Termination w.r.t. Q of the following Term Rewriting System could not be shown:

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

app'2(app'2(eq, 0), 0) -> true
app'2(app'2(eq, 0), app'2(s, x)) -> false
app'2(app'2(eq, app'2(s, x)), 0) -> false
app'2(app'2(eq, app'2(s, x)), app'2(s, y)) -> app'2(app'2(eq, x), y)
app'2(app'2(le, 0), y) -> true
app'2(app'2(le, app'2(s, x)), 0) -> false
app'2(app'2(le, app'2(s, x)), app'2(s, y)) -> app'2(app'2(le, x), y)
app'2(app'2(app, nil), y) -> y
app'2(app'2(app, app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(app, x), y))
app'2(min, app'2(app'2(add, n), nil)) -> n
app'2(min, app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(app'2(if_min, app'2(app'2(le, n), m)), app'2(app'2(add, n), app'2(app'2(add, m), x)))
app'2(app'2(if_min, true), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, n), x))
app'2(app'2(if_min, false), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, m), x))
app'2(app'2(rm, n), nil) -> nil
app'2(app'2(rm, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_rm, app'2(app'2(eq, n), m)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_rm, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(rm, n), x)
app'2(app'2(app'2(if_rm, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(rm, n), x))
app'2(app'2(minsort, nil), nil) -> nil
app'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> app'2(app'2(app'2(if_minsort, app'2(app'2(eq, n), app'2(min, app'2(app'2(add, n), x)))), app'2(app'2(add, n), x)), y)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(minsort, app'2(app'2(app, app'2(app'2(rm, n), x)), y)), nil))
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, n), x)), y) -> app'2(app'2(minsort, x), app'2(app'2(add, n), y))
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(eq, 0), 0) -> true
app'2(app'2(eq, 0), app'2(s, x)) -> false
app'2(app'2(eq, app'2(s, x)), 0) -> false
app'2(app'2(eq, app'2(s, x)), app'2(s, y)) -> app'2(app'2(eq, x), y)
app'2(app'2(le, 0), y) -> true
app'2(app'2(le, app'2(s, x)), 0) -> false
app'2(app'2(le, app'2(s, x)), app'2(s, y)) -> app'2(app'2(le, x), y)
app'2(app'2(app, nil), y) -> y
app'2(app'2(app, app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(app, x), y))
app'2(min, app'2(app'2(add, n), nil)) -> n
app'2(min, app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(app'2(if_min, app'2(app'2(le, n), m)), app'2(app'2(add, n), app'2(app'2(add, m), x)))
app'2(app'2(if_min, true), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, n), x))
app'2(app'2(if_min, false), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, m), x))
app'2(app'2(rm, n), nil) -> nil
app'2(app'2(rm, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_rm, app'2(app'2(eq, n), m)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_rm, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(rm, n), x)
app'2(app'2(app'2(if_rm, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(rm, n), x))
app'2(app'2(minsort, nil), nil) -> nil
app'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> app'2(app'2(app'2(if_minsort, app'2(app'2(eq, n), app'2(min, app'2(app'2(add, n), x)))), app'2(app'2(add, n), x)), y)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(minsort, app'2(app'2(app, app'2(app'2(rm, n), x)), y)), nil))
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, n), x)), y) -> app'2(app'2(minsort, x), app'2(app'2(add, n), y))
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_minsort, true), app'2(app'2(add, n), x)), y) -> APP'2(rm, 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(eq, app'2(s, x)), app'2(s, y)) -> APP'2(app'2(eq, x), y)
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(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> APP'2(app'2(app, app'2(app'2(rm, n), x)), y)
APP'2(app'2(app'2(if_minsort, false), app'2(app'2(add, n), x)), y) -> APP'2(app'2(add, n), y)
APP'2(app'2(if_min, false), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> APP'2(min, app'2(app'2(add, m), x))
APP'2(app'2(if_min, true), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> APP'2(app'2(add, n), x)
