Termination w.r.t. Q proof of /tmp/tmpEyEaty/PEANO_nokinds

Termination w.r.t. Q of the following Term Rewriting System could be proven:

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

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

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

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

Q is empty.

Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

U21¹(tt, M, N) → S(plus(activate(N), activate(M)))
ACTIVATE(n__s(X)) → S(activate(X))
PLUS(N, s(M)) → ISNAT(M)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__plus(V1, V2)) → AND(isNat(activate(V1)), n__isNat(activate(V2)))
AND(tt, X) → ACTIVATE(X)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ACTIVATE(V1)
PLUS(N, 0) → U11¹(isNat(N), N)
ACTIVATE(n__s(X)) → ACTIVATE(X)
U21¹(tt, M, N) → PLUS(activate(N), activate(M))
U21¹(tt, M, N) → ACTIVATE(M)
PLUS(N, s(M)) → U21¹(and(isNat(M), n__isNat(N)), M, N)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__0) → 0¹
U21¹(tt, M, N) → ACTIVATE(N)
ACTIVATE(n__isNat(X)) → ISNAT(X)
ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
PLUS(N, s(M)) → AND(isNat(M), n__isNat(N))
PLUS(N, 0) → ISNAT(N)
U11¹(tt, N) → ACTIVATE(N)
ISNAT(n__plus(V1, V2)) → ISNAT(activate(V1))

The TRS R consists of the following rules:

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

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

U21¹(tt, M, N) → S(plus(activate(N), activate(M)))
ACTIVATE(n__s(X)) → S(activate(X))
PLUS(N, s(M)) → ISNAT(M)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__plus(V1, V2)) → AND(isNat(activate(V1)), n__isNat(activate(V2)))
AND(tt, X) → ACTIVATE(X)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ACTIVATE(V1)
PLUS(N, 0) → U11¹(isNat(N), N)
ACTIVATE(n__s(X)) → ACTIVATE(X)
U21¹(tt, M, N) → PLUS(activate(N), activate(M))
U21¹(tt, M, N) → ACTIVATE(M)
PLUS(N, s(M)) → U21¹(and(isNat(M), n__isNat(N)), M, N)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__0) → 0¹
U21¹(tt, M, N) → ACTIVATE(N)
ACTIVATE(n__isNat(X)) → ISNAT(X)
ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
PLUS(N, s(M)) → AND(isNat(M), n__isNat(N))
PLUS(N, 0) → ISNAT(N)
U11¹(tt, N) → ACTIVATE(N)
ISNAT(n__plus(V1, V2)) → ISNAT(activate(V1))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 3 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

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

PLUS(N, s(M)) → ISNAT(M)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__plus(V1, V2)) → AND(isNat(activate(V1)), n__isNat(activate(V2)))
ISNAT(n__s(V1)) → ACTIVATE(V1)
PLUS(N, 0) → U11¹(isNat(N), N)
AND(tt, X) → ACTIVATE(X)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
U21¹(tt, M, N) → ACTIVATE(M)
U21¹(tt, M, N) → PLUS(activate(N), activate(M))
ACTIVATE(n__s(X)) → ACTIVATE(X)
PLUS(N, s(M)) → U21¹(and(isNat(M), n__isNat(N)), M, N)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
U21¹(tt, M, N) → ACTIVATE(N)
ACTIVATE(n__isNat(X)) → ISNAT(X)
ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
PLUS(N, s(M)) → AND(isNat(M), n__isNat(N))
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
PLUS(N, 0) → ISNAT(N)
U11¹(tt, N) → ACTIVATE(N)
ISNAT(n__plus(V1, V2)) → ISNAT(activate(V1))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

PLUS(N, 0) → U11¹(isNat(N), N)
PLUS(N, 0) → ISNAT(N)

The remaining pairs can at least be oriented weakly.

PLUS(N, s(M)) → ISNAT(M)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__plus(V1, V2)) → AND(isNat(activate(V1)), n__isNat(activate(V2)))
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
U21¹(tt, M, N) → ACTIVATE(M)
U21¹(tt, M, N) → PLUS(activate(N), activate(M))
ACTIVATE(n__s(X)) → ACTIVATE(X)
PLUS(N, s(M)) → U21¹(and(isNat(M), n__isNat(N)), M, N)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
U21¹(tt, M, N) → ACTIVATE(N)
ACTIVATE(n__isNat(X)) → ISNAT(X)
ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
PLUS(N, s(M)) → AND(isNat(M), n__isNat(N))
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
U11¹(tt, N) → ACTIVATE(N)
ISNAT(n__plus(V1, V2)) → ISNAT(activate(V1))

Used ordering: Polynomial interpretation [25,35]:

POL(n__plus(x₁, x₂)) = (2)x_1 + (4)x_2
POL(plus(x₁, x₂)) = (2)x_1 + (4)x_2
POL(n__isNat(x₁)) = x_1
POL(activate(x₁)) = x_1
POL(U11(x₁, x₂)) = x_2
POL(and(x₁, x₂)) = (4)x_2
POL(n__s(x₁)) = x_1
POL(0) = 1/2
POL(ISNAT(x₁)) = (1/4)x_1
POL(PLUS(x₁, x₂)) = (1/2)x_1 + x_2
POL(tt) = 0
POL(U11¹(x₁, x₂)) = (1/4)x_2
POL(AND(x₁, x₂)) = (1/2)x_2
POL(n__0) = 1/2
POL(s(x₁)) = x_1
POL(U21¹(x₁, x₂, x₃)) = x_2 + (1/2)x_3
POL(isNat(x₁)) = x_1
POL(U21(x₁, x₂, x₃)) = (4)x_2 + (2)x_3
POL(ACTIVATE(x₁)) = (1/4)x_1

The value of delta used in the strict ordering is 1/2.
The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

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

PLUS(N, s(M)) → ISNAT(M)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__plus(V1, V2)) → AND(isNat(activate(V1)), n__isNat(activate(V2)))
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
U21¹(tt, M, N) → ACTIVATE(M)
U21¹(tt, M, N) → PLUS(activate(N), activate(M))
ACTIVATE(n__s(X)) → ACTIVATE(X)
PLUS(N, s(M)) → U21¹(and(isNat(M), n__isNat(N)), M, N)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
U21¹(tt, M, N) → ACTIVATE(N)
ACTIVATE(n__isNat(X)) → ISNAT(X)
ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
PLUS(N, s(M)) → AND(isNat(M), n__isNat(N))
ISNAT(n__plus(V1, V2)) → ISNAT(activate(V1))
U11¹(tt, N) → ACTIVATE(N)

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

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

ISNAT(n__s(V1)) → ISNAT(activate(V1))
PLUS(N, s(M)) → ISNAT(M)
ISNAT(n__plus(V1, V2)) → AND(isNat(activate(V1)), n__isNat(activate(V2)))
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
U21¹(tt, M, N) → ACTIVATE(M)
ACTIVATE(n__s(X)) → ACTIVATE(X)
U21¹(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, s(M)) → U21¹(and(isNat(M), n__isNat(N)), M, N)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
U21¹(tt, M, N) → ACTIVATE(N)
ACTIVATE(n__isNat(X)) → ISNAT(X)
ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
PLUS(N, s(M)) → AND(isNat(M), n__isNat(N))
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
ISNAT(n__plus(V1, V2)) → ISNAT(activate(V1))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

ISNAT(n__plus(V1, V2)) → AND(isNat(activate(V1)), n__isNat(activate(V2)))
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
ISNAT(n__plus(V1, V2)) → ISNAT(activate(V1))

The remaining pairs can at least be oriented weakly.

ISNAT(n__s(V1)) → ISNAT(activate(V1))
PLUS(N, s(M)) → ISNAT(M)
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
U21¹(tt, M, N) → ACTIVATE(M)
ACTIVATE(n__s(X)) → ACTIVATE(X)
U21¹(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, s(M)) → U21¹(and(isNat(M), n__isNat(N)), M, N)
U21¹(tt, M, N) → ACTIVATE(N)
ACTIVATE(n__isNat(X)) → ISNAT(X)
PLUS(N, s(M)) → AND(isNat(M), n__isNat(N))

Used ordering: Polynomial interpretation [25,35]:

POL(n__plus(x₁, x₂)) = 2 + (4)x_1 + (4)x_2
POL(plus(x₁, x₂)) = 2 + (4)x_1 + (4)x_2
POL(n__isNat(x₁)) = x_1
POL(activate(x₁)) = x_1
POL(U11(x₁, x₂)) = (4)x_2
POL(n__s(x₁)) = x_1
POL(and(x₁, x₂)) = (4)x_2
POL(0) = 0
POL(ISNAT(x₁)) = (1/4)x_1
POL(PLUS(x₁, x₂)) = (1/2)x_1 + (1/4)x_2
POL(tt) = 0
POL(AND(x₁, x₂)) = (1/2)x_2
POL(n__0) = 0
POL(s(x₁)) = x_1
POL(U21¹(x₁, x₂, x₃)) = (1/4)x_2 + (1/2)x_3
POL(isNat(x₁)) = x_1
POL(U21(x₁, x₂, x₃)) = 2 + (4)x_2 + (4)x_3
POL(ACTIVATE(x₁)) = (1/4)x_1

The value of delta used in the strict ordering is 1/2.
The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

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

PLUS(N, s(M)) → U21¹(and(isNat(M), n__isNat(N)), M, N)
U21¹(tt, M, N) → ACTIVATE(N)
ACTIVATE(n__isNat(X)) → ISNAT(X)
PLUS(N, s(M)) → ISNAT(M)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
AND(tt, X) → ACTIVATE(X)
ISNAT(n__s(V1)) → ACTIVATE(V1)
PLUS(N, s(M)) → AND(isNat(M), n__isNat(N))
U21¹(tt, M, N) → PLUS(activate(N), activate(M))
ACTIVATE(n__s(X)) → ACTIVATE(X)
U21¹(tt, M, N) → ACTIVATE(M)

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 5 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ QDPOrderProof

                          ↳ QDP

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

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

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

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.

ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
ACTIVATE(n__s(X)) → ACTIVATE(X)

Used ordering: Polynomial interpretation [25,35]:

POL(n__plus(x₁, x₂)) = (4)x_1 + (2)x_2
POL(plus(x₁, x₂)) = (4)x_1 + (2)x_2
POL(n__isNat(x₁)) = 1/2 + (4)x_1
POL(activate(x₁)) = x_1
POL(U11(x₁, x₂)) = (4)x_2
POL(n__s(x₁)) = x_1
POL(and(x₁, x₂)) = x_2
POL(0) = 0
POL(ISNAT(x₁)) = (4)x_1
POL(tt) = 0
POL(n__0) = 0
POL(s(x₁)) = x_1
POL(isNat(x₁)) = 1/2 + (4)x_1
POL(U21(x₁, x₂, x₃)) = (2)x_2 + (4)x_3
POL(ACTIVATE(x₁)) = (4)x_1

The value of delta used in the strict ordering is 2.
The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                          ↳ QDP

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

ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
ACTIVATE(n__s(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 1 less node.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ AND

                                    ↳ QDP

                                      ↳ QDPOrderProof

                                    ↳ QDP

                          ↳ QDP

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

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.
none
Used ordering: Polynomial interpretation [25,35]:

POL(n__s(x₁)) = 4 + (4)x_1
POL(ACTIVATE(x₁)) = (2)x_1

The value of delta used in the strict ordering is 8.
The following usable rules [17] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ AND

                                    ↳ QDP

                                      ↳ QDPOrderProof

                                        ↳ QDP

                                          ↳ PisEmptyProof

                                    ↳ QDP

                          ↳ QDP

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

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ AND

                                    ↳ QDP

                                    ↳ QDP

                                      ↳ QDPOrderProof

                          ↳ QDP

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

ISNAT(n__s(V1)) → ISNAT(activate(V1))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

ISNAT(n__s(V1)) → ISNAT(activate(V1))

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(n__plus(x₁, x₂)) = (4)x_1 + (2)x_2
POL(plus(x₁, x₂)) = (4)x_1 + (2)x_2
POL(n__isNat(x₁)) = 0
POL(activate(x₁)) = x_1
POL(U11(x₁, x₂)) = x_2
POL(n__s(x₁)) = 1/4 + x_1
POL(and(x₁, x₂)) = (4)x_2
POL(0) = 0
POL(ISNAT(x₁)) = (1/4)x_1
POL(tt) = 0
POL(n__0) = 0
POL(s(x₁)) = 1/4 + x_1
POL(isNat(x₁)) = 0
POL(U21(x₁, x₂, x₃)) = 1/4 + (2)x_2 + (4)x_3

The value of delta used in the strict ordering is 1/16.
The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ AND

                                    ↳ QDP

                                    ↳ QDP

                                      ↳ QDPOrderProof

                                        ↳ QDP

                                          ↳ PisEmptyProof

                          ↳ QDP

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

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ QDPOrderProof

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

PLUS(N, s(M)) → U21¹(and(isNat(M), n__isNat(N)), M, N)
U21¹(tt, M, N) → PLUS(activate(N), activate(M))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

PLUS(N, s(M)) → U21¹(and(isNat(M), n__isNat(N)), M, N)

The remaining pairs can at least be oriented weakly.

U21¹(tt, M, N) → PLUS(activate(N), activate(M))

Used ordering: Polynomial interpretation [25,35]:

POL(n__plus(x₁, x₂)) = (4)x_1 + (4)x_2
POL(plus(x₁, x₂)) = (4)x_1 + (4)x_2
POL(n__isNat(x₁)) = 0
POL(activate(x₁)) = x_1
POL(U11(x₁, x₂)) = x_2
POL(n__s(x₁)) = 4 + x_1
POL(and(x₁, x₂)) = x_2
POL(0) = 0
POL(PLUS(x₁, x₂)) = x_2
POL(tt) = 0
POL(n__0) = 0
POL(s(x₁)) = 4 + x_1
POL(U21¹(x₁, x₂, x₃)) = x_2
POL(isNat(x₁)) = 0
POL(U21(x₁, x₂, x₃)) = 4 + (4)x_2 + (4)x_3

The value of delta used in the strict ordering is 4.
The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

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

U21¹(tt, M, N) → PLUS(activate(N), activate(M))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 1 less node.