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:

-2(-2(neg1(x), neg1(x)), -2(neg1(y), neg1(y))) -> -2(-2(x, y), -2(x, y))

Q is empty.


↳ QTRS
  ↳ Non-Overlap Check

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

-2(-2(neg1(x), neg1(x)), -2(neg1(y), neg1(y))) -> -2(-2(x, y), -2(x, y))

Q is empty.

The TRS is non-overlapping. Hence, we can switch to innermost.

↳ QTRS
  ↳ Non-Overlap Check
    ↳ QTRS
      ↳ DependencyPairsProof

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

-2(-2(neg1(x), neg1(x)), -2(neg1(y), neg1(y))) -> -2(-2(x, y), -2(x, y))

The set Q consists of the following terms:

-2(-2(neg1(x0), neg1(x0)), -2(neg1(x1), neg1(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:

-12(-2(neg1(x), neg1(x)), -2(neg1(y), neg1(y))) -> -12(-2(x, y), -2(x, y))
-12(-2(neg1(x), neg1(x)), -2(neg1(y), neg1(y))) -> -12(x, y)

The TRS R consists of the following rules:

-2(-2(neg1(x), neg1(x)), -2(neg1(y), neg1(y))) -> -2(-2(x, y), -2(x, y))

The set Q consists of the following terms:

-2(-2(neg1(x0), neg1(x0)), -2(neg1(x1), neg1(x1)))

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

↳ QTRS
  ↳ Non-Overlap Check
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ QDPOrderProof

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

-12(-2(neg1(x), neg1(x)), -2(neg1(y), neg1(y))) -> -12(-2(x, y), -2(x, y))
-12(-2(neg1(x), neg1(x)), -2(neg1(y), neg1(y))) -> -12(x, y)

The TRS R consists of the following rules:

-2(-2(neg1(x), neg1(x)), -2(neg1(y), neg1(y))) -> -2(-2(x, y), -2(x, y))

The set Q consists of the following terms:

-2(-2(neg1(x0), neg1(x0)), -2(neg1(x1), neg1(x1)))

We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be strictly oriented and are deleted.


-12(-2(neg1(x), neg1(x)), -2(neg1(y), neg1(y))) -> -12(-2(x, y), -2(x, y))
-12(-2(neg1(x), neg1(x)), -2(neg1(y), neg1(y))) -> -12(x, y)
The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
-12(x1, x2)  =  -12(x1, x2)
-2(x1, x2)  =  x2
neg1(x1)  =  neg1(x1)

Lexicographic Path Order [19].
Precedence:
neg1 > -^12


The following usable rules [14] were oriented:

-2(-2(neg1(x), neg1(x)), -2(neg1(y), neg1(y))) -> -2(-2(x, y), -2(x, y))



↳ QTRS
  ↳ Non-Overlap Check
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ PisEmptyProof

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

-2(-2(neg1(x), neg1(x)), -2(neg1(y), neg1(y))) -> -2(-2(x, y), -2(x, y))

The set Q consists of the following terms:

-2(-2(neg1(x0), neg1(x0)), -2(neg1(x1), neg1(x1)))

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