(0) Obligation:

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.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

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

+1(-(x, y), z) → -1(+(x, z), y)
+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.

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(4) Obligation:

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.

(5) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


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

Recursive path order with status [RPO].
Quasi-Precedence:
[+^12, -1, +1]

Status:
+^12: [1,2]
-1: [1]
+1: [1]


The following usable rules [FROCOS05] were oriented:

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

(6) Obligation:

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.

(7) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(8) TRUE