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(eq, 0), 0) -> true
app2(app2(eq, 0), app2(s, x)) -> false
app2(app2(eq, app2(s, x)), 0) -> false
app2(app2(eq, app2(s, x)), app2(s, y)) -> app2(app2(eq, x), y)
app2(app2(or, true), y) -> true
app2(app2(or, false), y) -> y
app2(app2(union, empty), h) -> h
app2(app2(union, app2(app2(app2(edge, x), y), i)), h) -> app2(app2(app2(edge, x), y), app2(app2(union, i), h))
app2(app2(app2(app2(reach, x), y), empty), h) -> false
app2(app2(app2(app2(reach, x), y), app2(app2(app2(edge, u), v), i)), h) -> app2(app2(app2(app2(app2(if_reach_1, app2(app2(eq, x), u)), x), y), app2(app2(app2(edge, u), v), i)), h)
app2(app2(app2(app2(app2(if_reach_1, true), x), y), app2(app2(app2(edge, u), v), i)), h) -> app2(app2(app2(app2(app2(if_reach_2, app2(app2(eq, y), v)), x), y), app2(app2(app2(edge, u), v), i)), h)
app2(app2(app2(app2(app2(if_reach_1, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> app2(app2(app2(app2(reach, x), y), i), app2(app2(app2(edge, u), v), h))
app2(app2(app2(app2(app2(if_reach_2, true), x), y), app2(app2(app2(edge, u), v), i)), h) -> true
app2(app2(app2(app2(app2(if_reach_2, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> app2(app2(or, app2(app2(app2(app2(reach, x), y), i), h)), app2(app2(app2(app2(reach, v), y), app2(app2(union, i), h)), empty))
app2(app2(map, f), nil) -> nil
app2(app2(map, f), app2(app2(cons, x), xs)) -> app2(app2(cons, app2(f, x)), app2(app2(map, f), xs))
app2(app2(filter, f), nil) -> nil
app2(app2(filter, f), app2(app2(cons, x), xs)) -> app2(app2(app2(app2(filter2, app2(f, x)), f), x), xs)
app2(app2(app2(app2(filter2, true), f), x), xs) -> app2(app2(cons, x), app2(app2(filter, f), xs))
app2(app2(app2(app2(filter2, false), f), x), xs) -> app2(app2(filter, f), xs)

