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(app2(f, 0), 1), x) -> app2(app2(app2(f, app2(s, x)), x), x)
app2(app2(app2(f, x), y), app2(s, z)) -> app2(s, app2(app2(app2(f, 0), 1), z))
app2(app2(map, fun), nil) -> nil
app2(app2(map, fun), app2(app2(cons, x), xs)) -> app2(app2(cons, app2(fun, x)), app2(app2(map, fun), xs))
app2(app2(filter, fun), nil) -> nil
app2(app2(filter, fun), app2(app2(cons, x), xs)) -> app2(app2(app2(app2(filter2, app2(fun, x)), fun), x), xs)
app2(app2(app2(app2(filter2, true), fun), x), xs) -> app2(app2(cons, x), app2(app2(filter, fun), xs))
app2(app2(app2(app2(filter2, false), fun), x), xs) -> app2(app2(filter, fun), xs)

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

app2(app2(app2(f, 0), 1), x) -> app2(app2(app2(f, app2(s, x)), x), x)
app2(app2(app2(f, x), y), app2(s, z)) -> app2(s, app2(app2(app2(f, 0), 1), z))
app2(app2(map, fun), nil) -> nil
app2(app2(map, fun), app2(app2(cons, x), xs)) -> app2(app2(cons, app2(fun, x)), app2(app2(map, fun), xs))
app2(app2(filter, fun), nil) -> nil
app2(app2(filter, fun), app2(app2(cons, x), xs)) -> app2(app2(app2(app2(filter2, app2(fun, x)), fun), x), xs)
app2(app2(app2(app2(filter2, true), fun), x), xs) -> app2(app2(cons, x), app2(app2(filter, fun), xs))
app2(app2(app2(app2(filter2, false), fun), x), xs) -> app2(app2(filter, fun), xs)

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(app2(app2(filter2, true), fun), x), xs) -> APP2(app2(filter, fun), xs)
APP2(app2(map, fun), app2(app2(cons, x), xs)) -> APP2(app2(cons, app2(fun, x)), app2(app2(map, fun), xs))
APP2(app2(filter, fun), app2(app2(cons, x), xs)) -> APP2(app2(filter2, app2(fun, x)), fun)
APP2(app2(app2(app2(filter2, false), fun), x), xs) -> APP2(filter, fun)
APP2(app2(app2(f, 0), 1), x) -> APP2(f, app2(s, x))
APP2(app2(filter, fun), app2(app2(cons, x), xs)) -> APP2(app2(app2(filter2, app2(fun, x)), fun), x)
APP2(app2(app2(f, x), y), app2(s, z)) -> APP2(app2(f, 0), 1)
APP2(app2(filter, fun), app2(app2(cons, x), xs)) -> APP2(app2(app2(app2(filter2, app2(fun, x)), fun), x), xs)
APP2(app2(filter, fun), app2(app2(cons, x), xs)) -> APP2(fun, x)
APP2(app2(app2(f, x), y), app2(s, z)) -> APP2(app2(app2(f, 0), 1), z)
APP2(app2(map, fun), app2(app2(cons, x), xs)) -> APP2(app2(map, fun), xs)
APP2(app2(app2(app2(filter2, true), fun), x), xs) -> APP2(app2(cons, x), app2(app2(filter, fun), xs))
APP2(app2(app2(f, x), y), app2(s, z)) -> APP2(s, app2(app2(app2(f, 0), 1), z))
APP2(app2(app2(app2(filter2, true), fun), x), xs) -> APP2(filter, fun)
APP2(app2(filter, fun), app2(app2(cons, x), xs)) -> APP2(filter2, app2(fun, x))
APP2(app2(map, fun), app2(app2(cons, x), xs)) -> APP2(fun, x)
APP2(app2(app2(app2(filter2, true), fun), x), xs) -> APP2(cons, x)
APP2(app2(app2(f, 0), 1), x) -> APP2(app2(app2(f, app2(s, x)), x), x)
APP2(app2(app2(f, 0), 1), x) -> APP2(app2(f, app2(s, x)), x)
APP2(app2(app2(app2(filter2, false), fun), x), xs) -> APP2(app2(filter, fun), xs)
APP2(app2(app2(f, x), y), app2(s, z)) -> APP2(f, 0)
APP2(app2(map, fun), app2(app2(cons, x), xs)) -> APP2(cons, app2(fun, x))
APP2(app2(app2(f, 0), 1), x) -> APP2(s, x)

The TRS R consists of the following rules:

app2(app2(app2(f, 0), 1), x) -> app2(app2(app2(f, app2(s, x)), x), x)
app2(app2(app2(f, x), y), app2(s, z)) -> app2(s, app2(app2(app2(f, 0), 1), z))
app2(app2(map, fun), nil) -> nil
app2(app2(map, fun), app2(app2(cons, x), xs)) -> app2(app2(cons, app2(fun, x)), app2(app2(map, fun), xs))
app2(app2(filter, fun), nil) -> nil
app2(app2(filter, fun), app2(app2(cons, x), xs)) -> app2(app2(app2(app2(filter2, app2(fun, x)), fun), x), xs)
app2(app2(app2(app2(filter2, true), fun), x), xs) -> app2(app2(cons, x), app2(app2(filter, fun), xs))
app2(app2(app2(app2(filter2, false), fun), x), xs) -> app2(app2(filter, fun), xs)

