₂)) = x₂
POL(A__U31(x₁, x₂)) = 1
POL(MARK(x₁)) = 1
POL(U11(x₁, x₂, x₃)) = x₁
POL(U12(x₁, x₂)) = x₁
POL(U13(x₁)) = x₁
POL(U21(x₁, x₂)) = x₁
POL(U22(x₁)) = x₁
POL(U31(x₁, x₂)) = 1 + x₂
POL(U41(x₁, x₂, x₃)) = x₃
POL(a__U11(x₁, x₂, x₃)) = x₁
POL(a__U12(x₁, x₂)) = x₁
POL(a__U13(x₁)) = x₁
POL(a__U21(x₁, x₂)) = x₁
POL(a__U22(x₁)) = x₁
POL(a__U31(x₁, x₂)) = 1 + x₂
POL(a__U41(x₁, x₂, x₃)) = x₃
POL(a__and(x₁, x₂)) = x₁ + x₂
POL(a__isNat(x₁)) = 0
POL(a__isNatKind(x₁)) = 0
POL(a__plus(x₁, x₂)) = x₁ + x₂
POL(and(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)) = 0
POL(tt) = 0

The following usable rules [FROCOS05] were oriented: none

(20) Obligation:

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

A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__U11(tt, V1, V2) → A__U12(a__isNat(V1), V2)
A__U12(tt, V2) → A__ISNAT(V2)
A__ISNAT(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__AND(tt, X) → MARK(X)
MARK(U11(X1, X2, X3)) → A__U11(mark(X1), X2, X3)
A__U11(tt, V1, V2) → A__ISNAT(V1)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U21(tt, V1) → A__ISNAT(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → A__U12(mark(X1), X2)
MARK(U12(X1, X2)) → MARK(X1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U13(X)) → MARK(X)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X)) → MARK(X)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(isNatKind(X)) → A__ISNATKIND(X)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, N) → mark(N)
a__U41(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__plus(N, 0) → a__U31(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U41(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
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) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2) → U31(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

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

(21) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__ISNAT(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(isNatKind(X)) → A__ISNATKIND(X)

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(x0, x1)
A__U11(x0, x1, x2, x3) = A__U11(x0, x2, x3)
A__U12(x0, x1, x2) = A__U12(x2)
A__AND(x0, x1, x2) = A__AND(x0)
MARK(x0, x1) = MARK(x0, x1)
A__ISNATKIND(x0, x1) = A__ISNATKIND(x1)
A__U21(x0, x1, x2) = A__U21(x0)

Tags:
A__ISNAT has argument tags [1,2] and root tag 0
A__U11 has argument tags [2,22,2,2] and root tag 0
A__U12 has argument tags [9,29,2] and root tag 0
A__AND has argument tags [2,29,2] and root tag 0
MARK has argument tags [2,0] and root tag 0
A__ISNATKIND has argument tags [16,1] and root tag 2
A__U21 has argument tags [2,9,1] 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__AND(x₁, x₂)) = x₂
POL(A__ISNAT(x₁)) = 0
POL(A__ISNATKIND(x₁)) = 1
POL(A__U11(x₁, x₂, x₃)) = 0
POL(A__U12(x₁, x₂)) = 1 + x₂
POL(A__U21(x₁, x₂)) = x₂
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U12(x₁, x₂)) = x₁ + x₂
POL(U13(x₁)) = x₁
POL(U21(x₁, x₂)) = x₁ + x₂
POL(U22(x₁)) = x₁
POL(U31(x₁, x₂)) = 1 + x₂
POL(U41(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(a__U11(x₁, x₂, x₃)) = 0
POL(a__U12(x₁, x₂)) = 1
POL(a__U13(x₁)) = 0
POL(a__U21(x₁, x₂)) = 1 + x₂
POL(a__U22(x₁)) = 0
POL(a__U31(x₁, x₂)) = x₁
POL(a__U41(x₁, x₂, x₃)) = x₁ + x₂
POL(a__and(x₁, x₂)) = 1
POL(a__isNat(x₁)) = 0
POL(a__isNatKind(x₁)) = 0
POL(a__plus(x₁, x₂)) = 1
POL(and(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = x₁
POL(isNatKind(x₁)) = x₁
POL(mark(x₁)) = 0
POL(plus(x₁, x₂)) = 1 + x₁ + x₂
POL(s(x₁)) = x₁
POL(tt) = 0

The following usable rules [FROCOS05] were oriented: none

(22) Obligation:

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

A__U11(tt, V1, V2) → A__U12(a__isNat(V1), V2)
A__U12(tt, V2) → A__ISNAT(V2)
A__AND(tt, X) → MARK(X)
MARK(U11(X1, X2, X3)) → A__U11(mark(X1), X2, X3)
A__U11(tt, V1, V2) → A__ISNAT(V1)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U21(tt, V1) → A__ISNAT(V1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → A__U12(mark(X1), X2)
MARK(U12(X1, X2)) → MARK(X1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U13(X)) → MARK(X)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X)) → MARK(X)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, N) → mark(N)
a__U41(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__plus(N, 0) → a__U31(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U41(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
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) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2) → U31(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

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

(23) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 7 less nodes.

(24) Complex Obligation (AND)

(25) Obligation:

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

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

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, N) → mark(N)
a__U41(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__plus(N, 0) → a__U31(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U41(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
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) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2) → U31(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

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

(26) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A__U21(tt, V1) → A__ISNAT(V1)
A__ISNAT(s(V1)) → A__U21(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__U21(x0, x1, x2) = A__U21(x2)
A__ISNAT(x0, x1) = A__ISNAT(x0)

Tags:
A__U21 has argument tags [1,3,0] and root tag 1
A__ISNAT has argument tags [0,5] 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)) = x₁
POL(A__U21(x₁, x₂)) = 1 + x₂
POL(U11(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U12(x₁, x₂)) = 1 + x₁
POL(U13(x₁)) = 1 + x₁
POL(U21(x₁, x₂)) = x₁
POL(U22(x₁)) = 1 + x₁
POL(U31(x₁, x₂)) = 1 + x₂
POL(U41(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(a__U11(x₁, x₂, x₃)) = 1
POL(a__U12(x₁, x₂)) = x₂
POL(a__U13(x₁)) = 0
POL(a__U21(x₁, x₂)) = 1 + x₂
POL(a__U22(x₁)) = 0
POL(a__U31(x₁, x₂)) = x₂
POL(a__U41(x₁, x₂, x₃)) = x₂
POL(a__and(x₁, x₂)) = 0
POL(a__isNat(x₁)) = 0
POL(a__isNatKind(x₁)) = 1 + x₁
POL(a__plus(x₁, x₂)) = 0
POL(and(x₁, x₂)) = 1 + x₂
POL(isNat(x₁)) = 0
POL(isNatKind(x₁)) = 1
POL(mark(x₁)) = 0
POL(plus(x₁, x₂)) = x₁
POL(s(x₁)) = 1 + x₁
POL(tt) = 0

The following usable rules [FROCOS05] were oriented: none

(27) Obligation:

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

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, N) → mark(N)
a__U41(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__plus(N, 0) → a__U31(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U41(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
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) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2) → U31(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

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

(28) PisEmptyProof (EQUIVALENT transformation)

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

(29) TRUE

(30) Obligation:

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

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

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, N) → mark(N)
a__U41(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__plus(N, 0) → a__U31(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U41(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
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) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2) → U31(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

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

(31) 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 [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(s(x₁)) = 1 + x₁

The following usable rules [FROCOS05] were oriented: none

(32) Obligation:

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

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, N) → mark(N)
a__U41(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__plus(N, 0) → a__U31(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U41(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
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) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2) → U31(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

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

(33) PisEmptyProof (EQUIVALENT transformation)

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

(34) TRUE

(35) Obligation:

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

MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → MARK(X1)
MARK(U13(X)) → MARK(X)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X)) → MARK(X)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__AND(tt, X) → MARK(X)
MARK(and(X1, X2)) → MARK(X1)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, N) → mark(N)
a__U41(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__plus(N, 0) → a__U31(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U41(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
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) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2) → U31(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

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

(36) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(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__AND(x0, x1, x2) = A__AND(x0)

Tags:
MARK has argument tags [0,4] and root tag 0
A__AND has argument tags [4,0,5] 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__AND(x₁, x₂)) = x₂
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂, x₃)) = 1 + x₁ + x₂
POL(U12(x₁, x₂)) = x₁ + x₂
POL(U13(x₁)) = x₁
POL(U21(x₁, x₂)) = x₁
POL(U22(x₁)) = x₁
POL(U31(x₁, x₂)) = 1 + x₁
POL(U41(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(a__U11(x₁, x₂, x₃)) = x₃
POL(a__U12(x₁, x₂)) = 0
POL(a__U13(x₁)) = 0
POL(a__U21(x₁, x₂)) = 1 + x₂
POL(a__U22(x₁)) = 1 + x₁
POL(a__U31(x₁, x₂)) = x₂
POL(a__U41(x₁, x₂, x₃)) = 1
POL(a__and(x₁, x₂)) = 0
POL(a__isNat(x₁)) = 1 + x₁
POL(a__isNatKind(x₁)) = 1
POL(a__plus(x₁, x₂)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = x₁
POL(isNatKind(x₁)) = 0
POL(mark(x₁)) = 0
POL(plus(x₁, x₂)) = 1
POL(s(x₁)) = 1
POL(tt) = 1

The following usable rules [FROCOS05] were oriented: none

(37) Obligation:

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

MARK(U12(X1, X2)) → MARK(X1)
MARK(U13(X)) → MARK(X)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X)) → MARK(X)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__AND(tt, X) → MARK(X)
MARK(and(X1, X2)) → MARK(X1)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, N) → mark(N)
a__U41(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__plus(N, 0) → a__U31(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U41(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
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) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2) → U31(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

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(U12(X1, X2)) → MARK(X1)
MARK(U13(X)) → MARK(X)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X)) → MARK(X)
MARK(and(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__AND(x0, x1, x2) = A__AND(x0, x2)

