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))))
app'2(app'2(map, f), nil) -> nil
app'2(app'2(map, f), app'2(app'2(add, x), xs)) -> app'2(app'2(add, app'2(f, x)), app'2(app'2(map, f), xs))
app'2(app'2(filter, f), nil) -> nil
app'2(app'2(filter, f), app'2(app'2(add, x), xs)) -> app'2(app'2(app'2(app'2(filter2, app'2(f, x)), f), x), xs)
app'2(app'2(app'2(app'2(filter2, true), f), x), xs) -> app'2(app'2(add, x), app'2(app'2(filter, f), xs))
app'2(app'2(app'2(app'2(filter2, false), f), x), xs) -> app'2(app'2(filter, f), xs)

Q is empty.


↳ 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))))
app'2(app'2(map, f), nil) -> nil
app'2(app'2(map, f), app'2(app'2(add, x), xs)) -> app'2(app'2(add, app'2(f, x)), app'2(app'2(map, f), xs))
app'2(app'2(filter, f), nil) -> nil
app'2(app'2(filter, f), app'2(app'2(add, x), xs)) -> app'2(app'2(app'2(app'2(filter2, app'2(f, x)), f), x), xs)
app'2(app'2(app'2(app'2(filter2, true), f), x), xs) -> app'2(app'2(add, x), app'2(app'2(filter, f), xs))
app'2(app'2(app'2(app'2(filter2, false), f), x), xs) -> app'2(app'2(filter, f), xs)

Q is empty.

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

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(low, n), app'2(app'2(add, m), x)) -> APP'2(if_low, app'2(app'2(le, m), n))
APP'2(app'2(filter, f), app'2(app'2(add, x), xs)) -> APP'2(app'2(app'2(filter2, app'2(f, x)), f), x)
APP'2(app'2(map, f), app'2(app'2(add, x), xs)) -> APP'2(app'2(add, app'2(f, x)), app'2(app'2(map, f), xs))
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, false), 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(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(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(quicksort, app'2(app'2(add, n), x)) -> APP'2(quicksort, app'2(app'2(low, n), x))
APP'2(quicksort, app'2(app'2(add, n), x)) -> APP'2(high, n)
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(low, n), 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(app'2(quot, app'2(s, x)), app'2(s, y)) -> APP'2(app'2(minus, x), y)
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(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(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(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(app'2(filter, f), app'2(app'2(add, x), xs)) -> APP'2(app'2(filter2, app'2(f, x)), f)
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, true), n), app'2(app'2(add, m), x)) -> APP'2(app'2(high, n), x)
APP'2(app'2(app'2(app'2(filter2, true), f), x), xs) -> APP'2(filter, f)
APP'2(quicksort, app'2(app'2(add, n), x)) -> APP'2(low, n)
APP'2(app'2(filter, f), app'2(app'2(add, x), xs)) -> APP'2(filter2, app'2(f, x))
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(map, f), app'2(app'2(add, x), xs)) -> APP'2(add, app'2(f, 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(map, f), app'2(app'2(add, x), xs)) -> APP'2(app'2(map, f), xs)
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(app'2(filter2, false), f), x), xs) -> APP'2(filter, f)
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(filter, f), app'2(app'2(add, x), xs)) -> APP'2(f, x)
APP'2(app'2(low, n), app'2(app'2(add, m), x)) -> APP'2(app'2(le, m), n)
APP'2(app'2(app'2(app'2(filter2, true), f), x), xs) -> APP'2(add, 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(app'2(high, n), app'2(app'2(add, m), x)) -> APP'2(le, m)
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(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(app'2(filter2, true), f), x), xs) -> APP'2(app'2(filter, f), xs)
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(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_high, false), n), app'2(app'2(add, m), 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(filter, f), app'2(app'2(add, x), xs)) -> APP'2(app'2(app'2(app'2(filter2, app'2(f, x)), f), x), xs)
APP'2(app'2(low, n), app'2(app'2(add, m), x)) -> APP'2(le, m)
APP'2(app'2(app, app'2(app'2(add, n), x)), y) -> APP'2(app, x)
APP'2(app'2(app'2(app'2(filter2, true), f), x), xs) -> APP'2(app'2(add, x), app'2(app'2(filter, f), xs))
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(app'2(filter2, false), f), x), xs) -> APP'2(app'2(filter, f), xs)
APP'2(app'2(le, app'2(s, x)), app'2(s, y)) -> APP'2(app'2(le, x), y)
APP'2(app'2(map, f), app'2(app'2(add, x), xs)) -> APP'2(f, x)
APP'2(app'2(le, app'2(s, x)), app'2(s, y)) -> APP'2(le, x)
APP'2(app'2(high, n), app'2(app'2(add, m), x)) -> APP'2(app'2(le, m), n)

