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:

app'2(app'2(eq, 0), 0) -> true
app'2(app'2(eq, 0), app'2(s, x)) -> false
app'2(app'2(eq, app'2(s, x)), 0) -> false
app'2(app'2(eq, app'2(s, x)), app'2(s, y)) -> app'2(app'2(eq, x), y)
app'2(app'2(le, 0), y) -> true
app'2(app'2(le, app'2(s, x)), 0) -> false
app'2(app'2(le, app'2(s, x)), app'2(s, y)) -> app'2(app'2(le, x), y)
app'2(app'2(app, nil), y) -> y
app'2(app'2(app, app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(app, x), y))
app'2(min, app'2(app'2(add, n), nil)) -> n
app'2(min, app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(app'2(if_min, app'2(app'2(le, n), m)), app'2(app'2(add, n), app'2(app'2(add, m), x)))
app'2(app'2(if_min, true), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, n), x))
app'2(app'2(if_min, false), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, m), x))
app'2(app'2(rm, n), nil) -> nil
app'2(app'2(rm, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_rm, app'2(app'2(eq, n), m)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_rm, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(rm, n), x)
app'2(app'2(app'2(if_rm, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(rm, n), x))
app'2(app'2(minsort, nil), nil) -> nil
app'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> app'2(app'2(app'2(if_minsort, app'2(app'2(eq, n), app'2(min, app'2(app'2(add, n), x)))), app'2(app'2(add, n), x)), y)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(minsort, app'2(app'2(app, app'2(app'2(rm, n), x)), y)), nil))
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, n), x)), y) -> app'2(app'2(minsort, x), app'2(app'2(add, n), y))

Q is empty.


↳ QTRS
  ↳ Non-Overlap Check

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

app'2(app'2(eq, 0), 0) -> true
app'2(app'2(eq, 0), app'2(s, x)) -> false
app'2(app'2(eq, app'2(s, x)), 0) -> false
app'2(app'2(eq, app'2(s, x)), app'2(s, y)) -> app'2(app'2(eq, x), y)
app'2(app'2(le, 0), y) -> true
app'2(app'2(le, app'2(s, x)), 0) -> false
app'2(app'2(le, app'2(s, x)), app'2(s, y)) -> app'2(app'2(le, x), y)
app'2(app'2(app, nil), y) -> y
app'2(app'2(app, app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(app, x), y))
app'2(min, app'2(app'2(add, n), nil)) -> n
app'2(min, app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(app'2(if_min, app'2(app'2(le, n), m)), app'2(app'2(add, n), app'2(app'2(add, m), x)))
app'2(app'2(if_min, true), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, n), x))
app'2(app'2(if_min, false), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, m), x))
app'2(app'2(rm, n), nil) -> nil
app'2(app'2(rm, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_rm, app'2(app'2(eq, n), m)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_rm, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(rm, n), x)
app'2(app'2(app'2(if_rm, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(rm, n), x))
app'2(app'2(minsort, nil), nil) -> nil
app'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> app'2(app'2(app'2(if_minsort, app'2(app'2(eq, n), app'2(min, app'2(app'2(add, n), x)))), app'2(app'2(add, n), x)), y)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(minsort, app'2(app'2(app, app'2(app'2(rm, n), x)), y)), nil))
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, n), x)), y) -> app'2(app'2(minsort, x), app'2(app'2(add, n), y))

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:

app'2(app'2(eq, 0), 0) -> true
app'2(app'2(eq, 0), app'2(s, x)) -> false
app'2(app'2(eq, app'2(s, x)), 0) -> false
app'2(app'2(eq, app'2(s, x)), app'2(s, y)) -> app'2(app'2(eq, x), y)
app'2(app'2(le, 0), y) -> true
app'2(app'2(le, app'2(s, x)), 0) -> false
app'2(app'2(le, app'2(s, x)), app'2(s, y)) -> app'2(app'2(le, x), y)
app'2(app'2(app, nil), y) -> y
app'2(app'2(app, app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(app, x), y))
app'2(min, app'2(app'2(add, n), nil)) -> n
app'2(min, app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(app'2(if_min, app'2(app'2(le, n), m)), app'2(app'2(add, n), app'2(app'2(add, m), x)))
app'2(app'2(if_min, true), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, n), x))
app'2(app'2(if_min, false), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, m), x))
app'2(app'2(rm, n), nil) -> nil
app'2(app'2(rm, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_rm, app'2(app'2(eq, n), m)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_rm, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(rm, n), x)
app'2(app'2(app'2(if_rm, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(rm, n), x))
app'2(app'2(minsort, nil), nil) -> nil
app'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> app'2(app'2(app'2(if_minsort, app'2(app'2(eq, n), app'2(min, app'2(app'2(add, n), x)))), app'2(app'2(add, n), x)), y)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(minsort, app'2(app'2(app, app'2(app'2(rm, n), x)), y)), nil))
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, n), x)), y) -> app'2(app'2(minsort, x), app'2(app'2(add, n), y))

The set Q consists of the following terms:

app'2(app'2(eq, 0), 0)
app'2(app'2(eq, 0), app'2(s, x0))
app'2(app'2(eq, app'2(s, x0)), 0)
app'2(app'2(eq, app'2(s, x0)), app'2(s, x1))
app'2(app'2(le, 0), x0)
app'2(app'2(le, app'2(s, x0)), 0)
app'2(app'2(le, app'2(s, x0)), app'2(s, x1))
app'2(app'2(app, nil), x0)
app'2(app'2(app, app'2(app'2(add, x0), x1)), x2)
app'2(min, app'2(app'2(add, x0), nil))
app'2(min, app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, true), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, false), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(rm, x0), nil)
app'2(app'2(rm, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, false), x0), app'2(app'2(add, x1), x2))
app'2(app'2(minsort, nil), nil)
app'2(app'2(minsort, app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, x0), x1)), x2)


