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

    ↳18 AND

      ↳19 QDP

        ↳20 QDPOrderProof (⇔)

        ↳21 QDP

        ↳22 DependencyGraphProof (⇔)

        ↳23 AND

          ↳24 QDP

            ↳25 QDPOrderProof (⇔)

            ↳26 QDP

            ↳27 PisEmptyProof (⇔)

            ↳28 TRUE

          ↳29 QDP

            ↳30 QDPOrderProof (⇔)

            ↳31 QDP

            ↳32 PisEmptyProof (⇔)

            ↳33 TRUE

      ↳34 QDP

        ↳35 QDPOrderProof (⇔)

        ↳36 QDP

        ↳37 PisEmptyProof (⇔)

        ↳38 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(n__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)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
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(n__isNat(N), n__isNatKind(N))), M, N)
PLUS(N, s(M)) → AND(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))
PLUS(N, s(M)) → AND(isNat(M), n__isNatKind(M))
PLUS(N, s(M)) → ISNAT(M)
ACTIVATE(n__0) → 0¹
ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ACTIVATE(n__s(X)) → S(activate(X))
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__isNat(X)) → ISNAT(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(n__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)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
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(activate(X1), activate(X2))
PLUS(N, 0) → U31¹(and(isNat(N), n__isNatKind(N)), N)
U31¹(tt, N) → ACTIVATE(N)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
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__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__isNat(X)) → ISNAT(X)
ISNAT(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
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)
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, 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(n__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(n__isNat(N), n__isNatKind(N)))
PLUS(N, s(M)) → AND(isNat(M), n__isNatKind(M))
PLUS(N, s(M)) → ISNAT(M)
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(n__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)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
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)
PLUS(N, 0) → U31¹(and(isNat(N), n__isNatKind(N)), N)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
ISNATKIND(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNAT(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
PLUS(N, 0) → AND(isNat(N), n__isNatKind(N))
PLUS(N, 0) → ISNAT(N)
PLUS(N, s(M)) → AND(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))
PLUS(N, s(M)) → AND(isNat(M), n__isNatKind(M))
PLUS(N, s(M)) → ISNAT(M)
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) = x1
activate(x1) = x1
n__plus(x1, x2) = n__plus(x1, x2)
U11¹(x1, x2, x3) = U11¹(x2, x3)
and(x1, x2) = and(x1, x2)
isNatKind(x1) = x1
n__isNatKind(x1) = x1
isNat(x1) = x1
ACTIVATE(x1) = x1
PLUS(x1, x2) = PLUS(x1, x2)
0 = 0
U31¹(x1, x2) = x2
ISNATKIND(x1) = x1
AND(x1, x2) = x2
n__s(x1) = x1
n__and(x1, x2) = n__and(x1, x2)
n__isNat(x1) = x1
U21¹(x1, x2) = 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) = x2
U11(x1, x2, x3) = x3
U12(x1, x2) = x2
U13(x1) = x1
U22(x1) = x1
U21(x1, x2) = x2

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

[0, n_₀] > tt > [n_{_plus₂}, PLUS₂, U41^1₂, plus₂, U41₂] > U11^1₂ > U12^1₂ > [and₂, n_{_and₂}]

Status:

n_{_plus₂}: [1,2]
plus₂: [1,2]
U12^1₂: [2,1]
and₂: multiset
0: multiset
n_{_and₂}: multiset
PLUS₂: [1,2]
tt: multiset
U41^1₂: [2,1]
U11^1₂: [1,2]
U41₂: [2,1]
n_₀: multiset

The following usable rules [FROCOS05] were oriented:

plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(n__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)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
activate(n__and(X1, X2)) → and(activate(X1), X2)
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__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__isNat(X)) → isNat(X)
activate(X) → X
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)))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNatKind(n__0) → tt

(6) Obligation:

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

ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
U31¹(tt, N) → ACTIVATE(N)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__isNat(X)) → ISNAT(X)
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(n__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(n__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)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
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 2 SCCs with 3 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(n__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(n__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)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
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))
PLUS(N, s(M)) → U41¹(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)

