(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,
x2)
Lexicographic path order with status [LPO].
Quasi-Precedence:
+2 > [+^12, -1]
Status:
+^12: [2,1]
-1: [1]
+2: [2,1]
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