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))))
app'2(app'2(map, f), nil) -> nil
app'2(app'2(map, f), app'2(app'2(add, x), xs)) -> app'2(app'2(add, app'2(f, x)), app'2(app'2(map, f), xs))
app'2(app'2(filter, f), nil) -> nil
app'2(app'2(filter, f), app'2(app'2(add, x), xs)) -> app'2(app'2(app'2(app'2(filter2, app'2(f, x)), f), x), xs)
app'2(app'2(app'2(app'2(filter2, true), f), x), xs) -> app'2(app'2(add, x), app'2(app'2(filter, f), xs))
app'2(app'2(app'2(app'2(filter2, false), f), x), xs) -> app'2(app'2(filter, f), xs)

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof

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

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(low, n), app'2(app'2(add, m), x)) -> APP'2(if_low, app'2(app'2(le, m), n))
APP'2(app'2(filter, f), app'2(app'2(add, x), xs)) -> APP'2(app'2(app'2(filter2, app'2(f, x)), f), x)
APP'2(app'2(map, f), app'2(app'2(add, x), xs)) -> APP'2(app'2(add, app'2(f, x)), app'2(app'2(map, f), xs))
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, false), 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(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(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(quicksort, app'2(app'2(add, n), x)) -> APP'2(quicksort, app'2(app'2(low, n), x))
APP'2(quicksort, app'2(app'2(add, n), x)) -> APP'2(high, n)
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(low, n), 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(app'2(quot, app'2(s, x)), app'2(s, y)) -> APP'2(app'2(minus, x), y)
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(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(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(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(app'2(filter, f), app'2(app'2(add, x), xs)) -> APP'2(app'2(filter2, app'2(f, x)), f)
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, true), n), app'2(app'2(add, m), x)) -> APP'2(app'2(high, n), x)
APP'2(app'2(app'2(app'2(filter2, true), f), x), xs) -> APP'2(filter, f)
APP'2(quicksort, app'2(app'2(add, n), x)) -> APP'2(low, n)
APP'2(app'2(filter, f), app'2(app'2(add, x), xs)) -> APP'2(filter2, app'2(f, x))
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(map, f), app'2(app'2(add, x), xs)) -> APP'2(add, app'2(f, 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(map, f), app'2(app'2(add, x), xs)) -> APP'2(app'2(map, f), xs)
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(app'2(filter2, false), f), x), xs) -> APP'2(filter, f)
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(filter, f), app'2(app'2(add, x), xs)) -> APP'2(f, x)
APP'2(app'2(low, n), app'2(app'2(add, m), x)) -> APP'2(app'2(le, m), n)
APP'2(app'2(app'2(app'2(filter2, true), f), x), xs) -> APP'2(add, 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(app'2(high, n), app'2(app'2(add, m), x)) -> APP'2(le, m)
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(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(app'2(filter2, true), f), x), xs) -> APP'2(app'2(filter, f), xs)
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(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_high, false), n), app'2(app'2(add, m), 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(filter, f), app'2(app'2(add, x), xs)) -> APP'2(app'2(app'2(app'2(filter2, app'2(f, x)), f), x), xs)
APP'2(app'2(low, n), app'2(app'2(add, m), x)) -> APP'2(le, m)
APP'2(app'2(app, app'2(app'2(add, n), x)), y) -> APP'2(app, x)
APP'2(app'2(app'2(app'2(filter2, true), f), x), xs) -> APP'2(app'2(add, x), app'2(app'2(filter, f), xs))
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(app'2(filter2, false), f), x), xs) -> APP'2(app'2(filter, f), xs)
APP'2(app'2(le, app'2(s, x)), app'2(s, y)) -> APP'2(app'2(le, x), y)
APP'2(app'2(map, f), app'2(app'2(add, x), xs)) -> APP'2(f, x)
APP'2(app'2(le, app'2(s, x)), app'2(s, y)) -> APP'2(le, x)
APP'2(app'2(high, n), app'2(app'2(add, m), x)) -> APP'2(app'2(le, m), n)

