Termination w.r.t. Q proof of /tmp/tmpNrdzg3/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 DependencyGraphProof (⇔)

    ↳13 TRUE

  ↳14 QDP

    ↳15 QDPOrderProof (⇔)

    ↳16 QDP

    ↳17 DependencyGraphProof (⇔)

    ↳18 AND

      ↳19 QDP

        ↳20 QDPOrderProof (⇔)

        ↳21 QDP

        ↳22 PisEmptyProof (⇔)

        ↳23 TRUE

      ↳24 QDP

        ↳25 QDPOrderProof (⇔)

        ↳26 QDP

        ↳27 DependencyGraphProof (⇔)

        ↳28 TRUE

  ↳29 QDP

    ↳30 QDPOrderProof (⇔)

    ↳31 QDP

    ↳32 PisEmptyProof (⇔)

    ↳33 TRUE

(0) Obligation:

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

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → 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:

U11¹(tt, V1, V2) → U12¹(isNat(activate(V1)), activate(V2))
U11¹(tt, V1, V2) → ISNAT(activate(V1))
U11¹(tt, V1, V2) → ACTIVATE(V1)
U11¹(tt, V1, V2) → ACTIVATE(V2)
U12¹(tt, V2) → U13¹(isNat(activate(V2)))
U12¹(tt, V2) → ISNAT(activate(V2))
U12¹(tt, V2) → ACTIVATE(V2)
U21¹(tt, V1) → U22¹(isNat(activate(V1)))
U21¹(tt, V1) → ISNAT(activate(V1))
U21¹(tt, V1) → ACTIVATE(V1)
U31¹(tt, N) → ACTIVATE(N)
U41¹(tt, M, N) → S(plus(activate(N), activate(M)))
U41¹(tt, M, N) → PLUS(activate(N), activate(M))
U41¹(tt, M, N) → ACTIVATE(N)
U41¹(tt, M, N) → ACTIVATE(M)
AND(tt, X) → ACTIVATE(X)
ISNAT(n__plus(V1, V2)) → U11¹(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
ISNAT(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
PLUS(N, 0) → U31¹(and(isNat(N), n__isNatKind(N)), N)
PLUS(N, 0) → AND(isNat(N), n__isNatKind(N))
PLUS(N, 0) → ISNAT(N)
PLUS(N, s(M)) → U41¹(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
PLUS(N, s(M)) → AND(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))
PLUS(N, s(M)) → AND(isNat(M), n__isNatKind(M))
PLUS(N, s(M)) → ISNAT(M)
PLUS(N, s(M)) → ISNAT(N)
ACTIVATE(n__0) → 0¹
ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ACTIVATE(n__s(X)) → S(X)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → 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 5 less nodes.

(4) Obligation:

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

U12¹(tt, V2) → ISNAT(activate(V2))
ISNAT(n__plus(V1, V2)) → U11¹(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
U11¹(tt, V1, V2) → U12¹(isNat(activate(V1)), activate(V2))
U12¹(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
PLUS(N, 0) → U31¹(and(isNat(N), n__isNatKind(N)), N)
U31¹(tt, N) → ACTIVATE(N)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATKIND(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
PLUS(N, 0) → AND(isNat(N), n__isNatKind(N))
PLUS(N, 0) → ISNAT(N)
ISNAT(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
U21¹(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
U21¹(tt, V1) → ACTIVATE(V1)
PLUS(N, s(M)) → U41¹(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
U41¹(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, s(M)) → AND(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))
PLUS(N, s(M)) → AND(isNat(M), n__isNatKind(M))
PLUS(N, s(M)) → ISNAT(M)
PLUS(N, s(M)) → ISNAT(N)
U41¹(tt, M, N) → ACTIVATE(N)
U41¹(tt, M, N) → ACTIVATE(M)
U11¹(tt, V1, V2) → ISNAT(activate(V1))
U11¹(tt, V1, V2) → ACTIVATE(V1)
U11¹(tt, V1, V2) → ACTIVATE(V2)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → 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.