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

app2(app2(eq, 0), 0) -> true
app2(app2(eq, 0), app2(s, x)) -> false
app2(app2(eq, app2(s, x)), 0) -> false
app2(app2(eq, app2(s, x)), app2(s, y)) -> app2(app2(eq, x), y)
app2(app2(or, true), y) -> true
app2(app2(or, false), y) -> y
app2(app2(union, empty), h) -> h
app2(app2(union, app2(app2(app2(edge, x), y), i)), h) -> app2(app2(app2(edge, x), y), app2(app2(union, i), h))
app2(app2(app2(app2(reach, x), y), empty), h) -> false
app2(app2(app2(app2(reach, x), y), app2(app2(app2(edge, u), v), i)), h) -> app2(app2(app2(app2(app2(if_reach_1, app2(app2(eq, x), u)), x), y), app2(app2(app2(edge, u), v), i)), h)
app2(app2(app2(app2(app2(if_reach_1, true), x), y), app2(app2(app2(edge, u), v), i)), h) -> app2(app2(app2(app2(app2(if_reach_2, app2(app2(eq, y), v)), x), y), app2(app2(app2(edge, u), v), i)), h)
app2(app2(app2(app2(app2(if_reach_1, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> app2(app2(app2(app2(reach, x), y), i), app2(app2(app2(edge, u), v), h))
app2(app2(app2(app2(app2(if_reach_2, true), x), y), app2(app2(app2(edge, u), v), i)), h) -> true
app2(app2(app2(app2(app2(if_reach_2, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> app2(app2(or, app2(app2(app2(app2(reach, x), y), i), h)), app2(app2(app2(app2(reach, v), y), app2(app2(union, i), h)), empty))
app2(app2(map, f), nil) -> nil
app2(app2(map, f), app2(app2(cons, x), xs)) -> app2(app2(cons, app2(f, x)), app2(app2(map, f), xs))
app2(app2(filter, f), nil) -> nil
app2(app2(filter, f), app2(app2(cons, x), xs)) -> app2(app2(app2(app2(filter2, app2(f, x)), f), x), xs)
app2(app2(app2(app2(filter2, true), f), x), xs) -> app2(app2(cons, x), app2(app2(filter, f), xs))
app2(app2(app2(app2(filter2, false), f), x), xs) -> app2(app2(filter, f), 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(app2(if_reach_1, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(reach, x), y)
APP2(app2(map, f), app2(app2(cons, x), xs)) -> APP2(app2(cons, app2(f, x)), app2(app2(map, f), xs))
APP2(app2(filter, f), app2(app2(cons, x), xs)) -> APP2(app2(app2(filter2, app2(f, x)), f), x)
APP2(app2(app2(app2(app2(if_reach_2, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(or, app2(app2(app2(app2(reach, x), y), i), h))
APP2(app2(filter, f), app2(app2(cons, x), xs)) -> APP2(app2(app2(app2(filter2, app2(f, x)), f), x), xs)
APP2(app2(app2(app2(app2(if_reach_2, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(reach, x)
APP2(app2(app2(app2(app2(if_reach_2, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(reach, x), y)
APP2(app2(app2(app2(app2(if_reach_2, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(app2(app2(reach, x), y), i), h)
APP2(app2(app2(app2(app2(if_reach_1, true), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(eq, y), v)
APP2(app2(app2(app2(app2(if_reach_1, true), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(app2(if_reach_2, app2(app2(eq, y), v)), x), y)
APP2(app2(app2(app2(reach, x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(if_reach_1, app2(app2(eq, x), u)), x)
APP2(app2(filter, f), app2(app2(cons, x), xs)) -> APP2(filter2, app2(f, x))
APP2(app2(union, app2(app2(app2(edge, x), y), i)), h) -> APP2(app2(app2(edge, x), y), app2(app2(union, i), h))
APP2(app2(app2(app2(app2(if_reach_1, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(reach, x)
APP2(app2(app2(app2(app2(if_reach_1, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(app2(reach, x), y), i)
APP2(app2(eq, app2(s, x)), app2(s, y)) -> APP2(eq, x)
APP2(app2(eq, app2(s, x)), app2(s, y)) -> APP2(app2(eq, x), y)
APP2(app2(app2(app2(reach, x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(app2(app2(app2(if_reach_1, app2(app2(eq, x), u)), x), y), app2(app2(app2(edge, u), v), i)), h)
APP2(app2(app2(app2(reach, x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(if_reach_1, app2(app2(eq, x), u))
APP2(app2(app2(app2(app2(if_reach_2, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(app2(reach, v), y), app2(app2(union, i), h))
APP2(app2(app2(app2(app2(if_reach_2, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(app2(app2(reach, v), y), app2(app2(union, i), h)), empty)
APP2(app2(app2(app2(app2(if_reach_2, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(reach, v), y)
APP2(app2(app2(app2(app2(if_reach_2, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(reach, v)
APP2(app2(app2(app2(filter2, false), f), x), xs) -> APP2(app2(filter, f), xs)
APP2(app2(app2(app2(reach, x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(app2(if_reach_1, app2(app2(eq, x), u)), x), y)
APP2(app2(map, f), app2(app2(cons, x), xs)) -> APP2(cons, app2(f, x))
APP2(app2(app2(app2(app2(if_reach_1, true), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(app2(app2(if_reach_2, app2(app2(eq, y), v)), x), y), app2(app2(app2(edge, u), v), i))
APP2(app2(app2(app2(app2(if_reach_1, true), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(eq, y)
APP2(app2(app2(app2(reach, x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(eq, x), u)
APP2(app2(app2(app2(filter2, true), f), x), xs) -> APP2(app2(cons, x), app2(app2(filter, f), xs))
APP2(app2(union, app2(app2(app2(edge, x), y), i)), h) -> APP2(union, i)
APP2(app2(map, f), app2(app2(cons, x), xs)) -> APP2(f, x)
APP2(app2(app2(app2(app2(if_reach_2, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(union, i)
APP2(app2(app2(app2(filter2, true), f), x), xs) -> APP2(app2(filter, f), xs)
APP2(app2(filter, f), app2(app2(cons, x), xs)) -> APP2(app2(filter2, app2(f, x)), f)
APP2(app2(app2(app2(filter2, false), f), x), xs) -> APP2(filter, f)
APP2(app2(app2(app2(app2(if_reach_2, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(or, app2(app2(app2(app2(reach, x), y), i), h)), app2(app2(app2(app2(reach, v), y), app2(app2(union, i), h)), empty))
APP2(app2(app2(app2(app2(if_reach_1, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(app2(app2(reach, x), y), i), app2(app2(app2(edge, u), v), h))
APP2(app2(app2(app2(app2(if_reach_1, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(app2(edge, u), v), h)
APP2(app2(map, f), app2(app2(cons, x), xs)) -> APP2(app2(map, f), xs)
APP2(app2(filter, f), app2(app2(cons, x), xs)) -> APP2(f, x)
APP2(app2(app2(app2(app2(if_reach_1, true), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(if_reach_2, app2(app2(eq, y), v))
APP2(app2(app2(app2(app2(if_reach_2, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(app2(reach, x), y), i)
APP2(app2(union, app2(app2(app2(edge, x), y), i)), h) -> APP2(app2(union, i), h)
APP2(app2(app2(app2(filter2, true), f), x), xs) -> APP2(filter, f)
APP2(app2(app2(app2(app2(if_reach_2, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(union, i), h)
APP2(app2(app2(app2(app2(if_reach_1, true), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(if_reach_2, app2(app2(eq, y), v)), x)
APP2(app2(app2(app2(filter2, true), f), x), xs) -> APP2(cons, x)
APP2(app2(app2(app2(reach, x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(app2(app2(if_reach_1, app2(app2(eq, x), u)), x), y), app2(app2(app2(edge, u), v), i))
APP2(app2(app2(app2(app2(if_reach_1, true), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(app2(app2(app2(if_reach_2, app2(app2(eq, y), v)), x), y), app2(app2(app2(edge, u), v), i)), h)
APP2(app2(app2(app2(reach, x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(eq, x)