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

APP'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> APP'2(rm, n)
APP'2(app'2(app'2(if_minsort, false), app'2(app'2(add, n), x)), y) -> APP'2(minsort, x)
APP'2(app'2(eq, app'2(s, x)), app'2(s, y)) -> APP'2(app'2(eq, x), y)
APP'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> APP'2(app'2(app, app'2(app'2(rm, n), x)), y)
APP'2(app'2(app'2(if_minsort, false), app'2(app'2(add, n), x)), y) -> APP'2(app'2(add, n), y)
APP'2(app'2(app'2(if_minsort, false), app'2(app'2(add, n), x)), y) -> APP'2(app'2(minsort, x), app'2(app'2(add, n), y))
APP'2(app'2(app, app'2(app'2(add, n), x)), y) -> APP'2(app'2(app, x), y)
APP'2(app'2(if_min, false), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> APP'2(min, app'2(app'2(add, m), x))
APP'2(app'2(rm, n), app'2(app'2(add, m), x)) -> APP'2(if_rm, app'2(app'2(eq, n), m))
APP'2(app'2(if_min, true), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> APP'2(app'2(add, n), x)
APP'2(app'2(eq, app'2(s, x)), app'2(s, y)) -> APP'2(eq, x)
APP'2(app'2(rm, n), app'2(app'2(add, m), x)) -> APP'2(app'2(if_rm, app'2(app'2(eq, n), m)), n)
APP'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> APP'2(app'2(rm, n), x)
APP'2(app'2(app'2(if_rm, true), n), app'2(app'2(add, m), x)) -> APP'2(rm, n)
APP'2(app'2(app, app'2(app'2(add, n), x)), y) -> APP'2(app, x)
APP'2(app'2(app'2(if_rm, false), n), app'2(app'2(add, m), x)) -> APP'2(rm, n)
APP'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> APP'2(app'2(add, n), app'2(app'2(minsort, app'2(app'2(app, app'2(app'2(rm, n), x)), y)), nil))
APP'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> APP'2(min, app'2(app'2(add, n), x))
APP'2(min, app'2(app'2(add, n), app'2(app'2(add, m), x))) -> APP'2(if_min, app'2(app'2(le, n), m))
APP'2(app'2(rm, n), app'2(app'2(add, m), x)) -> APP'2(eq, n)
APP'2(min, app'2(app'2(add, n), app'2(app'2(add, m), x))) -> APP'2(le, n)
APP'2(min, app'2(app'2(add, n), app'2(app'2(add, m), x))) -> APP'2(app'2(if_min, app'2(app'2(le, n), m)), app'2(app'2(add, n), app'2(app'2(add, m), x)))
APP'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> APP'2(app'2(eq, n), app'2(min, app'2(app'2(add, n), x)))
APP'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> APP'2(eq, n)
APP'2(min, app'2(app'2(add, n), app'2(app'2(add, m), x))) -> APP'2(app'2(le, n), m)
APP'2(app'2(le, app'2(s, x)), app'2(s, y)) -> APP'2(app'2(le, x), y)
APP'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> APP'2(minsort, app'2(app'2(app, app'2(app'2(rm, n), x)), y))
APP'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> APP'2(app'2(app'2(if_minsort, app'2(app'2(eq, n), app'2(min, app'2(app'2(add, n), x)))), app'2(app'2(add, n), x)), y)
APP'2(app'2(if_min, true), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> APP'2(min, app'2(app'2(add, n), x))
APP'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> APP'2(app'2(if_minsort, app'2(app'2(eq, n), app'2(min, app'2(app'2(add, n), x)))), app'2(app'2(add, n), x))
APP'2(app'2(app'2(if_rm, false), n), app'2(app'2(add, m), x)) -> APP'2(app'2(add, m), app'2(app'2(rm, n), x))
APP'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> APP'2(app'2(minsort, app'2(app'2(app, app'2(app'2(rm, n), x)), y)), nil)
APP'2(app'2(app, app'2(app'2(add, n), x)), y) -> APP'2(app'2(add, n), app'2(app'2(app, x), y))
APP'2(app'2(le, app'2(s, x)), app'2(s, y)) -> APP'2(le, x)
APP'2(app'2(rm, n), app'2(app'2(add, m), x)) -> APP'2(app'2(app'2(if_rm, app'2(app'2(eq, n), m)), n), app'2(app'2(add, m), x))
APP'2(app'2(app'2(if_rm, true), n), app'2(app'2(add, m), x)) -> APP'2(app'2(rm, n), x)
APP'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> APP'2(if_minsort, app'2(app'2(eq, n), app'2(min, app'2(app'2(add, n), x))))
APP'2(app'2(app'2(if_rm, false), n), app'2(app'2(add, m), x)) -> APP'2(app'2(rm, n), x)
APP'2(app'2(rm, n), app'2(app'2(add, m), x)) -> APP'2(app'2(eq, n), m)
APP'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> APP'2(app, app'2(app'2(rm, n), x))

The TRS R consists of the following rules:

app'2(app'2(eq, 0), 0) -> true
app'2(app'2(eq, 0), app'2(s, x)) -> false
app'2(app'2(eq, app'2(s, x)), 0) -> false
app'2(app'2(eq, app'2(s, x)), app'2(s, y)) -> app'2(app'2(eq, x), y)
app'2(app'2(le, 0), y) -> true
app'2(app'2(le, app'2(s, x)), 0) -> false
app'2(app'2(le, app'2(s, x)), app'2(s, y)) -> app'2(app'2(le, x), y)
app'2(app'2(app, nil), y) -> y
app'2(app'2(app, app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(app, x), y))
app'2(min, app'2(app'2(add, n), nil)) -> n
app'2(min, app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(app'2(if_min, app'2(app'2(le, n), m)), app'2(app'2(add, n), app'2(app'2(add, m), x)))
app'2(app'2(if_min, true), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, n), x))
app'2(app'2(if_min, false), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, m), x))
app'2(app'2(rm, n), nil) -> nil
app'2(app'2(rm, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_rm, app'2(app'2(eq, n), m)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_rm, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(rm, n), x)
app'2(app'2(app'2(if_rm, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(rm, n), x))
app'2(app'2(minsort, nil), nil) -> nil
app'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> app'2(app'2(app'2(if_minsort, app'2(app'2(eq, n), app'2(min, app'2(app'2(add, n), x)))), app'2(app'2(add, n), x)), y)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(minsort, app'2(app'2(app, app'2(app'2(rm, n), x)), y)), nil))
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, n), x)), y) -> app'2(app'2(minsort, x), app'2(app'2(add, n), y))

The set Q consists of the following terms:

app'2(app'2(eq, 0), 0)
app'2(app'2(eq, 0), app'2(s, x0))
app'2(app'2(eq, app'2(s, x0)), 0)
app'2(app'2(eq, app'2(s, x0)), app'2(s, x1))
app'2(app'2(le, 0), x0)
app'2(app'2(le, app'2(s, x0)), 0)
app'2(app'2(le, app'2(s, x0)), app'2(s, x1))
app'2(app'2(app, nil), x0)
app'2(app'2(app, app'2(app'2(add, x0), x1)), x2)
app'2(min, app'2(app'2(add, x0), nil))
app'2(min, app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, true), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, false), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(rm, x0), nil)
app'2(app'2(rm, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, false), x0), app'2(app'2(add, x1), x2))
app'2(app'2(minsort, nil), nil)
app'2(app'2(minsort, app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, x0), x1)), x2)

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:

