Termination w.r.t. Q proof of /tmp/tmpnhoVfu/MYNAT_nosorts-noand

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, M, N) → U12(tt, activate(M), activate(N))
U12(tt, M, N) → s(plus(activate(N), activate(M)))
U21(tt, M, N) → U22(tt, activate(M), activate(N))
U22(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
plus(N, 0) → N
plus(N, s(M)) → U11(tt, M, N)
x(N, 0) → 0
x(N, s(M)) → U21(tt, M, N)
activate(X) → X

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

U11(tt, M, N) → U12(tt, activate(M), activate(N))
U12(tt, M, N) → s(plus(activate(N), activate(M)))
U21(tt, M, N) → U22(tt, activate(M), activate(N))
U22(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
plus(N, 0) → N
plus(N, s(M)) → U11(tt, M, N)
x(N, 0) → 0
x(N, s(M)) → U21(tt, M, N)
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:

U11¹(tt, M, N) → ACTIVATE(N)
U22¹(tt, M, N) → ACTIVATE(N)
U21¹(tt, M, N) → U22¹(tt, activate(M), activate(N))
U22¹(tt, M, N) → ACTIVATE(M)
U12¹(tt, M, N) → PLUS(activate(N), activate(M))
U11¹(tt, M, N) → ACTIVATE(M)
U22¹(tt, M, N) → X(activate(N), activate(M))
X(N, s(M)) → U21¹(tt, M, N)
U11¹(tt, M, N) → U12¹(tt, activate(M), activate(N))
U21¹(tt, M, N) → ACTIVATE(M)
PLUS(N, s(M)) → U11¹(tt, M, N)
U22¹(tt, M, N) → PLUS(x(activate(N), activate(M)), activate(N))
U21¹(tt, M, N) → ACTIVATE(N)
U12¹(tt, M, N) → ACTIVATE(N)
U12¹(tt, M, N) → ACTIVATE(M)

The TRS R consists of the following rules:

U11(tt, M, N) → U12(tt, activate(M), activate(N))
U12(tt, M, N) → s(plus(activate(N), activate(M)))
U21(tt, M, N) → U22(tt, activate(M), activate(N))
U22(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
plus(N, 0) → N
plus(N, s(M)) → U11(tt, M, N)
x(N, 0) → 0
x(N, s(M)) → U21(tt, M, N)
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:

U11¹(tt, M, N) → ACTIVATE(N)
U22¹(tt, M, N) → ACTIVATE(N)
U21¹(tt, M, N) → U22¹(tt, activate(M), activate(N))
U22¹(tt, M, N) → ACTIVATE(M)
U12¹(tt, M, N) → PLUS(activate(N), activate(M))
U11¹(tt, M, N) → ACTIVATE(M)
U22¹(tt, M, N) → X(activate(N), activate(M))
X(N, s(M)) → U21¹(tt, M, N)
U11¹(tt, M, N) → U12¹(tt, activate(M), activate(N))
U21¹(tt, M, N) → ACTIVATE(M)
PLUS(N, s(M)) → U11¹(tt, M, N)
U22¹(tt, M, N) → PLUS(x(activate(N), activate(M)), activate(N))
U21¹(tt, M, N) → ACTIVATE(N)
U12¹(tt, M, N) → ACTIVATE(N)
U12¹(tt, M, N) → ACTIVATE(M)

The TRS R consists of the following rules:

U11(tt, M, N) → U12(tt, activate(M), activate(N))
U12(tt, M, N) → s(plus(activate(N), activate(M)))
U21(tt, M, N) → U22(tt, activate(M), activate(N))
U22(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
plus(N, 0) → N
plus(N, s(M)) → U11(tt, M, N)
x(N, 0) → 0
x(N, s(M)) → U21(tt, M, N)
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 9 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

          ↳ QDP

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

PLUS(N, s(M)) → U11¹(tt, M, N)
U12¹(tt, M, N) → PLUS(activate(N), activate(M))
U11¹(tt, M, N) → U12¹(tt, activate(M), activate(N))

The TRS R consists of the following rules:

U11(tt, M, N) → U12(tt, activate(M), activate(N))
U12(tt, M, N) → s(plus(activate(N), activate(M)))
U21(tt, M, N) → U22(tt, activate(M), activate(N))
U22(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
plus(N, 0) → N
plus(N, s(M)) → U11(tt, M, N)
x(N, 0) → 0
x(N, s(M)) → U21(tt, M, N)
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)) → U11¹(tt, M, N)
U12¹(tt, M, N) → PLUS(activate(N), activate(M))
U11¹(tt, M, N) → U12¹(tt, activate(M), activate(N))

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

POL(PLUS(x₁, x₂)) = 5/4 + (4)x_2
POL(U12¹(x₁, x₂, x₃)) = 7/4 + (2)x_1 + (4)x_2
POL(tt) = 9/4
POL(U11¹(x₁, x₂, x₃)) = 3 + (2)x_1 + (4)x_2
POL(s(x₁)) = 4 + x_1
POL(activate(x₁)) = x_1

The value of delta used in the strict ordering is 5/4.
The following usable rules [17] were oriented:

activate(X) → X

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ PisEmptyProof

          ↳ QDP

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

U11(tt, M, N) → U12(tt, activate(M), activate(N))
U12(tt, M, N) → s(plus(activate(N), activate(M)))
U21(tt, M, N) → U22(tt, activate(M), activate(N))
U22(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
plus(N, 0) → N
plus(N, s(M)) → U11(tt, M, N)
x(N, 0) → 0
x(N, s(M)) → U21(tt, M, N)
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

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

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

U21¹(tt, M, N) → U22¹(tt, activate(M), activate(N))
U22¹(tt, M, N) → X(activate(N), activate(M))
X(N, s(M)) → U21¹(tt, M, N)

The TRS R consists of the following rules:

U11(tt, M, N) → U12(tt, activate(M), activate(N))
U12(tt, M, N) → s(plus(activate(N), activate(M)))
U21(tt, M, N) → U22(tt, activate(M), activate(N))
U22(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
plus(N, 0) → N
plus(N, s(M)) → U11(tt, M, N)
x(N, 0) → 0
x(N, s(M)) → U21(tt, M, N)
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.

U22¹(tt, M, N) → X(activate(N), activate(M))

The remaining pairs can at least be oriented weakly.

U21¹(tt, M, N) → U22¹(tt, activate(M), activate(N))
X(N, s(M)) → U21¹(tt, M, N)

Used ordering: Polynomial interpretation [25,35]:

POL(X(x₁, x₂)) = (1/2)x_2
POL(tt) = 0
POL(U22¹(x₁, x₂, x₃)) = 1/4 + (15/4)x_1 + (1/2)x_2
POL(activate(x₁)) = x_1
POL(s(x₁)) = 1/2 + (4)x_1
POL(U21¹(x₁, x₂, x₃)) = 1/4 + (4)x_1 + (7/4)x_2

The value of delta used in the strict ordering is 1/4.
The following usable rules [17] were oriented:

activate(X) → X

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

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

U21¹(tt, M, N) → U22¹(tt, activate(M), activate(N))
X(N, s(M)) → U21¹(tt, M, N)

The TRS R consists of the following rules:

U11(tt, M, N) → U12(tt, activate(M), activate(N))
U12(tt, M, N) → s(plus(activate(N), activate(M)))
U21(tt, M, N) → U22(tt, activate(M), activate(N))
U22(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
plus(N, 0) → N
plus(N, s(M)) → U11(tt, M, N)
x(N, 0) → 0
x(N, s(M)) → U21(tt, M, N)
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 2 less nodes.