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:

+(-(x, y), z) → -(+(x, z), y)
-(+(x, y), y) → x

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

+(-(x, y), z) → -(+(x, z), y)
-(+(x, y), y) → 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:

+1(-(x, y), z) → +1(x, z)
+1(-(x, y), z) → -1(+(x, z), y)

The TRS R consists of the following rules:

+(-(x, y), z) → -(+(x, z), y)
-(+(x, y), y) → 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:

+1(-(x, y), z) → +1(x, z)
+1(-(x, y), z) → -1(+(x, z), y)

The TRS R consists of the following rules:

+(-(x, y), z) → -(+(x, z), y)
-(+(x, y), y) → 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

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

+1(-(x, y), z) → +1(x, z)

The TRS R consists of the following rules:

+(-(x, y), z) → -(+(x, z), y)
-(+(x, y), y) → 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.


+1(-(x, y), z) → +1(x, z)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(-(x1, x2)) = 1/4 + (7/2)x_1   
POL(+1(x1, x2)) = (2)x_1   
The value of delta used in the strict ordering is 1/2.
The following usable rules [17] were oriented: none



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

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

+(-(x, y), z) → -(+(x, z), y)
-(+(x, y), y) → 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.