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))))
app'2(app'2(map, f), nil) -> nil
app'2(app'2(map, f), app'2(app'2(add, x), xs)) -> app'2(app'2(add, app'2(f, x)), app'2(app'2(map, f), xs))
app'2(app'2(filter, f), nil) -> nil
app'2(app'2(filter, f), app'2(app'2(add, x), xs)) -> app'2(app'2(app'2(app'2(filter2, app'2(f, x)), f), x), xs)
app'2(app'2(app'2(app'2(filter2, true), f), x), xs) -> app'2(app'2(add, x), app'2(app'2(filter, f), xs))
app'2(app'2(app'2(app'2(filter2, false), f), x), xs) -> app'2(app'2(filter, f), xs)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 8 SCCs with 38 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ 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))))
app'2(app'2(map, f), nil) -> nil
app'2(app'2(map, f), app'2(app'2(add, x), xs)) -> app'2(app'2(add, app'2(f, x)), app'2(app'2(map, f), xs))
app'2(app'2(filter, f), nil) -> nil
app'2(app'2(filter, f), app'2(app'2(add, x), xs)) -> app'2(app'2(app'2(app'2(filter2, app'2(f, x)), f), x), xs)
app'2(app'2(app'2(app'2(filter2, true), f), x), xs) -> app'2(app'2(add, x), app'2(app'2(filter, f), xs))
app'2(app'2(app'2(app'2(filter2, false), f), x), xs) -> app'2(app'2(filter, f), xs)

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


The following pairs can be oriented strictly and are deleted.


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 be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(APP'2(x1, x2)) = x1   
POL(add) = 0   
POL(app) = 0   
POL(app'2(x1, x2)) = 2 + x2   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ PisEmptyProof
          ↳ QDP
          ↳ 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))))
app'2(app'2(map, f), nil) -> nil
app'2(app'2(map, f), app'2(app'2(add, x), xs)) -> app'2(app'2(add, app'2(f, x)), app'2(app'2(map, f), xs))
app'2(app'2(filter, f), nil) -> nil
app'2(app'2(filter, f), app'2(app'2(add, x), xs)) -> app'2(app'2(app'2(app'2(filter2, app'2(f, x)), f), x), xs)
app'2(app'2(app'2(app'2(filter2, true), f), x), xs) -> app'2(app'2(add, x), app'2(app'2(filter, f), xs))
app'2(app'2(app'2(app'2(filter2, false), f), x), xs) -> app'2(app'2(filter, f), xs)

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ 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))))
app'2(app'2(map, f), nil) -> nil
app'2(app'2(map, f), app'2(app'2(add, x), xs)) -> app'2(app'2(add, app'2(f, x)), app'2(app'2(map, f), xs))
app'2(app'2(filter, f), nil) -> nil
app'2(app'2(filter, f), app'2(app'2(add, x), xs)) -> app'2(app'2(app'2(app'2(filter2, app'2(f, x)), f), x), xs)
app'2(app'2(app'2(app'2(filter2, true), f), x), xs) -> app'2(app'2(add, x), app'2(app'2(filter, f), xs))
app'2(app'2(app'2(app'2(filter2, false), f), x), xs) -> app'2(app'2(filter, f), xs)

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


The following pairs can be oriented strictly and are deleted.