U12¹(tt, V2) → ISNAT(activate(V2))
ISNAT(n__plus(V1, V2)) → U11¹(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
U11¹(tt, V1, V2) → U12¹(isNat(activate(V1)), activate(V2))
U12¹(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
PLUS(N, 0) → U31¹(and(isNat(N), n__isNatKind(N)), N)
ISNATKIND(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
PLUS(N, 0) → AND(isNat(N), n__isNatKind(N))
PLUS(N, 0) → ISNAT(N)
ISNAT(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
PLUS(N, s(M)) → AND(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))
PLUS(N, s(M)) → AND(isNat(M), n__isNatKind(M))
PLUS(N, s(M)) → ISNAT(M)
PLUS(N, s(M)) → ISNAT(N)
U41¹(tt, M, N) → ACTIVATE(N)
U41¹(tt, M, N) → ACTIVATE(M)
U11¹(tt, V1, V2) → ISNAT(activate(V1))
U11¹(tt, V1, V2) → ACTIVATE(V1)
U11¹(tt, V1, V2) → ACTIVATE(V2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U12¹(x1, x2) = U12¹(x1, x2)
tt = tt
ISNAT(x1) = ISNAT(x1)
activate(x1) = x1
n__plus(x1, x2) = n__plus(x1, x2)
U11¹(x1, x2, x3) = U11¹(x1, x2, x3)
and(x1, x2) = x2
isNatKind(x1) = x1
n__isNatKind(x1) = x1
isNat(x1) = isNat
ACTIVATE(x1) = ACTIVATE(x1)
PLUS(x1, x2) = PLUS(x1, x2)
0 = 0
U31¹(x1, x2) = U31¹(x2)
ISNATKIND(x1) = ISNATKIND(x1)
AND(x1, x2) = AND(x2)
n__and(x1, x2) = x2
n__s(x1) = x1
U21¹(x1, x2) = U21¹(x2)
s(x1) = x1
U41¹(x1, x2, x3) = U41¹(x2, x3)
plus(x1, x2) = plus(x1, x2)
U41(x1, x2, x3) = U41(x2, x3)
n__0 = n__0
U31(x1, x2) = U31(x2)
U11(x1, x2, x3) = U11
U21(x1, x2) = U21
U12(x1, x2) = U12
U13(x1) = x1
U22(x1) = x1

Recursive Path Order [RPO].
Precedence:

[0, n_₀] > [tt, isNat, U11, U21, U12] > [U12^1₂, ISNAT₁, U11^1₃, ACTIVATE₁, PLUS₂, U31^1₁, ISNATKIND₁, AND₁, U21^1₁, U41^1₂] > U31₁
[0, n_₀] > [tt, isNat, U11, U21, U12] > [n_{_plus₂}, plus₂, U41₂] > U31₁

The following usable rules [FROCOS05] were oriented:

activate(X) → X
activate(n__s(X)) → s(X)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
isNatKind(n__s(V1)) → isNatKind(activate(V1))
and(tt, X) → activate(X)
U31(tt, N) → activate(N)
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__and(X1, X2)) → and(X1, X2)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U13(tt) → tt
U12(tt, V2) → U13(isNat(activate(V2)))
U22(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U41(tt, M, N) → s(plus(activate(N), activate(M)))

(6) Obligation:

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

U31¹(tt, N) → ACTIVATE(N)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
U21¹(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
U21¹(tt, V1) → ACTIVATE(V1)
PLUS(N, s(M)) → U41¹(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
U41¹(tt, M, N) → PLUS(activate(N), activate(M))

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → 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 3 SCCs with 4 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

U41¹(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, s(M)) → U41¹(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → 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.

U41¹(tt, M, N) → PLUS(activate(N), activate(M))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U41¹(x1, x2, x3) = U41¹(x2)
tt = tt
PLUS(x1, x2) = x2
activate(x1) = x1
s(x1) = s(x1)
and(x1, x2) = x2
isNat(x1) = isNat(x1)
n__isNatKind(x1) = n__isNatKind
n__and(x1, x2) = x2
plus(x1, x2) = plus(x1, x2)
U41(x1, x2, x3) = U41(x1, x2, x3)
0 = 0
n__0 = n__0
n__plus(x1, x2) = n__plus(x1, x2)
isNatKind(x1) = isNatKind
n__s(x1) = n__s(x1)
U31(x1, x2) = U31(x2)
U11(x1, x2, x3) = U11(x1, x2, x3)
U21(x1, x2) = U21(x1, x2)
U12(x1, x2) = U12
U13(x1) = U13
U22(x1) = U22

