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:

app2(app2(twice, f), x) -> app2(f, app2(f, x))
app2(app2(map, f), nil) -> nil
app2(app2(map, f), app2(app2(cons, h), t)) -> app2(app2(cons, app2(f, h)), app2(app2(map, f), t))
app2(app2(fmap, nil), x) -> nil
app2(app2(fmap, app2(app2(cons, f), t_f)), x) -> app2(app2(cons, app2(f, x)), app2(app2(fmap, t_f), x))

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

app2(app2(twice, f), x) -> app2(f, app2(f, x))
app2(app2(map, f), nil) -> nil
app2(app2(map, f), app2(app2(cons, h), t)) -> app2(app2(cons, app2(f, h)), app2(app2(map, f), t))
app2(app2(fmap, nil), x) -> nil
app2(app2(fmap, app2(app2(cons, f), t_f)), x) -> app2(app2(cons, app2(f, x)), app2(app2(fmap, t_f), x))

Q is empty.

Using Dependency Pairs [1,13] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

APP2(app2(fmap, app2(app2(cons, f), t_f)), x) -> APP2(f, x)
APP2(app2(twice, f), x) -> APP2(f, x)
APP2(app2(map, f), app2(app2(cons, h), t)) -> APP2(app2(cons, app2(f, h)), app2(app2(map, f), t))
APP2(app2(fmap, app2(app2(cons, f), t_f)), x) -> APP2(app2(cons, app2(f, x)), app2(app2(fmap, t_f), x))
APP2(app2(map, f), app2(app2(cons, h), t)) -> APP2(f, h)
APP2(app2(fmap, app2(app2(cons, f), t_f)), x) -> APP2(fmap, t_f)
APP2(app2(map, f), app2(app2(cons, h), t)) -> APP2(app2(map, f), t)
APP2(app2(twice, f), x) -> APP2(f, app2(f, x))
APP2(app2(fmap, app2(app2(cons, f), t_f)), x) -> APP2(app2(fmap, t_f), x)
APP2(app2(fmap, app2(app2(cons, f), t_f)), x) -> APP2(cons, app2(f, x))
APP2(app2(map, f), app2(app2(cons, h), t)) -> APP2(cons, app2(f, h))

The TRS R consists of the following rules:

app2(app2(twice, f), x) -> app2(f, app2(f, x))
app2(app2(map, f), nil) -> nil
app2(app2(map, f), app2(app2(cons, h), t)) -> app2(app2(cons, app2(f, h)), app2(app2(map, f), t))
app2(app2(fmap, nil), x) -> nil
app2(app2(fmap, app2(app2(cons, f), t_f)), x) -> app2(app2(cons, app2(f, x)), app2(app2(fmap, t_f), x))

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof

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

APP2(app2(fmap, app2(app2(cons, f), t_f)), x) -> APP2(f, x)
APP2(app2(twice, f), x) -> APP2(f, x)
APP2(app2(map, f), app2(app2(cons, h), t)) -> APP2(app2(cons, app2(f, h)), app2(app2(map, f), t))
APP2(app2(fmap, app2(app2(cons, f), t_f)), x) -> APP2(app2(cons, app2(f, x)), app2(app2(fmap, t_f), x))
APP2(app2(map, f), app2(app2(cons, h), t)) -> APP2(f, h)
APP2(app2(fmap, app2(app2(cons, f), t_f)), x) -> APP2(fmap, t_f)
APP2(app2(map, f), app2(app2(cons, h), t)) -> APP2(app2(map, f), t)
APP2(app2(twice, f), x) -> APP2(f, app2(f, x))
APP2(app2(fmap, app2(app2(cons, f), t_f)), x) -> APP2(app2(fmap, t_f), x)
APP2(app2(fmap, app2(app2(cons, f), t_f)), x) -> APP2(cons, app2(f, x))
APP2(app2(map, f), app2(app2(cons, h), t)) -> APP2(cons, app2(f, h))

The TRS R consists of the following rules:

app2(app2(twice, f), x) -> app2(f, app2(f, x))
app2(app2(map, f), nil) -> nil
app2(app2(map, f), app2(app2(cons, h), t)) -> app2(app2(cons, app2(f, h)), app2(app2(map, f), t))
app2(app2(fmap, nil), x) -> nil
app2(app2(fmap, app2(app2(cons, f), t_f)), x) -> app2(app2(cons, app2(f, x)), app2(app2(fmap, t_f), x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 1 SCC with 6 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof

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

APP2(app2(fmap, app2(app2(cons, f), t_f)), x) -> APP2(f, x)
APP2(app2(twice, f), x) -> APP2(f, x)
APP2(app2(map, f), app2(app2(cons, h), t)) -> APP2(f, h)
APP2(app2(map, f), app2(app2(cons, h), t)) -> APP2(app2(map, f), t)
APP2(app2(twice, f), x) -> APP2(f, app2(f, x))

