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

Q is empty.


↳ QTRS
  ↳ Overlay + Local Confluence

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

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

The set Q consists of the following terms:

f(f(f(a, f(a, a)), a), x0)


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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

f(f(f(a, f(a, a)), a), x0)

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

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

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

f(f(f(a, f(a, a)), a), x0)

We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule F(f(f(a, f(a, a)), a), x) → F(x, f(x, a)) at position [] we obtained the following new rules:

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



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

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

f(f(f(a, f(a, a)), a), x0)

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

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

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

F(f(f(a, f(a, a)), a), x) → F(x, a)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

f(f(f(a, f(a, a)), a), x0)

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
          ↳ Narrowing
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ QDP
                  ↳ UsableRulesProof
                    ↳ QDP
                      ↳ Instantiation

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

F(f(f(a, f(a, a)), a), x) → F(x, a)

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

f(f(f(a, f(a, a)), a), x0)

We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule F(f(f(a, f(a, a)), a), x) → F(x, a) we obtained the following new rules:

F(f(f(a, f(a, a)), a), a) → F(a, a)



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

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

F(f(f(a, f(a, a)), a), a) → F(a, a)

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

f(f(f(a, f(a, a)), a), x0)

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 1 less node.