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))

Q is empty.


↳ QTRS
  ↳ Non-Overlap Check

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))

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(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))

The set Q consists of the following terms:

app'2(app'2(eq, 0), 0)
app'2(app'2(eq, 0), app'2(s, x0))
app'2(app'2(eq, app'2(s, x0)), 0)
app'2(app'2(eq, 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(min, app'2(app'2(add, x0), nil))
app'2(min, app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, true), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, false), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(rm, x0), nil)
app'2(app'2(rm, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, false), x0), app'2(app'2(add, x1), x2))
app'2(app'2(minsort, nil), nil)
app'2(app'2(minsort, app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, x0), x1)), x2)


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

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))

The set Q consists of the following terms:

app'2(app'2(eq, 0), 0)
app'2(app'2(eq, 0), app'2(s, x0))
app'2(app'2(eq, app'2(s, x0)), 0)
app'2(app'2(eq, 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(min, app'2(app'2(add, x0), nil))
app'2(min, app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, true), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, false), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(rm, x0), nil)
app'2(app'2(rm, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, false), x0), app'2(app'2(add, x1), x2))
app'2(app'2(minsort, nil), nil)
app'2(app'2(minsort, app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, x0), x1)), x2)

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

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))

The set Q consists of the following terms:

app'2(app'2(eq, 0), 0)
app'2(app'2(eq, 0), app'2(s, x0))
app'2(app'2(eq, app'2(s, x0)), 0)
app'2(app'2(eq, 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(min, app'2(app'2(add, x0), nil))
app'2(min, app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, true), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, false), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(rm, x0), nil)
app'2(app'2(rm, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, false), x0), app'2(app'2(add, x1), x2))
app'2(app'2(minsort, nil), nil)
app'2(app'2(minsort, app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, x0), x1)), x2)

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 6 SCCs with 28 less nodes.

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

The set Q consists of the following terms:

app'2(app'2(eq, 0), 0)
app'2(app'2(eq, 0), app'2(s, x0))
app'2(app'2(eq, app'2(s, x0)), 0)
app'2(app'2(eq, 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(min, app'2(app'2(add, x0), nil))
app'2(min, app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, true), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, false), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(rm, x0), nil)
app'2(app'2(rm, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, false), x0), app'2(app'2(add, x1), x2))
app'2(app'2(minsort, nil), nil)
app'2(app'2(minsort, app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, x0), x1)), x2)

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


The following pairs can be strictly oriented and are deleted.


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

Lexicographic Path Order [19].
Precedence:
trivial

The following usable rules [14] were oriented: none



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

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))

The set Q consists of the following terms:

app'2(app'2(eq, 0), 0)
app'2(app'2(eq, 0), app'2(s, x0))
app'2(app'2(eq, app'2(s, x0)), 0)
app'2(app'2(eq, 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(min, app'2(app'2(add, x0), nil))
app'2(min, app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, true), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, false), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(rm, x0), nil)
app'2(app'2(rm, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, false), x0), app'2(app'2(add, x1), x2))
app'2(app'2(minsort, nil), nil)
app'2(app'2(minsort, app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, x0), x1)), x2)

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

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

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))

The set Q consists of the following terms:

app'2(app'2(eq, 0), 0)
app'2(app'2(eq, 0), app'2(s, x0))
app'2(app'2(eq, app'2(s, x0)), 0)
app'2(app'2(eq, 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(min, app'2(app'2(add, x0), nil))
app'2(min, app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, true), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, false), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(rm, x0), nil)
app'2(app'2(rm, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, false), x0), app'2(app'2(add, x1), x2))
app'2(app'2(minsort, nil), nil)
app'2(app'2(minsort, app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, x0), x1)), x2)

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


The following pairs can be strictly oriented and are deleted.


