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))

Q is empty.


↳ QTRS
  ↳ Non-Overlap Check

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))

Q is empty.

The TRS is non-overlapping. Hence, we can switch to innermost.

↳ QTRS
  ↳ Non-Overlap Check
    ↳ 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))

The set Q consists of the following terms:

app2(app2(eq, 0), 0)
app2(app2(eq, 0), app2(s, x0))
app2(app2(eq, app2(s, x0)), 0)
app2(app2(eq, app2(s, x0)), app2(s, x1))
app2(app2(or, true), x0)
app2(app2(or, false), x0)
app2(app2(union, empty), x0)
app2(app2(union, app2(app2(app2(edge, x0), x1), x2)), x3)
app2(app2(app2(app2(reach, x0), x1), empty), x2)
app2(app2(app2(app2(reach, x0), x1), app2(app2(app2(edge, x2), x3), x4)), x5)
app2(app2(app2(app2(app2(if_reach_1, true), x0), x1), app2(app2(app2(edge, x2), x3), x4)), x5)
app2(app2(app2(app2(app2(if_reach_1, false), x0), x1), app2(app2(app2(edge, x2), x3), x4)), x5)
app2(app2(app2(app2(app2(if_reach_2, true), x0), x1), app2(app2(app2(edge, x2), x3), x4)), x5)
app2(app2(app2(app2(app2(if_reach_2, false), x0), x1), app2(app2(app2(edge, x2), x3), x4)), x5)


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(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_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(app2(app2(app2(if_reach_2, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(reach, x)
APP2(app2(app2(app2(reach, x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(eq, x), u)
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(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(union, app2(app2(app2(edge, x), y), i)), h) -> APP2(union, i)
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(app2(if_reach_1, true), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(eq, y), v)
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(app2(app2(app2(if_reach_2, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(union, i)
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_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(reach, x), y), i)
APP2(app2(eq, app2(s, x)), app2(s, y)) -> APP2(eq, x)
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(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(eq, app2(s, x)), app2(s, y)) -> APP2(app2(eq, x), y)
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(union, app2(app2(app2(edge, x), y), i)), h) -> 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(union, 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, 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(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(reach, v)
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(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)
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(reach, x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(eq, x)
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)

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))

The set Q consists of the following terms:

app2(app2(eq, 0), 0)
app2(app2(eq, 0), app2(s, x0))
app2(app2(eq, app2(s, x0)), 0)
app2(app2(eq, app2(s, x0)), app2(s, x1))
app2(app2(or, true), x0)
app2(app2(or, false), x0)
app2(app2(union, empty), x0)
app2(app2(union, app2(app2(app2(edge, x0), x1), x2)), x3)
app2(app2(app2(app2(reach, x0), x1), empty), x2)
app2(app2(app2(app2(reach, x0), x1), app2(app2(app2(edge, x2), x3), x4)), x5)
app2(app2(app2(app2(app2(if_reach_1, true), x0), x1), app2(app2(app2(edge, x2), x3), x4)), x5)
app2(app2(app2(app2(app2(if_reach_1, false), x0), x1), app2(app2(app2(edge, x2), x3), x4)), x5)
app2(app2(app2(app2(app2(if_reach_2, true), x0), x1), app2(app2(app2(edge, x2), x3), x4)), x5)
app2(app2(app2(app2(app2(if_reach_2, false), x0), x1), app2(app2(app2(edge, x2), x3), x4)), x5)

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

↳ QTRS
  ↳ Non-Overlap Check
    ↳ 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(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_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(app2(app2(app2(if_reach_2, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(reach, x)
APP2(app2(app2(app2(reach, x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(eq, x), u)
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(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(union, app2(app2(app2(edge, x), y), i)), h) -> APP2(union, i)
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(app2(if_reach_1, true), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(app2(eq, y), v)
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(app2(app2(app2(if_reach_2, false), x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(union, i)
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_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(reach, x), y), i)
APP2(app2(eq, app2(s, x)), app2(s, y)) -> APP2(eq, x)
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(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(eq, app2(s, x)), app2(s, y)) -> APP2(app2(eq, x), y)
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(union, app2(app2(app2(edge, x), y), i)), h) -> 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(union, 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, 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(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(reach, v)
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(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)
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(reach, x), y), app2(app2(app2(edge, u), v), i)), h) -> APP2(eq, x)
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)

