(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
app'(app'(minus, x), 0) → x
app'(app'(minus, app'(s, x)), app'(s, y)) → app'(app'(minus, x), y)
app'(app'(quot, 0), app'(s, y)) → 0
app'(app'(quot, app'(s, x)), app'(s, y)) → app'(s, app'(app'(quot, app'(app'(minus, x), y)), app'(s, y)))
app'(app'(le, 0), y) → true
app'(app'(le, app'(s, x)), 0) → false
app'(app'(le, app'(s, x)), app'(s, y)) → app'(app'(le, x), y)
app'(app'(app, nil), y) → y
app'(app'(app, app'(app'(add, n), x)), y) → app'(app'(add, n), app'(app'(app, x), y))
app'(app'(low, n), nil) → nil
app'(app'(low, n), app'(app'(add, m), x)) → app'(app'(app'(if_low, app'(app'(le, m), n)), n), app'(app'(add, m), x))
app'(app'(app'(if_low, true), n), app'(app'(add, m), x)) → app'(app'(add, m), app'(app'(low, n), x))
app'(app'(app'(if_low, false), n), app'(app'(add, m), x)) → app'(app'(low, n), x)
app'(app'(high, n), nil) → nil
app'(app'(high, n), app'(app'(add, m), x)) → app'(app'(app'(if_high, app'(app'(le, m), n)), n), app'(app'(add, m), x))
app'(app'(app'(if_high, true), n), app'(app'(add, m), x)) → app'(app'(high, n), x)
app'(app'(app'(if_high, false), n), app'(app'(add, m), x)) → app'(app'(add, m), app'(app'(high, n), x))
app'(quicksort, nil) → nil
app'(quicksort, app'(app'(add, n), x)) → app'(app'(app, app'(quicksort, app'(app'(low, n), x))), app'(app'(add, n), app'(quicksort, app'(app'(high, n), x))))
app'(app'(map, f), nil) → nil
app'(app'(map, f), app'(app'(add, x), xs)) → app'(app'(add, app'(f, x)), app'(app'(map, f), xs))
app'(app'(filter, f), nil) → nil
app'(app'(filter, f), app'(app'(add, x), xs)) → app'(app'(app'(app'(filter2, app'(f, x)), f), x), xs)
app'(app'(app'(app'(filter2, true), f), x), xs) → app'(app'(add, x), app'(app'(filter, f), xs))
app'(app'(app'(app'(filter2, false), f), x), xs) → app'(app'(filter, f), xs)
Q is empty.