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(app2(app2(filter2, true), fun), x), xs) -> APP2(app2(filter, fun), xs)
APP2(app2(map, fun), app2(app2(cons, x), xs)) -> APP2(app2(cons, app2(fun, x)), app2(app2(map, fun), xs))
APP2(app2(filter, fun), app2(app2(cons, x), xs)) -> APP2(app2(filter2, app2(fun, x)), fun)
APP2(app2(app2(app2(filter2, false), fun), x), xs) -> APP2(filter, fun)
APP2(app2(app2(f, 0), 1), x) -> APP2(f, app2(s, x))
APP2(app2(filter, fun), app2(app2(cons, x), xs)) -> APP2(app2(app2(filter2, app2(fun, x)), fun), x)
APP2(app2(app2(f, x), y), app2(s, z)) -> APP2(app2(f, 0), 1)
APP2(app2(filter, fun), app2(app2(cons, x), xs)) -> APP2(app2(app2(app2(filter2, app2(fun, x)), fun), x), xs)
APP2(app2(filter, fun), app2(app2(cons, x), xs)) -> APP2(fun, x)
APP2(app2(app2(f, x), y), app2(s, z)) -> APP2(app2(app2(f, 0), 1), z)
APP2(app2(map, fun), app2(app2(cons, x), xs)) -> APP2(app2(map, fun), xs)
APP2(app2(app2(app2(filter2, true), fun), x), xs) -> APP2(app2(cons, x), app2(app2(filter, fun), xs))
APP2(app2(app2(f, x), y), app2(s, z)) -> APP2(s, app2(app2(app2(f, 0), 1), z))
APP2(app2(app2(app2(filter2, true), fun), x), xs) -> APP2(filter, fun)
APP2(app2(filter, fun), app2(app2(cons, x), xs)) -> APP2(filter2, app2(fun, x))
APP2(app2(map, fun), app2(app2(cons, x), xs)) -> APP2(fun, x)
APP2(app2(app2(app2(filter2, true), fun), x), xs) -> APP2(cons, x)
APP2(app2(app2(f, 0), 1), x) -> APP2(app2(app2(f, app2(s, x)), x), x)
APP2(app2(app2(f, 0), 1), x) -> APP2(app2(f, app2(s, x)), x)
APP2(app2(app2(app2(filter2, false), fun), x), xs) -> APP2(app2(filter, fun), xs)
APP2(app2(app2(f, x), y), app2(s, z)) -> APP2(f, 0)
APP2(app2(map, fun), app2(app2(cons, x), xs)) -> APP2(cons, app2(fun, x))
APP2(app2(app2(f, 0), 1), x) -> APP2(s, x)

The TRS R consists of the following rules:

app2(app2(app2(f, 0), 1), x) -> app2(app2(app2(f, app2(s, x)), x), x)
app2(app2(app2(f, x), y), app2(s, z)) -> app2(s, app2(app2(app2(f, 0), 1), z))
app2(app2(map, fun), nil) -> nil
app2(app2(map, fun), app2(app2(cons, x), xs)) -> app2(app2(cons, app2(fun, x)), app2(app2(map, fun), xs))
app2(app2(filter, fun), nil) -> nil
app2(app2(filter, fun), app2(app2(cons, x), xs)) -> app2(app2(app2(app2(filter2, app2(fun, x)), fun), x), xs)
app2(app2(app2(app2(filter2, true), fun), x), xs) -> app2(app2(cons, x), app2(app2(filter, fun), xs))
app2(app2(app2(app2(filter2, false), fun), x), xs) -> app2(app2(filter, fun), xs)

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 16 less nodes.

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

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

APP2(app2(app2(f, 0), 1), x) -> APP2(app2(app2(f, app2(s, x)), x), x)
APP2(app2(app2(f, x), y), app2(s, z)) -> APP2(app2(app2(f, 0), 1), z)

The TRS R consists of the following rules:

app2(app2(app2(f, 0), 1), x) -> app2(app2(app2(f, app2(s, x)), x), x)
app2(app2(app2(f, x), y), app2(s, z)) -> app2(s, app2(app2(app2(f, 0), 1), z))
app2(app2(map, fun), nil) -> nil
app2(app2(map, fun), app2(app2(cons, x), xs)) -> app2(app2(cons, app2(fun, x)), app2(app2(map, fun), xs))
app2(app2(filter, fun), nil) -> nil
app2(app2(filter, fun), app2(app2(cons, x), xs)) -> app2(app2(app2(app2(filter2, app2(fun, x)), fun), x), xs)
app2(app2(app2(app2(filter2, true), fun), x), xs) -> app2(app2(cons, x), app2(app2(filter, fun), xs))
app2(app2(app2(app2(filter2, false), fun), x), xs) -> app2(app2(filter, fun), xs)

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(app2(f, x), y), app2(s, z)) -> APP2(app2(app2(f, 0), 1), z)
The remaining pairs can at least be oriented weakly.

