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, x), y), a), h(a))

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

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

The TRS R consists of the following rules:

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

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

The TRS R consists of the following rules:

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

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
              ↳ ForwardInstantiation

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

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

The TRS R consists of the following rules:

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

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(x, f(y, a)) → F(a, x) we obtained the following new rules:

F(f(y_1, a), f(x1, a)) → F(a, f(y_1, a))



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

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

F(f(y_1, a), f(x1, a)) → F(a, f(y_1, a))
F(x, f(y, a)) → F(f(a, x), y)

The TRS R consists of the following rules:

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

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(x, f(y, a)) → F(f(a, x), y) we obtained the following new rules:

F(a, f(f(y_1, a), a)) → F(f(a, a), f(y_1, a))
F(x0, f(f(y_1, a), a)) → F(f(a, x0), f(y_1, a))



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

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

F(a, f(f(y_1, a), a)) → F(f(a, a), f(y_1, a))
F(f(y_1, a), f(x1, a)) → F(a, f(y_1, a))
F(x0, f(f(y_1, a), a)) → F(f(a, x0), f(y_1, a))

The TRS R consists of the following rules:

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

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(y_1, a), f(x1, a)) → F(a, f(y_1, a)) we obtained the following new rules:

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



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

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

F(a, f(f(y_1, a), a)) → F(f(a, a), f(y_1, a))
F(x0, f(f(y_1, a), a)) → F(f(a, x0), f(y_1, a))
F(f(f(y_0, a), a), f(x1, a)) → F(a, f(f(y_0, a), a))

The TRS R consists of the following rules:

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

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
              ↳ ForwardInstantiation
                ↳ QDP
                  ↳ ForwardInstantiation
                    ↳ QDP
                      ↳ ForwardInstantiation
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ QDPSizeChangeProof

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

F(x0, f(f(y_1, a), a)) → F(f(a, x0), f(y_1, a))

The TRS R consists of the following rules:

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

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: