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(minus, x), 0) -> x
app'2(app'2(minus, app'2(s, x)), app'2(s, y)) -> app'2(app'2(minus, x), y)
app'2(app'2(quot, 0), app'2(s, y)) -> 0
app'2(app'2(quot, app'2(s, x)), app'2(s, y)) -> app'2(s, app'2(app'2(quot, app'2(app'2(minus, x), y)), app'2(s, 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(app'2(low, n), nil) -> nil
app'2(app'2(low, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_low, app'2(app'2(le, m), n)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_low, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(low, n), x))
app'2(app'2(app'2(if_low, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(low, n), x)
app'2(app'2(high, n), nil) -> nil
app'2(app'2(high, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_high, app'2(app'2(le, m), n)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_high, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(high, n), x)
app'2(app'2(app'2(if_high, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(high, n), x))
app'2(quicksort, nil) -> nil
app'2(quicksort, app'2(app'2(add, n), x)) -> app'2(app'2(app, app'2(quicksort, app'2(app'2(low, n), x))), app'2(app'2(add, n), app'2(quicksort, app'2(app'2(high, n), x))))

Q is empty.


↳ QTRS
  ↳ Non-Overlap Check

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

app'2(app'2(minus, x), 0) -> x
app'2(app'2(minus, app'2(s, x)), app'2(s, y)) -> app'2(app'2(minus, x), y)
app'2(app'2(quot, 0), app'2(s, y)) -> 0
app'2(app'2(quot, app'2(s, x)), app'2(s, y)) -> app'2(s, app'2(app'2(quot, app'2(app'2(minus, x), y)), app'2(s, 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(app'2(low, n), nil) -> nil
app'2(app'2(low, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_low, app'2(app'2(le, m), n)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_low, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(low, n), x))
app'2(app'2(app'2(if_low, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(low, n), x)
app'2(app'2(high, n), nil) -> nil
app'2(app'2(high, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_high, app'2(app'2(le, m), n)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_high, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(high, n), x)
app'2(app'2(app'2(if_high, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(high, n), x))
app'2(quicksort, nil) -> nil
app'2(quicksort, app'2(app'2(add, n), x)) -> app'2(app'2(app, app'2(quicksort, app'2(app'2(low, n), x))), app'2(app'2(add, n), app'2(quicksort, app'2(app'2(high, n), x))))

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(minus, x), 0) -> x
app'2(app'2(minus, app'2(s, x)), app'2(s, y)) -> app'2(app'2(minus, x), y)
app'2(app'2(quot, 0), app'2(s, y)) -> 0
app'2(app'2(quot, app'2(s, x)), app'2(s, y)) -> app'2(s, app'2(app'2(quot, app'2(app'2(minus, x), y)), app'2(s, 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(app'2(low, n), nil) -> nil
app'2(app'2(low, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_low, app'2(app'2(le, m), n)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_low, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(low, n), x))
app'2(app'2(app'2(if_low, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(low, n), x)
app'2(app'2(high, n), nil) -> nil
app'2(app'2(high, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_high, app'2(app'2(le, m), n)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_high, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(high, n), x)
app'2(app'2(app'2(if_high, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(high, n), x))
app'2(quicksort, nil) -> nil
app'2(quicksort, app'2(app'2(add, n), x)) -> app'2(app'2(app, app'2(quicksort, app'2(app'2(low, n), x))), app'2(app'2(add, n), app'2(quicksort, app'2(app'2(high, n), x))))

The set Q consists of the following terms:

app'2(app'2(minus, x0), 0)
app'2(app'2(minus, app'2(s, x0)), app'2(s, x1))
app'2(app'2(quot, 0), app'2(s, x0))
app'2(app'2(quot, 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(app'2(low, x0), nil)
app'2(app'2(low, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_low, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_low, false), x0), app'2(app'2(add, x1), x2))
app'2(app'2(high, x0), nil)
app'2(app'2(high, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_high, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_high, false), x0), app'2(app'2(add, x1), x2))
app'2(quicksort, nil)
app'2(quicksort, app'2(app'2(add, x0), x1))


Using Dependency Pairs [1,13] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

