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:

app2(perfectp, 0) -> false
app2(perfectp, app2(s, x)) -> app2(app2(app2(app2(f, x), app2(s, 0)), app2(s, x)), app2(s, x))
app2(app2(app2(app2(f, 0), y), 0), u) -> true
app2(app2(app2(app2(f, 0), y), app2(s, z)), u) -> false
app2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> app2(app2(app2(app2(f, x), u), app2(app2(minus, z), app2(s, x))), u)
app2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> app2(app2(app2(if, app2(app2(le, x), y)), app2(app2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z), u)), app2(app2(app2(app2(f, x), u), z), u))
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(perfectp, 0) -> false
app2(perfectp, app2(s, x)) -> app2(app2(app2(app2(f, x), app2(s, 0)), app2(s, x)), app2(s, x))
app2(app2(app2(app2(f, 0), y), 0), u) -> true
app2(app2(app2(app2(f, 0), y), app2(s, z)), u) -> false
app2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> app2(app2(app2(app2(f, x), u), app2(app2(minus, z), app2(s, x))), u)
app2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> app2(app2(app2(if, app2(app2(le, x), y)), app2(app2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z), u)), app2(app2(app2(app2(f, x), u), z), u))
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(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(f, app2(s, x)), app2(app2(minus, y), x))
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(minus, y), x)
APP2(perfectp, app2(s, x)) -> APP2(app2(app2(f, x), app2(s, 0)), app2(s, x))
APP2(app2(map, fun), app2(app2(cons, x), xs)) -> APP2(app2(cons, app2(fun, x)), app2(app2(map, fun), xs))
APP2(perfectp, app2(s, x)) -> APP2(f, x)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(f, x), u)
APP2(app2(filter, fun), app2(app2(cons, x), xs)) -> APP2(app2(app2(filter2, app2(fun, x)), fun), x)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(le, x)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z), u)
APP2(app2(filter, fun), app2(app2(cons, x), xs)) -> APP2(app2(app2(app2(filter2, app2(fun, x)), fun), x), xs)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(if, app2(app2(le, x), y))
APP2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> APP2(minus, z)
APP2(app2(app2(app2(filter2, true), fun), x), xs) -> APP2(app2(cons, x), app2(app2(filter, fun), xs))
APP2(perfectp, app2(s, x)) -> APP2(app2(f, x), app2(s, 0))
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(minus, y)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(app2(f, x), u), z)
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(app2(filter, fun), xs)
APP2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> APP2(app2(f, x), u)
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(perfectp, app2(s, x)) -> APP2(app2(app2(app2(f, x), app2(s, 0)), app2(s, x)), app2(s, x))
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z)
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)
APP2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> APP2(f, x)
APP2(perfectp, app2(s, x)) -> APP2(s, 0)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(app2(app2(f, x), u), z), u)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(le, x), y)
APP2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> APP2(app2(app2(f, x), u), app2(app2(minus, z), app2(s, x)))
APP2(app2(app2(app2(filter2, true), fun), x), xs) -> APP2(filter, fun)
APP2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> APP2(app2(minus, z), app2(s, x))
APP2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> APP2(app2(app2(app2(f, x), u), app2(app2(minus, z), app2(s, x))), u)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(if, app2(app2(le, x), y)), app2(app2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z), u))
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(app2(if, app2(app2(le, x), y)), app2(app2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z), u)), app2(app2(app2(app2(f, x), u), z), u))
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(f, x)
APP2(app2(app2(app2(filter2, true), fun), x), xs) -> APP2(cons, x)
APP2(app2(app2(app2(filter2, false), fun), x), xs) -> APP2(app2(filter, fun), xs)
APP2(app2(map, fun), app2(app2(cons, x), xs)) -> APP2(cons, app2(fun, x))

The TRS R consists of the following rules:

app2(perfectp, 0) -> false
app2(perfectp, app2(s, x)) -> app2(app2(app2(app2(f, x), app2(s, 0)), app2(s, x)), app2(s, x))
app2(app2(app2(app2(f, 0), y), 0), u) -> true
app2(app2(app2(app2(f, 0), y), app2(s, z)), u) -> false
app2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> app2(app2(app2(app2(f, x), u), app2(app2(minus, z), app2(s, x))), u)
app2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> app2(app2(app2(if, app2(app2(le, x), y)), app2(app2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z), u)), app2(app2(app2(app2(f, x), u), z), u))
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(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(f, app2(s, x)), app2(app2(minus, y), x))
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(minus, y), x)
APP2(perfectp, app2(s, x)) -> APP2(app2(app2(f, x), app2(s, 0)), app2(s, x))
APP2(app2(map, fun), app2(app2(cons, x), xs)) -> APP2(app2(cons, app2(fun, x)), app2(app2(map, fun), xs))
APP2(perfectp, app2(s, x)) -> APP2(f, x)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(f, x), u)
APP2(app2(filter, fun), app2(app2(cons, x), xs)) -> APP2(app2(app2(filter2, app2(fun, x)), fun), x)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(le, x)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z), u)
APP2(app2(filter, fun), app2(app2(cons, x), xs)) -> APP2(app2(app2(app2(filter2, app2(fun, x)), fun), x), xs)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(if, app2(app2(le, x), y))
APP2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> APP2(minus, z)
APP2(app2(app2(app2(filter2, true), fun), x), xs) -> APP2(app2(cons, x), app2(app2(filter, fun), xs))
APP2(perfectp, app2(s, x)) -> APP2(app2(f, x), app2(s, 0))
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(minus, y)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(app2(f, x), u), z)
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(app2(filter, fun), xs)
APP2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> APP2(app2(f, x), u)
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(perfectp, app2(s, x)) -> APP2(app2(app2(app2(f, x), app2(s, 0)), app2(s, x)), app2(s, x))
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z)
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)
APP2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> APP2(f, x)
APP2(perfectp, app2(s, x)) -> APP2(s, 0)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(app2(app2(f, x), u), z), u)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(le, x), y)
APP2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> APP2(app2(app2(f, x), u), app2(app2(minus, z), app2(s, x)))
APP2(app2(app2(app2(filter2, true), fun), x), xs) -> APP2(filter, fun)
APP2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> APP2(app2(minus, z), app2(s, x))
APP2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> APP2(app2(app2(app2(f, x), u), app2(app2(minus, z), app2(s, x))), u)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(if, app2(app2(le, x), y)), app2(app2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z), u))
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(app2(if, app2(app2(le, x), y)), app2(app2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z), u)), app2(app2(app2(app2(f, x), u), z), u))
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(f, x)
APP2(app2(app2(app2(filter2, true), fun), x), xs) -> APP2(cons, x)
APP2(app2(app2(app2(filter2, false), fun), x), xs) -> APP2(app2(filter, fun), xs)
APP2(app2(map, fun), app2(app2(cons, x), xs)) -> APP2(cons, app2(fun, x))

The TRS R consists of the following rules:

app2(perfectp, 0) -> false
app2(perfectp, app2(s, x)) -> app2(app2(app2(app2(f, x), app2(s, 0)), app2(s, x)), app2(s, x))
app2(app2(app2(app2(f, 0), y), 0), u) -> true
app2(app2(app2(app2(f, 0), y), app2(s, z)), u) -> false
app2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> app2(app2(app2(app2(f, x), u), app2(app2(minus, z), app2(s, x))), u)
app2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> app2(app2(app2(if, app2(app2(le, x), y)), app2(app2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z), u)), app2(app2(app2(app2(f, x), u), z), u))
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 32 less nodes.

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

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

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

The TRS R consists of the following rules:

app2(perfectp, 0) -> false
app2(perfectp, app2(s, x)) -> app2(app2(app2(app2(f, x), app2(s, 0)), app2(s, x)), app2(s, x))
app2(app2(app2(app2(f, 0), y), 0), u) -> true
app2(app2(app2(app2(f, 0), y), app2(s, z)), u) -> false
app2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> app2(app2(app2(app2(f, x), u), app2(app2(minus, z), app2(s, x))), u)
app2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> app2(app2(app2(if, app2(app2(le, x), y)), app2(app2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z), u)), app2(app2(app2(app2(f, x), u), z), u))
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
        ↳ AND
          ↳ QDP
          ↳ QDP

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(filter, fun), app2(app2(cons, x), xs)) -> APP2(app2(app2(app2(filter2, app2(fun, x)), fun), x), xs)
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(perfectp, 0) -> false
app2(perfectp, app2(s, x)) -> app2(app2(app2(app2(f, x), app2(s, 0)), app2(s, x)), app2(s, x))
app2(app2(app2(app2(f, 0), y), 0), u) -> true
app2(app2(app2(app2(f, 0), y), app2(s, z)), u) -> false
app2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> app2(app2(app2(app2(f, x), u), app2(app2(minus, z), app2(s, x))), u)
app2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> app2(app2(app2(if, app2(app2(le, x), y)), app2(app2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z), u)), app2(app2(app2(app2(f, x), u), z), u))
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.