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:

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

Q is empty.


↳ QTRS
  ↳ Overlay + Local Confluence

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.

The TRS is overlay and locally confluent. By [15] we can switch to innermost.

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof

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

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

The set Q consists of the following terms:

-(-(neg(x0), neg(x0)), -(neg(x1), neg(x1)))


Using Dependency Pairs [1,13] we result in the following initial DP problem:
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))

The set Q consists of the following terms:

-(-(neg(x0), neg(x0)), -(neg(x1), neg(x1)))

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

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ QDPOrderProof

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))

The set Q consists of the following terms:

-(-(neg(x0), neg(x0)), -(neg(x1), neg(x1)))

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.


-1(-(neg(x), neg(x)), -(neg(y), neg(y))) → -1(x, y)
The remaining pairs can at least be oriented weakly.

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

Lexicographic path order with status [19].
Precedence:
-1 > -^11

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

The following usable rules [14] were oriented:

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



↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ QDPOrderProof

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))

The set Q consists of the following terms:

-(-(neg(x0), neg(x0)), -(neg(x1), neg(x1)))

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.


-1(-(neg(x), neg(x)), -(neg(y), neg(y))) → -1(-(x, y), -(x, y))
The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
-1(x1, x2)  =  -1(x1)
-(x1, x2)  =  -(x1)
neg(x1)  =  neg(x1)

Lexicographic path order with status [19].
Precedence:
trivial

Status:
neg1: multiset
-1: [1]
-^11: [1]

The following usable rules [14] were oriented:

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



↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ QDPOrderProof
                ↳ QDP
                  ↳ PisEmptyProof

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))

The set Q consists of the following terms:

-(-(neg(x0), neg(x0)), -(neg(x1), neg(x1)))

We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.