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

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

[plus₂, U41₃, n_{_plus₂}] > [s₁, n_{_s₁}, U21] > U22 > [U41^1₃, tt, PLUS₂]
[0, n_₀] > [isNat, n_{_isNat}] > [s₁, n_{_s₁}, U21] > U22 > [U41^1₃, tt, PLUS₂]
[0, n_₀] > [isNat, n_{_isNat}] > U11 > [U12, U13] > [U41^1₃, tt, PLUS₂]

Status:

n_{_plus₂}: [2,1]
U13: multiset
plus₂: [2,1]
U21: []
U22: multiset
U41₃: [2,3,1]
U12: multiset
U11: []
n_{_s₁}: multiset
U41^1₃: [3,2,1]
0: multiset
isNat: multiset
PLUS₂: [1,2]
tt: multiset
n_{_isNat}: multiset
n_₀: multiset
s₁: multiset

The following usable rules [FROCOS05] were oriented:

plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(n__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)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
activate(n__and(X1, X2)) → and(activate(X1), X2)
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__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__isNat(X)) → isNat(X)
activate(X) → X
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)))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNatKind(n__0) → tt

(11) 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(n__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)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(X) → 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:

ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__isNat(X)) → ISNAT(X)
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)

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(n__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)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
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.

ACTIVATE(n__isNat(X)) → ISNAT(X)

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

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

[n_{_isNat₁}, plus₂, U41₂, isNat₁, n_{_plus₂}, U11₂] > U21₂ > U22 > [tt, 0, n_₀, U12, U13]

Status:

U13: multiset
n_{_plus₂}: multiset
plus₂: multiset
U22: []
n_{_isNat₁}: [1]
U12: multiset
U11₂: [2,1]
0: multiset
U21₂: multiset
tt: multiset
U41₂: multiset
n_₀: multiset
isNat₁: [1]

The following usable rules [FROCOS05] were oriented:

plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(n__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)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
activate(n__and(X1, X2)) → and(activate(X1), X2)
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__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__isNat(X)) → isNat(X)
activate(X) → X
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)))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNatKind(n__0) → tt

(16) Obligation:

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

ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
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)

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(n__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)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
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 3 less nodes.

(18) Complex Obligation (AND)

(19) 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__s(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(n__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)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
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.

ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)

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

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

[n_{_isNatKind₁}, isNatKind₁] > [tt, U11, U12, U13, U22, U21]
[plus₂, U41₂, n_{_plus₂}] > [tt, U11, U12, U13, U22, U21]
[0, n_₀] > [tt, U11, U12, U13, U22, U21]

Status:

U13: []
n_{_plus₂}: multiset
U21: []
plus₂: multiset
U22: []
n_{_isNatKind₁}: [1]
U12: []
U11: []
isNatKind₁: [1]
0: multiset
tt: multiset
U41₂: multiset
n_₀: multiset

The following usable rules [FROCOS05] were oriented:

plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(n__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)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
activate(n__and(X1, X2)) → and(activate(X1), X2)
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__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__isNat(X)) → isNat(X)
activate(X) → X
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)))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNatKind(n__0) → tt

(21) 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__s(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(n__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)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

(22) DependencyGraphProof (EQUIVALENT transformation)

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

(23) Complex Obligation (AND)

(24) Obligation:

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

ACTIVATE(n__s(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(n__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)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
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__s(X)) → ACTIVATE(X)

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

n_{_s₁} > ACTIVATE₁

Status:

n_{_s₁}: multiset
ACTIVATE₁: multiset

The following usable rules [FROCOS05] were oriented: none

(26) 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(n__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)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

(27) PisEmptyProof (EQUIVALENT transformation)

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

(28) TRUE

(29) 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(n__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)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
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.

