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(app2(minus, 0), y) -> 0
app2(app2(minus, app2(s, x)), 0) -> app2(s, x)
app2(app2(minus, app2(s, x)), app2(s, y)) -> app2(app2(minus, x), y)
app2(app2(le, 0), y) -> true
app2(app2(le, app2(s, x)), 0) -> false
app2(app2(le, app2(s, x)), app2(s, y)) -> app2(app2(le, x), y)
app2(app2(app2(if, true), x), y) -> x
app2(app2(app2(if, false), x), y) -> y
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(app2(minus, 0), y) -> 0
app2(app2(minus, app2(s, x)), 0) -> app2(s, x)
app2(app2(minus, app2(s, x)), app2(s, y)) -> app2(app2(minus, x), y)
app2(app2(le, 0), y) -> true
app2(app2(le, app2(s, x)), 0) -> false
app2(app2(le, app2(s, x)), app2(s, y)) -> app2(app2(le, x), y)
app2(app2(app2(if, true), x), y) -> x
app2(app2(app2(if, false), x), y) -> y
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(app2(minus, app2(s, x)), app2(s, y)) -> APP2(minus, 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(le, app2(s, x)), app2(s, y)) -> APP2(le, x)
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(minus, app2(s, x)), app2(s, y)) -> APP2(app2(minus, x), y)
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(le, app2(s, x)), app2(s, y)) -> APP2(app2(le, x), y)
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)), app2(s, y)), z), u) -> APP2(app2(app2(app2(f, x), u), z), u)
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(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(f, x)
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(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(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(app2(minus, 0), y) -> 0
app2(app2(minus, app2(s, x)), 0) -> app2(s, x)
app2(app2(minus, app2(s, x)), app2(s, y)) -> app2(app2(minus, x), y)
app2(app2(le, 0), y) -> true
app2(app2(le, app2(s, x)), 0) -> false
app2(app2(le, app2(s, x)), app2(s, y)) -> app2(app2(le, x), y)
app2(app2(app2(if, true), x), y) -> x
app2(app2(app2(if, false), x), y) -> y
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(app2(minus, app2(s, x)), app2(s, y)) -> APP2(minus, 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(le, app2(s, x)), app2(s, y)) -> APP2(le, x)
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(minus, app2(s, x)), app2(s, y)) -> APP2(app2(minus, x), y)
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(le, app2(s, x)), app2(s, y)) -> APP2(app2(le, x), y)
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)), app2(s, y)), z), u) -> APP2(app2(app2(app2(f, x), u), z), u)
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(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(f, x)
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(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(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(app2(minus, 0), y) -> 0
app2(app2(minus, app2(s, x)), 0) -> app2(s, x)
app2(app2(minus, app2(s, x)), app2(s, y)) -> app2(app2(minus, x), y)
app2(app2(le, 0), y) -> true
app2(app2(le, app2(s, x)), 0) -> false
app2(app2(le, app2(s, x)), app2(s, y)) -> app2(app2(le, x), y)
app2(app2(app2(if, true), x), y) -> x
app2(app2(app2(if, false), x), y) -> y
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 4 SCCs with 33 less nodes.

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

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

APP2(app2(le, app2(s, x)), app2(s, y)) -> APP2(app2(le, x), y)

The TRS R consists of the following rules:

app2(app2(minus, 0), y) -> 0
app2(app2(minus, app2(s, x)), 0) -> app2(s, x)
app2(app2(minus, app2(s, x)), app2(s, y)) -> app2(app2(minus, x), y)
app2(app2(le, 0), y) -> true
app2(app2(le, app2(s, x)), 0) -> false
app2(app2(le, app2(s, x)), app2(s, y)) -> app2(app2(le, x), y)
app2(app2(app2(if, true), x), y) -> x
app2(app2(app2(if, false), x), y) -> y
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.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


APP2(app2(le, app2(s, x)), app2(s, y)) -> APP2(app2(le, x), y)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( le ) = 3


POL( s ) = max{0, -2}


POL( APP2(x1, x2) ) = x1 + 1


POL( app2(x1, x2) ) = 3x1 + 2x2 + 3



The following usable rules [14] were oriented: none



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

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

app2(app2(minus, 0), y) -> 0
app2(app2(minus, app2(s, x)), 0) -> app2(s, x)
app2(app2(minus, app2(s, x)), app2(s, y)) -> app2(app2(minus, x), y)
app2(app2(le, 0), y) -> true
app2(app2(le, app2(s, x)), 0) -> false
app2(app2(le, app2(s, x)), app2(s, y)) -> app2(app2(le, x), y)
app2(app2(app2(if, true), x), y) -> x
app2(app2(app2(if, false), x), y) -> y
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 TRS P is empty. Hence, there is no (P,Q,R) chain.