APP'2(app'2(minus, app'2(s, x)), app'2(s, y)) -> APP'2(app'2(minus, x), y)
APP'2(quicksort, app'2(app'2(add, n), x)) -> APP'2(quicksort, app'2(app'2(high, n), x))
APP'2(app'2(app'2(if_low, true), n), app'2(app'2(add, m), x)) -> APP'2(low, n)
APP'2(app'2(app'2(if_low, true), n), app'2(app'2(add, m), x)) -> APP'2(app'2(add, m), app'2(app'2(low, n), x))
APP'2(app'2(quot, app'2(s, x)), app'2(s, y)) -> APP'2(quot, app'2(app'2(minus, x), y))
APP'2(app'2(low, n), app'2(app'2(add, m), x)) -> APP'2(app'2(le, m), n)
APP'2(app'2(low, n), app'2(app'2(add, m), x)) -> APP'2(if_low, app'2(app'2(le, m), n))
APP'2(app'2(app'2(if_high, false), n), app'2(app'2(add, m), x)) -> APP'2(app'2(add, m), app'2(app'2(high, n), x))
APP'2(app'2(app'2(if_high, true), n), app'2(app'2(add, m), x)) -> APP'2(high, n)
APP'2(app'2(high, n), app'2(app'2(add, m), x)) -> APP'2(le, m)
APP'2(app'2(quot, app'2(s, x)), app'2(s, y)) -> APP'2(minus, x)
APP'2(app'2(app'2(if_low, true), n), app'2(app'2(add, m), x)) -> APP'2(app'2(low, n), x)
APP'2(quicksort, app'2(app'2(add, n), x)) -> APP'2(app'2(high, n), x)
APP'2(app'2(app'2(if_low, false), n), app'2(app'2(add, m), x)) -> APP'2(app'2(low, n), x)
APP'2(app'2(app'2(if_low, false), n), app'2(app'2(add, m), x)) -> APP'2(low, n)
APP'2(app'2(quot, app'2(s, x)), app'2(s, y)) -> APP'2(s, app'2(app'2(quot, app'2(app'2(minus, x), y)), app'2(s, y)))
APP'2(app'2(high, n), app'2(app'2(add, m), x)) -> APP'2(app'2(app'2(if_high, app'2(app'2(le, m), n)), n), app'2(app'2(add, m), x))
APP'2(quicksort, app'2(app'2(add, n), x)) -> APP'2(quicksort, app'2(app'2(low, n), x))
APP'2(app'2(app'2(if_high, false), n), app'2(app'2(add, m), x)) -> APP'2(high, n)
APP'2(app'2(low, n), app'2(app'2(add, m), x)) -> APP'2(app'2(app'2(if_low, app'2(app'2(le, m), n)), n), app'2(app'2(add, m), x))
APP'2(quicksort, app'2(app'2(add, n), x)) -> APP'2(high, n)
APP'2(app'2(app, app'2(app'2(add, n), x)), y) -> APP'2(app'2(app, x), y)
APP'2(app'2(minus, app'2(s, x)), app'2(s, y)) -> APP'2(minus, x)
APP'2(quicksort, app'2(app'2(add, n), x)) -> APP'2(app'2(add, n), app'2(quicksort, app'2(app'2(high, n), x)))
APP'2(quicksort, app'2(app'2(add, n), x)) -> APP'2(app'2(low, n), x)
APP'2(app'2(quot, app'2(s, x)), app'2(s, y)) -> APP'2(app'2(minus, x), y)
APP'2(app'2(low, n), app'2(app'2(add, m), x)) -> APP'2(le, m)
APP'2(app'2(high, n), app'2(app'2(add, m), x)) -> APP'2(app'2(if_high, app'2(app'2(le, m), n)), n)
APP'2(app'2(high, n), app'2(app'2(add, m), x)) -> APP'2(if_high, app'2(app'2(le, m), n))
APP'2(app'2(quot, app'2(s, x)), app'2(s, y)) -> APP'2(app'2(quot, app'2(app'2(minus, x), y)), app'2(s, y))
APP'2(app'2(low, n), app'2(app'2(add, m), x)) -> APP'2(app'2(if_low, app'2(app'2(le, m), n)), n)
APP'2(app'2(app, app'2(app'2(add, n), x)), y) -> APP'2(app, x)
APP'2(quicksort, app'2(app'2(add, n), x)) -> APP'2(app'2(app, app'2(quicksort, app'2(app'2(low, n), x))), app'2(app'2(add, n), app'2(quicksort, app'2(app'2(high, n), x))))
APP'2(quicksort, app'2(app'2(add, n), x)) -> APP'2(app, app'2(quicksort, app'2(app'2(low, n), x)))
APP'2(app'2(app'2(if_high, false), n), app'2(app'2(add, m), x)) -> APP'2(app'2(high, n), x)
APP'2(app'2(app'2(if_high, true), n), app'2(app'2(add, m), x)) -> APP'2(app'2(high, n), x)
APP'2(app'2(le, app'2(s, x)), app'2(s, y)) -> APP'2(app'2(le, x), y)
APP'2(quicksort, app'2(app'2(add, n), x)) -> APP'2(low, n)
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(high, n), app'2(app'2(add, m), x)) -> APP'2(app'2(le, m), n)
APP'2(app'2(le, app'2(s, x)), app'2(s, y)) -> APP'2(le, x)

The TRS R consists of the following rules:

app'2(app'2(minus, x), 0) -> x
app'2(app'2(minus, app'2(s, x)), app'2(s, y)) -> app'2(app'2(minus, x), y)
app'2(app'2(quot, 0), app'2(s, y)) -> 0
app'2(app'2(quot, app'2(s, x)), app'2(s, y)) -> app'2(s, app'2(app'2(quot, app'2(app'2(minus, x), y)), app'2(s, 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(app'2(low, n), nil) -> nil
app'2(app'2(low, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_low, app'2(app'2(le, m), n)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_low, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(low, n), x))
app'2(app'2(app'2(if_low, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(low, n), x)
app'2(app'2(high, n), nil) -> nil
app'2(app'2(high, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_high, app'2(app'2(le, m), n)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_high, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(high, n), x)
app'2(app'2(app'2(if_high, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(high, n), x))
app'2(quicksort, nil) -> nil
app'2(quicksort, app'2(app'2(add, n), x)) -> app'2(app'2(app, app'2(quicksort, app'2(app'2(low, n), x))), app'2(app'2(add, n), app'2(quicksort, app'2(app'2(high, n), x))))

The set Q consists of the following terms:

app'2(app'2(minus, x0), 0)
app'2(app'2(minus, app'2(s, x0)), app'2(s, x1))
app'2(app'2(quot, 0), app'2(s, x0))
app'2(app'2(quot, 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(app'2(low, x0), nil)
app'2(app'2(low, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_low, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_low, false), x0), app'2(app'2(add, x1), x2))
app'2(app'2(high, x0), nil)
app'2(app'2(high, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_high, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_high, false), x0), app'2(app'2(add, x1), x2))
app'2(quicksort, nil)
app'2(quicksort, app'2(app'2(add, x0), x1))

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(minus, app'2(s, x)), app'2(s, y)) -> APP'2(app'2(minus, x), y)
APP'2(quicksort, app'2(app'2(add, n), x)) -> APP'2(quicksort, app'2(app'2(high, n), x))
APP'2(app'2(app'2(if_low, true), n), app'2(app'2(add, m), x)) -> APP'2(low, n)
APP'2(app'2(app'2(if_low, true), n), app'2(app'2(add, m), x)) -> APP'2(app'2(add, m), app'2(app'2(low, n), x))
APP'2(app'2(quot, app'2(s, x)), app'2(s, y)) -> APP'2(quot, app'2(app'2(minus, x), y))
APP'2(app'2(low, n), app'2(app'2(add, m), x)) -> APP'2(app'2(le, m), n)
APP'2(app'2(low, n), app'2(app'2(add, m), x)) -> APP'2(if_low, app'2(app'2(le, m), n))
APP'2(app'2(app'2(if_high, false), n), app'2(app'2(add, m), x)) -> APP'2(app'2(add, m), app'2(app'2(high, n), x))
APP'2(app'2(app'2(if_high, true), n), app'2(app'2(add, m), x)) -> APP'2(high, n)
APP'2(app'2(high, n), app'2(app'2(add, m), x)) -> APP'2(le, m)
APP'2(app'2(quot, app'2(s, x)), app'2(s, y)) -> APP'2(minus, x)
APP'2(app'2(app'2(if_low, true), n), app'2(app'2(add, m), x)) -> APP'2(app'2(low, n), x)
APP'2(quicksort, app'2(app'2(add, n), x)) -> APP'2(app'2(high, n), x)
APP'2(app'2(app'2(if_low, false), n), app'2(app'2(add, m), x)) -> APP'2(app'2(low, n), x)
APP'2(app'2(app'2(if_low, false), n), app'2(app'2(add, m), x)) -> APP'2(low, n)
APP'2(app'2(quot, app'2(s, x)), app'2(s, y)) -> APP'2(s, app'2(app'2(quot, app'2(app'2(minus, x), y)), app'2(s, y)))
APP'2(app'2(high, n), app'2(app'2(add, m), x)) -> APP'2(app'2(app'2(if_high, app'2(app'2(le, m), n)), n), app'2(app'2(add, m), x))
APP'2(quicksort, app'2(app'2(add, n), x)) -> APP'2(quicksort, app'2(app'2(low, n), x))
APP'2(app'2(app'2(if_high, false), n), app'2(app'2(add, m), x)) -> APP'2(high, n)
APP'2(app'2(low, n), app'2(app'2(add, m), x)) -> APP'2(app'2(app'2(if_low, app'2(app'2(le, m), n)), n), app'2(app'2(add, m), x))
APP'2(quicksort, app'2(app'2(add, n), x)) -> APP'2(high, n)
APP'2(app'2(app, app'2(app'2(add, n), x)), y) -> APP'2(app'2(app, x), y)
APP'2(app'2(minus, app'2(s, x)), app'2(s, y)) -> APP'2(minus, x)
APP'2(quicksort, app'2(app'2(add, n), x)) -> APP'2(app'2(add, n), app'2(quicksort, app'2(app'2(high, n), x)))
APP'2(quicksort, app'2(app'2(add, n), x)) -> APP'2(app'2(low, n), x)
APP'2(app'2(quot, app'2(s, x)), app'2(s, y)) -> APP'2(app'2(minus, x), y)
APP'2(app'2(low, n), app'2(app'2(add, m), x)) -> APP'2(le, m)
APP'2(app'2(high, n), app'2(app'2(add, m), x)) -> APP'2(app'2(if_high, app'2(app'2(le, m), n)), n)
APP'2(app'2(high, n), app'2(app'2(add, m), x)) -> APP'2(if_high, app'2(app'2(le, m), n))
APP'2(app'2(quot, app'2(s, x)), app'2(s, y)) -> APP'2(app'2(quot, app'2(app'2(minus, x), y)), app'2(s, y))
APP'2(app'2(low, n), app'2(app'2(add, m), x)) -> APP'2(app'2(if_low, app'2(app'2(le, m), n)), n)
APP'2(app'2(app, app'2(app'2(add, n), x)), y) -> APP'2(app, x)
APP'2(quicksort, app'2(app'2(add, n), x)) -> APP'2(app'2(app, app'2(quicksort, app'2(app'2(low, n), x))), app'2(app'2(add, n), app'2(quicksort, app'2(app'2(high, n), x))))
APP'2(quicksort, app'2(app'2(add, n), x)) -> APP'2(app, app'2(quicksort, app'2(app'2(low, n), x)))
APP'2(app'2(app'2(if_high, false), n), app'2(app'2(add, m), x)) -> APP'2(app'2(high, n), x)
APP'2(app'2(app'2(if_high, true), n), app'2(app'2(add, m), x)) -> APP'2(app'2(high, n), x)
APP'2(app'2(le, app'2(s, x)), app'2(s, y)) -> APP'2(app'2(le, x), y)
APP'2(quicksort, app'2(app'2(add, n), x)) -> APP'2(low, n)
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(high, n), app'2(app'2(add, m), x)) -> APP'2(app'2(le, m), n)
APP'2(app'2(le, app'2(s, x)), app'2(s, y)) -> APP'2(le, x)

The TRS R consists of the following rules:

app'2(app'2(minus, x), 0) -> x
app'2(app'2(minus, app'2(s, x)), app'2(s, y)) -> app'2(app'2(minus, x), y)
app'2(app'2(quot, 0), app'2(s, y)) -> 0
app'2(app'2(quot, app'2(s, x)), app'2(s, y)) -> app'2(s, app'2(app'2(quot, app'2(app'2(minus, x), y)), app'2(s, 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(app'2(low, n), nil) -> nil
app'2(app'2(low, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_low, app'2(app'2(le, m), n)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_low, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(low, n), x))
app'2(app'2(app'2(if_low, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(low, n), x)
app'2(app'2(high, n), nil) -> nil
app'2(app'2(high, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_high, app'2(app'2(le, m), n)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_high, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(high, n), x)
app'2(app'2(app'2(if_high, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(high, n), x))
app'2(quicksort, nil) -> nil
app'2(quicksort, app'2(app'2(add, n), x)) -> app'2(app'2(app, app'2(quicksort, app'2(app'2(low, n), x))), app'2(app'2(add, n), app'2(quicksort, app'2(app'2(high, n), x))))

The set Q consists of the following terms:

app'2(app'2(minus, x0), 0)
app'2(app'2(minus, app'2(s, x0)), app'2(s, x1))
app'2(app'2(quot, 0), app'2(s, x0))
app'2(app'2(quot, 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(app'2(low, x0), nil)
app'2(app'2(low, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_low, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_low, false), x0), app'2(app'2(add, x1), x2))
app'2(app'2(high, x0), nil)
app'2(app'2(high, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_high, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_high, false), x0), app'2(app'2(add, x1), x2))
app'2(quicksort, nil)
app'2(quicksort, app'2(app'2(add, x0), x1))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 7 SCCs with 29 less nodes.

↳ QTRS
  ↳ Non-Overlap Check
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
                ↳ QDPOrderProof
              ↳ QDP
              ↳ 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(minus, x), 0) -> x
app'2(app'2(minus, app'2(s, x)), app'2(s, y)) -> app'2(app'2(minus, x), y)
app'2(app'2(quot, 0), app'2(s, y)) -> 0
app'2(app'2(quot, app'2(s, x)), app'2(s, y)) -> app'2(s, app'2(app'2(quot, app'2(app'2(minus, x), y)), app'2(s, 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(app'2(low, n), nil) -> nil
app'2(app'2(low, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_low, app'2(app'2(le, m), n)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_low, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(low, n), x))
app'2(app'2(app'2(if_low, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(low, n), x)
app'2(app'2(high, n), nil) -> nil
app'2(app'2(high, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_high, app'2(app'2(le, m), n)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_high, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(high, n), x)
app'2(app'2(app'2(if_high, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(high, n), x))
app'2(quicksort, nil) -> nil
app'2(quicksort, app'2(app'2(add, n), x)) -> app'2(app'2(app, app'2(quicksort, app'2(app'2(low, n), x))), app'2(app'2(add, n), app'2(quicksort, app'2(app'2(high, n), x))))

The set Q consists of the following terms:

app'2(app'2(minus, x0), 0)
app'2(app'2(minus, app'2(s, x0)), app'2(s, x1))
app'2(app'2(quot, 0), app'2(s, x0))
app'2(app'2(quot, 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(app'2(low, x0), nil)
app'2(app'2(low, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_low, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_low, false), x0), app'2(app'2(add, x1), x2))
app'2(app'2(high, x0), nil)
app'2(app'2(high, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_high, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_high, false), x0), app'2(app'2(add, x1), x2))
app'2(quicksort, nil)
app'2(quicksort, app'2(app'2(add, x0), x1))

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.


APP'2(app'2(app, app'2(app'2(add, n), x)), y) -> APP'2(app'2(app, x), y)
The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
APP'2(x1, x2)  =  x1
app'2(x1, x2)  =  app'1(x2)
app  =  app
add  =  add

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
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP

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

app'2(app'2(minus, x), 0) -> x
app'2(app'2(minus, app'2(s, x)), app'2(s, y)) -> app'2(app'2(minus, x), y)
app'2(app'2(quot, 0), app'2(s, y)) -> 0
app'2(app'2(quot, app'2(s, x)), app'2(s, y)) -> app'2(s, app'2(app'2(quot, app'2(app'2(minus, x), y)), app'2(s, 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(app'2(low, n), nil) -> nil
app'2(app'2(low, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_low, app'2(app'2(le, m), n)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_low, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(low, n), x))
app'2(app'2(app'2(if_low, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(low, n), x)
app'2(app'2(high, n), nil) -> nil
app'2(app'2(high, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_high, app'2(app'2(le, m), n)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_high, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(high, n), x)
app'2(app'2(app'2(if_high, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(high, n), x))
app'2(quicksort, nil) -> nil
app'2(quicksort, app'2(app'2(add, n), x)) -> app'2(app'2(app, app'2(quicksort, app'2(app'2(low, n), x))), app'2(app'2(add, n), app'2(quicksort, app'2(app'2(high, n), x))))

The set Q consists of the following terms:

app'2(app'2(minus, x0), 0)
app'2(app'2(minus, app'2(s, x0)), app'2(s, x1))
app'2(app'2(quot, 0), app'2(s, x0))
app'2(app'2(quot, 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(app'2(low, x0), nil)
app'2(app'2(low, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_low, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_low, false), x0), app'2(app'2(add, x1), x2))
app'2(app'2(high, x0), nil)
app'2(app'2(high, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_high, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_high, false), x0), app'2(app'2(add, x1), x2))
app'2(quicksort, nil)
app'2(quicksort, app'2(app'2(add, x0), x1))

