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(x, f(y, a)) → f(f(f(f(a, a), y), h(a)), x)

Q is empty.


↳ QTRS
  ↳ Overlay + Local Confluence

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

f(x, f(y, a)) → f(f(f(f(a, a), y), h(a)), x)

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(x, f(y, a)) → f(f(f(f(a, a), y), h(a)), x)

The set Q consists of the following terms:

f(x0, f(x1, 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(x, f(y, a)) → F(a, a)
F(x, f(y, a)) → F(f(f(f(a, a), y), h(a)), x)
F(x, f(y, a)) → F(f(a, a), y)
F(x, f(y, a)) → F(f(f(a, a), y), h(a))

The TRS R consists of the following rules:

f(x, f(y, a)) → f(f(f(f(a, a), y), h(a)), x)

The set Q consists of the following terms:

f(x0, f(x1, 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(x, f(y, a)) → F(a, a)
F(x, f(y, a)) → F(f(f(f(a, a), y), h(a)), x)
F(x, f(y, a)) → F(f(a, a), y)
F(x, f(y, a)) → F(f(f(a, a), y), h(a))

The TRS R consists of the following rules:

f(x, f(y, a)) → f(f(f(f(a, a), y), h(a)), x)

The set Q consists of the following terms:

f(x0, f(x1, a))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.

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

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

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

The TRS R consists of the following rules:

f(x, f(y, a)) → f(f(f(f(a, a), y), h(a)), x)

The set Q consists of the following terms:

f(x0, f(x1, a))

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

F(f(y_0, h(a)), f(x1, a)) → F(f(a, a), x1)
F(f(a, a), f(x1, a)) → F(f(a, a), x1)



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

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

F(x, f(y, a)) → F(f(f(f(a, a), y), h(a)), x)
F(f(y_0, h(a)), f(x1, a)) → F(f(a, a), x1)
F(f(a, a), f(x1, a)) → F(f(a, a), x1)

The TRS R consists of the following rules:

f(x, f(y, a)) → f(f(f(f(a, a), y), h(a)), x)

The set Q consists of the following terms:

f(x0, f(x1, a))

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

F(f(y_0, h(a)), f(x1, a)) → F(f(f(f(a, a), x1), h(a)), f(y_0, h(a)))
F(f(a, a), f(x1, a)) → F(f(f(f(a, a), x1), h(a)), f(a, a))



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

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

F(f(y_0, h(a)), f(x1, a)) → F(f(f(f(a, a), x1), h(a)), f(y_0, h(a)))
F(f(y_0, h(a)), f(x1, a)) → F(f(a, a), x1)
F(f(a, a), f(x1, a)) → F(f(a, a), x1)
F(f(a, a), f(x1, a)) → F(f(f(f(a, a), x1), h(a)), f(a, a))

The TRS R consists of the following rules:

f(x, f(y, a)) → f(f(f(f(a, a), y), h(a)), x)

The set Q consists of the following terms:

f(x0, f(x1, a))

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
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Instantiation
                ↳ QDP
                  ↳ Instantiation
                    ↳ QDP
                      ↳ DependencyGraphProof
                        ↳ QDP
                          ↳ Instantiation

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

F(f(y_0, h(a)), f(x1, a)) → F(f(a, a), x1)
F(f(a, a), f(x1, a)) → F(f(a, a), x1)
F(f(a, a), f(x1, a)) → F(f(f(f(a, a), x1), h(a)), f(a, a))

The TRS R consists of the following rules:

f(x, f(y, a)) → f(f(f(f(a, a), y), h(a)), x)

The set Q consists of the following terms:

f(x0, f(x1, a))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule F(f(y_0, h(a)), f(x1, a)) → F(f(a, a), x1) we obtained the following new rules:

F(f(y_0, h(a)), f(a, a)) → F(f(a, a), a)



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

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

F(f(a, a), f(x1, a)) → F(f(a, a), x1)
F(f(a, a), f(x1, a)) → F(f(f(f(a, a), x1), h(a)), f(a, a))
F(f(y_0, h(a)), f(a, a)) → F(f(a, a), a)

The TRS R consists of the following rules:

f(x, f(y, a)) → f(f(f(f(a, a), y), h(a)), x)

The set Q consists of the following terms:

f(x0, f(x1, a))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.

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

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

F(f(a, a), f(x1, a)) → F(f(a, a), x1)

The TRS R consists of the following rules:

f(x, f(y, a)) → f(f(f(f(a, a), y), h(a)), x)

The set Q consists of the following terms:

f(x0, f(x1, 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
              ↳ Instantiation
                ↳ QDP
                  ↳ Instantiation
                    ↳ QDP
                      ↳ DependencyGraphProof
                        ↳ QDP
                          ↳ Instantiation
                            ↳ QDP
                              ↳ DependencyGraphProof
                                ↳ QDP
                                  ↳ UsableRulesProof
                                    ↳ QDP
                                      ↳ ForwardInstantiation

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

F(f(a, a), f(x1, a)) → F(f(a, a), x1)

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

f(x0, f(x1, a))

We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule F(f(a, a), f(x1, a)) → F(f(a, a), x1) we obtained the following new rules:

F(f(a, a), f(f(y_0, a), a)) → F(f(a, a), f(y_0, a))



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

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

F(f(a, a), f(f(y_0, a), a)) → F(f(a, a), f(y_0, a))

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

f(x0, f(x1, 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: