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(append, nil), l) -> l
app2(app2(append, app2(app2(cons, h), t)), l) -> app2(app2(cons, h), app2(app2(append, t), l))
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(append, app2(app2(append, l1), l2)), l3) -> app2(app2(append, l1), app2(app2(append, l2), l3))
app2(app2(map, f), app2(app2(append, l1), l2)) -> app2(app2(append, app2(app2(map, f), l1)), app2(app2(map, f), l2))

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

app2(app2(append, nil), l) -> l
app2(app2(append, app2(app2(cons, h), t)), l) -> app2(app2(cons, h), app2(app2(append, t), l))
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(append, app2(app2(append, l1), l2)), l3) -> app2(app2(append, l1), app2(app2(append, l2), l3))
app2(app2(map, f), app2(app2(append, l1), l2)) -> app2(app2(append, app2(app2(map, f), l1)), app2(app2(map, f), l2))

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(append, app2(app2(append, l1), l2)), l3) -> APP2(app2(append, l2), l3)
APP2(app2(map, f), app2(app2(cons, h), t)) -> APP2(app2(cons, app2(f, h)), app2(app2(map, f), t))
APP2(app2(append, app2(app2(cons, h), t)), l) -> APP2(append, t)
APP2(app2(map, f), app2(app2(append, l1), l2)) -> APP2(app2(append, app2(app2(map, f), l1)), app2(app2(map, f), l2))
APP2(app2(map, f), app2(app2(append, l1), l2)) -> APP2(append, app2(app2(map, f), l1))
APP2(app2(append, app2(app2(append, l1), l2)), l3) -> APP2(app2(append, l1), app2(app2(append, l2), l3))
APP2(app2(map, f), app2(app2(cons, h), t)) -> APP2(app2(map, f), t)
APP2(app2(append, app2(app2(append, l1), l2)), l3) -> APP2(append, l2)
APP2(app2(append, app2(app2(cons, h), t)), l) -> APP2(app2(append, t), l)
APP2(app2(map, f), app2(app2(append, l1), l2)) -> APP2(app2(map, f), l1)
APP2(app2(map, f), app2(app2(append, l1), l2)) -> APP2(app2(map, f), l2)
APP2(app2(map, f), app2(app2(cons, h), t)) -> APP2(f, h)
APP2(app2(append, app2(app2(cons, h), t)), l) -> APP2(app2(cons, h), app2(app2(append, t), l))
APP2(app2(map, f), app2(app2(cons, h), t)) -> APP2(cons, app2(f, h))

The TRS R consists of the following rules:

app2(app2(append, nil), l) -> l
app2(app2(append, app2(app2(cons, h), t)), l) -> app2(app2(cons, h), app2(app2(append, t), l))
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(append, app2(app2(append, l1), l2)), l3) -> app2(app2(append, l1), app2(app2(append, l2), l3))
app2(app2(map, f), app2(app2(append, l1), l2)) -> app2(app2(append, app2(app2(map, f), l1)), app2(app2(map, f), l2))

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(append, app2(app2(append, l1), l2)), l3) -> APP2(app2(append, l2), l3)
APP2(app2(map, f), app2(app2(cons, h), t)) -> APP2(app2(cons, app2(f, h)), app2(app2(map, f), t))
APP2(app2(append, app2(app2(cons, h), t)), l) -> APP2(append, t)
APP2(app2(map, f), app2(app2(append, l1), l2)) -> APP2(app2(append, app2(app2(map, f), l1)), app2(app2(map, f), l2))
APP2(app2(map, f), app2(app2(append, l1), l2)) -> APP2(append, app2(app2(map, f), l1))
APP2(app2(append, app2(app2(append, l1), l2)), l3) -> APP2(app2(append, l1), app2(app2(append, l2), l3))
APP2(app2(map, f), app2(app2(cons, h), t)) -> APP2(app2(map, f), t)
APP2(app2(append, app2(app2(append, l1), l2)), l3) -> APP2(append, l2)
APP2(app2(append, app2(app2(cons, h), t)), l) -> APP2(app2(append, t), l)
APP2(app2(map, f), app2(app2(append, l1), l2)) -> APP2(app2(map, f), l1)
APP2(app2(map, f), app2(app2(append, l1), l2)) -> APP2(app2(map, f), l2)
APP2(app2(map, f), app2(app2(cons, h), t)) -> APP2(f, h)
APP2(app2(append, app2(app2(cons, h), t)), l) -> APP2(app2(cons, h), app2(app2(append, t), l))
APP2(app2(map, f), app2(app2(cons, h), t)) -> APP2(cons, app2(f, h))

The TRS R consists of the following rules:

app2(app2(append, nil), l) -> l
app2(app2(append, app2(app2(cons, h), t)), l) -> app2(app2(cons, h), app2(app2(append, t), l))
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(append, app2(app2(append, l1), l2)), l3) -> app2(app2(append, l1), app2(app2(append, l2), l3))
app2(app2(map, f), app2(app2(append, l1), l2)) -> app2(app2(append, app2(app2(map, f), l1)), app2(app2(map, f), l2))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 2 SCCs with 7 less nodes.

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

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

APP2(app2(append, app2(app2(append, l1), l2)), l3) -> APP2(app2(append, l2), l3)
APP2(app2(append, app2(app2(append, l1), l2)), l3) -> APP2(app2(append, l1), app2(app2(append, l2), l3))
APP2(app2(append, app2(app2(cons, h), t)), l) -> APP2(app2(append, t), l)

The TRS R consists of the following rules:

app2(app2(append, nil), l) -> l
app2(app2(append, app2(app2(cons, h), t)), l) -> app2(app2(cons, h), app2(app2(append, t), l))
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(append, app2(app2(append, l1), l2)), l3) -> app2(app2(append, l1), app2(app2(append, l2), l3))
app2(app2(map, f), app2(app2(append, l1), l2)) -> app2(app2(append, app2(app2(map, f), l1)), app2(app2(map, f), l2))

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(append, app2(app2(append, l1), l2)), l3) -> APP2(app2(append, l2), l3)
APP2(app2(append, app2(app2(append, l1), l2)), l3) -> APP2(app2(append, l1), app2(app2(append, l2), l3))
APP2(app2(append, app2(app2(cons, h), t)), l) -> APP2(app2(append, t), l)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( APP2(x1, x2) ) = max{0, x1 - 2}


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


POL( append ) = 2


POL( cons ) = 1



The following usable rules [14] were oriented: none



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

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

app2(app2(append, nil), l) -> l
app2(app2(append, app2(app2(cons, h), t)), l) -> app2(app2(cons, h), app2(app2(append, t), l))
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(append, app2(app2(append, l1), l2)), l3) -> app2(app2(append, l1), app2(app2(append, l2), l3))
app2(app2(map, f), app2(app2(append, l1), l2)) -> app2(app2(append, app2(app2(map, f), l1)), app2(app2(map, f), l2))

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.

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

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

APP2(app2(map, f), app2(app2(append, l1), l2)) -> APP2(app2(map, f), l1)
APP2(app2(map, f), app2(app2(append, l1), l2)) -> APP2(app2(map, f), l2)
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(append, nil), l) -> l
app2(app2(append, app2(app2(cons, h), t)), l) -> app2(app2(cons, h), app2(app2(append, t), l))
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(append, app2(app2(append, l1), l2)), l3) -> app2(app2(append, l1), app2(app2(append, l2), l3))
app2(app2(map, f), app2(app2(append, l1), l2)) -> app2(app2(append, app2(app2(map, f), l1)), app2(app2(map, f), l2))

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(append, l1), l2)) -> APP2(app2(map, f), l1)
APP2(app2(map, f), app2(app2(append, l1), l2)) -> APP2(app2(map, f), l2)
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, x2 - 2}


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


POL( append ) = 1


POL( cons ) = 0



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ 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(append, nil), l) -> l
app2(app2(append, app2(app2(cons, h), t)), l) -> app2(app2(cons, h), app2(app2(append, t), l))
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(append, app2(app2(append, l1), l2)), l3) -> app2(app2(append, l1), app2(app2(append, l2), l3))
app2(app2(map, f), app2(app2(append, l1), l2)) -> app2(app2(append, app2(app2(map, f), l1)), app2(app2(map, f), l2))

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 - 2}


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


POL( cons ) = 1



The following usable rules [14] were oriented: none



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

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

app2(app2(append, nil), l) -> l
app2(app2(append, app2(app2(cons, h), t)), l) -> app2(app2(cons, h), app2(app2(append, t), l))
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(append, app2(app2(append, l1), l2)), l3) -> app2(app2(append, l1), app2(app2(append, l2), l3))
app2(app2(map, f), app2(app2(append, l1), l2)) -> app2(app2(append, app2(app2(map, f), l1)), app2(app2(map, f), l2))

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.