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
              ↳ 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(minus, x), 0) -> x
app'2(app'2(minus, app'2(s, x)), app'2(s, y)) -> app'2(app'2(minus, x), y)
app'2(app'2(quot, 0), app'2(s, y)) -> 0
app'2(app'2(quot, app'2(s, x)), app'2(s, y)) -> app'2(s, app'2(app'2(quot, app'2(app'2(minus, x), y)), app'2(s, 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(app'2(low, n), nil) -> nil
app'2(app'2(low, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_low, app'2(app'2(le, m), n)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_low, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(low, n), x))
app'2(app'2(app'2(if_low, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(low, n), x)
app'2(app'2(high, n), nil) -> nil
app'2(app'2(high, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_high, app'2(app'2(le, m), n)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_high, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(high, n), x)
app'2(app'2(app'2(if_high, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(high, n), x))
app'2(quicksort, nil) -> nil
app'2(quicksort, app'2(app'2(add, n), x)) -> app'2(app'2(app, app'2(quicksort, app'2(app'2(low, n), x))), app'2(app'2(add, n), app'2(quicksort, app'2(app'2(high, n), x))))

The set Q consists of the following terms:

app'2(app'2(minus, x0), 0)
app'2(app'2(minus, app'2(s, x0)), app'2(s, x1))
app'2(app'2(quot, 0), app'2(s, x0))
app'2(app'2(quot, 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(app'2(low, x0), nil)
app'2(app'2(low, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_low, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_low, false), x0), app'2(app'2(add, x1), x2))
app'2(app'2(high, x0), nil)
app'2(app'2(high, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_high, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_high, false), x0), app'2(app'2(add, x1), x2))
app'2(quicksort, nil)
app'2(quicksort, app'2(app'2(add, x0), x1))

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.


APP'2(app'2(le, app'2(s, x)), app'2(s, y)) -> APP'2(app'2(le, x), y)
The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
APP'2(x1, x2)  =  APP'1(x2)
app'2(x1, x2)  =  app'1(x2)
le  =  le
s  =  s

Lexicographic Path Order [19].
Precedence:
APP'1 > app'1
APP'1 > le
s > app'1
s > le

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ Non-Overlap Check
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ 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(minus, x), 0) -> x
app'2(app'2(minus, app'2(s, x)), app'2(s, y)) -> app'2(app'2(minus, x), y)
app'2(app'2(quot, 0), app'2(s, y)) -> 0
app'2(app'2(quot, app'2(s, x)), app'2(s, y)) -> app'2(s, app'2(app'2(quot, app'2(app'2(minus, x), y)), app'2(s, 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(app'2(low, n), nil) -> nil
app'2(app'2(low, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_low, app'2(app'2(le, m), n)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_low, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(low, n), x))
app'2(app'2(app'2(if_low, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(low, n), x)
app'2(app'2(high, n), nil) -> nil
app'2(app'2(high, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_high, app'2(app'2(le, m), n)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_high, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(high, n), x)
app'2(app'2(app'2(if_high, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(high, n), x))
app'2(quicksort, nil) -> nil
app'2(quicksort, app'2(app'2(add, n), x)) -> app'2(app'2(app, app'2(quicksort, app'2(app'2(low, n), x))), app'2(app'2(add, n), app'2(quicksort, app'2(app'2(high, n), x))))

The set Q consists of the following terms:

app'2(app'2(minus, x0), 0)
app'2(app'2(minus, app'2(s, x0)), app'2(s, x1))
app'2(app'2(quot, 0), app'2(s, x0))
app'2(app'2(quot, 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(app'2(low, x0), nil)
app'2(app'2(low, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_low, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_low, false), x0), app'2(app'2(add, x1), x2))
app'2(app'2(high, x0), nil)
app'2(app'2(high, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_high, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_high, false), x0), app'2(app'2(add, x1), x2))
app'2(quicksort, nil)
app'2(quicksort, app'2(app'2(add, x0), x1))

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
                ↳ QDPOrderProof
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP

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

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

The TRS R consists of the following rules:

app'2(app'2(minus, x), 0) -> x
app'2(app'2(minus, app'2(s, x)), app'2(s, y)) -> app'2(app'2(minus, x), y)
app'2(app'2(quot, 0), app'2(s, y)) -> 0
app'2(app'2(quot, app'2(s, x)), app'2(s, y)) -> app'2(s, app'2(app'2(quot, app'2(app'2(minus, x), y)), app'2(s, 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(app'2(low, n), nil) -> nil
app'2(app'2(low, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_low, app'2(app'2(le, m), n)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_low, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(low, n), x))
app'2(app'2(app'2(if_low, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(low, n), x)
app'2(app'2(high, n), nil) -> nil
app'2(app'2(high, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_high, app'2(app'2(le, m), n)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_high, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(high, n), x)
app'2(app'2(app'2(if_high, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(high, n), x))
app'2(quicksort, nil) -> nil
app'2(quicksort, app'2(app'2(add, n), x)) -> app'2(app'2(app, app'2(quicksort, app'2(app'2(low, n), x))), app'2(app'2(add, n), app'2(quicksort, app'2(app'2(high, n), x))))

The set Q consists of the following terms:

app'2(app'2(minus, x0), 0)
app'2(app'2(minus, app'2(s, x0)), app'2(s, x1))
app'2(app'2(quot, 0), app'2(s, x0))
app'2(app'2(quot, 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(app'2(low, x0), nil)
app'2(app'2(low, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_low, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_low, false), x0), app'2(app'2(add, x1), x2))
app'2(app'2(high, x0), nil)
app'2(app'2(high, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_high, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_high, false), x0), app'2(app'2(add, x1), x2))
app'2(quicksort, nil)
app'2(quicksort, app'2(app'2(add, x0), x1))

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.


