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:

f(a, f(x, a)) → f(a, f(f(a, x), f(a, a)))

Q is empty.


↳ QTRS
  ↳ Overlay + Local Confluence

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

f(a, f(x, a)) → f(a, f(f(a, x), f(a, a)))

Q is empty.

The TRS is overlay and locally confluent. By [19] we can switch to innermost.

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof

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

f(a, f(x, a)) → f(a, f(f(a, x), f(a, a)))

The set Q consists of the following terms:

f(a, f(x0, a))


Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

F(a, f(x, a)) → F(a, f(f(a, x), f(a, a)))
F(a, f(x, a)) → F(a, a)
F(a, f(x, a)) → F(f(a, x), f(a, a))
F(a, f(x, a)) → F(a, x)

The TRS R consists of the following rules:

f(a, f(x, a)) → f(a, f(f(a, x), f(a, a)))

The set Q consists of the following terms:

f(a, f(x0, a))

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

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

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

F(a, f(x, a)) → F(a, f(f(a, x), f(a, a)))
F(a, f(x, a)) → F(a, a)
F(a, f(x, a)) → F(f(a, x), f(a, a))
F(a, f(x, a)) → F(a, x)

The TRS R consists of the following rules:

f(a, f(x, a)) → f(a, f(f(a, x), f(a, a)))

The set Q consists of the following terms:

f(a, f(x0, a))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 3 less nodes.

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ UsableRulesProof

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

F(a, f(x, a)) → F(a, x)

The TRS R consists of the following rules:

f(a, f(x, a)) → f(a, f(f(a, x), f(a, a)))

The set Q consists of the following terms:

f(a, f(x0, a))

We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ UsableRulesProof
                ↳ QDP
                  ↳ QDPSizeChangeProof

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

F(a, f(x, a)) → F(a, x)

R is empty.
The set Q consists of the following terms:

f(a, f(x0, a))

We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs: