Termination w.r.t. Q proof of TRS/SK904.19.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:

-(-(neg(x), neg(x)), -(neg(y), neg(y))) → -(-(x, y), -(x, y))

Q is empty.

↳ QTRS

  ↳ Overlay + Local Confluence

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

-(-(neg(x), neg(x)), -(neg(y), neg(y))) → -(-(x, y), -(x, y))

Q is empty.

The TRS is overlay and locally confluent. By [15] we can switch to innermost.

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

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

-(-(neg(x), neg(x)), -(neg(y), neg(y))) → -(-(x, y), -(x, y))

The set Q consists of the following terms:

-(-(neg(x0), neg(x0)), -(neg(x1), neg(x1)))

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

-¹(-(neg(x), neg(x)), -(neg(y), neg(y))) → -¹(-(x, y), -(x, y))
-¹(-(neg(x), neg(x)), -(neg(y), neg(y))) → -¹(x, y)

The TRS R consists of the following rules:

-(-(neg(x), neg(x)), -(neg(y), neg(y))) → -(-(x, y), -(x, y))

The set Q consists of the following terms:

-(-(neg(x0), neg(x0)), -(neg(x1), neg(x1)))

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

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ QDPOrderProof

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

-¹(-(neg(x), neg(x)), -(neg(y), neg(y))) → -¹(-(x, y), -(x, y))
-¹(-(neg(x), neg(x)), -(neg(y), neg(y))) → -¹(x, y)

The TRS R consists of the following rules:

-(-(neg(x), neg(x)), -(neg(y), neg(y))) → -(-(x, y), -(x, y))

The set Q consists of the following terms:

-(-(neg(x0), neg(x0)), -(neg(x1), neg(x1)))

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.

-¹(-(neg(x), neg(x)), -(neg(y), neg(y))) → -¹(x, y)

The remaining pairs can at least be oriented weakly.

-¹(-(neg(x), neg(x)), -(neg(y), neg(y))) → -¹(-(x, y), -(x, y))

Used ordering: Combined order from the following AFS and order.
-¹(x1, x2) = -¹(x2)
-(x1, x2) = -(x2)
neg(x1) = x1

Recursive path order with status [2].
Quasi-Precedence:

-^1₁ > -₁

Status:

-₁: [1]
-^1₁: multiset

The following usable rules [14] were oriented:

-(-(neg(x), neg(x)), -(neg(y), neg(y))) → -(-(x, y), -(x, y))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

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

-¹(-(neg(x), neg(x)), -(neg(y), neg(y))) → -¹(-(x, y), -(x, y))

The TRS R consists of the following rules:

-(-(neg(x), neg(x)), -(neg(y), neg(y))) → -(-(x, y), -(x, y))

The set Q consists of the following terms:

-(-(neg(x0), neg(x0)), -(neg(x1), neg(x1)))

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.

-¹(-(neg(x), neg(x)), -(neg(y), neg(y))) → -¹(-(x, y), -(x, y))

The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
-¹(x1, x2) = -¹(x2)
-(x1, x2) = x2
neg(x1) = neg(x1)

Recursive path order with status [2].
Quasi-Precedence:

[-^1₁, neg_1]

Status:

neg₁: multiset
-^1₁: multiset

The following usable rules [14] were oriented:

-(-(neg(x), neg(x)), -(neg(y), neg(y))) → -(-(x, y), -(x, y))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ PisEmptyProof

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

-(-(neg(x), neg(x)), -(neg(y), neg(y))) → -(-(x, y), -(x, y))

The set Q consists of the following terms:

-(-(neg(x0), neg(x0)), -(neg(x1), neg(x1)))

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