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))

The set Q consists of the following terms:

app2(app2(eq, 0), 0)
app2(app2(eq, 0), app2(s, x0))
app2(app2(eq, app2(s, x0)), 0)
app2(app2(eq, app2(s, x0)), app2(s, x1))
app2(app2(or, true), x0)
app2(app2(or, false), x0)
app2(app2(union, empty), x0)
app2(app2(union, app2(app2(app2(edge, x0), x1), x2)), x3)
app2(app2(app2(app2(reach, x0), x1), empty), x2)
app2(app2(app2(app2(reach, x0), x1), app2(app2(app2(edge, x2), x3), x4)), x5)
app2(app2(app2(app2(app2(if_reach_1, true), x0), x1), app2(app2(app2(edge, x2), x3), x4)), x5)
app2(app2(app2(app2(app2(if_reach_1, false), x0), x1), app2(app2(app2(edge, x2), x3), x4)), x5)
app2(app2(app2(app2(app2(if_reach_2, true), x0), x1), app2(app2(app2(edge, x2), x3), x4)), x5)
app2(app2(app2(app2(app2(if_reach_2, false), x0), x1), app2(app2(app2(edge, x2), x3), x4)), x5)

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 3 SCCs with 29 less nodes.

↳ QTRS
  ↳ Non-Overlap Check
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
                ↳ QDPOrderProof
              ↳ 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))

The set Q consists of the following terms:

app2(app2(eq, 0), 0)
app2(app2(eq, 0), app2(s, x0))
app2(app2(eq, app2(s, x0)), 0)
app2(app2(eq, app2(s, x0)), app2(s, x1))
app2(app2(or, true), x0)
app2(app2(or, false), x0)
app2(app2(union, empty), x0)
app2(app2(union, app2(app2(app2(edge, x0), x1), x2)), x3)
app2(app2(app2(app2(reach, x0), x1), empty), x2)
app2(app2(app2(app2(reach, x0), x1), app2(app2(app2(edge, x2), x3), x4)), x5)
app2(app2(app2(app2(app2(if_reach_1, true), x0), x1), app2(app2(app2(edge, x2), x3), x4)), x5)
app2(app2(app2(app2(app2(if_reach_1, false), x0), x1), app2(app2(app2(edge, x2), x3), x4)), x5)
app2(app2(app2(app2(app2(if_reach_2, true), x0), x1), app2(app2(app2(edge, x2), x3), x4)), x5)
app2(app2(app2(app2(app2(if_reach_2, false), x0), x1), app2(app2(app2(edge, x2), x3), x4)), x5)

We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be strictly oriented 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 by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
APP2(x1, x2)  =  x1
app2(x1, x2)  =  app1(x2)
union  =  union
edge  =  edge

Lexicographic Path Order [19].
Precedence:
trivial

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ Non-Overlap Check
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ 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))

The set Q consists of the following terms:

app2(app2(eq, 0), 0)
app2(app2(eq, 0), app2(s, x0))
app2(app2(eq, app2(s, x0)), 0)
app2(app2(eq, app2(s, x0)), app2(s, x1))
app2(app2(or, true), x0)
app2(app2(or, false), x0)
app2(app2(union, empty), x0)
app2(app2(union, app2(app2(app2(edge, x0), x1), x2)), x3)
app2(app2(app2(app2(reach, x0), x1), empty), x2)
app2(app2(app2(app2(reach, x0), x1), app2(app2(app2(edge, x2), x3), x4)), x5)
app2(app2(app2(app2(app2(if_reach_1, true), x0), x1), app2(app2(app2(edge, x2), x3), x4)), x5)
app2(app2(app2(app2(app2(if_reach_1, false), x0), x1), app2(app2(app2(edge, x2), x3), x4)), x5)
app2(app2(app2(app2(app2(if_reach_2, true), x0), x1), app2(app2(app2(edge, x2), x3), x4)), x5)
app2(app2(app2(app2(app2(if_reach_2, false), x0), x1), app2(app2(app2(edge, x2), x3), x4)), x5)