The TRS R consists of the following rules:

app2(app2(eq, 0), 0) -> true
app2(app2(eq, 0), app2(s, x)) -> false
app2(app2(eq, app2(s, x)), 0) -> false
app2(app2(eq, app2(s, x)), app2(s, y)) -> app2(app2(eq, x), y)
app2(app2(or, true), y) -> true
app2(app2(or, false), y) -> y
app2(app2(union, empty), h) -> h
app2(app2(union, app2(app2(app2(edge, x), y), i)), h) -> app2(app2(app2(edge, x), y), app2(app2(union, i), h))
app2(app2(app2(app2(reach, x), y), empty), h) -> false
app2(app2(app2(app2(reach, x), y), app2(app2(app2(edge, u), v), i)), h) -> app2(app2(app2(app2(app2(if_reach_1, app2(app2(eq, x), u)), x), y), app2(app2(app2(edge, u), v), i)), h)
app2(app2(app2(app2(app2(if_reach_1, true), x), y), app2(app2(app2(edge, u), v), i)), h) -> app2(app2(app2(app2(app2(if_reach_2, app2(app2(eq, y), v)), x), y), app2(app2(app2(edge, u), v), i)), h)
app2(app2(app2(app2(app2(if_reach_1, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> app2(app2(app2(app2(reach, x), y), i), app2(app2(app2(edge, u), v), h))
app2(app2(app2(app2(app2(if_reach_2, true), x), y), app2(app2(app2(edge, u), v), i)), h) -> true
app2(app2(app2(app2(app2(if_reach_2, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> app2(app2(or, app2(app2(app2(app2(reach, x), y), i), h)), app2(app2(app2(app2(reach, v), y), app2(app2(union, i), h)), empty))
app2(app2(map, f), nil) -> nil
app2(app2(map, f), app2(app2(cons, x), xs)) -> app2(app2(cons, app2(f, x)), app2(app2(map, f), xs))
app2(app2(filter, f), nil) -> nil
app2(app2(filter, f), app2(app2(cons, x), xs)) -> app2(app2(app2(app2(filter2, app2(f, x)), f), x), xs)
app2(app2(app2(app2(filter2, true), f), x), xs) -> app2(app2(cons, x), app2(app2(filter, f), xs))
app2(app2(app2(app2(filter2, false), f), x), xs) -> app2(app2(filter, f), 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(app2(if_reach_1, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(reach, x), y)
APP2(app2(map, f), app2(app2(cons, x), xs)) -> APP2(app2(cons, app2(f, x)), app2(app2(map, f), xs))
APP2(app2(filter, f), app2(app2(cons, x), xs)) -> APP2(app2(app2(filter2, app2(f, x)), f), x)
APP2(app2(app2(app2(app2(if_reach_2, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(or, app2(app2(app2(app2(reach, x), y), i), h))
APP2(app2(filter, f), app2(app2(cons, x), xs)) -> APP2(app2(app2(app2(filter2, app2(f, x)), f), x), xs)
APP2(app2(app2(app2(app2(if_reach_2, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(reach, x)
APP2(app2(app2(app2(app2(if_reach_2, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(reach, x), y)
APP2(app2(app2(app2(app2(if_reach_2, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(app2(app2(reach, x), y), i), h)
APP2(app2(app2(app2(app2(if_reach_1, true), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(eq, y), v)
APP2(app2(app2(app2(app2(if_reach_1, true), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(app2(if_reach_2, app2(app2(eq, y), v)), x), y)
APP2(app2(app2(app2(reach, x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(if_reach_1, app2(app2(eq, x), u)), x)
APP2(app2(filter, f), app2(app2(cons, x), xs)) -> APP2(filter2, app2(f, x))
APP2(app2(union, app2(app2(app2(edge, x), y), i)), h) -> APP2(app2(app2(edge, x), y), app2(app2(union, i), h))
APP2(app2(app2(app2(app2(if_reach_1, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(reach, x)
APP2(app2(app2(app2(app2(if_reach_1, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(app2(reach, x), y), i)
APP2(app2(eq, app2(s, x)), app2(s, y)) -> APP2(eq, x)
APP2(app2(eq, app2(s, x)), app2(s, y)) -> APP2(app2(eq, x), y)
APP2(app2(app2(app2(reach, x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(app2(app2(app2(if_reach_1, app2(app2(eq, x), u)), x), y), app2(app2(app2(edge, u), v), i)), h)
