(0) Obligation:

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.

(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(-(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))

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

(3) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


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

Lexicographic path order with status [LPO].
Quasi-Precedence:
-^11 > -1

Status:
-^11: [1]
-1: [1]


The following usable rules [FROCOS05] were oriented:

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

(4) Obligation:

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))

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(-(neg(x), neg(x)), -(neg(y), neg(y))) → -1(-(x, y), -(x, y))
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)  =  x2
neg(x1)  =  neg(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
neg1 > -^12

Status:
-^12: [1,2]
neg1: [1]


The following usable rules [FROCOS05] were oriented:

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

(6) Obligation:

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))

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