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 be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(APP'2(x1, x2)) = 2·x2   
POL(app'2(x1, x2)) = 2 + x2   
POL(le) = 0   
POL(s) = 0   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ 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))))
app'2(app'2(map, f), nil) -> nil
app'2(app'2(map, f), app'2(app'2(add, x), xs)) -> app'2(app'2(add, app'2(f, x)), app'2(app'2(map, f), xs))
app'2(app'2(filter, f), nil) -> nil
app'2(app'2(filter, f), app'2(app'2(add, x), xs)) -> app'2(app'2(app'2(app'2(filter2, app'2(f, x)), f), x), xs)
app'2(app'2(app'2(app'2(filter2, true), f), x), xs) -> app'2(app'2(add, x), app'2(app'2(filter, f), xs))
app'2(app'2(app'2(app'2(filter2, false), f), x), xs) -> app'2(app'2(filter, f), xs)

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ 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))))
app'2(app'2(map, f), nil) -> nil
app'2(app'2(map, f), app'2(app'2(add, x), xs)) -> app'2(app'2(add, app'2(f, x)), app'2(app'2(map, f), xs))
app'2(app'2(filter, f), nil) -> nil
app'2(app'2(filter, f), app'2(app'2(add, x), xs)) -> app'2(app'2(app'2(app'2(filter2, app'2(f, x)), f), x), xs)
app'2(app'2(app'2(app'2(filter2, true), f), x), xs) -> app'2(app'2(add, x), app'2(app'2(filter, f), xs))
app'2(app'2(app'2(app'2(filter2, false), f), x), xs) -> app'2(app'2(filter, f), xs)

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


The following pairs can be oriented strictly and are deleted.


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 be oriented weakly.

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: Polynomial interpretation [21]:

POL(0) = 1   
POL(APP'2(x1, x2)) = 2·x2   
POL(add) = 0   
POL(app'2(x1, x2)) = 1 + x2   
POL(false) = 0   
POL(high) = 0   
POL(if_high) = 0   
POL(le) = 0   
POL(s) = 0   
POL(true) = 0   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof
          ↳ QDP
          ↳ 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))))
app'2(app'2(map, f), nil) -> nil
app'2(app'2(map, f), app'2(app'2(add, x), xs)) -> app'2(app'2(add, app'2(f, x)), app'2(app'2(map, f), xs))
app'2(app'2(filter, f), nil) -> nil
app'2(app'2(filter, f), app'2(app'2(add, x), xs)) -> app'2(app'2(app'2(app'2(filter2, app'2(f, x)), f), x), xs)
app'2(app'2(app'2(app'2(filter2, true), f), x), xs) -> app'2(app'2(add, x), app'2(app'2(filter, f), xs))
app'2(app'2(app'2(app'2(filter2, false), f), x), xs) -> app'2(app'2(filter, f), xs)

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ 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))))
app'2(app'2(map, f), nil) -> nil
app'2(app'2(map, f), app'2(app'2(add, x), xs)) -> app'2(app'2(add, app'2(f, x)), app'2(app'2(map, f), xs))
app'2(app'2(filter, f), nil) -> nil
app'2(app'2(filter, f), app'2(app'2(add, x), xs)) -> app'2(app'2(app'2(app'2(filter2, app'2(f, x)), f), x), xs)
app'2(app'2(app'2(app'2(filter2, true), f), x), xs) -> app'2(app'2(add, x), app'2(app'2(filter, f), xs))
app'2(app'2(app'2(app'2(filter2, false), f), x), xs) -> app'2(app'2(filter, f), xs)

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


The following pairs can be oriented strictly and are deleted.


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 be oriented weakly.

