Termination w.r.t. Q of the following Term Rewriting System could not be shown:

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

f2(f2(a, a), x) -> f2(a, f2(b, f2(a, x)))
f2(x, f2(y, z)) -> f2(f2(x, y), z)

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

f2(f2(a, a), x) -> f2(a, f2(b, f2(a, x)))
f2(x, f2(y, z)) -> f2(f2(x, y), z)

Q is empty.

Using Dependency Pairs [1,13] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

F2(x, f2(y, z)) -> F2(x, y)
F2(x, f2(y, z)) -> F2(f2(x, y), z)
F2(f2(a, a), x) -> F2(b, f2(a, x))
F2(f2(a, a), x) -> F2(a, f2(b, f2(a, x)))
F2(f2(a, a), x) -> F2(a, x)

The TRS R consists of the following rules:

f2(f2(a, a), x) -> f2(a, f2(b, f2(a, x)))
f2(x, f2(y, z)) -> f2(f2(x, y), z)

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ QDPOrderProof

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

F2(x, f2(y, z)) -> F2(x, y)
F2(x, f2(y, z)) -> F2(f2(x, y), z)
F2(f2(a, a), x) -> F2(b, f2(a, x))
F2(f2(a, a), x) -> F2(a, f2(b, f2(a, x)))
F2(f2(a, a), x) -> F2(a, x)

The TRS R consists of the following rules:

f2(f2(a, a), x) -> f2(a, f2(b, f2(a, x)))
f2(x, f2(y, z)) -> f2(f2(x, y), z)

Q is empty.
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.


F2(f2(a, a), x) -> F2(b, f2(a, x))
The remaining pairs can at least be oriented weakly.

F2(x, f2(y, z)) -> F2(x, y)
F2(x, f2(y, z)) -> F2(f2(x, y), z)
F2(f2(a, a), x) -> F2(a, f2(b, f2(a, x)))
F2(f2(a, a), x) -> F2(a, x)
Used ordering: Polynomial interpretation [21]:

POL(F2(x1, x2)) = 2·x1   
POL(a) = 2   
POL(b) = 0   
POL(f2(x1, x2)) = x1   

The following usable rules [14] were oriented:

f2(x, f2(y, z)) -> f2(f2(x, y), z)
f2(f2(a, a), x) -> f2(a, f2(b, f2(a, x)))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ QDPOrderProof
        ↳ QDP
          ↳ QDPOrderProof

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

F2(x, f2(y, z)) -> F2(x, y)
F2(x, f2(y, z)) -> F2(f2(x, y), z)
F2(f2(a, a), x) -> F2(a, f2(b, f2(a, x)))
F2(f2(a, a), x) -> F2(a, x)

The TRS R consists of the following rules:

f2(f2(a, a), x) -> f2(a, f2(b, f2(a, x)))
f2(x, f2(y, z)) -> f2(f2(x, y), z)

Q is empty.
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.


F2(f2(a, a), x) -> F2(a, x)
The remaining pairs can at least be oriented weakly.

F2(x, f2(y, z)) -> F2(x, y)
F2(x, f2(y, z)) -> F2(f2(x, y), z)
F2(f2(a, a), x) -> F2(a, f2(b, f2(a, x)))
Used ordering: Polynomial interpretation [21]:

POL(F2(x1, x2)) = 2·x1 + 2·x2   
POL(a) = 1   
POL(b) = 0   
POL(f2(x1, x2)) = x1 + x2   

The following usable rules [14] were oriented:

f2(x, f2(y, z)) -> f2(f2(x, y), z)
f2(f2(a, a), x) -> f2(a, f2(b, f2(a, x)))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ QDPOrderProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP

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

F2(x, f2(y, z)) -> F2(x, y)
F2(x, f2(y, z)) -> F2(f2(x, y), z)
F2(f2(a, a), x) -> F2(a, f2(b, f2(a, x)))

The TRS R consists of the following rules:

f2(f2(a, a), x) -> f2(a, f2(b, f2(a, x)))
f2(x, f2(y, z)) -> f2(f2(x, y), z)

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