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:

f2(f2(a, x), a) -> f2(f2(f2(a, a), f2(x, a)), a)

Q is empty.


↳ QTRS
  ↳ Non-Overlap Check

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

f2(f2(a, x), a) -> f2(f2(f2(a, a), f2(x, a)), a)

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:

f2(f2(a, x), a) -> f2(f2(f2(a, a), f2(x, a)), a)

The set Q consists of the following terms:

f2(f2(a, x0), a)


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(f2(a, x), a) -> F2(f2(f2(a, a), f2(x, a)), a)
F2(f2(a, x), a) -> F2(a, a)
F2(f2(a, x), a) -> F2(f2(a, a), f2(x, a))
F2(f2(a, x), a) -> F2(x, a)

The TRS R consists of the following rules:

f2(f2(a, x), a) -> f2(f2(f2(a, a), f2(x, a)), a)

The set Q consists of the following terms:

f2(f2(a, x0), a)

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

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

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

F2(f2(a, x), a) -> F2(f2(f2(a, a), f2(x, a)), a)
F2(f2(a, x), a) -> F2(a, a)
F2(f2(a, x), a) -> F2(f2(a, a), f2(x, a))
F2(f2(a, x), a) -> F2(x, a)

The TRS R consists of the following rules:

f2(f2(a, x), a) -> f2(f2(f2(a, a), f2(x, a)), a)

The set Q consists of the following terms:

f2(f2(a, x0), a)

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 1 SCC with 3 less nodes.

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

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

F2(f2(a, x), a) -> F2(x, a)

The TRS R consists of the following rules:

f2(f2(a, x), a) -> f2(f2(f2(a, a), f2(x, a)), a)

The set Q consists of the following terms:

f2(f2(a, x0), a)

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.


F2(f2(a, x), a) -> F2(x, a)
The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
F2(x1, x2)  =  F1(x1)
f2(x1, x2)  =  f2(x1, x2)
a  =  a

Lexicographic Path Order [19].
Precedence:
f2 > F1 > a

The following usable rules [14] were oriented: none



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

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

f2(f2(a, x), a) -> f2(f2(f2(a, a), f2(x, a)), a)

The set Q consists of the following terms:

f2(f2(a, x0), a)

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