APP'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> APP'2(rm, n)
APP'2(app'2(app'2(if_minsort, false), app'2(app'2(add, n), x)), y) -> APP'2(minsort, x)
APP'2(app'2(eq, app'2(s, x)), app'2(s, y)) -> APP'2(app'2(eq, x), y)
APP'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> APP'2(app'2(app, app'2(app'2(rm, n), x)), y)
APP'2(app'2(app'2(if_minsort, false), app'2(app'2(add, n), x)), y) -> APP'2(app'2(add, n), y)
APP'2(app'2(app'2(if_minsort, false), app'2(app'2(add, n), x)), y) -> APP'2(app'2(minsort, x), app'2(app'2(add, n), y))
APP'2(app'2(app, app'2(app'2(add, n), x)), y) -> APP'2(app'2(app, x), y)
APP'2(app'2(if_min, false), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> APP'2(min, app'2(app'2(add, m), x))
APP'2(app'2(rm, n), app'2(app'2(add, m), x)) -> APP'2(if_rm, app'2(app'2(eq, n), m))
APP'2(app'2(if_min, true), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> APP'2(app'2(add, n), x)
APP'2(app'2(eq, app'2(s, x)), app'2(s, y)) -> APP'2(eq, x)
APP'2(app'2(rm, n), app'2(app'2(add, m), x)) -> APP'2(app'2(if_rm, app'2(app'2(eq, n), m)), n)
APP'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> APP'2(app'2(rm, n), x)
APP'2(app'2(app'2(if_rm, true), n), app'2(app'2(add, m), x)) -> APP'2(rm, n)
APP'2(app'2(app, app'2(app'2(add, n), x)), y) -> APP'2(app, x)
APP'2(app'2(app'2(if_rm, false), n), app'2(app'2(add, m), x)) -> APP'2(rm, n)
APP'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> APP'2(app'2(add, n), app'2(app'2(minsort, app'2(app'2(app, app'2(app'2(rm, n), x)), y)), nil))
APP'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> APP'2(min, app'2(app'2(add, n), x))
APP'2(min, app'2(app'2(add, n), app'2(app'2(add, m), x))) -> APP'2(if_min, app'2(app'2(le, n), m))
APP'2(app'2(rm, n), app'2(app'2(add, m), x)) -> APP'2(eq, n)
APP'2(min, app'2(app'2(add, n), app'2(app'2(add, m), x))) -> APP'2(le, n)
APP'2(min, app'2(app'2(add, n), app'2(app'2(add, m), x))) -> APP'2(app'2(if_min, app'2(app'2(le, n), m)), app'2(app'2(add, n), app'2(app'2(add, m), x)))
APP'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> APP'2(app'2(eq, n), app'2(min, app'2(app'2(add, n), x)))
APP'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> APP'2(eq, n)
APP'2(min, app'2(app'2(add, n), app'2(app'2(add, m), x))) -> APP'2(app'2(le, n), m)
APP'2(app'2(le, app'2(s, x)), app'2(s, y)) -> APP'2(app'2(le, x), y)
APP'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> APP'2(minsort, app'2(app'2(app, app'2(app'2(rm, n), x)), y))
APP'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> APP'2(app'2(app'2(if_minsort, app'2(app'2(eq, n), app'2(min, app'2(app'2(add, n), x)))), app'2(app'2(add, n), x)), y)
APP'2(app'2(if_min, true), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> APP'2(min, app'2(app'2(add, n), x))
APP'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> APP'2(app'2(if_minsort, app'2(app'2(eq, n), app'2(min, app'2(app'2(add, n), x)))), app'2(app'2(add, n), x))
APP'2(app'2(app'2(if_rm, false), n), app'2(app'2(add, m), x)) -> APP'2(app'2(add, m), app'2(app'2(rm, n), x))
APP'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> APP'2(app'2(minsort, app'2(app'2(app, app'2(app'2(rm, n), x)), y)), nil)
APP'2(app'2(app, app'2(app'2(add, n), x)), y) -> APP'2(app'2(add, n), app'2(app'2(app, x), y))
APP'2(app'2(le, app'2(s, x)), app'2(s, y)) -> APP'2(le, x)
APP'2(app'2(rm, n), app'2(app'2(add, m), x)) -> APP'2(app'2(app'2(if_rm, app'2(app'2(eq, n), m)), n), app'2(app'2(add, m), x))
APP'2(app'2(app'2(if_rm, true), n), app'2(app'2(add, m), x)) -> APP'2(app'2(rm, n), x)
APP'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> APP'2(if_minsort, app'2(app'2(eq, n), app'2(min, app'2(app'2(add, n), x))))
APP'2(app'2(app'2(if_rm, false), n), app'2(app'2(add, m), x)) -> APP'2(app'2(rm, n), x)
APP'2(app'2(rm, n), app'2(app'2(add, m), x)) -> APP'2(app'2(eq, n), m)
APP'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> APP'2(app, app'2(app'2(rm, n), x))

The TRS R consists of the following rules:

app'2(app'2(eq, 0), 0) -> true
app'2(app'2(eq, 0), app'2(s, x)) -> false
app'2(app'2(eq, app'2(s, x)), 0) -> false
app'2(app'2(eq, app'2(s, x)), app'2(s, y)) -> app'2(app'2(eq, x), y)
app'2(app'2(le, 0), y) -> true
app'2(app'2(le, app'2(s, x)), 0) -> false
app'2(app'2(le, app'2(s, x)), app'2(s, y)) -> app'2(app'2(le, x), y)
app'2(app'2(app, nil), y) -> y
app'2(app'2(app, app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(app, x), y))
app'2(min, app'2(app'2(add, n), nil)) -> n
app'2(min, app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(app'2(if_min, app'2(app'2(le, n), m)), app'2(app'2(add, n), app'2(app'2(add, m), x)))
app'2(app'2(if_min, true), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, n), x))
app'2(app'2(if_min, false), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, m), x))
app'2(app'2(rm, n), nil) -> nil
app'2(app'2(rm, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_rm, app'2(app'2(eq, n), m)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_rm, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(rm, n), x)
app'2(app'2(app'2(if_rm, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(rm, n), x))
app'2(app'2(minsort, nil), nil) -> nil
app'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> app'2(app'2(app'2(if_minsort, app'2(app'2(eq, n), app'2(min, app'2(app'2(add, n), x)))), app'2(app'2(add, n), x)), y)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(minsort, app'2(app'2(app, app'2(app'2(rm, n), x)), y)), nil))
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, n), x)), y) -> app'2(app'2(minsort, x), app'2(app'2(add, n), y))

The set Q consists of the following terms:

app'2(app'2(eq, 0), 0)
app'2(app'2(eq, 0), app'2(s, x0))
app'2(app'2(eq, app'2(s, x0)), 0)
app'2(app'2(eq, app'2(s, x0)), app'2(s, x1))
app'2(app'2(le, 0), x0)
app'2(app'2(le, app'2(s, x0)), 0)
app'2(app'2(le, app'2(s, x0)), app'2(s, x1))
app'2(app'2(app, nil), x0)
app'2(app'2(app, app'2(app'2(add, x0), x1)), x2)
app'2(min, app'2(app'2(add, x0), nil))
app'2(min, app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, true), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, false), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(rm, x0), nil)
app'2(app'2(rm, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, false), x0), app'2(app'2(add, x1), x2))
app'2(app'2(minsort, nil), nil)
app'2(app'2(minsort, app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, x0), x1)), x2)

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph contains 6 SCCs with 28 less nodes.