Recursive Path Order [RPO].
Precedence:

isNat₁ > [plus₂, U41₃, n_{_plus₂}, U11₃, U12, U13] > [U41^1₁, tt, s₁, n_{_isNatKind}, isNatKind, n_{_s₁}]
isNat₁ > [plus₂, U41₃, n_{_plus₂}, U11₃, U12, U13] > U31₁
isNat₁ > U21₂ > U22 > [U41^1₁, tt, s₁, n_{_isNatKind}, isNatKind, n_{_s₁}]
[0, n_₀] > [U41^1₁, tt, s₁, n_{_isNatKind}, isNatKind, n_{_s₁}]

The following usable rules [FROCOS05] were oriented:

plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
isNatKind(n__s(V1)) → isNatKind(activate(V1))
and(tt, X) → activate(X)
U31(tt, N) → activate(N)
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__and(X1, X2)) → and(X1, X2)
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U13(tt) → tt
U12(tt, V2) → U13(isNat(activate(V2)))
U22(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U41(tt, M, N) → s(plus(activate(N), activate(M)))
activate(X) → X
activate(n__s(X)) → s(X)

(11) Obligation:

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

PLUS(N, s(M)) → U41¹(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

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

(12) DependencyGraphProof (EQUIVALENT transformation)

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

(13) TRUE

(14) Obligation:

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

ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
AND(tt, X) → ACTIVATE(X)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → 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.

ISNATKIND(n__s(V1)) → ACTIVATE(V1)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNATKIND(x1) = ISNATKIND(x1)
n__s(x1) = x1
activate(x1) = x1
ACTIVATE(x1) = x1
n__isNatKind(x1) = n__isNatKind(x1)
n__and(x1, x2) = x2
AND(x1, x2) = x2
tt = tt
plus(x1, x2) = plus(x1, x2)
s(x1) = x1
U41(x1, x2, x3) = U41(x2, x3)
and(x1, x2) = x2
isNat(x1) = x1
0 = 0
n__0 = n__0
n__plus(x1, x2) = n__plus(x1, x2)
isNatKind(x1) = isNatKind(x1)
U31(x1, x2) = U31(x1, x2)
U11(x1, x2, x3) = U11(x1, x2, x3)
U21(x1, x2) = x2
U12(x1, x2) = U12(x1, x2)
U13(x1) = x1
U22(x1) = x1

Recursive Path Order [RPO].
Precedence:

[plus₂, U41₂, n_{_plus₂}] > [ISNATKIND₁, n_{_isNatKind₁}, isNatKind₁]
[plus₂, U41₂, n_{_plus₂}] > U31₂
[plus₂, U41₂, n_{_plus₂}] > U11₃ > U12₂
[0, n_₀] > [ISNATKIND₁, n_{_isNatKind₁}, isNatKind₁]
[0, n_₀] > tt > U12₂
[0, n_₀] > U31₂

The following usable rules [FROCOS05] were oriented:

plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
isNatKind(n__s(V1)) → isNatKind(activate(V1))
and(tt, X) → activate(X)
U31(tt, N) → activate(N)
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__and(X1, X2)) → and(X1, X2)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U13(tt) → tt
U12(tt, V2) → U13(isNat(activate(V2)))
U22(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U41(tt, M, N) → s(plus(activate(N), activate(M)))
activate(X) → X
activate(n__s(X)) → s(X)

(16) Obligation:

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

ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
AND(tt, X) → ACTIVATE(X)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

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

(17) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.

(18) Complex Obligation (AND)

(19) Obligation:

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

ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

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

(20) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNATKIND(x1) = ISNATKIND(x1)
n__s(x1) = n__s(x1)
activate(x1) = x1
plus(x1, x2) = plus(x1, x2)
s(x1) = s(x1)
U41(x1, x2, x3) = U41(x1, x2, x3)
and(x1, x2) = x2
isNat(x1) = isNat
n__isNatKind(x1) = n__isNatKind
n__and(x1, x2) = x2
0 = 0
n__0 = n__0
n__plus(x1, x2) = n__plus(x1, x2)
isNatKind(x1) = isNatKind
tt = tt
U31(x1, x2) = x2
U11(x1, x2, x3) = U11
U21(x1, x2) = x1
U12(x1, x2) = x1
U13(x1) = U13
U22(x1) = U22

Recursive Path Order [RPO].
Precedence:

[plus₂, U41₃, n_{_plus₂}] > [isNat, U11] > [n_{_s₁}, s₁, n_{_isNatKind}, isNatKind] > [tt, U13, U22]
[0, n_₀] > [isNat, U11] > [n_{_s₁}, s₁, n_{_isNatKind}, isNatKind] > [tt, U13, U22]

The following usable rules [FROCOS05] were oriented:

plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
isNatKind(n__s(V1)) → isNatKind(activate(V1))
and(tt, X) → activate(X)
U31(tt, N) → activate(N)
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__and(X1, X2)) → and(X1, X2)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U13(tt) → tt
U12(tt, V2) → U13(isNat(activate(V2)))
U22(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U41(tt, M, N) → s(plus(activate(N), activate(M)))
activate(X) → X
activate(n__s(X)) → s(X)

(21) Obligation:

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

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

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

(22) PisEmptyProof (EQUIVALENT transformation)

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

(23) TRUE

(24) Obligation:

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

ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
AND(tt, X) → ACTIVATE(X)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

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

(25) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVATE(n__and(X1, X2)) → AND(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVATE(x1) = ACTIVATE(x1)
n__and(x1, x2) = n__and(x2)
AND(x1, x2) = AND(x2)
tt = tt

Recursive Path Order [RPO].
Precedence:

n_{_and₁} > [ACTIVATE₁, AND₁, tt]

The following usable rules [FROCOS05] were oriented: none

(26) Obligation:

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

AND(tt, X) → ACTIVATE(X)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

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

(27) DependencyGraphProof (EQUIVALENT transformation)

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

(28) TRUE

(29) Obligation:

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

ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
U21¹(tt, V1) → ISNAT(activate(V1))

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

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

(30) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
U21¹(tt, V1) → ISNAT(activate(V1))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNAT(x1) = ISNAT(x1)
n__s(x1) = n__s(x1)
U21¹(x1, x2) = U21¹(x1, x2)
isNatKind(x1) = isNatKind
activate(x1) = x1
tt = tt
plus(x1, x2) = plus(x1, x2)
s(x1) = s(x1)
U41(x1, x2, x3) = U41(x1, x2, x3)
and(x1, x2) = x2
isNat(x1) = isNat(x1)
n__isNatKind(x1) = n__isNatKind
n__and(x1, x2) = x2
0 = 0
n__0 = n__0
n__plus(x1, x2) = n__plus(x1, x2)
U31(x1, x2) = x2
U11(x1, x2, x3) = U11(x1, x2, x3)
U21(x1, x2) = x2
U12(x1, x2) = U12(x2)
U13(x1) = U13
U22(x1) = U22

Recursive Path Order [RPO].
Precedence:

[ISNAT₁, U21^1₂] > [isNatKind, n_{_isNatKind}] > [tt, 0, n_₀, U22]
[plus₂, U41₃, n_{_plus₂}] > [n_{_s₁}, s₁] > [isNatKind, n_{_isNatKind}] > [tt, 0, n_₀, U22]
isNat₁ > [isNatKind, n_{_isNatKind}] > [tt, 0, n_₀, U22]
isNat₁ > U11₃ > U12₁ > U13 > [tt, 0, n_₀, U22]

The following usable rules [FROCOS05] were oriented:

plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
isNatKind(n__s(V1)) → isNatKind(activate(V1))
and(tt, X) → activate(X)
U31(tt, N) → activate(N)
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__and(X1, X2)) → and(X1, X2)
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U13(tt) → tt
U12(tt, V2) → U13(isNat(activate(V2)))
U22(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U41(tt, M, N) → s(plus(activate(N), activate(M)))
activate(X) → X
activate(n__s(X)) → s(X)

(31) Obligation:

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

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

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

(32) 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) DependencyGraphProof (EQUIVALENT transformation)

(13) TRUE

(14) Obligation:

(15) QDPOrderProof (EQUIVALENT transformation)

(16) Obligation:

(17) DependencyGraphProof (EQUIVALENT transformation)

(18) Complex Obligation (AND)

(19) Obligation:

(20) QDPOrderProof (EQUIVALENT transformation)

(21) Obligation:

(22) PisEmptyProof (EQUIVALENT transformation)

(23) TRUE

(24) Obligation:

(25) QDPOrderProof (EQUIVALENT transformation)

(26) Obligation:

(27) DependencyGraphProof (EQUIVALENT transformation)

(28) TRUE

(29) Obligation:

(30) QDPOrderProof (EQUIVALENT transformation)

(31) Obligation:

(32) PisEmptyProof (EQUIVALENT transformation)

(33) TRUE