Termination w.r.t. Q of the following Term Rewriting System could not be shown:

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

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

Q is empty.


QTRS
  ↳ Overlay + Local Confluence

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

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

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)) → h(f(a, f(f(x, a), y)))

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

The TRS R consists of the following rules:

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

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

The TRS R consists of the following rules:

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

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
              ↳ MNOCProof

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

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

The TRS R consists of the following rules:

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

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

F(f(z0, a), f(x1, a)) → F(f(f(z0, a), a), x1)
F(a, f(x1, a)) → F(f(a, a), x1)



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

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

F(f(z0, a), f(x1, a)) → F(f(f(z0, a), a), x1)
F(a, f(x1, a)) → F(f(a, a), x1)
F(x, f(y, a)) → F(a, f(f(x, a), y))

The TRS R consists of the following rules:

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

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

F(f(f(z0, a), a), f(x1, a)) → F(a, f(f(f(f(z0, a), a), a), x1))
F(a, f(x1, a)) → F(a, f(f(a, a), x1))
F(f(a, a), f(x1, a)) → F(a, f(f(f(a, a), a), x1))



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

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

F(f(f(z0, a), a), f(x1, a)) → F(a, f(f(f(f(z0, a), a), a), x1))
F(f(z0, a), f(x1, a)) → F(f(f(z0, a), a), x1)
F(a, f(x1, a)) → F(f(a, a), x1)
F(a, f(x1, a)) → F(a, f(f(a, a), x1))
F(f(a, a), f(x1, a)) → F(a, f(f(f(a, a), a), x1))

The TRS R consists of the following rules:

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

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(z0, a), f(x1, a)) → F(f(f(z0, a), a), x1) we obtained the following new rules:

F(f(a, a), f(x1, a)) → F(f(f(a, a), a), x1)
F(f(f(z0, a), a), f(x1, a)) → F(f(f(f(z0, a), a), a), x1)



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

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

F(f(f(z0, a), a), f(x1, a)) → F(a, f(f(f(f(z0, a), a), a), x1))
F(f(a, a), f(x1, a)) → F(f(f(a, a), a), x1)
F(a, f(x1, a)) → F(f(a, a), x1)
F(f(f(z0, a), a), f(x1, a)) → F(f(f(f(z0, a), a), a), x1)
F(a, f(x1, a)) → F(a, f(f(a, a), x1))
F(f(a, a), f(x1, a)) → F(a, f(f(f(a, a), a), x1))

The TRS R consists of the following rules:

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

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(f(z0, a), a), f(x1, a)) → F(a, f(f(f(f(z0, a), a), a), x1)) we obtained the following new rules:

F(f(f(f(z0, a), a), a), f(x1, a)) → F(a, f(f(f(f(f(z0, a), a), a), a), x1))
F(f(f(a, a), a), f(x1, a)) → F(a, f(f(f(f(a, a), a), a), x1))



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

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

F(f(f(f(z0, a), a), a), f(x1, a)) → F(a, f(f(f(f(f(z0, a), a), a), a), x1))
F(f(a, a), f(x1, a)) → F(f(f(a, a), a), x1)
F(a, f(x1, a)) → F(f(a, a), x1)
F(f(f(a, a), a), f(x1, a)) → F(a, f(f(f(f(a, a), a), a), x1))
F(f(f(z0, a), a), f(x1, a)) → F(f(f(f(z0, a), a), a), x1)
F(a, f(x1, a)) → F(a, f(f(a, a), x1))
F(f(a, a), f(x1, a)) → F(a, f(f(f(a, a), a), x1))

The TRS R consists of the following rules:

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

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

F(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
                      ↳ Instantiation
                        ↳ QDP
                          ↳ Instantiation
                            ↳ QDP
                              ↳ ForwardInstantiation
QDP
                                  ↳ ForwardInstantiation
              ↳ MNOCProof

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

F(f(f(f(z0, a), a), a), f(x1, a)) → F(a, f(f(f(f(f(z0, a), a), a), a), x1))
F(f(a, a), f(x1, a)) → F(f(f(a, a), a), x1)
F(a, f(f(y_0, a), a)) → F(f(a, a), f(y_0, a))
F(f(f(a, a), a), f(x1, a)) → F(a, f(f(f(f(a, a), a), a), x1))
F(f(f(z0, a), a), f(x1, a)) → F(f(f(f(z0, a), a), a), x1)
F(a, f(x1, a)) → F(a, f(f(a, a), x1))
F(f(a, a), f(x1, a)) → F(a, f(f(f(a, a), a), x1))

The TRS R consists of the following rules:

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

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

F(a, f(a, a)) → F(a, f(f(a, a), a))



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

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

F(f(f(f(z0, a), a), a), f(x1, a)) → F(a, f(f(f(f(f(z0, a), a), a), a), x1))
F(f(a, a), f(x1, a)) → F(f(f(a, a), a), x1)
F(a, f(f(y_0, a), a)) → F(f(a, a), f(y_0, a))
F(f(f(a, a), a), f(x1, a)) → F(a, f(f(f(f(a, a), a), a), x1))
F(f(f(z0, a), a), f(x1, a)) → F(f(f(f(z0, a), a), a), x1)
F(a, f(a, a)) → F(a, f(f(a, a), a))
F(f(a, a), f(x1, a)) → F(a, f(f(f(a, a), a), x1))

The TRS R consists of the following rules:

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

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
                      ↳ Instantiation
                        ↳ QDP
                          ↳ Instantiation
                            ↳ QDP
                              ↳ ForwardInstantiation
                                ↳ QDP
                                  ↳ ForwardInstantiation
                                    ↳ QDP
                                      ↳ DependencyGraphProof
QDP
                                          ↳ ForwardInstantiation
              ↳ MNOCProof

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

F(f(f(f(z0, a), a), a), f(x1, a)) → F(a, f(f(f(f(f(z0, a), a), a), a), x1))
F(f(a, a), f(x1, a)) → F(f(f(a, a), a), x1)
F(a, f(f(y_0, a), a)) → F(f(a, a), f(y_0, a))
F(f(f(a, a), a), f(x1, a)) → F(a, f(f(f(f(a, a), a), a), x1))
F(f(f(z0, a), a), f(x1, a)) → F(f(f(f(z0, a), a), a), x1)
F(f(a, a), f(x1, a)) → F(a, f(f(f(a, a), a), x1))

The TRS R consists of the following rules:

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

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

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



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

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

F(f(f(f(z0, a), a), a), f(x1, a)) → F(a, f(f(f(f(f(z0, a), a), a), a), x1))
F(f(a, a), f(x1, a)) → F(f(f(a, a), a), x1)
F(a, f(f(y_0, a), a)) → F(f(a, a), f(y_0, a))
F(f(f(a, a), a), f(x1, a)) → F(a, f(f(f(f(a, a), a), a), x1))
F(f(f(z0, a), a), f(x1, a)) → F(f(f(f(z0, a), a), a), x1)
F(f(a, a), f(a, a)) → F(a, f(f(f(a, a), a), a))

The TRS R consists of the following rules:

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

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

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



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

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

F(f(f(f(z0, a), a), a), f(x1, a)) → F(a, f(f(f(f(f(z0, a), a), a), a), x1))
F(f(a, a), f(f(y_1, a), a)) → F(f(f(a, a), a), f(y_1, a))
F(a, f(f(y_0, a), a)) → F(f(a, a), f(y_0, a))
F(f(f(a, a), a), f(x1, a)) → F(a, f(f(f(f(a, a), a), a), x1))
F(f(f(z0, a), a), f(x1, a)) → F(f(f(f(z0, a), a), a), x1)
F(f(a, a), f(a, a)) → F(a, f(f(f(a, a), a), a))

The TRS R consists of the following rules:

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

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(a, f(f(y_0, a), a)) → F(f(a, a), f(y_0, a)) we obtained the following new rules:

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



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

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

F(f(f(f(z0, a), a), a), f(x1, a)) → F(a, f(f(f(f(f(z0, a), a), a), a), x1))
F(f(a, a), f(f(y_1, a), a)) → F(f(f(a, a), a), f(y_1, a))
F(a, f(f(f(y_0, a), a), a)) → F(f(a, a), f(f(y_0, a), a))
F(f(f(a, a), a), f(x1, a)) → F(a, f(f(f(f(a, a), a), a), x1))
F(f(f(z0, a), a), f(x1, a)) → F(f(f(f(z0, a), a), a), x1)
F(a, f(f(a, a), a)) → F(f(a, a), f(a, a))
F(f(a, a), f(a, a)) → F(a, f(f(f(a, a), a), a))