APP2(app2(app2(app2(reach, x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(if_reach_1, app2(app2(eq, x), u))
APP2(app2(app2(app2(app2(if_reach_2, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(app2(reach, v), y), app2(app2(union, i), h))
APP2(app2(app2(app2(app2(if_reach_2, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(app2(app2(reach, v), y), app2(app2(union, i), h)), empty)
APP2(app2(app2(app2(app2(if_reach_2, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(reach, v), y)
APP2(app2(app2(app2(app2(if_reach_2, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(reach, v)
APP2(app2(app2(app2(filter2, false), f), x), xs) -> APP2(app2(filter, f), xs)
APP2(app2(app2(app2(reach, x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(app2(if_reach_1, app2(app2(eq, x), u)), x), y)
APP2(app2(map, f), app2(app2(cons, x), xs)) -> APP2(cons, app2(f, x))
APP2(app2(app2(app2(app2(if_reach_1, true), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(app2(app2(if_reach_2, app2(app2(eq, y), v)), x), y), app2(app2(app2(edge, u), v), i))
APP2(app2(app2(app2(app2(if_reach_1, true), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(eq, y)
APP2(app2(app2(app2(reach, x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(eq, x), u)
APP2(app2(app2(app2(filter2, true), f), x), xs) -> APP2(app2(cons, x), app2(app2(filter, f), xs))
APP2(app2(union, app2(app2(app2(edge, x), y), i)), h) -> APP2(union, i)
APP2(app2(map, f), app2(app2(cons, x), xs)) -> APP2(f, x)
APP2(app2(app2(app2(app2(if_reach_2, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(union, i)
APP2(app2(app2(app2(filter2, true), f), x), xs) -> APP2(app2(filter, f), xs)
APP2(app2(filter, f), app2(app2(cons, x), xs)) -> APP2(app2(filter2, app2(f, x)), f)
APP2(app2(app2(app2(filter2, false), f), x), xs) -> APP2(filter, f)
APP2(app2(app2(app2(app2(if_reach_2, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(or, app2(app2(app2(app2(reach, x), y), i), h)), app2(app2(app2(app2(reach, v), y), app2(app2(union, i), h)), empty))
APP2(app2(app2(app2(app2(if_reach_1, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(app2(app2(reach, x), y), i), app2(app2(app2(edge, u), v), h))
APP2(app2(app2(app2(app2(if_reach_1, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(app2(edge, u), v), h)
APP2(app2(map, f), app2(app2(cons, x), xs)) -> APP2(app2(map, f), xs)
APP2(app2(filter, f), app2(app2(cons, x), xs)) -> APP2(f, x)
APP2(app2(app2(app2(app2(if_reach_1, true), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(if_reach_2, app2(app2(eq, y), v))
APP2(app2(app2(app2(app2(if_reach_2, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(app2(reach, x), y), i)
APP2(app2(union, app2(app2(app2(edge, x), y), i)), h) -> APP2(app2(union, i), h)
APP2(app2(app2(app2(filter2, true), f), x), xs) -> APP2(filter, f)
APP2(app2(app2(app2(app2(if_reach_2, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(union, i), h)
APP2(app2(app2(app2(app2(if_reach_1, true), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(if_reach_2, app2(app2(eq, y), v)), x)
APP2(app2(app2(app2(filter2, true), f), x), xs) -> APP2(cons, x)
APP2(app2(app2(app2(reach, x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(app2(app2(if_reach_1, app2(app2(eq, x), u)), x), y), app2(app2(app2(edge, u), v), i))
APP2(app2(app2(app2(app2(if_reach_1, true), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(app2(app2(app2(if_reach_2, app2(app2(eq, y), v)), x), y), app2(app2(app2(edge, u), v), i)), h)
APP2(app2(app2(app2(reach, x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(eq, x)