APP'2(app'2(app'2(if_high, false), n), app'2(app'2(add, m), x)) -> APP'2(app'2(high, n), x)
APP'2(app'2(app'2(if_high, true), n), app'2(app'2(add, m), x)) -> APP'2(app'2(high, n), x)
The remaining pairs can at least by weakly be oriented.

APP'2(app'2(high, n), app'2(app'2(add, m), x)) -> APP'2(app'2(app'2(if_high, app'2(app'2(le, m), n)), n), app'2(app'2(add, m), x))
Used ordering: Combined order from the following AFS and order.
APP'2(x1, x2)  =  APP'1(x2)
app'2(x1, x2)  =  app'1(x2)
high  =  high
add  =  add
if_high  =  if_high
le  =  le
false  =  false
true  =  true
0  =  0
s  =  s

Lexicographic Path Order [19].
Precedence:
ifhigh > high > APP'1 > add > true
ifhigh > high > app'1 > true
le > false > true
0 > false > true
s > true

The following usable rules [14] were oriented:

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)



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

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

APP'2(app'2(high, n), app'2(app'2(add, m), x)) -> APP'2(app'2(app'2(if_high, app'2(app'2(le, m), n)), n), app'2(app'2(add, m), x))

The TRS R consists of the following rules:

app'2(app'2(minus, x), 0) -> x
app'2(app'2(minus, app'2(s, x)), app'2(s, y)) -> app'2(app'2(minus, x), y)
app'2(app'2(quot, 0), app'2(s, y)) -> 0
app'2(app'2(quot, app'2(s, x)), app'2(s, y)) -> app'2(s, app'2(app'2(quot, app'2(app'2(minus, x), y)), app'2(s, 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(app'2(low, n), nil) -> nil
app'2(app'2(low, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_low, app'2(app'2(le, m), n)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_low, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(low, n), x))
app'2(app'2(app'2(if_low, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(low, n), x)
app'2(app'2(high, n), nil) -> nil
app'2(app'2(high, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_high, app'2(app'2(le, m), n)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_high, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(high, n), x)
app'2(app'2(app'2(if_high, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(high, n), x))
app'2(quicksort, nil) -> nil
app'2(quicksort, app'2(app'2(add, n), x)) -> app'2(app'2(app, app'2(quicksort, app'2(app'2(low, n), x))), app'2(app'2(add, n), app'2(quicksort, app'2(app'2(high, n), x))))

The set Q consists of the following terms:

app'2(app'2(minus, x0), 0)
app'2(app'2(minus, app'2(s, x0)), app'2(s, x1))
app'2(app'2(quot, 0), app'2(s, x0))
app'2(app'2(quot, 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(app'2(low, x0), nil)
app'2(app'2(low, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_low, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_low, false), x0), app'2(app'2(add, x1), x2))
app'2(app'2(high, x0), nil)
app'2(app'2(high, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_high, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_high, false), x0), app'2(app'2(add, x1), x2))
app'2(quicksort, nil)
app'2(quicksort, app'2(app'2(add, x0), x1))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 0 SCCs with 1 less node.

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

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

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

The TRS R consists of the following rules:

app'2(app'2(minus, x), 0) -> x
app'2(app'2(minus, app'2(s, x)), app'2(s, y)) -> app'2(app'2(minus, x), y)
app'2(app'2(quot, 0), app'2(s, y)) -> 0
app'2(app'2(quot, app'2(s, x)), app'2(s, y)) -> app'2(s, app'2(app'2(quot, app'2(app'2(minus, x), y)), app'2(s, 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(app'2(low, n), nil) -> nil
app'2(app'2(low, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_low, app'2(app'2(le, m), n)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_low, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(low, n), x))
app'2(app'2(app'2(if_low, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(low, n), x)
app'2(app'2(high, n), nil) -> nil
app'2(app'2(high, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_high, app'2(app'2(le, m), n)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_high, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(high, n), x)
app'2(app'2(app'2(if_high, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(high, n), x))
app'2(quicksort, nil) -> nil
app'2(quicksort, app'2(app'2(add, n), x)) -> app'2(app'2(app, app'2(quicksort, app'2(app'2(low, n), x))), app'2(app'2(add, n), app'2(quicksort, app'2(app'2(high, n), x))))

The set Q consists of the following terms:

app'2(app'2(minus, x0), 0)
app'2(app'2(minus, app'2(s, x0)), app'2(s, x1))
app'2(app'2(quot, 0), app'2(s, x0))
app'2(app'2(quot, 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(app'2(low, x0), nil)
app'2(app'2(low, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_low, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_low, false), x0), app'2(app'2(add, x1), x2))
app'2(app'2(high, x0), nil)
app'2(app'2(high, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_high, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_high, false), x0), app'2(app'2(add, x1), x2))
app'2(quicksort, nil)
app'2(quicksort, app'2(app'2(add, x0), x1))

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.