The TRS R consists of the following rules:

app2(app2(twice, f), x) -> app2(f, app2(f, x))
app2(app2(map, f), nil) -> nil
app2(app2(map, f), app2(app2(cons, h), t)) -> app2(app2(cons, app2(f, h)), app2(app2(map, f), t))
app2(app2(fmap, nil), x) -> nil
app2(app2(fmap, app2(app2(cons, f), t_f)), x) -> app2(app2(cons, app2(f, x)), app2(app2(fmap, t_f), x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


APP2(app2(fmap, app2(app2(cons, f), t_f)), x) -> APP2(f, x)
The remaining pairs can at least be oriented weakly.

APP2(app2(twice, f), x) -> APP2(f, x)
APP2(app2(map, f), app2(app2(cons, h), t)) -> APP2(f, h)
APP2(app2(map, f), app2(app2(cons, h), t)) -> APP2(app2(map, f), t)
APP2(app2(twice, f), x) -> APP2(f, app2(f, x))
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( APP2(x1, x2) ) = max{0, x1 - 3}


POL( app2(x1, x2) ) = max{0, x1 + x2 - 1}


POL( fmap ) = 1


POL( cons ) = 3


POL( t_f ) = 3


POL( twice ) = 1


POL( map ) = 1



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ QDPOrderProof

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

APP2(app2(twice, f), x) -> APP2(f, x)
APP2(app2(map, f), app2(app2(cons, h), t)) -> APP2(f, h)
APP2(app2(twice, f), x) -> APP2(f, app2(f, x))
APP2(app2(map, f), app2(app2(cons, h), t)) -> APP2(app2(map, f), t)

The TRS R consists of the following rules:

app2(app2(twice, f), x) -> app2(f, app2(f, x))
app2(app2(map, f), nil) -> nil
app2(app2(map, f), app2(app2(cons, h), t)) -> app2(app2(cons, app2(f, h)), app2(app2(map, f), t))
app2(app2(fmap, nil), x) -> nil
app2(app2(fmap, app2(app2(cons, f), t_f)), x) -> app2(app2(cons, app2(f, x)), app2(app2(fmap, t_f), x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


APP2(app2(twice, f), x) -> APP2(f, x)
APP2(app2(twice, f), x) -> APP2(f, app2(f, x))
The remaining pairs can at least be oriented weakly.

APP2(app2(map, f), app2(app2(cons, h), t)) -> APP2(f, h)
APP2(app2(map, f), app2(app2(cons, h), t)) -> APP2(app2(map, f), t)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( APP2(x1, x2) ) = max{0, x1 - 3}


POL( app2(x1, x2) ) = x1 + x2 + 1


POL( twice ) = 3


POL( map ) = 2



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ QDPOrderProof
                ↳ QDP
                  ↳ QDPOrderProof

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

APP2(app2(map, f), app2(app2(cons, h), t)) -> APP2(f, h)
APP2(app2(map, f), app2(app2(cons, h), t)) -> APP2(app2(map, f), t)

The TRS R consists of the following rules:

app2(app2(twice, f), x) -> app2(f, app2(f, x))
app2(app2(map, f), nil) -> nil
app2(app2(map, f), app2(app2(cons, h), t)) -> app2(app2(cons, app2(f, h)), app2(app2(map, f), t))
app2(app2(fmap, nil), x) -> nil
app2(app2(fmap, app2(app2(cons, f), t_f)), x) -> app2(app2(cons, app2(f, x)), app2(app2(fmap, t_f), x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


APP2(app2(map, f), app2(app2(cons, h), t)) -> APP2(f, h)
APP2(app2(map, f), app2(app2(cons, h), t)) -> APP2(app2(map, f), t)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( APP2(x1, x2) ) = max{0, x2 - 3}


POL( app2(x1, x2) ) = x1 + x2 + 1


POL( cons ) = 2



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ QDPOrderProof
                ↳ QDP
                  ↳ QDPOrderProof
                    ↳ QDP
                      ↳ PisEmptyProof

Q DP problem:
P is empty.
The TRS R consists of the following rules:

app2(app2(twice, f), x) -> app2(f, app2(f, x))
app2(app2(map, f), nil) -> nil
app2(app2(map, f), app2(app2(cons, h), t)) -> app2(app2(cons, app2(f, h)), app2(app2(map, f), t))
app2(app2(fmap, nil), x) -> nil
app2(app2(fmap, app2(app2(cons, f), t_f)), x) -> app2(app2(cons, app2(f, x)), app2(app2(fmap, t_f), x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.