The TRS R consists of the following rules:

app2(app2(eq, 0), 0) -> true
app2(app2(eq, 0), app2(s, x)) -> false
app2(app2(eq, app2(s, x)), 0) -> false
app2(app2(eq, app2(s, x)), app2(s, y)) -> app2(app2(eq, x), y)
app2(app2(or, true), y) -> true
app2(app2(or, false), y) -> y
app2(app2(union, empty), h) -> h
app2(app2(union, app2(app2(app2(edge, x), y), i)), h) -> app2(app2(app2(edge, x), y), app2(app2(union, i), h))
app2(app2(app2(app2(reach, x), y), empty), h) -> false
app2(app2(app2(app2(reach, x), y), app2(app2(app2(edge, u), v), i)), h) -> app2(app2(app2(app2(app2(if_reach_1, app2(app2(eq, x), u)), x), y), app2(app2(app2(edge, u), v), i)), h)
app2(app2(app2(app2(app2(if_reach_1, true), x), y), app2(app2(app2(edge, u), v), i)), h) -> app2(app2(app2(app2(app2(if_reach_2, app2(app2(eq, y), v)), x), y), app2(app2(app2(edge, u), v), i)), h)
app2(app2(app2(app2(app2(if_reach_1, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> app2(app2(app2(app2(reach, x), y), i), app2(app2(app2(edge, u), v), h))
app2(app2(app2(app2(app2(if_reach_2, true), x), y), app2(app2(app2(edge, u), v), i)), h) -> true
app2(app2(app2(app2(app2(if_reach_2, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> app2(app2(or, app2(app2(app2(app2(reach, x), y), i), h)), app2(app2(app2(app2(reach, v), y), app2(app2(union, i), h)), empty))
app2(app2(map, f), nil) -> nil
app2(app2(map, f), app2(app2(cons, x), xs)) -> app2(app2(cons, app2(f, x)), app2(app2(map, f), xs))
app2(app2(filter, f), nil) -> nil
app2(app2(filter, f), app2(app2(cons, x), xs)) -> app2(app2(app2(app2(filter2, app2(f, x)), f), x), xs)
app2(app2(app2(app2(filter2, true), f), x), xs) -> app2(app2(cons, x), app2(app2(filter, f), xs))
app2(app2(app2(app2(filter2, false), f), x), xs) -> app2(app2(filter, f), 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 38 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(union, app2(app2(app2(edge, x), y), i)), h) -> APP2(app2(union, i), h)

The TRS R consists of the following rules:

app2(app2(eq, 0), 0) -> true
app2(app2(eq, 0), app2(s, x)) -> false
app2(app2(eq, app2(s, x)), 0) -> false
app2(app2(eq, app2(s, x)), app2(s, y)) -> app2(app2(eq, x), y)
app2(app2(or, true), y) -> true
app2(app2(or, false), y) -> y
app2(app2(union, empty), h) -> h
app2(app2(union, app2(app2(app2(edge, x), y), i)), h) -> app2(app2(app2(edge, x), y), app2(app2(union, i), h))
app2(app2(app2(app2(reach, x), y), empty), h) -> false
app2(app2(app2(app2(reach, x), y), app2(app2(app2(edge, u), v), i)), h) -> app2(app2(app2(app2(app2(if_reach_1, app2(app2(eq, x), u)), x), y), app2(app2(app2(edge, u), v), i)), h)
app2(app2(app2(app2(app2(if_reach_1, true), x), y), app2(app2(app2(edge, u), v), i)), h) -> app2(app2(app2(app2(app2(if_reach_2, app2(app2(eq, y), v)), x), y), app2(app2(app2(edge, u), v), i)), h)
app2(app2(app2(app2(app2(if_reach_1, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> app2(app2(app2(app2(reach, x), y), i), app2(app2(app2(edge, u), v), h))
app2(app2(app2(app2(app2(if_reach_2, true), x), y), app2(app2(app2(edge, u), v), i)), h) -> true
app2(app2(app2(app2(app2(if_reach_2, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> app2(app2(or, app2(app2(app2(app2(reach, x), y), i), h)), app2(app2(app2(app2(reach, v), y), app2(app2(union, i), h)), empty))
app2(app2(map, f), nil) -> nil
app2(app2(map, f), app2(app2(cons, x), xs)) -> app2(app2(cons, app2(f, x)), app2(app2(map, f), xs))
app2(app2(filter, f), nil) -> nil
app2(app2(filter, f), app2(app2(cons, x), xs)) -> app2(app2(app2(app2(filter2, app2(f, x)), f), x), xs)
app2(app2(app2(app2(filter2, true), f), x), xs) -> app2(app2(cons, x), app2(app2(filter, f), xs))
app2(app2(app2(app2(filter2, false), f), x), xs) -> app2(app2(filter, f), 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(union, app2(app2(app2(edge, x), y), i)), h) -> APP2(app2(union, i), h)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(APP2(x1, x2)) = x1 + 3·x1·x2   
POL(app2(x1, x2)) = 2 + x1 + 3·x1·x2 + 3·x2   
POL(edge) = 3   
POL(union) = 2   