APP'2(app'2(rm, n), app'2(app'2(add, m), x)) -> APP'2(app'2(if_rm, app'2(app'2(eq, n), m)), n)
APP'2(app'2(app'2(if_rm, true), n), app'2(app'2(add, m), x)) -> APP'2(rm, n)
APP'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> APP'2(app'2(rm, n), x)
APP'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> APP'2(min, app'2(app'2(add, n), x))
APP'2(app'2(filter, f), app'2(app'2(add, x), xs)) -> APP'2(app'2(filter2, app'2(f, x)), f)
APP'2(min, app'2(app'2(add, n), app'2(app'2(add, m), x))) -> APP'2(le, n)
APP'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> APP'2(eq, n)
APP'2(app'2(app'2(app'2(filter2, true), f), x), xs) -> APP'2(filter, f)
APP'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> APP'2(app'2(app'2(if_minsort, app'2(app'2(eq, n), app'2(min, app'2(app'2(add, n), x)))), app'2(app'2(add, n), x)), y)
APP'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> APP'2(minsort, app'2(app'2(app, app'2(app'2(rm, n), x)), y))
APP'2(app'2(app'2(if_rm, false), n), app'2(app'2(add, m), x)) -> APP'2(app'2(add, m), app'2(app'2(rm, n), x))
APP'2(app'2(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(app'2(if_rm, true), n), app'2(app'2(add, m), x)) -> APP'2(app'2(rm, n), x)
APP'2(app'2(app'2(if_rm, false), n), app'2(app'2(add, m), x)) -> APP'2(app'2(rm, n), x)
APP'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> APP'2(if_minsort, app'2(app'2(eq, n), app'2(min, app'2(app'2(add, n), x))))
APP'2(app'2(map, f), app'2(app'2(add, x), xs)) -> APP'2(add, app'2(f, x))
APP'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> APP'2(app, app'2(app'2(rm, n), x))
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, false), f), x), xs) -> APP'2(filter, f)
APP'2(app'2(filter, f), app'2(app'2(add, x), xs)) -> APP'2(f, x)
APP'2(app'2(app'2(app'2(filter2, true), f), x), xs) -> APP'2(add, x)
APP'2(app'2(app'2(if_minsort, false), app'2(app'2(add, n), x)), y) -> APP'2(minsort, x)
APP'2(app'2(app'2(app'2(filter2, true), f), x), xs) -> APP'2(app'2(filter, f), xs)
APP'2(app'2(app, app'2(app'2(add, n), x)), y) -> APP'2(app'2(app, x), y)
APP'2(app'2(app'2(if_minsort, false), app'2(app'2(add, n), x)), y) -> APP'2(app'2(minsort, x), app'2(app'2(add, n), y))
APP'2(app'2(rm, n), app'2(app'2(add, m), x)) -> APP'2(if_rm, app'2(app'2(eq, n), m))
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(eq, app'2(s, x)), app'2(s, y)) -> APP'2(eq, x)
APP'2(app'2(app, app'2(app'2(add, n), x)), y) -> APP'2(app, x)
APP'2(app'2(app'2(if_rm, false), n), app'2(app'2(add, m), x)) -> APP'2(rm, n)
APP'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> APP'2(app'2(add, n), app'2(app'2(minsort, app'2(app'2(app, app'2(app'2(rm, n), x)), y)), nil))
APP'2(min, app'2(app'2(add, n), app'2(app'2(add, m), x))) -> APP'2(if_min, app'2(app'2(le, n), m))
APP'2(app'2(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(rm, n), app'2(app'2(add, m), x)) -> APP'2(eq, n)
APP'2(min, app'2(app'2(add, n), app'2(app'2(add, m), x))) -> APP'2(app'2(if_min, app'2(app'2(le, n), m)), app'2(app'2(add, n), app'2(app'2(add, m), x)))
APP'2(app'2(app'2(app'2(filter2, false), f), x), xs) -> APP'2(app'2(filter, f), xs)
APP'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> APP'2(app'2(eq, n), app'2(min, app'2(app'2(add, n), x)))
APP'2(min, app'2(app'2(add, n), app'2(app'2(add, m), x))) -> APP'2(app'2(le, n), m)
APP'2(app'2(le, app'2(s, x)), app'2(s, y)) -> APP'2(app'2(le, x), y)
APP'2(app'2(if_min, true), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> APP'2(min, app'2(app'2(add, n), x))
APP'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> APP'2(app'2(if_minsort, app'2(app'2(eq, n), app'2(min, app'2(app'2(add, n), x)))), app'2(app'2(add, n), x))
APP'2(app'2(map, f), app'2(app'2(add, x), xs)) -> APP'2(f, x)
APP'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> APP'2(app'2(minsort, app'2(app'2(app, app'2(app'2(rm, n), x)), y)), nil)
APP'2(app'2(rm, n), app'2(app'2(add, m), x)) -> APP'2(app'2(app'2(if_rm, app'2(app'2(eq, n), m)), n), app'2(app'2(add, m), x))
APP'2(app'2(le, app'2(s, x)), app'2(s, y)) -> APP'2(le, x)
APP'2(app'2(rm, n), app'2(app'2(add, m), x)) -> APP'2(app'2(eq, n), m)