APP'2(app'2(app'2(if_low, true), n), app'2(app'2(add, m), x)) -> APP'2(app'2(low, n), x)
APP'2(app'2(app'2(if_low, false), n), app'2(app'2(add, m), x)) -> APP'2(app'2(low, n), x)
The remaining pairs can at least by weakly be oriented.

APP'2(app'2(low, n), app'2(app'2(add, m), x)) -> APP'2(app'2(app'2(if_low, app'2(app'2(le, m), n)), n), app'2(app'2(add, m), x))
Used ordering: Combined order from the following AFS and order.
APP'2(x1, x2)  =  APP'2(x1, x2)
app'2(x1, x2)  =  app'1(x2)
if_low  =  if_low
true  =  true
add  =  add
low  =  low
false  =  false
le  =  le
0  =  0
s  =  s

Lexicographic Path Order [19].
Precedence:
APP'2 > le > iflow
app'1 > true > iflow
app'1 > false > iflow
app'1 > le > iflow
add > le > iflow
low > le > iflow
0 > true > iflow
0 > false > iflow
s > false > iflow
s > le > iflow

The following usable rules [14] were oriented:

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)



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

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

APP'2(app'2(low, n), app'2(app'2(add, m), x)) -> APP'2(app'2(app'2(if_low, app'2(app'2(le, m), n)), n), app'2(app'2(add, m), x))

The TRS R consists of the following rules:

app'2(app'2(minus, x), 0) -> x
app'2(app'2(minus, app'2(s, x)), app'2(s, y)) -> app'2(app'2(minus, x), y)
app'2(app'2(quot, 0), app'2(s, y)) -> 0
app'2(app'2(quot, app'2(s, x)), app'2(s, y)) -> app'2(s, app'2(app'2(quot, app'2(app'2(minus, x), y)), app'2(s, 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(app'2(low, n), nil) -> nil
app'2(app'2(low, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_low, app'2(app'2(le, m), n)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_low, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(low, n), x))
app'2(app'2(app'2(if_low, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(low, n), x)
app'2(app'2(high, n), nil) -> nil
app'2(app'2(high, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_high, app'2(app'2(le, m), n)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_high, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(high, n), x)
app'2(app'2(app'2(if_high, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(high, n), x))
app'2(quicksort, nil) -> nil
app'2(quicksort, app'2(app'2(add, n), x)) -> app'2(app'2(app, app'2(quicksort, app'2(app'2(low, n), x))), app'2(app'2(add, n), app'2(quicksort, app'2(app'2(high, n), x))))

The set Q consists of the following terms:

app'2(app'2(minus, x0), 0)
app'2(app'2(minus, app'2(s, x0)), app'2(s, x1))
app'2(app'2(quot, 0), app'2(s, x0))
app'2(app'2(quot, 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(app'2(low, x0), nil)
app'2(app'2(low, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_low, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_low, false), x0), app'2(app'2(add, x1), x2))
app'2(app'2(high, x0), nil)
app'2(app'2(high, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_high, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_high, false), x0), app'2(app'2(add, x1), x2))
app'2(quicksort, nil)
app'2(quicksort, app'2(app'2(add, x0), x1))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 0 SCCs with 1 less node.

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

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

APP'2(quicksort, app'2(app'2(add, n), x)) -> APP'2(quicksort, app'2(app'2(high, n), x))
APP'2(quicksort, app'2(app'2(add, n), x)) -> APP'2(quicksort, app'2(app'2(low, n), x))

The TRS R consists of the following rules:

app'2(app'2(minus, x), 0) -> x
app'2(app'2(minus, app'2(s, x)), app'2(s, y)) -> app'2(app'2(minus, x), y)
app'2(app'2(quot, 0), app'2(s, y)) -> 0
app'2(app'2(quot, app'2(s, x)), app'2(s, y)) -> app'2(s, app'2(app'2(quot, app'2(app'2(minus, x), y)), app'2(s, 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(app'2(low, n), nil) -> nil
app'2(app'2(low, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_low, app'2(app'2(le, m), n)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_low, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(low, n), x))
app'2(app'2(app'2(if_low, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(low, n), x)
app'2(app'2(high, n), nil) -> nil
app'2(app'2(high, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_high, app'2(app'2(le, m), n)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_high, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(high, n), x)
app'2(app'2(app'2(if_high, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(high, n), x))
app'2(quicksort, nil) -> nil
app'2(quicksort, app'2(app'2(add, n), x)) -> app'2(app'2(app, app'2(quicksort, app'2(app'2(low, n), x))), app'2(app'2(add, n), app'2(quicksort, app'2(app'2(high, n), x))))

The set Q consists of the following terms:

app'2(app'2(minus, x0), 0)
app'2(app'2(minus, app'2(s, x0)), app'2(s, x1))
app'2(app'2(quot, 0), app'2(s, x0))
app'2(app'2(quot, 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(app'2(low, x0), nil)
app'2(app'2(low, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_low, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_low, false), x0), app'2(app'2(add, x1), x2))
app'2(app'2(high, x0), nil)
app'2(app'2(high, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_high, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_high, false), x0), app'2(app'2(add, x1), x2))
app'2(quicksort, nil)
app'2(quicksort, app'2(app'2(add, x0), x1))