The TRS R consists of the following rules:

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

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
                      ↳ Instantiation
                        ↳ QDP
                          ↳ Instantiation
                            ↳ QDP
                              ↳ ForwardInstantiation
                                ↳ QDP
                                  ↳ ForwardInstantiation
                                    ↳ QDP
                                      ↳ DependencyGraphProof
                                        ↳ QDP
                                          ↳ ForwardInstantiation
                                            ↳ QDP
                                              ↳ ForwardInstantiation
                                                ↳ QDP
                                                  ↳ ForwardInstantiation
                                                    ↳ QDP
                                                      ↳ DependencyGraphProof
QDP
                                                          ↳ ForwardInstantiation
              ↳ MNOCProof

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

F(f(f(f(z0, a), a), a), f(x1, a)) → F(a, f(f(f(f(f(z0, a), a), a), a), x1))
F(f(a, a), f(f(y_1, a), a)) → F(f(f(a, a), a), f(y_1, a))
F(f(f(a, a), a), f(x1, a)) → F(a, f(f(f(f(a, a), a), a), x1))
F(a, f(f(f(y_0, a), a), a)) → F(f(a, a), f(f(y_0, a), a))
F(f(f(z0, a), a), f(x1, a)) → F(f(f(f(z0, a), a), a), x1)

The TRS R consists of the following rules:

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

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(f(z0, a), a), f(x1, a)) → F(f(f(f(z0, a), a), a), x1) we obtained the following new rules:

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



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

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

F(f(f(f(z0, a), a), a), f(x1, a)) → F(a, f(f(f(f(f(z0, a), a), a), a), x1))
F(f(a, a), f(f(y_1, a), a)) → F(f(f(a, a), a), f(y_1, a))
F(a, f(f(f(y_0, a), a), a)) → F(f(a, a), f(f(y_0, a), a))
F(f(f(a, a), a), f(x1, a)) → F(a, f(f(f(f(a, a), a), a), x1))
F(f(f(x0, a), a), f(f(y_1, a), a)) → F(f(f(f(x0, a), a), a), f(y_1, a))

The TRS R consists of the following rules:

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

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(f(f(z0, a), a), a), f(x1, a)) → F(a, f(f(f(f(f(z0, a), a), a), a), x1)) we obtained the following new rules:

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



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

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

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

The TRS R consists of the following rules:

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

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

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



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

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

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

The TRS R consists of the following rules:

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

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
                      ↳ Instantiation
                        ↳ QDP
                          ↳ Instantiation
                            ↳ QDP
                              ↳ ForwardInstantiation
                                ↳ QDP
                                  ↳ ForwardInstantiation
                                    ↳ QDP
                                      ↳ DependencyGraphProof
                                        ↳ QDP
                                          ↳ ForwardInstantiation
                                            ↳ QDP
                                              ↳ ForwardInstantiation
                                                ↳ QDP
                                                  ↳ ForwardInstantiation
                                                    ↳ QDP
                                                      ↳ DependencyGraphProof
                                                        ↳ QDP
                                                          ↳ ForwardInstantiation
                                                            ↳ QDP
                                                              ↳ ForwardInstantiation
                                                                ↳ QDP
                                                                  ↳ ForwardInstantiation
                                                                    ↳ QDP
                                                                      ↳ UsableRulesProof
QDP
                                                                          ↳ Instantiation
              ↳ MNOCProof

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

F(f(a, a), f(f(y_1, a), a)) → F(f(f(a, a), a), f(y_1, a))
F(f(f(a, a), a), f(a, a)) → F(a, f(f(f(f(a, a), a), a), a))
F(f(f(f(x0, a), a), a), f(a, a)) → F(a, f(f(f(f(f(x0, a), a), a), a), a))
F(a, f(f(f(y_0, a), a), a)) → F(f(a, a), f(f(y_0, a), a))
F(f(f(x0, a), a), f(f(y_1, a), a)) → F(f(f(f(x0, a), a), a), f(y_1, 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 instantiating [15] the rule F(a, f(f(f(y_0, a), a), a)) → F(f(a, a), f(f(y_0, a), a)) we obtained the following new rules:

F(a, f(f(f(f(f(z0, a), a), a), a), a)) → F(f(a, a), f(f(f(f(z0, a), a), a), a))
F(a, f(f(f(f(a, a), a), a), a)) → F(f(a, a), f(f(f(a, a), a), a))



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

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

F(a, f(f(f(f(f(z0, a), a), a), a), a)) → F(f(a, a), f(f(f(f(z0, a), a), a), a))
F(a, f(f(f(f(a, a), a), a), a)) → F(f(a, a), f(f(f(a, a), a), a))
F(f(a, a), f(f(y_1, a), a)) → F(f(f(a, a), a), f(y_1, a))
F(f(f(a, a), a), f(a, a)) → F(a, f(f(f(f(a, a), a), a), a))
F(f(f(f(x0, a), a), a), f(a, a)) → F(a, f(f(f(f(f(x0, a), a), a), a), a))
F(f(f(x0, a), a), f(f(y_1, a), a)) → F(f(f(f(x0, a), a), a), f(y_1, 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 instantiating [15] the rule F(f(a, a), f(f(y_1, a), a)) → F(f(f(a, a), a), f(y_1, a)) we obtained the following new rules:

F(f(a, a), f(f(f(a, a), a), a)) → F(f(f(a, a), a), f(f(a, a), a))
F(f(a, a), f(f(f(f(z0, a), a), a), a)) → F(f(f(a, a), a), f(f(f(z0, a), a), a))



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

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

F(a, f(f(f(f(f(z0, a), a), a), a), a)) → F(f(a, a), f(f(f(f(z0, a), a), a), a))
F(a, f(f(f(f(a, a), a), a), a)) → F(f(a, a), f(f(f(a, a), a), a))
F(f(f(a, a), a), f(a, a)) → F(a, f(f(f(f(a, a), a), a), a))
F(f(f(f(x0, a), a), a), f(a, a)) → F(a, f(f(f(f(f(x0, a), a), a), a), a))
F(f(a, a), f(f(f(a, a), a), a)) → F(f(f(a, a), a), f(f(a, a), a))
F(f(a, a), f(f(f(f(z0, a), a), a), a)) → F(f(f(a, a), a), f(f(f(z0, a), a), a))
F(f(f(x0, a), a), f(f(y_1, a), a)) → F(f(f(f(x0, a), a), a), f(y_1, 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.
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
              ↳ Instantiation
                ↳ QDP
                  ↳ Instantiation
                    ↳ QDP
                      ↳ Instantiation
                        ↳ QDP
                          ↳ Instantiation
                            ↳ QDP
                              ↳ ForwardInstantiation
                                ↳ QDP
                                  ↳ ForwardInstantiation
                                    ↳ QDP
                                      ↳ DependencyGraphProof
                                        ↳ QDP
                                          ↳ ForwardInstantiation
                                            ↳ QDP
                                              ↳ ForwardInstantiation
                                                ↳ QDP
                                                  ↳ ForwardInstantiation
                                                    ↳ QDP
                                                      ↳ DependencyGraphProof
                                                        ↳ QDP
                                                          ↳ ForwardInstantiation
                                                            ↳ QDP
                                                              ↳ ForwardInstantiation
                                                                ↳ QDP
                                                                  ↳ ForwardInstantiation
                                                                    ↳ QDP
                                                                      ↳ UsableRulesProof
                                                                        ↳ QDP
                                                                          ↳ Instantiation
                                                                            ↳ QDP
                                                                              ↳ Instantiation
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
QDP
              ↳ MNOCProof

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

F(a, f(f(f(f(f(z0, a), a), a), a), a)) → F(f(a, a), f(f(f(f(z0, a), a), a), a))
F(f(f(f(x0, a), a), a), f(a, a)) → F(a, f(f(f(f(f(x0, a), a), a), a), a))
F(f(a, a), f(f(f(f(z0, a), a), a), a)) → F(f(f(a, a), a), f(f(f(z0, a), a), a))
F(f(f(x0, a), a), f(f(y_1, a), a)) → F(f(f(f(x0, a), a), a), f(y_1, 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.
We use the modular non-overlap check [17] to decrease Q to the empty set.

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

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

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

The TRS R consists of the following rules:

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

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