The TRS R consists of the following rules:

app'2(app'2(eq, 0), 0) -> true
app'2(app'2(eq, 0), app'2(s, x)) -> false
app'2(app'2(eq, app'2(s, x)), 0) -> false
app'2(app'2(eq, app'2(s, x)), app'2(s, y)) -> app'2(app'2(eq, x), y)
app'2(app'2(le, 0), y) -> true
app'2(app'2(le, app'2(s, x)), 0) -> false
app'2(app'2(le, app'2(s, x)), app'2(s, y)) -> app'2(app'2(le, x), y)
app'2(app'2(app, nil), y) -> y
app'2(app'2(app, app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(app, x), y))
app'2(min, app'2(app'2(add, n), nil)) -> n
app'2(min, app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(app'2(if_min, app'2(app'2(le, n), m)), app'2(app'2(add, n), app'2(app'2(add, m), x)))
app'2(app'2(if_min, true), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, n), x))
app'2(app'2(if_min, false), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, m), x))
app'2(app'2(rm, n), nil) -> nil
app'2(app'2(rm, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_rm, app'2(app'2(eq, n), m)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_rm, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(rm, n), x)
app'2(app'2(app'2(if_rm, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(rm, n), x))
app'2(app'2(minsort, nil), nil) -> nil
app'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> app'2(app'2(app'2(if_minsort, app'2(app'2(eq, n), app'2(min, app'2(app'2(add, n), x)))), app'2(app'2(add, n), x)), y)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(minsort, app'2(app'2(app, app'2(app'2(rm, n), x)), y)), nil))
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, n), x)), y) -> app'2(app'2(minsort, x), app'2(app'2(add, n), y))
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_minsort, true), app'2(app'2(add, n), x)), y) -> APP'2(rm, 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(eq, app'2(s, x)), app'2(s, y)) -> APP'2(app'2(eq, x), y)
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(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> APP'2(app'2(app, app'2(app'2(rm, n), x)), y)
APP'2(app'2(app'2(if_minsort, false), app'2(app'2(add, n), x)), y) -> APP'2(app'2(add, n), y)
APP'2(app'2(if_min, false), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> APP'2(min, app'2(app'2(add, m), x))
APP'2(app'2(if_min, true), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> APP'2(app'2(add, n), x)
APP'2(app'2(rm, n), app'2(app'2(add, m), x)) -> APP'2(app'2(if_rm, app'2(app'2(eq, n), m)), n)
APP'2(app'2(app'2(if_rm, true), n), app'2(app'2(add, m), x)) -> APP'2(rm, n)
APP'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> APP'2(app'2(rm, n), x)
APP'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> APP'2(min, app'2(app'2(add, n), x))
APP'2(app'2(filter, f), app'2(app'2(add, x), xs)) -> APP'2(app'2(filter2, app'2(f, x)), f)
APP'2(min, app'2(app'2(add, n), app'2(app'2(add, m), x))) -> APP'2(le, n)
APP'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> APP'2(eq, n)
APP'2(app'2(app'2(app'2(filter2, true), f), x), xs) -> APP'2(filter, f)
APP'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> APP'2(app'2(app'2(if_minsort, app'2(app'2(eq, n), app'2(min, app'2(app'2(add, n), x)))), app'2(app'2(add, n), x)), y)
APP'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> APP'2(minsort, app'2(app'2(app, app'2(app'2(rm, n), x)), y))
APP'2(app'2(app'2(if_rm, false), n), app'2(app'2(add, m), x)) -> APP'2(app'2(add, m), app'2(app'2(rm, n), x))
APP'2(app'2(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(app'2(if_rm, true), n), app'2(app'2(add, m), x)) -> APP'2(app'2(rm, n), x)
APP'2(app'2(app'2(if_rm, false), n), app'2(app'2(add, m), x)) -> APP'2(app'2(rm, n), x)
APP'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> APP'2(if_minsort, app'2(app'2(eq, n), app'2(min, app'2(app'2(add, n), x))))
APP'2(app'2(map, f), app'2(app'2(add, x), xs)) -> APP'2(add, app'2(f, x))
APP'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> APP'2(app, app'2(app'2(rm, n), x))
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, false), f), x), xs) -> APP'2(filter, f)
APP'2(app'2(filter, f), app'2(app'2(add, x), xs)) -> APP'2(f, x)
APP'2(app'2(app'2(app'2(filter2, true), f), x), xs) -> APP'2(add, x)
APP'2(app'2(app'2(if_minsort, false), app'2(app'2(add, n), x)), y) -> APP'2(minsort, x)
APP'2(app'2(app'2(app'2(filter2, true), f), x), xs) -> APP'2(app'2(filter, f), xs)
APP'2(app'2(app, app'2(app'2(add, n), x)), y) -> APP'2(app'2(app, x), y)
APP'2(app'2(app'2(if_minsort, false), app'2(app'2(add, n), x)), y) -> APP'2(app'2(minsort, x), app'2(app'2(add, n), y))
APP'2(app'2(rm, n), app'2(app'2(add, m), x)) -> APP'2(if_rm, app'2(app'2(eq, n), m))
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(eq, app'2(s, x)), app'2(s, y)) -> APP'2(eq, x)
APP'2(app'2(app, app'2(app'2(add, n), x)), y) -> APP'2(app, x)
APP'2(app'2(app'2(if_rm, false), n), app'2(app'2(add, m), x)) -> APP'2(rm, n)
APP'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> APP'2(app'2(add, n), app'2(app'2(minsort, app'2(app'2(app, app'2(app'2(rm, n), x)), y)), nil))
APP'2(min, app'2(app'2(add, n), app'2(app'2(add, m), x))) -> APP'2(if_min, app'2(app'2(le, n), m))
APP'2(app'2(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(rm, n), app'2(app'2(add, m), x)) -> APP'2(eq, n)
APP'2(min, app'2(app'2(add, n), app'2(app'2(add, m), x))) -> APP'2(app'2(if_min, app'2(app'2(le, n), m)), app'2(app'2(add, n), app'2(app'2(add, m), x)))
APP'2(app'2(app'2(app'2(filter2, false), f), x), xs) -> APP'2(app'2(filter, f), xs)
APP'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> APP'2(app'2(eq, n), app'2(min, app'2(app'2(add, n), x)))
APP'2(min, app'2(app'2(add, n), app'2(app'2(add, m), x))) -> APP'2(app'2(le, n), m)
APP'2(app'2(le, app'2(s, x)), app'2(s, y)) -> APP'2(app'2(le, x), y)
APP'2(app'2(if_min, true), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> APP'2(min, app'2(app'2(add, n), x))
APP'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> APP'2(app'2(if_minsort, app'2(app'2(eq, n), app'2(min, app'2(app'2(add, n), x)))), app'2(app'2(add, n), x))
APP'2(app'2(map, f), app'2(app'2(add, x), xs)) -> APP'2(f, x)
APP'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> APP'2(app'2(minsort, app'2(app'2(app, app'2(app'2(rm, n), x)), y)), nil)
APP'2(app'2(rm, n), app'2(app'2(add, m), x)) -> APP'2(app'2(app'2(if_rm, app'2(app'2(eq, n), m)), n), app'2(app'2(add, m), x))
APP'2(app'2(le, app'2(s, x)), app'2(s, y)) -> APP'2(le, x)
APP'2(app'2(rm, n), app'2(app'2(add, m), x)) -> APP'2(app'2(eq, n), m)