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

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

APP'2(app'2(app, app'2(app'2(add, n), x)), y) -> APP'2(app'2(app, x), y)

The TRS R consists of the following rules:

app'2(app'2(eq, 0), 0) -> true
app'2(app'2(eq, 0), app'2(s, x)) -> false
app'2(app'2(eq, app'2(s, x)), 0) -> false
app'2(app'2(eq, app'2(s, x)), app'2(s, y)) -> app'2(app'2(eq, x), y)
app'2(app'2(le, 0), y) -> true
app'2(app'2(le, app'2(s, x)), 0) -> false
app'2(app'2(le, app'2(s, x)), app'2(s, y)) -> app'2(app'2(le, x), y)
app'2(app'2(app, nil), y) -> y
app'2(app'2(app, app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(app, x), y))
app'2(min, app'2(app'2(add, n), nil)) -> n
app'2(min, app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(app'2(if_min, app'2(app'2(le, n), m)), app'2(app'2(add, n), app'2(app'2(add, m), x)))
app'2(app'2(if_min, true), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, n), x))
app'2(app'2(if_min, false), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, m), x))
app'2(app'2(rm, n), nil) -> nil
app'2(app'2(rm, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_rm, app'2(app'2(eq, n), m)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_rm, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(rm, n), x)
app'2(app'2(app'2(if_rm, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(rm, n), x))
app'2(app'2(minsort, nil), nil) -> nil
app'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> app'2(app'2(app'2(if_minsort, app'2(app'2(eq, n), app'2(min, app'2(app'2(add, n), x)))), app'2(app'2(add, n), x)), y)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(minsort, app'2(app'2(app, app'2(app'2(rm, n), x)), y)), nil))
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, n), x)), y) -> app'2(app'2(minsort, x), app'2(app'2(add, n), y))

The set Q consists of the following terms:

app'2(app'2(eq, 0), 0)
app'2(app'2(eq, 0), app'2(s, x0))
app'2(app'2(eq, app'2(s, x0)), 0)
app'2(app'2(eq, app'2(s, x0)), app'2(s, x1))
app'2(app'2(le, 0), x0)
app'2(app'2(le, app'2(s, x0)), 0)
app'2(app'2(le, app'2(s, x0)), app'2(s, x1))
app'2(app'2(app, nil), x0)
app'2(app'2(app, app'2(app'2(add, x0), x1)), x2)
app'2(min, app'2(app'2(add, x0), nil))
app'2(min, app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, true), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, false), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(rm, x0), nil)
app'2(app'2(rm, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, false), x0), app'2(app'2(add, x1), x2))
app'2(app'2(minsort, nil), nil)
app'2(app'2(minsort, app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, x0), x1)), x2)

We have to consider all minimal (P,Q,R)-chains.
By using an argument filtering and a montonic ordering, at least one Dependency Pair of this SCC can be strictly oriented.

APP'2(app'2(app, app'2(app'2(add, n), x)), y) -> APP'2(app'2(app, x), y)
Used argument filtering: APP'2(x1, x2)  =  x1
app'2(x1, x2)  =  app'1(x2)
Used ordering: Quasi Precedence: trivial 


↳ QTRS
  ↳ Non-Overlap Check
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
                ↳ QDPAfsSolverProof
                  ↳ QDP
                    ↳ PisEmptyProof
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP

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

app'2(app'2(eq, 0), 0) -> true
app'2(app'2(eq, 0), app'2(s, x)) -> false
app'2(app'2(eq, app'2(s, x)), 0) -> false
app'2(app'2(eq, app'2(s, x)), app'2(s, y)) -> app'2(app'2(eq, x), y)
app'2(app'2(le, 0), y) -> true
app'2(app'2(le, app'2(s, x)), 0) -> false
app'2(app'2(le, app'2(s, x)), app'2(s, y)) -> app'2(app'2(le, x), y)
app'2(app'2(app, nil), y) -> y
app'2(app'2(app, app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(app, x), y))
app'2(min, app'2(app'2(add, n), nil)) -> n
app'2(min, app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(app'2(if_min, app'2(app'2(le, n), m)), app'2(app'2(add, n), app'2(app'2(add, m), x)))
app'2(app'2(if_min, true), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, n), x))
app'2(app'2(if_min, false), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, m), x))
app'2(app'2(rm, n), nil) -> nil
app'2(app'2(rm, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_rm, app'2(app'2(eq, n), m)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_rm, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(rm, n), x)
app'2(app'2(app'2(if_rm, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(rm, n), x))
app'2(app'2(minsort, nil), nil) -> nil
app'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> app'2(app'2(app'2(if_minsort, app'2(app'2(eq, n), app'2(min, app'2(app'2(add, n), x)))), app'2(app'2(add, n), x)), y)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(minsort, app'2(app'2(app, app'2(app'2(rm, n), x)), y)), nil))
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, n), x)), y) -> app'2(app'2(minsort, x), app'2(app'2(add, n), y))

The set Q consists of the following terms:

app'2(app'2(eq, 0), 0)
app'2(app'2(eq, 0), app'2(s, x0))
app'2(app'2(eq, app'2(s, x0)), 0)
app'2(app'2(eq, app'2(s, x0)), app'2(s, x1))
app'2(app'2(le, 0), x0)
app'2(app'2(le, app'2(s, x0)), 0)
app'2(app'2(le, app'2(s, x0)), app'2(s, x1))
app'2(app'2(app, nil), x0)
app'2(app'2(app, app'2(app'2(add, x0), x1)), x2)
app'2(min, app'2(app'2(add, x0), nil))
app'2(min, app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, true), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, false), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(rm, x0), nil)
app'2(app'2(rm, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, false), x0), app'2(app'2(add, x1), x2))
app'2(app'2(minsort, nil), nil)
app'2(app'2(minsort, app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, x0), x1)), x2)

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
                ↳ QDPAfsSolverProof
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP

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

APP'2(app'2(le, app'2(s, x)), app'2(s, y)) -> APP'2(app'2(le, x), y)