We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ Non-Overlap Check
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
                ↳ QDPOrderProof
              ↳ 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))

The set Q consists of the following terms:

app2(app2(eq, 0), 0)
app2(app2(eq, 0), app2(s, x0))
app2(app2(eq, app2(s, x0)), 0)
app2(app2(eq, app2(s, x0)), app2(s, x1))
app2(app2(or, true), x0)
app2(app2(or, false), x0)
app2(app2(union, empty), x0)
app2(app2(union, app2(app2(app2(edge, x0), x1), x2)), x3)
app2(app2(app2(app2(reach, x0), x1), empty), x2)
app2(app2(app2(app2(reach, x0), x1), app2(app2(app2(edge, x2), x3), x4)), x5)
app2(app2(app2(app2(app2(if_reach_1, true), x0), x1), app2(app2(app2(edge, x2), x3), x4)), x5)
app2(app2(app2(app2(app2(if_reach_1, false), x0), x1), app2(app2(app2(edge, x2), x3), x4)), x5)
app2(app2(app2(app2(app2(if_reach_2, true), x0), x1), app2(app2(app2(edge, x2), x3), x4)), x5)
app2(app2(app2(app2(app2(if_reach_2, false), x0), x1), app2(app2(app2(edge, x2), x3), x4)), x5)

We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be strictly oriented and are deleted.


APP2(app2(eq, app2(s, x)), app2(s, y)) -> APP2(app2(eq, x), y)
The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
APP2(x1, x2)  =  APP1(x2)
app2(x1, x2)  =  app1(x2)
eq  =  eq
s  =  s

Lexicographic Path Order [19].
Precedence:
APP1 > app1
APP1 > eq
s > app1
s > eq

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ Non-Overlap Check
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ PisEmptyProof
              ↳ 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))

The set Q consists of the following terms:

app2(app2(eq, 0), 0)
app2(app2(eq, 0), app2(s, x0))
app2(app2(eq, app2(s, x0)), 0)
app2(app2(eq, app2(s, x0)), app2(s, x1))
app2(app2(or, true), x0)
app2(app2(or, false), x0)
app2(app2(union, empty), x0)
app2(app2(union, app2(app2(app2(edge, x0), x1), x2)), x3)
app2(app2(app2(app2(reach, x0), x1), empty), x2)
app2(app2(app2(app2(reach, x0), x1), app2(app2(app2(edge, x2), x3), x4)), x5)
app2(app2(app2(app2(app2(if_reach_1, true), x0), x1), app2(app2(app2(edge, x2), x3), x4)), x5)
app2(app2(app2(app2(app2(if_reach_1, false), x0), x1), app2(app2(app2(edge, x2), x3), x4)), x5)
app2(app2(app2(app2(app2(if_reach_2, true), x0), x1), app2(app2(app2(edge, x2), x3), x4)), x5)
app2(app2(app2(app2(app2(if_reach_2, false), x0), x1), app2(app2(app2(edge, x2), x3), x4)), x5)

We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ Non-Overlap Check
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ 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))

The set Q consists of the following terms:

app2(app2(eq, 0), 0)
app2(app2(eq, 0), app2(s, x0))
app2(app2(eq, app2(s, x0)), 0)
app2(app2(eq, app2(s, x0)), app2(s, x1))
app2(app2(or, true), x0)
app2(app2(or, false), x0)
app2(app2(union, empty), x0)
app2(app2(union, app2(app2(app2(edge, x0), x1), x2)), x3)
app2(app2(app2(app2(reach, x0), x1), empty), x2)
app2(app2(app2(app2(reach, x0), x1), app2(app2(app2(edge, x2), x3), x4)), x5)
app2(app2(app2(app2(app2(if_reach_1, true), x0), x1), app2(app2(app2(edge, x2), x3), x4)), x5)
app2(app2(app2(app2(app2(if_reach_1, false), x0), x1), app2(app2(app2(edge, x2), x3), x4)), x5)
app2(app2(app2(app2(app2(if_reach_2, true), x0), x1), app2(app2(app2(edge, x2), x3), x4)), x5)
app2(app2(app2(app2(app2(if_reach_2, false), x0), x1), app2(app2(app2(edge, x2), x3), x4)), x5)

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