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(eq, 0), 0) -> true
app2(app2(eq, 0), app2(s, x)) -> false
app2(app2(eq, app2(s, x)), 0) -> false
app2(app2(eq, app2(s, x)), app2(s, y)) -> app2(app2(eq, x), y)
app2(app2(or, true), y) -> true
app2(app2(or, false), y) -> y
app2(app2(union, empty), h) -> h
app2(app2(union, app2(app2(app2(edge, x), y), i)), h) -> app2(app2(app2(edge, x), y), app2(app2(union, i), h))
app2(app2(app2(app2(reach, x), y), empty), h) -> false
app2(app2(app2(app2(reach, x), y), app2(app2(app2(edge, u), v), i)), h) -> app2(app2(app2(app2(app2(if_reach_1, app2(app2(eq, x), u)), x), y), app2(app2(app2(edge, u), v), i)), h)
app2(app2(app2(app2(app2(if_reach_1, true), x), y), app2(app2(app2(edge, u), v), i)), h) -> app2(app2(app2(app2(app2(if_reach_2, app2(app2(eq, y), v)), x), y), app2(app2(app2(edge, u), v), i)), h)
app2(app2(app2(app2(app2(if_reach_1, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> app2(app2(app2(app2(reach, x), y), i), app2(app2(app2(edge, u), v), h))
app2(app2(app2(app2(app2(if_reach_2, true), x), y), app2(app2(app2(edge, u), v), i)), h) -> true
app2(app2(app2(app2(app2(if_reach_2, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> app2(app2(or, app2(app2(app2(app2(reach, x), y), i), h)), app2(app2(app2(app2(reach, v), y), app2(app2(union, i), h)), empty))
app2(app2(map, f), nil) -> nil
app2(app2(map, f), app2(app2(cons, x), xs)) -> app2(app2(cons, app2(f, x)), app2(app2(map, f), xs))
app2(app2(filter, f), nil) -> nil
app2(app2(filter, f), app2(app2(cons, x), xs)) -> app2(app2(app2(app2(filter2, app2(f, x)), f), x), xs)
app2(app2(app2(app2(filter2, true), f), x), xs) -> app2(app2(cons, x), app2(app2(filter, f), xs))
app2(app2(app2(app2(filter2, false), f), x), xs) -> app2(app2(filter, f), 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(eq, app2(s, x)), app2(s, y)) -> APP2(app2(eq, x), y)

The TRS R consists of the following rules:

app2(app2(eq, 0), 0) -> true
app2(app2(eq, 0), app2(s, x)) -> false
app2(app2(eq, app2(s, x)), 0) -> false
app2(app2(eq, app2(s, x)), app2(s, y)) -> app2(app2(eq, x), y)
app2(app2(or, true), y) -> true
app2(app2(or, false), y) -> y
app2(app2(union, empty), h) -> h
app2(app2(union, app2(app2(app2(edge, x), y), i)), h) -> app2(app2(app2(edge, x), y), app2(app2(union, i), h))
app2(app2(app2(app2(reach, x), y), empty), h) -> false
app2(app2(app2(app2(reach, x), y), app2(app2(app2(edge, u), v), i)), h) -> app2(app2(app2(app2(app2(if_reach_1, app2(app2(eq, x), u)), x), y), app2(app2(app2(edge, u), v), i)), h)
app2(app2(app2(app2(app2(if_reach_1, true), x), y), app2(app2(app2(edge, u), v), i)), h) -> app2(app2(app2(app2(app2(if_reach_2, app2(app2(eq, y), v)), x), y), app2(app2(app2(edge, u), v), i)), h)
app2(app2(app2(app2(app2(if_reach_1, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> app2(app2(app2(app2(reach, x), y), i), app2(app2(app2(edge, u), v), h))
app2(app2(app2(app2(app2(if_reach_2, true), x), y), app2(app2(app2(edge, u), v), i)), h) -> true
app2(app2(app2(app2(app2(if_reach_2, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> app2(app2(or, app2(app2(app2(app2(reach, x), y), i), h)), app2(app2(app2(app2(reach, v), y), app2(app2(union, i), h)), empty))
app2(app2(map, f), nil) -> nil
app2(app2(map, f), app2(app2(cons, x), xs)) -> app2(app2(cons, app2(f, x)), app2(app2(map, f), xs))
app2(app2(filter, f), nil) -> nil
app2(app2(filter, f), app2(app2(cons, x), xs)) -> app2(app2(app2(app2(filter2, app2(f, x)), f), x), xs)
app2(app2(app2(app2(filter2, true), f), x), xs) -> app2(app2(cons, x), app2(app2(filter, f), xs))
app2(app2(app2(app2(filter2, false), f), x), xs) -> app2(app2(filter, f), 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(eq, app2(s, x)), app2(s, y)) -> APP2(app2(eq, x), y)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(APP2(x1, x2)) = 3·x1·x2   
POL(app2(x1, x2)) = 2·x1 + 3·x1·x2 + x2   
POL(eq) = 0   
POL(s) = 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(eq, 0), 0) -> true
app2(app2(eq, 0), app2(s, x)) -> false
app2(app2(eq, app2(s, x)), 0) -> false
app2(app2(eq, app2(s, x)), app2(s, y)) -> app2(app2(eq, x), y)
app2(app2(or, true), y) -> true
app2(app2(or, false), y) -> y
app2(app2(union, empty), h) -> h
app2(app2(union, app2(app2(app2(edge, x), y), i)), h) -> app2(app2(app2(edge, x), y), app2(app2(union, i), h))
app2(app2(app2(app2(reach, x), y), empty), h) -> false
app2(app2(app2(app2(reach, x), y), app2(app2(app2(edge, u), v), i)), h) -> app2(app2(app2(app2(app2(if_reach_1, app2(app2(eq, x), u)), x), y), app2(app2(app2(edge, u), v), i)), h)
app2(app2(app2(app2(app2(if_reach_1, true), x), y), app2(app2(app2(edge, u), v), i)), h) -> app2(app2(app2(app2(app2(if_reach_2, app2(app2(eq, y), v)), x), y), app2(app2(app2(edge, u), v), i)), h)
app2(app2(app2(app2(app2(if_reach_1, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> app2(app2(app2(app2(reach, x), y), i), app2(app2(app2(edge, u), v), h))
app2(app2(app2(app2(app2(if_reach_2, true), x), y), app2(app2(app2(edge, u), v), i)), h) -> true
app2(app2(app2(app2(app2(if_reach_2, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> app2(app2(or, app2(app2(app2(app2(reach, x), y), i), h)), app2(app2(app2(app2(reach, v), y), app2(app2(union, i), h)), empty))
app2(app2(map, f), nil) -> nil
app2(app2(map, f), app2(app2(cons, x), xs)) -> app2(app2(cons, app2(f, x)), app2(app2(map, f), xs))
app2(app2(filter, f), nil) -> nil
app2(app2(filter, f), app2(app2(cons, x), xs)) -> app2(app2(app2(app2(filter2, app2(f, x)), f), x), xs)
app2(app2(app2(app2(filter2, true), f), x), xs) -> app2(app2(cons, x), app2(app2(filter, f), xs))
app2(app2(app2(app2(filter2, false), f), x), xs) -> app2(app2(filter, f), 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(app2(if_reach_2, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(app2(app2(reach, x), y), i), h)
APP2(app2(app2(app2(reach, x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(app2(app2(app2(if_reach_1, app2(app2(eq, x), u)), x), y), app2(app2(app2(edge, u), v), i)), h)
APP2(app2(app2(app2(app2(if_reach_1, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(app2(app2(reach, x), y), i), app2(app2(app2(edge, u), v), h))
APP2(app2(app2(app2(app2(if_reach_2, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(app2(app2(reach, v), y), app2(app2(union, i), h)), empty)
APP2(app2(app2(app2(app2(if_reach_1, true), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(app2(app2(app2(if_reach_2, app2(app2(eq, y), v)), x), y), app2(app2(app2(edge, u), v), i)), h)