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

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

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

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

The TRS R consists of the following rules:

app'2(app'2(minus, x), 0) -> x
app'2(app'2(minus, app'2(s, x)), app'2(s, y)) -> app'2(app'2(minus, x), y)
app'2(app'2(quot, 0), app'2(s, y)) -> 0
app'2(app'2(quot, app'2(s, x)), app'2(s, y)) -> app'2(s, app'2(app'2(quot, app'2(app'2(minus, x), y)), app'2(s, 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(app'2(low, n), nil) -> nil
app'2(app'2(low, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_low, app'2(app'2(le, m), n)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_low, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(low, n), x))
app'2(app'2(app'2(if_low, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(low, n), x)
app'2(app'2(high, n), nil) -> nil
app'2(app'2(high, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_high, app'2(app'2(le, m), n)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_high, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(high, n), x)
app'2(app'2(app'2(if_high, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(high, n), x))
app'2(quicksort, nil) -> nil
app'2(quicksort, app'2(app'2(add, n), x)) -> app'2(app'2(app, app'2(quicksort, app'2(app'2(low, n), x))), app'2(app'2(add, n), app'2(quicksort, app'2(app'2(high, n), x))))

The set Q consists of the following terms:

app'2(app'2(minus, x0), 0)
app'2(app'2(minus, app'2(s, x0)), app'2(s, x1))
app'2(app'2(quot, 0), app'2(s, x0))
app'2(app'2(quot, 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(app'2(low, x0), nil)
app'2(app'2(low, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_low, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_low, false), x0), app'2(app'2(add, x1), x2))
app'2(app'2(high, x0), nil)
app'2(app'2(high, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_high, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_high, false), x0), app'2(app'2(add, x1), x2))
app'2(quicksort, nil)
app'2(quicksort, app'2(app'2(add, x0), x1))

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.


APP'2(app'2(minus, app'2(s, x)), app'2(s, y)) -> APP'2(app'2(minus, x), y)
The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
APP'2(x1, x2)  =  APP'1(x2)
app'2(x1, x2)  =  app'1(x2)
minus  =  minus
s  =  s

Lexicographic Path Order [19].
Precedence:
APP'1 > app'1
APP'1 > minus
s > app'1
s > minus

The following usable rules [14] were oriented: none



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

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

app'2(app'2(minus, x), 0) -> x
app'2(app'2(minus, app'2(s, x)), app'2(s, y)) -> app'2(app'2(minus, x), y)
app'2(app'2(quot, 0), app'2(s, y)) -> 0
app'2(app'2(quot, app'2(s, x)), app'2(s, y)) -> app'2(s, app'2(app'2(quot, app'2(app'2(minus, x), y)), app'2(s, 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(app'2(low, n), nil) -> nil
app'2(app'2(low, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_low, app'2(app'2(le, m), n)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_low, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(low, n), x))
app'2(app'2(app'2(if_low, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(low, n), x)
app'2(app'2(high, n), nil) -> nil
app'2(app'2(high, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_high, app'2(app'2(le, m), n)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_high, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(high, n), x)
app'2(app'2(app'2(if_high, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(high, n), x))
app'2(quicksort, nil) -> nil
app'2(quicksort, app'2(app'2(add, n), x)) -> app'2(app'2(app, app'2(quicksort, app'2(app'2(low, n), x))), app'2(app'2(add, n), app'2(quicksort, app'2(app'2(high, n), x))))

The set Q consists of the following terms:

app'2(app'2(minus, x0), 0)
app'2(app'2(minus, app'2(s, x0)), app'2(s, x1))
app'2(app'2(quot, 0), app'2(s, x0))
app'2(app'2(quot, 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(app'2(low, x0), nil)
app'2(app'2(low, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_low, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_low, false), x0), app'2(app'2(add, x1), x2))
app'2(app'2(high, x0), nil)
app'2(app'2(high, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_high, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_high, false), x0), app'2(app'2(add, x1), x2))
app'2(quicksort, nil)
app'2(quicksort, app'2(app'2(add, x0), x1))

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
              ↳ QDP
              ↳ QDP

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

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

The TRS R consists of the following rules:

app'2(app'2(minus, x), 0) -> x
app'2(app'2(minus, app'2(s, x)), app'2(s, y)) -> app'2(app'2(minus, x), y)
app'2(app'2(quot, 0), app'2(s, y)) -> 0
app'2(app'2(quot, app'2(s, x)), app'2(s, y)) -> app'2(s, app'2(app'2(quot, app'2(app'2(minus, x), y)), app'2(s, 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(app'2(low, n), nil) -> nil
app'2(app'2(low, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_low, app'2(app'2(le, m), n)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_low, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(low, n), x))
app'2(app'2(app'2(if_low, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(low, n), x)
app'2(app'2(high, n), nil) -> nil
app'2(app'2(high, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_high, app'2(app'2(le, m), n)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_high, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(high, n), x)
app'2(app'2(app'2(if_high, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(high, n), x))
app'2(quicksort, nil) -> nil
app'2(quicksort, app'2(app'2(add, n), x)) -> app'2(app'2(app, app'2(quicksort, app'2(app'2(low, n), x))), app'2(app'2(add, n), app'2(quicksort, app'2(app'2(high, n), x))))

The set Q consists of the following terms:

app'2(app'2(minus, x0), 0)
app'2(app'2(minus, app'2(s, x0)), app'2(s, x1))
app'2(app'2(quot, 0), app'2(s, x0))
app'2(app'2(quot, 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(app'2(low, x0), nil)
app'2(app'2(low, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_low, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_low, false), x0), app'2(app'2(add, x1), x2))
app'2(app'2(high, x0), nil)
app'2(app'2(high, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_high, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_high, false), x0), app'2(app'2(add, x1), x2))
app'2(quicksort, nil)
app'2(quicksort, app'2(app'2(add, x0), x1))

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