The TRS R consists of the following rules:

app'2(app'2(eq, 0), 0) -> true
app'2(app'2(eq, 0), app'2(s, x)) -> false
app'2(app'2(eq, app'2(s, x)), 0) -> false
app'2(app'2(eq, app'2(s, x)), app'2(s, y)) -> app'2(app'2(eq, x), y)
app'2(app'2(le, 0), y) -> true
app'2(app'2(le, app'2(s, x)), 0) -> false
app'2(app'2(le, app'2(s, x)), app'2(s, y)) -> app'2(app'2(le, x), y)
app'2(app'2(app, nil), y) -> y
app'2(app'2(app, app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(app, x), y))
app'2(min, app'2(app'2(add, n), nil)) -> n
app'2(min, app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(app'2(if_min, app'2(app'2(le, n), m)), app'2(app'2(add, n), app'2(app'2(add, m), x)))
app'2(app'2(if_min, true), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, n), x))
app'2(app'2(if_min, false), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, m), x))
app'2(app'2(rm, n), nil) -> nil
app'2(app'2(rm, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_rm, app'2(app'2(eq, n), m)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_rm, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(rm, n), x)
app'2(app'2(app'2(if_rm, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(rm, n), x))
app'2(app'2(minsort, nil), nil) -> nil
app'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> app'2(app'2(app'2(if_minsort, app'2(app'2(eq, n), app'2(min, app'2(app'2(add, n), x)))), app'2(app'2(add, n), x)), y)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(minsort, app'2(app'2(app, app'2(app'2(rm, n), x)), y)), nil))
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, n), x)), y) -> app'2(app'2(minsort, x), app'2(app'2(add, n), y))
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 7 SCCs with 37 less nodes.

