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))
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 contains 7 SCCs with 29 less nodes.
↳ QTRS
↳ Non-Overlap Check
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPAfsSolverProof
↳ 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.
By using an argument filtering and a montonic ordering, at least one Dependency Pair of this SCC can be strictly oriented.
APP'2(app'2(app, app'2(app'2(add, n), x)), y) -> APP'2(app'2(app, x), y)
Used argument filtering: APP'2(x1, x2) = x1
app'2(x1, x2) = app'1(x2)
Used ordering: Quasi Precedence:
trivial
↳ QTRS
↳ Non-Overlap Check
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPAfsSolverProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ 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
↳ QDPAfsSolverProof
↳ 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.
By using an argument filtering and a montonic ordering, at least one Dependency Pair of this SCC can be strictly oriented.
APP'2(app'2(le, app'2(s, x)), app'2(s, y)) -> APP'2(app'2(le, x), y)
Used argument filtering: APP'2(x1, x2) = x2
app'2(x1, x2) = app'1(x2)
s = s
Used ordering: Quasi Precedence:
trivial
↳ QTRS
↳ Non-Overlap Check
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPAfsSolverProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ 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
↳ QDPAfsSolverProof
↳ 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.
By using an argument filtering and a montonic ordering, at least one Dependency Pair of this SCC can be strictly oriented.
APP'2(app'2(app'2(if_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)
Used argument filtering: APP'2(x1, x2) = x2
app'2(x1, x2) = app'1(x2)
true = true
0 = 0
false = false
Used ordering: Quasi Precedence:
[app'_1, false] > true
↳ QTRS
↳ Non-Overlap Check
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPAfsSolverProof
↳ 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 contains 0 SCCs with 1 less node.
↳ QTRS
↳ Non-Overlap Check
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPAfsSolverProof
↳ 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.
By using an argument filtering and a montonic ordering, at least one Dependency Pair of this SCC can be strictly oriented.
APP'2(app'2(app'2(if_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)
Used argument filtering: APP'2(x1, x2) = x2
app'2(x1, x2) = app'1(x2)
true = true
0 = 0
false = false
Used ordering: Quasi Precedence:
[app'_1, false] > true
↳ QTRS
↳ Non-Overlap Check
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPAfsSolverProof
↳ 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 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
↳ QDPAfsSolverProof
↳ 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.
By using an argument filtering and a montonic ordering, at least one Dependency Pair of this SCC can be strictly oriented.
APP'2(app'2(minus, app'2(s, x)), app'2(s, y)) -> APP'2(app'2(minus, x), y)
Used argument filtering: APP'2(x1, x2) = x2
app'2(x1, x2) = app'1(x2)
s = s
Used ordering: Quasi Precedence:
trivial
↳ QTRS
↳ Non-Overlap Check
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPAfsSolverProof
↳ 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.