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) = x1
n__isNatKind(x1) = x1
n__and(x1, x2) = x2
n__isNat(x1) = x1
0 = 0
n__0 = n__0
n__plus(x1, x2) = n__plus(x1, x2)
isNatKind(x1) = x1
tt = tt
U31(x1, x2) = U31(x1, x2)
U11(x1, x2, x3) = U11(x3)
U12(x1, x2) = U12(x2)
U13(x1) = U13(x1)
U22(x1) = x1
U21(x1, x2) = x2

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

[plus₂, U41₃, n_{_plus₂}] > [ISNATKIND₁, n_{_s₁}, s₁] > [0, n_₀, tt, U31₂]
[plus₂, U41₃, n_{_plus₂}] > U11₁ > U12₁ > U13₁ > [0, n_₀, tt, U31₂]

Status:

n_{_plus₂}: [2,1]
plus₂: [2,1]
U31₂: multiset
U41₃: [2,3,1]
ISNATKIND₁: multiset
n_{_s₁}: [1]
0: multiset
tt: multiset
n_₀: multiset
U13₁: multiset
U12₁: [1]
U11₁: [1]
s₁: [1]

The following usable rules [FROCOS05] were oriented:

plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(n__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)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
activate(n__and(X1, X2)) → and(activate(X1), X2)
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__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__isNat(X)) → isNat(X)
activate(X) → X
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)))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNatKind(n__0) → tt

(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(n__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)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
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.

(33) TRUE

(34) 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(n__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)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

(35) 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(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
n__isNatKind(x1) = n__isNatKind
n__and(x1, x2) = x2
n__isNat(x1) = n__isNat
0 = 0
n__0 = n__0
n__plus(x1, x2) = n__plus(x1, x2)
U31(x1, x2) = U31(x1, x2)
U11(x1, x2, x3) = U11
U12(x1, x2) = U12
U13(x1) = U13
U22(x1) = U22
U21(x1, x2) = U21(x1)

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

[plus₂, U41₃, n_{_plus₂}] > [n_{_s₁}, U21^1₁, s₁, U21₁] > ISNAT₁ > [isNatKind, tt, n_{_isNatKind}, 0, n_₀, U22]
[plus₂, U41₃, n_{_plus₂}] > U31₂ > [isNatKind, tt, n_{_isNatKind}, 0, n_₀, U22]
[plus₂, U41₃, n_{_plus₂}] > U11 > [U12, U13] > [isNatKind, tt, n_{_isNatKind}, 0, n_₀, U22]
[isNat, n_{_isNat}] > [n_{_s₁}, U21^1₁, s₁, U21₁] > ISNAT₁ > [isNatKind, tt, n_{_isNatKind}, 0, n_₀, U22]
[isNat, n_{_isNat}] > U11 > [U12, U13] > [isNatKind, tt, n_{_isNatKind}, 0, n_₀, U22]

Status:

n_{_plus₂}: [1,2]
U13: []
plus₂: [1,2]
U31₂: [1,2]
U22: multiset
U41₃: [3,2,1]
U12: []
n_{_isNatKind}: multiset
U11: multiset
n_{_s₁}: multiset
0: multiset
ISNAT₁: multiset
isNat: []
tt: multiset
n_{_isNat}: []
n_₀: multiset
U21^1₁: multiset
isNatKind: multiset
s₁: multiset
U21₁: multiset

The following usable rules [FROCOS05] were oriented:

plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(n__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)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
activate(n__and(X1, X2)) → and(activate(X1), X2)
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__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__isNat(X)) → isNat(X)
activate(X) → X
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)))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNatKind(n__0) → tt

(36) 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(n__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)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

(37) 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) DependencyGraphProof (EQUIVALENT transformation)

(18) Complex Obligation (AND)

(19) Obligation:

(20) QDPOrderProof (EQUIVALENT transformation)

(21) Obligation:

(22) DependencyGraphProof (EQUIVALENT transformation)

(23) Complex Obligation (AND)

(24) Obligation:

(25) QDPOrderProof (EQUIVALENT transformation)

(26) Obligation:

(27) PisEmptyProof (EQUIVALENT transformation)

(28) TRUE

(29) Obligation:

(30) QDPOrderProof (EQUIVALENT transformation)

(31) Obligation:

(32) PisEmptyProof (EQUIVALENT transformation)

(33) TRUE

(34) Obligation:

(35) QDPOrderProof (EQUIVALENT transformation)

(36) Obligation:

(37) PisEmptyProof (EQUIVALENT transformation)

(38) TRUE