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

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

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

The TRS R consists of the following rules:

app'2(app'2(eq, 0), 0) -> true
app'2(app'2(eq, 0), app'2(s, x)) -> false
app'2(app'2(eq, app'2(s, x)), 0) -> false
app'2(app'2(eq, app'2(s, x)), app'2(s, y)) -> app'2(app'2(eq, x), y)
app'2(app'2(le, 0), y) -> true
app'2(app'2(le, app'2(s, x)), 0) -> false
app'2(app'2(le, app'2(s, x)), app'2(s, y)) -> app'2(app'2(le, x), y)
app'2(app'2(app, nil), y) -> y
app'2(app'2(app, app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(app, x), y))
app'2(min, app'2(app'2(add, n), nil)) -> n
app'2(min, app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(app'2(if_min, app'2(app'2(le, n), m)), app'2(app'2(add, n), app'2(app'2(add, m), x)))
app'2(app'2(if_min, true), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, n), x))
app'2(app'2(if_min, false), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, m), x))
app'2(app'2(rm, n), nil) -> nil
app'2(app'2(rm, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_rm, app'2(app'2(eq, n), m)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_rm, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(rm, n), x)
app'2(app'2(app'2(if_rm, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(rm, n), x))
app'2(app'2(minsort, nil), nil) -> nil
app'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> app'2(app'2(app'2(if_minsort, app'2(app'2(eq, n), app'2(min, app'2(app'2(add, n), x)))), app'2(app'2(add, n), x)), y)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(minsort, app'2(app'2(app, app'2(app'2(rm, n), x)), y)), nil))
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, n), x)), y) -> app'2(app'2(minsort, x), app'2(app'2(add, n), y))
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 Order [17,21] with Interpretation:

POL( app ) = 0


POL( app'2(x1, x2) ) = x1 + x2 + 3


POL( APP'2(x1, x2) ) = x1 + 3x2 + 1


POL( add ) = 2



The following usable rules [14] were oriented: none



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

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

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

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

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

The TRS R consists of the following rules:

app'2(app'2(eq, 0), 0) -> true
app'2(app'2(eq, 0), app'2(s, x)) -> false
app'2(app'2(eq, app'2(s, x)), 0) -> false
app'2(app'2(eq, app'2(s, x)), app'2(s, y)) -> app'2(app'2(eq, x), y)
app'2(app'2(le, 0), y) -> true
app'2(app'2(le, app'2(s, x)), 0) -> false
app'2(app'2(le, app'2(s, x)), app'2(s, y)) -> app'2(app'2(le, x), y)
app'2(app'2(app, nil), y) -> y
app'2(app'2(app, app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(app, x), y))
app'2(min, app'2(app'2(add, n), nil)) -> n
app'2(min, app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(app'2(if_min, app'2(app'2(le, n), m)), app'2(app'2(add, n), app'2(app'2(add, m), x)))
app'2(app'2(if_min, true), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, n), x))
app'2(app'2(if_min, false), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, m), x))
app'2(app'2(rm, n), nil) -> nil
app'2(app'2(rm, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_rm, app'2(app'2(eq, n), m)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_rm, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(rm, n), x)
app'2(app'2(app'2(if_rm, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(rm, n), x))
app'2(app'2(minsort, nil), nil) -> nil
app'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> app'2(app'2(app'2(if_minsort, app'2(app'2(eq, n), app'2(min, app'2(app'2(add, n), x)))), app'2(app'2(add, n), x)), y)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(minsort, app'2(app'2(app, app'2(app'2(rm, n), x)), y)), nil))
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, n), x)), y) -> app'2(app'2(minsort, x), app'2(app'2(add, n), y))
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 Order [17,21] with Interpretation:

POL( le ) = 3


POL( s ) = max{0, -2}


POL( app'2(x1, x2) ) = 3x1 + 2x2 + 3


POL( APP'2(x1, x2) ) = x1 + 1



The following usable rules [14] were oriented: none



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

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

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

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

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

The TRS R consists of the following rules:

app'2(app'2(eq, 0), 0) -> true
app'2(app'2(eq, 0), app'2(s, x)) -> false
app'2(app'2(eq, app'2(s, x)), 0) -> false
app'2(app'2(eq, app'2(s, x)), app'2(s, y)) -> app'2(app'2(eq, x), y)
app'2(app'2(le, 0), y) -> true
app'2(app'2(le, app'2(s, x)), 0) -> false
app'2(app'2(le, app'2(s, x)), app'2(s, y)) -> app'2(app'2(le, x), y)
app'2(app'2(app, nil), y) -> y
app'2(app'2(app, app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(app, x), y))
app'2(min, app'2(app'2(add, n), nil)) -> n
app'2(min, app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(app'2(if_min, app'2(app'2(le, n), m)), app'2(app'2(add, n), app'2(app'2(add, m), x)))
app'2(app'2(if_min, true), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, n), x))
app'2(app'2(if_min, false), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, m), x))
app'2(app'2(rm, n), nil) -> nil
app'2(app'2(rm, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_rm, app'2(app'2(eq, n), m)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_rm, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(rm, n), x)
app'2(app'2(app'2(if_rm, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(rm, n), x))
app'2(app'2(minsort, nil), nil) -> nil
app'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> app'2(app'2(app'2(if_minsort, app'2(app'2(eq, n), app'2(min, app'2(app'2(add, n), x)))), app'2(app'2(add, n), x)), y)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(minsort, app'2(app'2(app, app'2(app'2(rm, n), x)), y)), nil))
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, n), x)), y) -> app'2(app'2(minsort, x), app'2(app'2(add, n), y))
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(if_min, true), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> APP'2(min, app'2(app'2(add, n), x))
APP'2(app'2(if_min, false), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> APP'2(min, app'2(app'2(add, m), x))
The remaining pairs can at least be oriented weakly.

APP'2(min, app'2(app'2(add, n), app'2(app'2(add, m), x))) -> APP'2(app'2(if_min, app'2(app'2(le, n), m)), app'2(app'2(add, n), app'2(app'2(add, m), x)))
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( true ) = 1


POL( app'2(x1, x2) ) = max{0, 2x1 + x2 - 2}


POL( APP'2(x1, x2) ) = max{0, x2 - 3}


POL( false ) = 0


POL( le ) = max{0, -3}


POL( min ) = 1


POL( 0 ) = 3


POL( s ) = 1


POL( add ) = 2


POL( if_min ) = 0



The following usable rules [14] were oriented: none



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

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

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

The TRS R consists of the following rules:

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

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

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

The TRS R consists of the following rules:

app'2(app'2(eq, 0), 0) -> true
app'2(app'2(eq, 0), app'2(s, x)) -> false
app'2(app'2(eq, app'2(s, x)), 0) -> false
app'2(app'2(eq, app'2(s, x)), app'2(s, y)) -> app'2(app'2(eq, x), y)
app'2(app'2(le, 0), y) -> true
app'2(app'2(le, app'2(s, x)), 0) -> false
app'2(app'2(le, app'2(s, x)), app'2(s, y)) -> app'2(app'2(le, x), y)
app'2(app'2(app, nil), y) -> y
app'2(app'2(app, app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(app, x), y))
app'2(min, app'2(app'2(add, n), nil)) -> n
app'2(min, app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(app'2(if_min, app'2(app'2(le, n), m)), app'2(app'2(add, n), app'2(app'2(add, m), x)))
app'2(app'2(if_min, true), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, n), x))
app'2(app'2(if_min, false), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, m), x))
app'2(app'2(rm, n), nil) -> nil
app'2(app'2(rm, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_rm, app'2(app'2(eq, n), m)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_rm, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(rm, n), x)
app'2(app'2(app'2(if_rm, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(rm, n), x))
app'2(app'2(minsort, nil), nil) -> nil
app'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> app'2(app'2(app'2(if_minsort, app'2(app'2(eq, n), app'2(min, app'2(app'2(add, n), x)))), app'2(app'2(add, n), x)), y)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(minsort, app'2(app'2(app, app'2(app'2(rm, n), x)), y)), nil))
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, n), x)), y) -> app'2(app'2(minsort, x), app'2(app'2(add, n), y))
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(eq, app'2(s, x)), app'2(s, y)) -> APP'2(app'2(eq, x), y)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( s ) = max{0, -2}