The TRS R consists of the following rules:

app'2(app'2(eq, 0), 0) -> true
app'2(app'2(eq, 0), app'2(s, x)) -> false
app'2(app'2(eq, app'2(s, x)), 0) -> false
app'2(app'2(eq, app'2(s, x)), app'2(s, y)) -> app'2(app'2(eq, x), y)
app'2(app'2(le, 0), y) -> true
app'2(app'2(le, app'2(s, x)), 0) -> false
app'2(app'2(le, app'2(s, x)), app'2(s, y)) -> app'2(app'2(le, x), y)
app'2(app'2(app, nil), y) -> y
app'2(app'2(app, app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(app, x), y))
app'2(min, app'2(app'2(add, n), nil)) -> n
app'2(min, app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(app'2(if_min, app'2(app'2(le, n), m)), app'2(app'2(add, n), app'2(app'2(add, m), x)))
app'2(app'2(if_min, true), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, n), x))
app'2(app'2(if_min, false), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, m), x))
app'2(app'2(rm, n), nil) -> nil
app'2(app'2(rm, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_rm, app'2(app'2(eq, n), m)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_rm, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(rm, n), x)
app'2(app'2(app'2(if_rm, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(rm, n), x))
app'2(app'2(minsort, nil), nil) -> nil
app'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> app'2(app'2(app'2(if_minsort, app'2(app'2(eq, n), app'2(min, app'2(app'2(add, n), x)))), app'2(app'2(add, n), x)), y)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(minsort, app'2(app'2(app, app'2(app'2(rm, n), x)), y)), nil))
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, n), x)), y) -> app'2(app'2(minsort, x), app'2(app'2(add, n), y))

The set Q consists of the following terms:

app'2(app'2(eq, 0), 0)
app'2(app'2(eq, 0), app'2(s, x0))
app'2(app'2(eq, app'2(s, x0)), 0)
app'2(app'2(eq, app'2(s, x0)), app'2(s, x1))
app'2(app'2(le, 0), x0)
app'2(app'2(le, app'2(s, x0)), 0)
app'2(app'2(le, app'2(s, x0)), app'2(s, x1))
app'2(app'2(app, nil), x0)
app'2(app'2(app, app'2(app'2(add, x0), x1)), x2)
app'2(min, app'2(app'2(add, x0), nil))
app'2(min, app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, true), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, false), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(rm, x0), nil)
app'2(app'2(rm, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, false), x0), app'2(app'2(add, x1), x2))
app'2(app'2(minsort, nil), nil)
app'2(app'2(minsort, app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, x0), x1)), x2)

We have to consider all minimal (P,Q,R)-chains.
By using an argument filtering and a montonic ordering, at least one Dependency Pair of this SCC can be strictly oriented.

APP'2(app'2(le, app'2(s, x)), app'2(s, y)) -> APP'2(app'2(le, x), y)
Used argument filtering: APP'2(x1, x2)  =  x2
app'2(x1, x2)  =  app'1(x2)
s  =  s
Used ordering: Quasi Precedence: trivial 


↳ QTRS
  ↳ Non-Overlap Check
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
                ↳ QDPAfsSolverProof
                  ↳ QDP
                    ↳ PisEmptyProof
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP

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

app'2(app'2(eq, 0), 0) -> true
app'2(app'2(eq, 0), app'2(s, x)) -> false
app'2(app'2(eq, app'2(s, x)), 0) -> false
app'2(app'2(eq, app'2(s, x)), app'2(s, y)) -> app'2(app'2(eq, x), y)
app'2(app'2(le, 0), y) -> true
app'2(app'2(le, app'2(s, x)), 0) -> false
app'2(app'2(le, app'2(s, x)), app'2(s, y)) -> app'2(app'2(le, x), y)
app'2(app'2(app, nil), y) -> y
app'2(app'2(app, app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(app, x), y))
app'2(min, app'2(app'2(add, n), nil)) -> n
app'2(min, app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(app'2(if_min, app'2(app'2(le, n), m)), app'2(app'2(add, n), app'2(app'2(add, m), x)))
app'2(app'2(if_min, true), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, n), x))
app'2(app'2(if_min, false), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, m), x))
app'2(app'2(rm, n), nil) -> nil
app'2(app'2(rm, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_rm, app'2(app'2(eq, n), m)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_rm, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(rm, n), x)
app'2(app'2(app'2(if_rm, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(rm, n), x))
app'2(app'2(minsort, nil), nil) -> nil
app'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> app'2(app'2(app'2(if_minsort, app'2(app'2(eq, n), app'2(min, app'2(app'2(add, n), x)))), app'2(app'2(add, n), x)), y)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(minsort, app'2(app'2(app, app'2(app'2(rm, n), x)), y)), nil))
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, n), x)), y) -> app'2(app'2(minsort, x), app'2(app'2(add, n), y))

The set Q consists of the following terms:

app'2(app'2(eq, 0), 0)
app'2(app'2(eq, 0), app'2(s, x0))
app'2(app'2(eq, app'2(s, x0)), 0)
app'2(app'2(eq, app'2(s, x0)), app'2(s, x1))
app'2(app'2(le, 0), x0)
app'2(app'2(le, app'2(s, x0)), 0)
app'2(app'2(le, app'2(s, x0)), app'2(s, x1))
app'2(app'2(app, nil), x0)
app'2(app'2(app, app'2(app'2(add, x0), x1)), x2)
app'2(min, app'2(app'2(add, x0), nil))
app'2(min, app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, true), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, false), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(rm, x0), nil)
app'2(app'2(rm, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, false), x0), app'2(app'2(add, x1), x2))
app'2(app'2(minsort, nil), nil)
app'2(app'2(minsort, app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, x0), x1)), x2)

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
                ↳ QDPAfsSolverProof
              ↳ QDP
              ↳ QDP
              ↳ QDP

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

APP'2(app'2(if_min, true), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> APP'2(min, app'2(app'2(add, n), x))
APP'2(app'2(if_min, false), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> APP'2(min, app'2(app'2(add, m), x))
APP'2(min, app'2(app'2(add, n), app'2(app'2(add, m), x))) -> APP'2(app'2(if_min, app'2(app'2(le, n), m)), app'2(app'2(add, n), app'2(app'2(add, m), x)))

The TRS R consists of the following rules:

app'2(app'2(eq, 0), 0) -> true
app'2(app'2(eq, 0), app'2(s, x)) -> false
app'2(app'2(eq, app'2(s, x)), 0) -> false
app'2(app'2(eq, app'2(s, x)), app'2(s, y)) -> app'2(app'2(eq, x), y)
app'2(app'2(le, 0), y) -> true
app'2(app'2(le, app'2(s, x)), 0) -> false
app'2(app'2(le, app'2(s, x)), app'2(s, y)) -> app'2(app'2(le, x), y)
app'2(app'2(app, nil), y) -> y
app'2(app'2(app, app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(app, x), y))
app'2(min, app'2(app'2(add, n), nil)) -> n
app'2(min, app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(app'2(if_min, app'2(app'2(le, n), m)), app'2(app'2(add, n), app'2(app'2(add, m), x)))
app'2(app'2(if_min, true), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, n), x))
app'2(app'2(if_min, false), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, m), x))
app'2(app'2(rm, n), nil) -> nil
app'2(app'2(rm, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_rm, app'2(app'2(eq, n), m)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_rm, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(rm, n), x)
app'2(app'2(app'2(if_rm, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(rm, n), x))
app'2(app'2(minsort, nil), nil) -> nil
app'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> app'2(app'2(app'2(if_minsort, app'2(app'2(eq, n), app'2(min, app'2(app'2(add, n), x)))), app'2(app'2(add, n), x)), y)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(minsort, app'2(app'2(app, app'2(app'2(rm, n), x)), y)), nil))
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, n), x)), y) -> app'2(app'2(minsort, x), app'2(app'2(add, n), y))