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

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

APP2(app2(minus, app2(s, x)), app2(s, y)) -> APP2(app2(minus, x), y)

The TRS R consists of the following rules:

app2(app2(minus, 0), y) -> 0
app2(app2(minus, app2(s, x)), 0) -> app2(s, x)
app2(app2(minus, app2(s, x)), app2(s, y)) -> app2(app2(minus, x), y)
app2(app2(le, 0), y) -> true
app2(app2(le, app2(s, x)), 0) -> false
app2(app2(le, app2(s, x)), app2(s, y)) -> app2(app2(le, x), y)
app2(app2(app2(if, true), x), y) -> x
app2(app2(app2(if, false), x), y) -> y
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.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


APP2(app2(minus, app2(s, x)), app2(s, y)) -> APP2(app2(minus, x), y)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( s ) = max{0, -2}


POL( APP2(x1, x2) ) = x1 + 1


POL( minus ) = 3


POL( app2(x1, x2) ) = 3x1 + 2x2 + 3



The following usable rules [14] were oriented: none



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

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

app2(app2(minus, 0), y) -> 0
app2(app2(minus, app2(s, x)), 0) -> app2(s, x)
app2(app2(minus, app2(s, x)), app2(s, y)) -> app2(app2(minus, x), y)
app2(app2(le, 0), y) -> true
app2(app2(le, app2(s, x)), 0) -> false
app2(app2(le, app2(s, x)), app2(s, y)) -> app2(app2(le, x), y)
app2(app2(app2(if, true), x), y) -> x
app2(app2(app2(if, false), x), y) -> y
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 TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ 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, 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(app2(f, x), u), z), u)

The TRS R consists of the following rules:

app2(app2(minus, 0), y) -> 0
app2(app2(minus, app2(s, x)), 0) -> app2(s, x)
app2(app2(minus, app2(s, x)), app2(s, y)) -> app2(app2(minus, x), y)
app2(app2(le, 0), y) -> true
app2(app2(le, app2(s, x)), 0) -> false
app2(app2(le, app2(s, x)), app2(s, y)) -> app2(app2(le, x), y)
app2(app2(app2(if, true), x), y) -> x
app2(app2(app2(if, false), x), y) -> y
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
          ↳ 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(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(app2(minus, 0), y) -> 0
app2(app2(minus, app2(s, x)), 0) -> app2(s, x)
app2(app2(minus, app2(s, x)), app2(s, y)) -> app2(app2(minus, x), y)
app2(app2(le, 0), y) -> true
app2(app2(le, app2(s, x)), 0) -> false
app2(app2(le, app2(s, x)), app2(s, y)) -> app2(app2(le, x), y)
app2(app2(app2(if, true), x), y) -> x
app2(app2(app2(if, false), x), y) -> y
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.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


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(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.

APP2(app2(app2(app2(filter2, true), fun), x), xs) -> APP2(app2(filter, fun), xs)
APP2(app2(app2(app2(filter2, false), fun), x), xs) -> APP2(app2(filter, fun), xs)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( map ) = 0


POL( 0 ) = 1


POL( perfectp ) = 1


POL( APP2(x1, x2) ) = 2x2


POL( s ) = 2


POL( minus ) = 0


POL( nil ) = 1


POL( cons ) = 3


POL( filter ) = 1


POL( true ) = 1


POL( if ) = 0


POL( false ) = 1


POL( app2(x1, x2) ) = max{0, 2x1 + 2x2 - 3}


POL( f ) = max{0, -2}


POL( le ) = 1


POL( filter2 ) = 0



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ 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(app2(app2(filter2, false), fun), x), xs) -> APP2(app2(filter, fun), xs)

The TRS R consists of the following rules:

app2(app2(minus, 0), y) -> 0
app2(app2(minus, app2(s, x)), 0) -> app2(s, x)
app2(app2(minus, app2(s, x)), app2(s, y)) -> app2(app2(minus, x), y)
app2(app2(le, 0), y) -> true
app2(app2(le, app2(s, x)), 0) -> false
app2(app2(le, app2(s, x)), app2(s, y)) -> app2(app2(le, x), y)
app2(app2(app2(if, true), x), y) -> x
app2(app2(app2(if, false), x), y) -> y
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 0 SCCs with 2 less nodes.