POL( app'2(x1, x2) ) = 3x1 + 2x2 + 3


POL( APP'2(x1, x2) ) = x1 + 1


POL( eq ) = 3



The following usable rules [14] were oriented: none



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

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

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

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

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

The TRS R consists of the following rules:

app'2(app'2(eq, 0), 0) -> true
app'2(app'2(eq, 0), app'2(s, x)) -> false
app'2(app'2(eq, app'2(s, x)), 0) -> false
app'2(app'2(eq, app'2(s, x)), app'2(s, y)) -> app'2(app'2(eq, x), y)
app'2(app'2(le, 0), y) -> true
app'2(app'2(le, app'2(s, x)), 0) -> false
app'2(app'2(le, app'2(s, x)), app'2(s, y)) -> app'2(app'2(le, x), y)
app'2(app'2(app, nil), y) -> y
app'2(app'2(app, app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(app, x), y))
app'2(min, app'2(app'2(add, n), nil)) -> n
app'2(min, app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(app'2(if_min, app'2(app'2(le, n), m)), app'2(app'2(add, n), app'2(app'2(add, m), x)))
app'2(app'2(if_min, true), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, n), x))
app'2(app'2(if_min, false), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, m), x))
app'2(app'2(rm, n), nil) -> nil
app'2(app'2(rm, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_rm, app'2(app'2(eq, n), m)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_rm, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(rm, n), x)
app'2(app'2(app'2(if_rm, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(rm, n), x))
app'2(app'2(minsort, nil), nil) -> nil
app'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> app'2(app'2(app'2(if_minsort, app'2(app'2(eq, n), app'2(min, app'2(app'2(add, n), x)))), app'2(app'2(add, n), x)), y)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(minsort, app'2(app'2(app, app'2(app'2(rm, n), x)), y)), nil))
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, n), x)), y) -> app'2(app'2(minsort, x), app'2(app'2(add, n), y))
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_rm, true), n), app'2(app'2(add, m), x)) -> APP'2(app'2(rm, n), x)
APP'2(app'2(app'2(if_rm, false), n), app'2(app'2(add, m), x)) -> APP'2(app'2(rm, n), x)
The remaining pairs can at least be oriented weakly.

APP'2(app'2(rm, n), app'2(app'2(add, m), x)) -> APP'2(app'2(app'2(if_rm, app'2(app'2(eq, n), m)), n), app'2(app'2(add, m), x))
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( true ) = 3


POL( app'2(x1, x2) ) = 3x2 + 2


POL( if_rm ) = 0


POL( APP'2(x1, x2) ) = x2 + 3


POL( false ) = 3


POL( eq ) = 1


POL( 0 ) = 3


POL( s ) = max{0, -1}


POL( rm ) = 0


POL( add ) = 1



The following usable rules [14] were oriented: none



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

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

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

The TRS R consists of the following rules:

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

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

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

The TRS R consists of the following rules:

app'2(app'2(eq, 0), 0) -> true
app'2(app'2(eq, 0), app'2(s, x)) -> false
app'2(app'2(eq, app'2(s, x)), 0) -> false
app'2(app'2(eq, app'2(s, x)), app'2(s, y)) -> app'2(app'2(eq, x), y)
app'2(app'2(le, 0), y) -> true
app'2(app'2(le, app'2(s, x)), 0) -> false
app'2(app'2(le, app'2(s, x)), app'2(s, y)) -> app'2(app'2(le, x), y)
app'2(app'2(app, nil), y) -> y
app'2(app'2(app, app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(app, x), y))
app'2(min, app'2(app'2(add, n), nil)) -> n
app'2(min, app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(app'2(if_min, app'2(app'2(le, n), m)), app'2(app'2(add, n), app'2(app'2(add, m), x)))
app'2(app'2(if_min, true), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, n), x))
app'2(app'2(if_min, false), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, m), x))
app'2(app'2(rm, n), nil) -> nil
app'2(app'2(rm, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_rm, app'2(app'2(eq, n), m)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_rm, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(rm, n), x)
app'2(app'2(app'2(if_rm, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(rm, n), x))
app'2(app'2(minsort, nil), nil) -> nil
app'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> app'2(app'2(app'2(if_minsort, app'2(app'2(eq, n), app'2(min, app'2(app'2(add, n), x)))), app'2(app'2(add, n), x)), y)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(minsort, app'2(app'2(app, app'2(app'2(rm, n), x)), y)), nil))
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, n), x)), y) -> app'2(app'2(minsort, x), app'2(app'2(add, n), y))
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
            ↳ 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(eq, 0), 0) -> true
app'2(app'2(eq, 0), app'2(s, x)) -> false
app'2(app'2(eq, app'2(s, x)), 0) -> false
app'2(app'2(eq, app'2(s, x)), app'2(s, y)) -> app'2(app'2(eq, x), y)
app'2(app'2(le, 0), y) -> true
app'2(app'2(le, app'2(s, x)), 0) -> false
app'2(app'2(le, app'2(s, x)), app'2(s, y)) -> app'2(app'2(le, x), y)
app'2(app'2(app, nil), y) -> y
app'2(app'2(app, app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(app, x), y))
app'2(min, app'2(app'2(add, n), nil)) -> n
app'2(min, app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(app'2(if_min, app'2(app'2(le, n), m)), app'2(app'2(add, n), app'2(app'2(add, m), x)))
app'2(app'2(if_min, true), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, n), x))
app'2(app'2(if_min, false), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, m), x))
app'2(app'2(rm, n), nil) -> nil
app'2(app'2(rm, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_rm, app'2(app'2(eq, n), m)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_rm, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(rm, n), x)
app'2(app'2(app'2(if_rm, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(rm, n), x))
app'2(app'2(minsort, nil), nil) -> nil
app'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> app'2(app'2(app'2(if_minsort, app'2(app'2(eq, n), app'2(min, app'2(app'2(add, n), x)))), app'2(app'2(add, n), x)), y)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(minsort, app'2(app'2(app, app'2(app'2(rm, n), x)), y)), nil))
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, n), x)), y) -> app'2(app'2(minsort, x), app'2(app'2(add, n), y))
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 Order [17,21] with Interpretation:

POL( app'2(x1, x2) ) = max{0, 2x1 + x2 - 3}


POL( map ) = 0


POL( eq ) = 1


POL( app ) = 2


POL( min ) = max{0, -3}


POL( 0 ) = 1


POL( rm ) = 3


POL( s ) = max{0, -3}


POL( nil ) = 1


POL( add ) = 3


POL( if_min ) = 2


POL( true ) = 2


POL( filter ) = 2


POL( if_rm ) = 3


POL( if_minsort ) = 2


POL( APP'2(x1, x2) ) = x2


POL( false ) = 0


POL( le ) = 1


POL( minsort ) = 1


POL( filter2 ) = 1



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ 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(eq, 0), 0) -> true
app'2(app'2(eq, 0), app'2(s, x)) -> false
app'2(app'2(eq, app'2(s, x)), 0) -> false
app'2(app'2(eq, app'2(s, x)), app'2(s, y)) -> app'2(app'2(eq, x), y)
app'2(app'2(le, 0), y) -> true
app'2(app'2(le, app'2(s, x)), 0) -> false
app'2(app'2(le, app'2(s, x)), app'2(s, y)) -> app'2(app'2(le, x), y)
app'2(app'2(app, nil), y) -> y
app'2(app'2(app, app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(app, x), y))
app'2(min, app'2(app'2(add, n), nil)) -> n
app'2(min, app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(app'2(if_min, app'2(app'2(le, n), m)), app'2(app'2(add, n), app'2(app'2(add, m), x)))
app'2(app'2(if_min, true), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, n), x))
app'2(app'2(if_min, false), app'2(app'2(add, n), app'2(app'2(add, m), x))) -> app'2(min, app'2(app'2(add, m), x))
app'2(app'2(rm, n), nil) -> nil
app'2(app'2(rm, n), app'2(app'2(add, m), x)) -> app'2(app'2(app'2(if_rm, app'2(app'2(eq, n), m)), n), app'2(app'2(add, m), x))
app'2(app'2(app'2(if_rm, true), n), app'2(app'2(add, m), x)) -> app'2(app'2(rm, n), x)
app'2(app'2(app'2(if_rm, false), n), app'2(app'2(add, m), x)) -> app'2(app'2(add, m), app'2(app'2(rm, n), x))
app'2(app'2(minsort, nil), nil) -> nil
app'2(app'2(minsort, app'2(app'2(add, n), x)), y) -> app'2(app'2(app'2(if_minsort, app'2(app'2(eq, n), app'2(min, app'2(app'2(add, n), x)))), app'2(app'2(add, n), x)), y)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, n), x)), y) -> app'2(app'2(add, n), app'2(app'2(minsort, app'2(app'2(app, app'2(app'2(rm, n), x)), y)), nil))
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, n), x)), y) -> app'2(app'2(minsort, x), app'2(app'2(add, n), y))
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.