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:

-2(-2(neg1(x), neg1(x)), -2(neg1(y), neg1(y))) -> -2(-2(x, y), -2(x, y))

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

-2(-2(neg1(x), neg1(x)), -2(neg1(y), neg1(y))) -> -2(-2(x, y), -2(x, y))

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:

-12(-2(neg1(x), neg1(x)), -2(neg1(y), neg1(y))) -> -12(-2(x, y), -2(x, y))
-12(-2(neg1(x), neg1(x)), -2(neg1(y), neg1(y))) -> -12(x, y)

The TRS R consists of the following rules:

-2(-2(neg1(x), neg1(x)), -2(neg1(y), neg1(y))) -> -2(-2(x, y), -2(x, y))

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ QDPOrderProof

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

-12(-2(neg1(x), neg1(x)), -2(neg1(y), neg1(y))) -> -12(-2(x, y), -2(x, y))
-12(-2(neg1(x), neg1(x)), -2(neg1(y), neg1(y))) -> -12(x, y)

The TRS R consists of the following rules:

-2(-2(neg1(x), neg1(x)), -2(neg1(y), neg1(y))) -> -2(-2(x, y), -2(x, y))

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.


-12(-2(neg1(x), neg1(x)), -2(neg1(y), neg1(y))) -> -12(x, y)
The remaining pairs can at least be oriented weakly.

-12(-2(neg1(x), neg1(x)), -2(neg1(y), neg1(y))) -> -12(-2(x, y), -2(x, y))
Used ordering: Polynomial interpretation [21]:

POL(-2(x1, x2)) = 1 + x2   
POL(-12(x1, x2)) = x1·x2   
POL(neg1(x1)) = 2·x1 + x12   

The following usable rules [14] were oriented:

-2(-2(neg1(x), neg1(x)), -2(neg1(y), neg1(y))) -> -2(-2(x, y), -2(x, y))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ QDPOrderProof
        ↳ QDP
          ↳ QDPOrderProof

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

-12(-2(neg1(x), neg1(x)), -2(neg1(y), neg1(y))) -> -12(-2(x, y), -2(x, y))

The TRS R consists of the following rules:

-2(-2(neg1(x), neg1(x)), -2(neg1(y), neg1(y))) -> -2(-2(x, y), -2(x, y))

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.


-12(-2(neg1(x), neg1(x)), -2(neg1(y), neg1(y))) -> -12(-2(x, y), -2(x, y))
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(-2(x1, x2)) = x2   
POL(-12(x1, x2)) = x1·x2   
POL(neg1(x1)) = 1 + x1 + x12   

The following usable rules [14] were oriented:

-2(-2(neg1(x), neg1(x)), -2(neg1(y), neg1(y))) -> -2(-2(x, y), -2(x, y))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ QDPOrderProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ PisEmptyProof

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

-2(-2(neg1(x), neg1(x)), -2(neg1(y), neg1(y))) -> -2(-2(x, y), -2(x, y))

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.