APP2(app2(app2(f, 0), 1), x) -> APP2(app2(app2(f, app2(s, x)), x), x)
Used ordering: Polynomial interpretation [21]:

POL(0) = 0   
POL(1) = 0   
POL(APP2(x1, x2)) = 2·x2   
POL(app2(x1, x2)) = 1 + 2·x1 + 2·x2   
POL(f) = 0   
POL(s) = 2   

The following usable rules [14] were oriented: none



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

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

APP2(app2(app2(f, 0), 1), x) -> APP2(app2(app2(f, app2(s, x)), x), x)

The TRS R consists of the following rules:

app2(app2(app2(f, 0), 1), x) -> app2(app2(app2(f, app2(s, x)), x), x)
app2(app2(app2(f, x), y), app2(s, z)) -> app2(s, app2(app2(app2(f, 0), 1), z))
app2(app2(map, fun), nil) -> nil
app2(app2(map, fun), app2(app2(cons, x), xs)) -> app2(app2(cons, app2(fun, x)), app2(app2(map, fun), xs))
app2(app2(filter, fun), nil) -> nil
app2(app2(filter, fun), app2(app2(cons, x), xs)) -> app2(app2(app2(app2(filter2, app2(fun, x)), fun), x), xs)
app2(app2(app2(app2(filter2, true), fun), x), xs) -> app2(app2(cons, x), app2(app2(filter, fun), xs))
app2(app2(app2(app2(filter2, false), fun), x), xs) -> app2(app2(filter, fun), xs)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 0 SCCs with 1 less node.

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

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

APP2(app2(app2(app2(filter2, true), fun), x), xs) -> APP2(app2(filter, fun), xs)
APP2(app2(map, fun), app2(app2(cons, x), xs)) -> APP2(fun, x)
APP2(app2(app2(app2(filter2, false), fun), x), xs) -> APP2(app2(filter, fun), xs)
APP2(app2(filter, fun), app2(app2(cons, x), xs)) -> APP2(fun, x)
APP2(app2(map, fun), app2(app2(cons, x), xs)) -> APP2(app2(map, fun), xs)

The TRS R consists of the following rules:

app2(app2(app2(f, 0), 1), x) -> app2(app2(app2(f, app2(s, x)), x), x)
app2(app2(app2(f, x), y), app2(s, z)) -> app2(s, app2(app2(app2(f, 0), 1), z))
app2(app2(map, fun), nil) -> nil
app2(app2(map, fun), app2(app2(cons, x), xs)) -> app2(app2(cons, app2(fun, x)), app2(app2(map, fun), xs))
app2(app2(filter, fun), nil) -> nil
app2(app2(filter, fun), app2(app2(cons, x), xs)) -> app2(app2(app2(app2(filter2, app2(fun, x)), fun), x), xs)
app2(app2(app2(app2(filter2, true), fun), x), xs) -> app2(app2(cons, x), app2(app2(filter, fun), xs))
app2(app2(app2(app2(filter2, false), fun), x), xs) -> app2(app2(filter, fun), xs)

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(app2(app2(filter2, true), fun), x), xs) -> APP2(app2(filter, fun), xs)
APP2(app2(map, fun), app2(app2(cons, x), xs)) -> APP2(fun, x)
APP2(app2(app2(app2(filter2, false), fun), x), xs) -> APP2(app2(filter, fun), xs)
APP2(app2(filter, fun), app2(app2(cons, x), xs)) -> APP2(fun, x)
APP2(app2(map, fun), app2(app2(cons, x), xs)) -> APP2(app2(map, fun), xs)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(APP2(x1, x2)) = 2·x1 + 2·x2   
POL(app2(x1, x2)) = 2 + 2·x1 + x2   
POL(cons) = 0   
POL(false) = 1   
POL(filter) = 1   
POL(filter2) = 2   
POL(map) = 2   
POL(true) = 1   

The following usable rules [14] were oriented: none



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

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

app2(app2(app2(f, 0), 1), x) -> app2(app2(app2(f, app2(s, x)), x), x)
app2(app2(app2(f, x), y), app2(s, z)) -> app2(s, app2(app2(app2(f, 0), 1), z))
app2(app2(map, fun), nil) -> nil
app2(app2(map, fun), app2(app2(cons, x), xs)) -> app2(app2(cons, app2(fun, x)), app2(app2(map, fun), xs))
app2(app2(filter, fun), nil) -> nil
app2(app2(filter, fun), app2(app2(cons, x), xs)) -> app2(app2(app2(app2(filter2, app2(fun, x)), fun), x), xs)
app2(app2(app2(app2(filter2, true), fun), x), xs) -> app2(app2(cons, x), app2(app2(filter, fun), xs))
app2(app2(app2(app2(filter2, false), fun), x), xs) -> app2(app2(filter, fun), xs)

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.