Tags:
MARK has argument tags [1,1] and root tag 0
A__AND has argument tags [1,0,1] 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__AND(x₁, x₂)) = 1
POL(MARK(x₁)) = 1
POL(U11(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U12(x₁, x₂)) = 1 + x₁ + x₂
POL(U13(x₁)) = 1 + x₁
POL(U21(x₁, x₂)) = 1 + x₁ + x₂
POL(U22(x₁)) = 1 + x₁
POL(U31(x₁, x₂)) = 1 + x₁ + x₂
POL(U41(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(a__U11(x₁, x₂, x₃)) = 0
POL(a__U12(x₁, x₂)) = 0
POL(a__U13(x₁)) = 0
POL(a__U21(x₁, x₂)) = 0
POL(a__U22(x₁)) = 0
POL(a__U31(x₁, x₂)) = 0
POL(a__U41(x₁, x₂, x₃)) = 0
POL(a__and(x₁, x₂)) = x₁ + x₂
POL(a__isNat(x₁)) = 0
POL(a__isNatKind(x₁)) = 1
POL(a__plus(x₁, x₂)) = 0
POL(and(x₁, x₂)) = 1 + x₁ + x₂
POL(isNat(x₁)) = 1 + x₁
POL(isNatKind(x₁)) = 1
POL(mark(x₁)) = 0
POL(plus(x₁, x₂)) = 1 + x₁ + x₂
POL(s(x₁)) = 1
POL(tt) = 1

The following usable rules [FROCOS05] were oriented: none

(39) Obligation:

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

MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__AND(tt, X) → MARK(X)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, N) → mark(N)
a__U41(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__plus(N, 0) → a__U31(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U41(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
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) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2) → U31(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

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(and(X1, X2)) → A__AND(mark(X1), X2)
A__AND(tt, 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__AND(x0, x1, x2) = A__AND(x2)

Tags:
MARK has argument tags [1,6] and root tag 0
A__AND has argument tags [5,0,6] 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__AND(x₁, x₂)) = 1
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U12(x₁, x₂)) = 1 + x₁ + x₂
POL(U13(x₁)) = 1 + x₁
POL(U21(x₁, x₂)) = 1 + x₁ + x₂
POL(U22(x₁)) = 1 + x₁
POL(U31(x₁, x₂)) = 1 + x₁ + x₂
POL(U41(x₁, x₂, x₃)) = x₁ + x₂
POL(a__U11(x₁, x₂, x₃)) = 0
POL(a__U12(x₁, x₂)) = 0
POL(a__U13(x₁)) = 0
POL(a__U21(x₁, x₂)) = 0
POL(a__U22(x₁)) = 0
POL(a__U31(x₁, x₂)) = 0
POL(a__U41(x₁, x₂, x₃)) = 0
POL(a__and(x₁, x₂)) = x₂
POL(a__isNat(x₁)) = 0
POL(a__isNatKind(x₁)) = 0
POL(a__plus(x₁, x₂)) = 0
POL(and(x₁, x₂)) = 1 + x₂
POL(isNat(x₁)) = 1 + x₁
POL(isNatKind(x₁)) = 1
POL(mark(x₁)) = 1 + x₁
POL(plus(x₁, x₂)) = 1 + x₁ + x₂
POL(s(x₁)) = x₁
POL(tt) = 1

The following usable rules [FROCOS05] were oriented: none

(41) Obligation:

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

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, N) → mark(N)
a__U41(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__plus(N, 0) → a__U31(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U41(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
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) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2) → U31(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

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

(42) 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) Obligation:

(5) QDPOrderProof (EQUIVALENT transformation)

(6) Obligation:

(7) DependencyGraphProof (EQUIVALENT transformation)

(8) Complex Obligation (AND)

(9) Obligation:

(10) QDPOrderProof (EQUIVALENT transformation)

(11) Obligation:

(12) PisEmptyProof (EQUIVALENT transformation)

(13) TRUE

(14) Obligation:

(15) QDPOrderProof (EQUIVALENT transformation)

(16) Obligation:

(17) QDPOrderProof (EQUIVALENT transformation)

(18) Obligation:

(19) QDPOrderProof (EQUIVALENT transformation)

(20) Obligation:

(21) QDPOrderProof (EQUIVALENT transformation)

(22) Obligation:

(23) DependencyGraphProof (EQUIVALENT transformation)

(24) Complex Obligation (AND)

(25) Obligation:

(26) QDPOrderProof (EQUIVALENT transformation)

(27) Obligation:

(28) PisEmptyProof (EQUIVALENT transformation)

(29) TRUE

(30) Obligation:

(31) QDPOrderProof (EQUIVALENT transformation)

(32) Obligation:

(33) PisEmptyProof (EQUIVALENT transformation)

(34) TRUE

(35) Obligation:

(36) QDPOrderProof (EQUIVALENT transformation)

(37) Obligation:

(38) QDPOrderProof (EQUIVALENT transformation)

(39) Obligation:

(40) QDPOrderProof (EQUIVALENT transformation)

(41) Obligation:

(42) PisEmptyProof (EQUIVALENT transformation)

(43) TRUE