The set Q consists of the following terms:

app'2(app'2(eq, 0), 0)
app'2(app'2(eq, 0), app'2(s, x0))
app'2(app'2(eq, app'2(s, x0)), 0)
app'2(app'2(eq, app'2(s, x0)), app'2(s, x1))
app'2(app'2(le, 0), x0)
app'2(app'2(le, app'2(s, x0)), 0)
app'2(app'2(le, app'2(s, x0)), app'2(s, x1))
app'2(app'2(app, nil), x0)
app'2(app'2(app, app'2(app'2(add, x0), x1)), x2)
app'2(min, app'2(app'2(add, x0), nil))
app'2(min, app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, true), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, false), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(rm, x0), nil)
app'2(app'2(rm, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, false), x0), app'2(app'2(add, x1), x2))
app'2(app'2(minsort, nil), nil)
app'2(app'2(minsort, app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, x0), x1)), x2)

We have to consider all minimal (P,Q,R)-chains.
By using an argument filtering and a montonic ordering, at least one Dependency Pair of this SCC can be strictly oriented.

APP'2(app'2(if_min, true), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> APP'2(min, app'2(app'2(add, n), x))
APP'2(app'2(if_min, false), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> APP'2(min, app'2(app'2(add, m), x))
Used argument filtering: APP'2(x1, x2)  =  x2
app'2(x1, x2)  =  app'1(x2)
Used ordering: Quasi Precedence: trivial 


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

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

APP'2(min, app'2(app'2(add, n), app'2(app'2(add, m), x))) -> APP'2(app'2(if_min, app'2(app'2(le, n), m)), app'2(app'2(add, n), app'2(app'2(add, m), x)))

The TRS R consists of the following rules:

app'2(app'2(eq, 0), 0) -> true
app'2(app'2(eq, 0), app'2(s, x)) -> false
app'2(app'2(eq, app'2(s, x)), 0) -> false
app'2(app'2(eq, app'2(s, x)), app'2(s, y)) -> app'2(app'2(eq, x), y)
app'2(app'2(le, 0), y) -> true
app'2(app'2(le, app'2(s, x)), 0) -> false
app'2(app'2(le, app'2(s, x)), app'2(s, y)) -> app'2(app'2(le, x), y)
app'2(app'2(app, nil), y) -> y
app'2(app'2(app, app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(app, x), y))
app'2(min, app'2(app'2(add, n), nil)) -> n
app'2(min, app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(app'2(if_min, app'2(app'2(le, n), m)), app'2(app'2(add, n), app'2(app'2(add, m), x)))
app'2(app'2(if_min, true), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, n), x))
app'2(app'2(if_min, false), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, m), x))
app'2(app'2(rm, n), nil) -> nil
app'2(app'2(rm, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_rm, app'2(app'2(eq, n), m)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_rm, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(rm, n), x)
app'2(app'2(app'2(if_rm, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(rm, n), x))
app'2(app'2(minsort, nil), nil) -> nil
app'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> app'2(app'2(app'2(if_minsort, app'2(app'2(eq, n), app'2(min, app'2(app'2(add, n), x)))), app'2(app'2(add, n), x)), y)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(minsort, app'2(app'2(app, app'2(app'2(rm, n), x)), y)), nil))
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, n), x)), y) -> app'2(app'2(minsort, x), app'2(app'2(add, n), y))

The set Q consists of the following terms:

app'2(app'2(eq, 0), 0)
app'2(app'2(eq, 0), app'2(s, x0))
app'2(app'2(eq, app'2(s, x0)), 0)
app'2(app'2(eq, app'2(s, x0)), app'2(s, x1))
app'2(app'2(le, 0), x0)
app'2(app'2(le, app'2(s, x0)), 0)
app'2(app'2(le, app'2(s, x0)), app'2(s, x1))
app'2(app'2(app, nil), x0)
app'2(app'2(app, app'2(app'2(add, x0), x1)), x2)
app'2(min, app'2(app'2(add, x0), nil))
app'2(min, app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, true), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, false), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(rm, x0), nil)
app'2(app'2(rm, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, false), x0), app'2(app'2(add, x1), x2))
app'2(app'2(minsort, nil), nil)
app'2(app'2(minsort, app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, x0), x1)), x2)

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

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

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

APP'2(app'2(eq, app'2(s, x)), app'2(s, y)) -> APP'2(app'2(eq, x), y)

The TRS R consists of the following rules:

app'2(app'2(eq, 0), 0) -> true
app'2(app'2(eq, 0), app'2(s, x)) -> false
app'2(app'2(eq, app'2(s, x)), 0) -> false
app'2(app'2(eq, app'2(s, x)), app'2(s, y)) -> app'2(app'2(eq, x), y)
app'2(app'2(le, 0), y) -> true
app'2(app'2(le, app'2(s, x)), 0) -> false
app'2(app'2(le, app'2(s, x)), app'2(s, y)) -> app'2(app'2(le, x), y)
app'2(app'2(app, nil), y) -> y
app'2(app'2(app, app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(app, x), y))
app'2(min, app'2(app'2(add, n), nil)) -> n
app'2(min, app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(app'2(if_min, app'2(app'2(le, n), m)), app'2(app'2(add, n), app'2(app'2(add, m), x)))
app'2(app'2(if_min, true), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, n), x))
app'2(app'2(if_min, false), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, m), x))
app'2(app'2(rm, n), nil) -> nil
app'2(app'2(rm, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_rm, app'2(app'2(eq, n), m)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_rm, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(rm, n), x)
app'2(app'2(app'2(if_rm, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(rm, n), x))
app'2(app'2(minsort, nil), nil) -> nil
app'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> app'2(app'2(app'2(if_minsort, app'2(app'2(eq, n), app'2(min, app'2(app'2(add, n), x)))), app'2(app'2(add, n), x)), y)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(minsort, app'2(app'2(app, app'2(app'2(rm, n), x)), y)), nil))
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, n), x)), y) -> app'2(app'2(minsort, x), app'2(app'2(add, n), y))

The set Q consists of the following terms:

app'2(app'2(eq, 0), 0)
app'2(app'2(eq, 0), app'2(s, x0))
app'2(app'2(eq, app'2(s, x0)), 0)
app'2(app'2(eq, app'2(s, x0)), app'2(s, x1))
app'2(app'2(le, 0), x0)
app'2(app'2(le, app'2(s, x0)), 0)
app'2(app'2(le, app'2(s, x0)), app'2(s, x1))
app'2(app'2(app, nil), x0)
app'2(app'2(app, app'2(app'2(add, x0), x1)), x2)
app'2(min, app'2(app'2(add, x0), nil))
app'2(min, app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, true), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, false), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(rm, x0), nil)
app'2(app'2(rm, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, false), x0), app'2(app'2(add, x1), x2))
app'2(app'2(minsort, nil), nil)
app'2(app'2(minsort, app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, x0), x1)), x2)

We have to consider all minimal (P,Q,R)-chains.
By using an argument filtering and a montonic ordering, at least one Dependency Pair of this SCC can be strictly oriented.

