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:

h(f(x, y)) → f(f(a, h(h(y))), x)

Q is empty.


↳ QTRS
  ↳ Overlay + Local Confluence

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

h(f(x, y)) → f(f(a, h(h(y))), 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:

h(f(x, y)) → f(f(a, h(h(y))), x)

The set Q consists of the following terms:

h(f(x0, x1))


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

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

The TRS R consists of the following rules:

h(f(x, y)) → f(f(a, h(h(y))), x)

The set Q consists of the following terms:

h(f(x0, x1))

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

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

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

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

The TRS R consists of the following rules:

h(f(x, y)) → f(f(a, h(h(y))), x)

The set Q consists of the following terms:

h(f(x0, x1))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule H(f(x, y)) → H(h(y)) at position [0] we obtained the following new rules:

H(f(y0, f(x0, x1))) → H(f(f(a, h(h(x1))), x0))



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

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

H(f(y0, f(x0, x1))) → H(f(f(a, h(h(x1))), x0))
H(f(x, y)) → H(y)

The TRS R consists of the following rules:

h(f(x, y)) → f(f(a, h(h(y))), x)

The set Q consists of the following terms:

h(f(x0, x1))

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

H(f(x0, f(y_0, f(y_1, y_2)))) → H(f(y_0, f(y_1, y_2)))
H(f(x0, f(y_0, y_1))) → H(f(y_0, y_1))



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

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

H(f(y0, f(x0, x1))) → H(f(f(a, h(h(x1))), x0))
H(f(x0, f(y_0, f(y_1, y_2)))) → H(f(y_0, f(y_1, y_2)))
H(f(x0, f(y_0, y_1))) → H(f(y_0, y_1))

The TRS R consists of the following rules:

h(f(x, y)) → f(f(a, h(h(y))), x)

The set Q consists of the following terms:

h(f(x0, x1))

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
          ↳ Narrowing
            ↳ QDP
              ↳ ForwardInstantiation
                ↳ QDP
                  ↳ MNOCProof
                    ↳ QDP
          ↳ MNOCProof

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

H(f(y0, f(x0, x1))) → H(f(f(a, h(h(x1))), x0))
H(f(x0, f(y_0, f(y_1, y_2)))) → H(f(y_0, f(y_1, y_2)))
H(f(x0, f(y_0, y_1))) → H(f(y_0, y_1))

The TRS R consists of the following rules:

h(f(x, y)) → f(f(a, h(h(y))), x)

Q is empty.
We have to consider all (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
          ↳ Narrowing
          ↳ MNOCProof
            ↳ QDP

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

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

The TRS R consists of the following rules:

h(f(x, y)) → f(f(a, h(h(y))), x)

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