The TRS R consists of the following rules:

app2(app2(eq, 0), 0) -> true
app2(app2(eq, 0), app2(s, x)) -> false
app2(app2(eq, app2(s, x)), 0) -> false
app2(app2(eq, app2(s, x)), app2(s, y)) -> app2(app2(eq, x), y)
app2(app2(or, true), y) -> true
app2(app2(or, false), y) -> y
app2(app2(union, empty), h) -> h
app2(app2(union, app2(app2(app2(edge, x), y), i)), h) -> app2(app2(app2(edge, x), y), app2(app2(union, i), h))
app2(app2(app2(app2(reach, x), y), empty), h) -> false
app2(app2(app2(app2(reach, x), y), app2(app2(app2(edge, u), v), i)), h) -> app2(app2(app2(app2(app2(if_reach_1, app2(app2(eq, x), u)), x), y), app2(app2(app2(edge, u), v), i)), h)
app2(app2(app2(app2(app2(if_reach_1, true), x), y), app2(app2(app2(edge, u), v), i)), h) -> app2(app2(app2(app2(app2(if_reach_2, app2(app2(eq, y), v)), x), y), app2(app2(app2(edge, u), v), i)), h)
app2(app2(app2(app2(app2(if_reach_1, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> app2(app2(app2(app2(reach, x), y), i), app2(app2(app2(edge, u), v), h))
app2(app2(app2(app2(app2(if_reach_2, true), x), y), app2(app2(app2(edge, u), v), i)), h) -> true
app2(app2(app2(app2(app2(if_reach_2, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> app2(app2(or, app2(app2(app2(app2(reach, x), y), i), h)), app2(app2(app2(app2(reach, v), y), app2(app2(union, i), h)), empty))
app2(app2(map, f), nil) -> nil
app2(app2(map, f), app2(app2(cons, x), xs)) -> app2(app2(cons, app2(f, x)), app2(app2(map, f), xs))
app2(app2(filter, f), nil) -> nil
app2(app2(filter, f), app2(app2(cons, x), xs)) -> app2(app2(app2(app2(filter2, app2(f, x)), f), x), xs)
app2(app2(app2(app2(filter2, true), f), x), xs) -> app2(app2(cons, x), app2(app2(filter, f), xs))
app2(app2(app2(app2(filter2, false), f), x), xs) -> app2(app2(filter, f), xs)

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

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

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

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

The TRS R consists of the following rules:

app2(app2(eq, 0), 0) -> true
app2(app2(eq, 0), app2(s, x)) -> false
app2(app2(eq, app2(s, x)), 0) -> false
app2(app2(eq, app2(s, x)), app2(s, y)) -> app2(app2(eq, x), y)
app2(app2(or, true), y) -> true
app2(app2(or, false), y) -> y
app2(app2(union, empty), h) -> h
app2(app2(union, app2(app2(app2(edge, x), y), i)), h) -> app2(app2(app2(edge, x), y), app2(app2(union, i), h))
app2(app2(app2(app2(reach, x), y), empty), h) -> false
app2(app2(app2(app2(reach, x), y), app2(app2(app2(edge, u), v), i)), h) -> app2(app2(app2(app2(app2(if_reach_1, app2(app2(eq, x), u)), x), y), app2(app2(app2(edge, u), v), i)), h)
app2(app2(app2(app2(app2(if_reach_1, true), x), y), app2(app2(app2(edge, u), v), i)), h) -> app2(app2(app2(app2(app2(if_reach_2, app2(app2(eq, y), v)), x), y), app2(app2(app2(edge, u), v), i)), h)
app2(app2(app2(app2(app2(if_reach_1, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> app2(app2(app2(app2(reach, x), y), i), app2(app2(app2(edge, u), v), h))
app2(app2(app2(app2(app2(if_reach_2, true), x), y), app2(app2(app2(edge, u), v), i)), h) -> true
app2(app2(app2(app2(app2(if_reach_2, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> app2(app2(or, app2(app2(app2(app2(reach, x), y), i), h)), app2(app2(app2(app2(reach, v), y), app2(app2(union, i), h)), empty))
app2(app2(map, f), nil) -> nil
app2(app2(map, f), app2(app2(cons, x), xs)) -> app2(app2(cons, app2(f, x)), app2(app2(map, f), xs))
app2(app2(filter, f), nil) -> nil
app2(app2(filter, f), app2(app2(cons, x), xs)) -> app2(app2(app2(app2(filter2, app2(f, x)), f), x), xs)
app2(app2(app2(app2(filter2, true), f), x), xs) -> app2(app2(cons, x), app2(app2(filter, f), xs))
app2(app2(app2(app2(filter2, false), f), x), xs) -> app2(app2(filter, f), xs)

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