APP'2(app'2(eq, app'2(s, x)), app'2(s, y)) -> APP'2(app'2(eq, x), y)
Used argument filtering: APP'2(x1, x2)  =  x2
app'2(x1, x2)  =  app'1(x2)
s  =  s
Used ordering: Quasi Precedence: trivial 


↳ QTRS
  ↳ Non-Overlap Check
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ QDPAfsSolverProof
                  ↳ QDP
                    ↳ PisEmptyProof
              ↳ QDP
              ↳ QDP

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

app'2(app'2(eq, 0), 0) -> true
app'2(app'2(eq, 0), app'2(s, x)) -> false
app'2(app'2(eq, app'2(s, x)), 0) -> false
app'2(app'2(eq, app'2(s, x)), app'2(s, y)) -> app'2(app'2(eq, x), y)
app'2(app'2(le, 0), y) -> true
app'2(app'2(le, app'2(s, x)), 0) -> false
app'2(app'2(le, app'2(s, x)), app'2(s, y)) -> app'2(app'2(le, x), y)
app'2(app'2(app, nil), y) -> y
app'2(app'2(app, app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(app, x), y))
app'2(min, app'2(app'2(add, n), nil)) -> n
app'2(min, app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(app'2(if_min, app'2(app'2(le, n), m)), app'2(app'2(add, n), app'2(app'2(add, m), x)))
app'2(app'2(if_min, true), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, n), x))
app'2(app'2(if_min, false), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, m), x))
app'2(app'2(rm, n), nil) -> nil
app'2(app'2(rm, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_rm, app'2(app'2(eq, n), m)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_rm, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(rm, n), x)
app'2(app'2(app'2(if_rm, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(rm, n), x))
app'2(app'2(minsort, nil), nil) -> nil
app'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> app'2(app'2(app'2(if_minsort, app'2(app'2(eq, n), app'2(min, app'2(app'2(add, n), x)))), app'2(app'2(add, n), x)), y)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(minsort, app'2(app'2(app, app'2(app'2(rm, n), x)), y)), nil))
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, n), x)), y) -> app'2(app'2(minsort, x), app'2(app'2(add, n), y))

The set Q consists of the following terms:

app'2(app'2(eq, 0), 0)
app'2(app'2(eq, 0), app'2(s, x0))
app'2(app'2(eq, app'2(s, x0)), 0)
app'2(app'2(eq, app'2(s, x0)), app'2(s, x1))
app'2(app'2(le, 0), x0)
app'2(app'2(le, app'2(s, x0)), 0)
app'2(app'2(le, app'2(s, x0)), app'2(s, x1))
app'2(app'2(app, nil), x0)
app'2(app'2(app, app'2(app'2(add, x0), x1)), x2)
app'2(min, app'2(app'2(add, x0), nil))
app'2(min, app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, true), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, false), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(rm, x0), nil)
app'2(app'2(rm, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, false), x0), app'2(app'2(add, x1), x2))
app'2(app'2(minsort, nil), nil)
app'2(app'2(minsort, app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, x0), x1)), x2)

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
              ↳ QDP
              ↳ QDP
                ↳ QDPAfsSolverProof
              ↳ QDP

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

APP'2(app'2(app'2(if_rm, true), n), app'2(app'2(add, m), x)) -> APP'2(app'2(rm, n), x)
APP'2(app'2(rm, n), app'2(app'2(add, m), x)) -> APP'2(app'2(app'2(if_rm, app'2(app'2(eq, n), m)), n), app'2(app'2(add, m), x))
APP'2(app'2(app'2(if_rm, false), n), app'2(app'2(add, m), x)) -> APP'2(app'2(rm, n), x)

The TRS R consists of the following rules:

app'2(app'2(eq, 0), 0) -> true
app'2(app'2(eq, 0), app'2(s, x)) -> false
app'2(app'2(eq, app'2(s, x)), 0) -> false
app'2(app'2(eq, app'2(s, x)), app'2(s, y)) -> app'2(app'2(eq, x), y)
app'2(app'2(le, 0), y) -> true
app'2(app'2(le, app'2(s, x)), 0) -> false
app'2(app'2(le, app'2(s, x)), app'2(s, y)) -> app'2(app'2(le, x), y)
app'2(app'2(app, nil), y) -> y
app'2(app'2(app, app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(app, x), y))
app'2(min, app'2(app'2(add, n), nil)) -> n
app'2(min, app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(app'2(if_min, app'2(app'2(le, n), m)), app'2(app'2(add, n), app'2(app'2(add, m), x)))
app'2(app'2(if_min, true), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, n), x))
app'2(app'2(if_min, false), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, m), x))
app'2(app'2(rm, n), nil) -> nil
app'2(app'2(rm, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_rm, app'2(app'2(eq, n), m)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_rm, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(rm, n), x)
app'2(app'2(app'2(if_rm, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(rm, n), x))
app'2(app'2(minsort, nil), nil) -> nil
app'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> app'2(app'2(app'2(if_minsort, app'2(app'2(eq, n), app'2(min, app'2(app'2(add, n), x)))), app'2(app'2(add, n), x)), y)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(minsort, app'2(app'2(app, app'2(app'2(rm, n), x)), y)), nil))
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, n), x)), y) -> app'2(app'2(minsort, x), app'2(app'2(add, n), y))

The set Q consists of the following terms:

app'2(app'2(eq, 0), 0)
app'2(app'2(eq, 0), app'2(s, x0))
app'2(app'2(eq, app'2(s, x0)), 0)
app'2(app'2(eq, app'2(s, x0)), app'2(s, x1))
app'2(app'2(le, 0), x0)
app'2(app'2(le, app'2(s, x0)), 0)
app'2(app'2(le, app'2(s, x0)), app'2(s, x1))
app'2(app'2(app, nil), x0)
app'2(app'2(app, app'2(app'2(add, x0), x1)), x2)
app'2(min, app'2(app'2(add, x0), nil))
app'2(min, app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, true), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, false), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(rm, x0), nil)
app'2(app'2(rm, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, false), x0), app'2(app'2(add, x1), x2))
app'2(app'2(minsort, nil), nil)
app'2(app'2(minsort, app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, x0), x1)), x2)

We have to consider all minimal (P,Q,R)-chains.
By using an argument filtering and a montonic ordering, at least one Dependency Pair of this SCC can be strictly oriented.