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: Polynomial interpretation [21]:

POL(0) = 0   
POL(APP'2(x1, x2)) = 2·x2   
POL(add) = 2   
POL(app'2(x1, x2)) = 1 + x1 + x2   
POL(false) = 0   
POL(if_low) = 0   
POL(le) = 1   
POL(low) = 0   
POL(s) = 0   
POL(true) = 0   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ 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(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))))
app'2(app'2(map, f), nil) -> nil
app'2(app'2(map, f), app'2(app'2(add, x), xs)) -> app'2(app'2(add, app'2(f, x)), app'2(app'2(map, f), xs))
app'2(app'2(filter, f), nil) -> nil
app'2(app'2(filter, f), app'2(app'2(add, x), xs)) -> app'2(app'2(app'2(app'2(filter2, app'2(f, x)), f), x), xs)
app'2(app'2(app'2(app'2(filter2, true), f), x), xs) -> app'2(app'2(add, x), app'2(app'2(filter, f), xs))
app'2(app'2(app'2(app'2(filter2, false), f), x), xs) -> app'2(app'2(filter, f), xs)

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ 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))))
app'2(app'2(map, f), nil) -> nil
app'2(app'2(map, f), app'2(app'2(add, x), xs)) -> app'2(app'2(add, app'2(f, x)), app'2(app'2(map, f), xs))
app'2(app'2(filter, f), nil) -> nil
app'2(app'2(filter, f), app'2(app'2(add, x), xs)) -> app'2(app'2(app'2(app'2(filter2, app'2(f, x)), f), x), xs)
app'2(app'2(app'2(app'2(filter2, true), f), x), xs) -> app'2(app'2(add, x), app'2(app'2(filter, f), xs))
app'2(app'2(app'2(app'2(filter2, false), f), x), xs) -> app'2(app'2(filter, f), xs)

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ 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))))
app'2(app'2(map, f), nil) -> nil
app'2(app'2(map, f), app'2(app'2(add, x), xs)) -> app'2(app'2(add, app'2(f, x)), app'2(app'2(map, f), xs))
app'2(app'2(filter, f), nil) -> nil
app'2(app'2(filter, f), app'2(app'2(add, x), xs)) -> app'2(app'2(app'2(app'2(filter2, app'2(f, x)), f), x), xs)
app'2(app'2(app'2(app'2(filter2, true), f), x), xs) -> app'2(app'2(add, x), app'2(app'2(filter, f), xs))
app'2(app'2(app'2(app'2(filter2, false), f), x), xs) -> app'2(app'2(filter, f), xs)

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


The following pairs can be oriented strictly and are deleted.


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 be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(APP'2(x1, x2)) = 2·x2   
POL(app'2(x1, x2)) = 2 + x2   
POL(minus) = 0   
POL(s) = 0   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ PisEmptyProof
          ↳ 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))))
app'2(app'2(map, f), nil) -> nil
app'2(app'2(map, f), app'2(app'2(add, x), xs)) -> app'2(app'2(add, app'2(f, x)), app'2(app'2(map, f), xs))
app'2(app'2(filter, f), nil) -> nil
app'2(app'2(filter, f), app'2(app'2(add, x), xs)) -> app'2(app'2(app'2(app'2(filter2, app'2(f, x)), f), x), xs)
app'2(app'2(app'2(app'2(filter2, true), f), x), xs) -> app'2(app'2(add, x), app'2(app'2(filter, f), xs))
app'2(app'2(app'2(app'2(filter2, false), f), x), xs) -> app'2(app'2(filter, f), xs)

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ 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))))
app'2(app'2(map, f), nil) -> nil
app'2(app'2(map, f), app'2(app'2(add, x), xs)) -> app'2(app'2(add, app'2(f, x)), app'2(app'2(map, f), xs))
app'2(app'2(filter, f), nil) -> nil
app'2(app'2(filter, f), app'2(app'2(add, x), xs)) -> app'2(app'2(app'2(app'2(filter2, app'2(f, x)), f), x), xs)
app'2(app'2(app'2(app'2(filter2, true), f), x), xs) -> app'2(app'2(add, x), app'2(app'2(filter, f), xs))
app'2(app'2(app'2(app'2(filter2, false), f), x), xs) -> app'2(app'2(filter, f), xs)

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

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

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

APP'2(app'2(map, f), app'2(app'2(add, x), xs)) -> APP'2(app'2(map, f), xs)
APP'2(app'2(app'2(app'2(filter2, true), f), x), xs) -> APP'2(app'2(filter, f), xs)
APP'2(app'2(filter, f), app'2(app'2(add, x), xs)) -> APP'2(f, x)
APP'2(app'2(map, f), app'2(app'2(add, x), xs)) -> APP'2(f, x)
APP'2(app'2(filter, f), app'2(app'2(add, x), xs)) -> APP'2(app'2(app'2(app'2(filter2, app'2(f, x)), f), x), xs)
APP'2(app'2(app'2(app'2(filter2, false), f), x), xs) -> APP'2(app'2(filter, f), xs)

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))))
app'2(app'2(map, f), nil) -> nil
app'2(app'2(map, f), app'2(app'2(add, x), xs)) -> app'2(app'2(add, app'2(f, x)), app'2(app'2(map, f), xs))
app'2(app'2(filter, f), nil) -> nil
app'2(app'2(filter, f), app'2(app'2(add, x), xs)) -> app'2(app'2(app'2(app'2(filter2, app'2(f, x)), f), x), xs)
app'2(app'2(app'2(app'2(filter2, true), f), x), xs) -> app'2(app'2(add, x), app'2(app'2(filter, f), xs))
app'2(app'2(app'2(app'2(filter2, false), f), x), xs) -> app'2(app'2(filter, f), xs)

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


The following pairs can be oriented strictly and are deleted.


APP'2(app'2(map, f), app'2(app'2(add, x), xs)) -> APP'2(app'2(map, f), xs)
APP'2(app'2(filter, f), app'2(app'2(add, x), xs)) -> APP'2(f, x)
APP'2(app'2(map, f), app'2(app'2(add, x), xs)) -> APP'2(f, x)
APP'2(app'2(filter, f), app'2(app'2(add, x), xs)) -> APP'2(app'2(app'2(app'2(filter2, app'2(f, x)), f), x), xs)
The remaining pairs can at least be oriented weakly.

APP'2(app'2(app'2(app'2(filter2, true), f), x), xs) -> APP'2(app'2(filter, f), xs)
APP'2(app'2(app'2(app'2(filter2, false), f), x), xs) -> APP'2(app'2(filter, f), xs)
Used ordering: Polynomial interpretation [21]:

POL(0) = 0   
POL(APP'2(x1, x2)) = 2·x2   
POL(add) = 0   
POL(app) = 0   
POL(app'2(x1, x2)) = 2 + x1 + x2   
POL(false) = 0   
POL(filter) = 0   
POL(filter2) = 0   
POL(high) = 0   
POL(if_high) = 0   
POL(if_low) = 0   
POL(le) = 0   
POL(low) = 0   
POL(map) = 0   
POL(minus) = 0   
POL(nil) = 0   
POL(quicksort) = 0   
POL(quot) = 0   
POL(s) = 0   
POL(true) = 0   

The following usable rules [14] were oriented: none



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

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

APP'2(app'2(app'2(app'2(filter2, true), f), x), xs) -> APP'2(app'2(filter, f), xs)
APP'2(app'2(app'2(app'2(filter2, false), f), x), xs) -> APP'2(app'2(filter, f), xs)

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))))
app'2(app'2(map, f), nil) -> nil
app'2(app'2(map, f), app'2(app'2(add, x), xs)) -> app'2(app'2(add, app'2(f, x)), app'2(app'2(map, f), xs))
app'2(app'2(filter, f), nil) -> nil
app'2(app'2(filter, f), app'2(app'2(add, x), xs)) -> app'2(app'2(app'2(app'2(filter2, app'2(f, x)), f), x), xs)
app'2(app'2(app'2(app'2(filter2, true), f), x), xs) -> app'2(app'2(add, x), app'2(app'2(filter, f), xs))
app'2(app'2(app'2(app'2(filter2, false), f), x), xs) -> app'2(app'2(filter, f), xs)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 0 SCCs with 2 less nodes.