Termination w.r.t. Q proof of TRS/Endrullispair3rotate.trs

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:

p₂(a₁(x0), p₂(b₁(x1), p₂(a₁(x2), x3))) -> p₂(x2, p₂(a₁(a₁(x0)), p₂(b₁(x1), x3)))

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

p₂(a₁(x0), p₂(b₁(x1), p₂(a₁(x2), x3))) -> p₂(x2, p₂(a₁(a₁(x0)), p₂(b₁(x1), x3)))

Q is empty.

Using Dependency Pairs [1,13] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

P₂(a₁(x0), p₂(b₁(x1), p₂(a₁(x2), x3))) -> P₂(x2, p₂(a₁(a₁(x0)), p₂(b₁(x1), x3)))
P₂(a₁(x0), p₂(b₁(x1), p₂(a₁(x2), x3))) -> P₂(a₁(a₁(x0)), p₂(b₁(x1), x3))
P₂(a₁(x0), p₂(b₁(x1), p₂(a₁(x2), x3))) -> P₂(b₁(x1), x3)

The TRS R consists of the following rules:

p₂(a₁(x0), p₂(b₁(x1), p₂(a₁(x2), x3))) -> p₂(x2, p₂(a₁(a₁(x0)), p₂(b₁(x1), x3)))

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:

P₂(a₁(x0), p₂(b₁(x1), p₂(a₁(x2), x3))) -> P₂(x2, p₂(a₁(a₁(x0)), p₂(b₁(x1), x3)))
P₂(a₁(x0), p₂(b₁(x1), p₂(a₁(x2), x3))) -> P₂(a₁(a₁(x0)), p₂(b₁(x1), x3))
P₂(a₁(x0), p₂(b₁(x1), p₂(a₁(x2), x3))) -> P₂(b₁(x1), x3)

The TRS R consists of the following rules:

p₂(a₁(x0), p₂(b₁(x1), p₂(a₁(x2), x3))) -> p₂(x2, p₂(a₁(a₁(x0)), p₂(b₁(x1), x3)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 1 SCC with 1 less node.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

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

P₂(a₁(x0), p₂(b₁(x1), p₂(a₁(x2), x3))) -> P₂(x2, p₂(a₁(a₁(x0)), p₂(b₁(x1), x3)))
P₂(a₁(x0), p₂(b₁(x1), p₂(a₁(x2), x3))) -> P₂(a₁(a₁(x0)), p₂(b₁(x1), x3))

The TRS R consists of the following rules:

p₂(a₁(x0), p₂(b₁(x1), p₂(a₁(x2), x3))) -> p₂(x2, p₂(a₁(a₁(x0)), p₂(b₁(x1), x3)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

P₂(a₁(x0), p₂(b₁(x1), p₂(a₁(x2), x3))) -> P₂(a₁(a₁(x0)), p₂(b₁(x1), x3))

The remaining pairs can at least be oriented weakly.

P₂(a₁(x0), p₂(b₁(x1), p₂(a₁(x2), x3))) -> P₂(x2, p₂(a₁(a₁(x0)), p₂(b₁(x1), x3)))

Used ordering: Polynomial interpretation [21]:

POL(P₂(x₁, x₂)) = x₂
POL(a₁(x₁)) = 0
POL(b₁(x₁)) = 0
POL(p₂(x₁, x₂)) = 1 + x₂

The following usable rules [14] were oriented:

p₂(a₁(x0), p₂(b₁(x1), p₂(a₁(x2), x3))) -> p₂(x2, p₂(a₁(a₁(x0)), p₂(b₁(x1), x3)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

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

P₂(a₁(x0), p₂(b₁(x1), p₂(a₁(x2), x3))) -> P₂(x2, p₂(a₁(a₁(x0)), p₂(b₁(x1), x3)))

The TRS R consists of the following rules:

p₂(a₁(x0), p₂(b₁(x1), p₂(a₁(x2), x3))) -> p₂(x2, p₂(a₁(a₁(x0)), p₂(b₁(x1), x3)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

P₂(a₁(x0), p₂(b₁(x1), p₂(a₁(x2), x3))) -> P₂(x2, p₂(a₁(a₁(x0)), p₂(b₁(x1), x3)))

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(P₂(x₁, x₂)) = 2 + x₁·x₂ + x₂
POL(a₁(x₁)) = 1 + 2·x₁
POL(b₁(x₁)) = 0
POL(p₂(x₁, x₂)) = x₁ + 2·x₁·x₂ + 2·x₂

The following usable rules [14] were oriented:

p₂(a₁(x0), p₂(b₁(x1), p₂(a₁(x2), x3))) -> p₂(x2, p₂(a₁(a₁(x0)), p₂(b₁(x1), x3)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ PisEmptyProof

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

p₂(a₁(x0), p₂(b₁(x1), p₂(a₁(x2), x3))) -> p₂(x2, p₂(a₁(a₁(x0)), p₂(b₁(x1), x3)))

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.