APP'2(app'2(app'2(if_rm, true), n), app'2(app'2(add, m), x)) -> APP'2(app'2(rm, n), x)
APP'2(app'2(app'2(if_rm, false), n), app'2(app'2(add, m), x)) -> APP'2(app'2(rm, n), x)
Used argument filtering: APP'2(x1, x2)  =  x2
app'2(x1, x2)  =  app'1(x2)
Used ordering: Quasi Precedence: trivial 


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

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

APP'2(app'2(rm, n), app'2(app'2(add, m), x)) -> APP'2(app'2(app'2(if_rm, app'2(app'2(eq, n), m)), n), app'2(app'2(add, m), x))

The TRS R consists of the following rules:

app'2(app'2(eq, 0), 0) -> true
app'2(app'2(eq, 0), app'2(s, x)) -> false
app'2(app'2(eq, app'2(s, x)), 0) -> false
app'2(app'2(eq, app'2(s, x)), app'2(s, y)) -> app'2(app'2(eq, x), y)
app'2(app'2(le, 0), y) -> true
app'2(app'2(le, app'2(s, x)), 0) -> false
app'2(app'2(le, app'2(s, x)), app'2(s, y)) -> app'2(app'2(le, x), y)
app'2(app'2(app, nil), y) -> y
app'2(app'2(app, app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(app, x), y))
app'2(min, app'2(app'2(add, n), nil)) -> n
app'2(min, app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(app'2(if_min, app'2(app'2(le, n), m)), app'2(app'2(add, n), app'2(app'2(add, m), x)))
app'2(app'2(if_min, true), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, n), x))
app'2(app'2(if_min, false), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, m), x))
app'2(app'2(rm, n), nil) -> nil
app'2(app'2(rm, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_rm, app'2(app'2(eq, n), m)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_rm, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(rm, n), x)
app'2(app'2(app'2(if_rm, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(rm, n), x))
app'2(app'2(minsort, nil), nil) -> nil
app'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> app'2(app'2(app'2(if_minsort, app'2(app'2(eq, n), app'2(min, app'2(app'2(add, n), x)))), app'2(app'2(add, n), x)), y)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(minsort, app'2(app'2(app, app'2(app'2(rm, n), x)), y)), nil))
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, n), x)), y) -> app'2(app'2(minsort, x), app'2(app'2(add, n), y))

The set Q consists of the following terms:

app'2(app'2(eq, 0), 0)
app'2(app'2(eq, 0), app'2(s, x0))
app'2(app'2(eq, app'2(s, x0)), 0)
app'2(app'2(eq, app'2(s, x0)), app'2(s, x1))
app'2(app'2(le, 0), x0)
app'2(app'2(le, app'2(s, x0)), 0)
app'2(app'2(le, app'2(s, x0)), app'2(s, x1))
app'2(app'2(app, nil), x0)
app'2(app'2(app, app'2(app'2(add, x0), x1)), x2)
app'2(min, app'2(app'2(add, x0), nil))
app'2(min, app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, true), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, false), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(rm, x0), nil)
app'2(app'2(rm, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, false), x0), app'2(app'2(add, x1), x2))
app'2(app'2(minsort, nil), nil)
app'2(app'2(minsort, app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, x0), x1)), x2)

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

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

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

APP'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> APP'2(app'2(app'2(if_minsort, app'2(app'2(eq, n), app'2(min, app'2(app'2(add, n), x)))), app'2(app'2(add, n), x)), y)
APP'2(app'2(app'2(if_minsort, false), app'2(app'2(add, n), x)), y) -> APP'2(app'2(minsort, x), app'2(app'2(add, n), y))
APP'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> APP'2(app'2(minsort, app'2(app'2(app, app'2(app'2(rm, n), x)), y)), nil)

The TRS R consists of the following rules:

app'2(app'2(eq, 0), 0) -> true
app'2(app'2(eq, 0), app'2(s, x)) -> false
app'2(app'2(eq, app'2(s, x)), 0) -> false
app'2(app'2(eq, app'2(s, x)), app'2(s, y)) -> app'2(app'2(eq, x), y)
app'2(app'2(le, 0), y) -> true
app'2(app'2(le, app'2(s, x)), 0) -> false
app'2(app'2(le, app'2(s, x)), app'2(s, y)) -> app'2(app'2(le, x), y)
app'2(app'2(app, nil), y) -> y
app'2(app'2(app, app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(app, x), y))
app'2(min, app'2(app'2(add, n), nil)) -> n
app'2(min, app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(app'2(if_min, app'2(app'2(le, n), m)), app'2(app'2(add, n), app'2(app'2(add, m), x)))
app'2(app'2(if_min, true), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, n), x))
app'2(app'2(if_min, false), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, m), x))
app'2(app'2(rm, n), nil) -> nil
app'2(app'2(rm, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_rm, app'2(app'2(eq, n), m)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_rm, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(rm, n), x)
app'2(app'2(app'2(if_rm, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(rm, n), x))
app'2(app'2(minsort, nil), nil) -> nil
app'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> app'2(app'2(app'2(if_minsort, app'2(app'2(eq, n), app'2(min, app'2(app'2(add, n), x)))), app'2(app'2(add, n), x)), y)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(minsort, app'2(app'2(app, app'2(app'2(rm, n), x)), y)), nil))
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, n), x)), y) -> app'2(app'2(minsort, x), app'2(app'2(add, n), y))

The set Q consists of the following terms:

app'2(app'2(eq, 0), 0)
app'2(app'2(eq, 0), app'2(s, x0))
app'2(app'2(eq, app'2(s, x0)), 0)
app'2(app'2(eq, app'2(s, x0)), app'2(s, x1))
app'2(app'2(le, 0), x0)
app'2(app'2(le, app'2(s, x0)), 0)
app'2(app'2(le, app'2(s, x0)), app'2(s, x1))
app'2(app'2(app, nil), x0)
app'2(app'2(app, app'2(app'2(add, x0), x1)), x2)
app'2(min, app'2(app'2(add, x0), nil))
app'2(min, app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, true), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, false), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(rm, x0), nil)
app'2(app'2(rm, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, false), x0), app'2(app'2(add, x1), x2))
app'2(app'2(minsort, nil), nil)
app'2(app'2(minsort, app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, x0), x1)), x2)

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