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

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

The following usable rules [14] were oriented: none



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

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))

The set Q consists of the following terms:

app'2(app'2(eq, 0), 0)
app'2(app'2(eq, 0), app'2(s, x0))
app'2(app'2(eq, app'2(s, x0)), 0)
app'2(app'2(eq, 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(min, app'2(app'2(add, x0), nil))
app'2(min, app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, true), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, false), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(rm, x0), nil)
app'2(app'2(rm, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, false), x0), app'2(app'2(add, x1), x2))
app'2(app'2(minsort, nil), nil)
app'2(app'2(minsort, app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, x0), x1)), x2)

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

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

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))

The set Q consists of the following terms:

app'2(app'2(eq, 0), 0)
app'2(app'2(eq, 0), app'2(s, x0))
app'2(app'2(eq, app'2(s, x0)), 0)
app'2(app'2(eq, 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(min, app'2(app'2(add, x0), nil))
app'2(min, app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, true), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, false), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(rm, x0), nil)
app'2(app'2(rm, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, false), x0), app'2(app'2(add, x1), x2))
app'2(app'2(minsort, nil), nil)
app'2(app'2(minsort, app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, x0), x1)), x2)

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


The following pairs can be strictly oriented and are deleted.


APP'2(app'2(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 by weakly be oriented.

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: Combined order from the following AFS and order.
APP'2(x1, x2)  =  x2
app'2(x1, x2)  =  app'2(x1, x2)
if_min  =  if_min
true  =  true
add  =  add
min  =  min
false  =  false
le  =  le
0  =  0
s  =  s

Lexicographic Path Order [19].
Precedence:
ifmin > app'2 > false > true
ifmin > app'2 > le > true
ifmin > add > true
min > app'2 > false > true
min > app'2 > le > true
min > add > true
0 > true
s > true

The following usable rules [14] were oriented:

app'2(app'2(le, 0), y) -> true
app'2(app'2(le, app'2(s, x)), 0) -> false
app'2(app'2(le, app'2(s, x)), app'2(s, y)) -> app'2(app'2(le, x), y)



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

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))

The set Q consists of the following terms:

app'2(app'2(eq, 0), 0)
app'2(app'2(eq, 0), app'2(s, x0))
app'2(app'2(eq, app'2(s, x0)), 0)
app'2(app'2(eq, 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(min, app'2(app'2(add, x0), nil))
app'2(min, app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, true), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, false), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(rm, x0), nil)
app'2(app'2(rm, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, false), x0), app'2(app'2(add, x1), x2))
app'2(app'2(minsort, nil), nil)
app'2(app'2(minsort, app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, x0), x1)), x2)

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

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

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))

The set Q consists of the following terms:

app'2(app'2(eq, 0), 0)
app'2(app'2(eq, 0), app'2(s, x0))
app'2(app'2(eq, app'2(s, x0)), 0)
app'2(app'2(eq, 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(min, app'2(app'2(add, x0), nil))
app'2(min, app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, true), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, false), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(rm, x0), nil)
app'2(app'2(rm, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, false), x0), app'2(app'2(add, x1), x2))
app'2(app'2(minsort, nil), nil)
app'2(app'2(minsort, app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, x0), x1)), x2)

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


The following pairs can be strictly oriented and are deleted.


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

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

The following usable rules [14] were oriented: none



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

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

app'2(app'2(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))

The set Q consists of the following terms:

app'2(app'2(eq, 0), 0)
app'2(app'2(eq, 0), app'2(s, x0))
app'2(app'2(eq, app'2(s, x0)), 0)
app'2(app'2(eq, 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(min, app'2(app'2(add, x0), nil))
app'2(min, app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, true), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, false), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(rm, x0), nil)
app'2(app'2(rm, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, false), x0), app'2(app'2(add, x1), x2))
app'2(app'2(minsort, nil), nil)
app'2(app'2(minsort, app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, x0), x1)), x2)

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
                ↳ QDPOrderProof
              ↳ 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))

The set Q consists of the following terms:

app'2(app'2(eq, 0), 0)
app'2(app'2(eq, 0), app'2(s, x0))
app'2(app'2(eq, app'2(s, x0)), 0)
app'2(app'2(eq, 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(min, app'2(app'2(add, x0), nil))
app'2(min, app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, true), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, false), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(rm, x0), nil)
app'2(app'2(rm, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, false), x0), app'2(app'2(add, x1), x2))
app'2(app'2(minsort, nil), nil)
app'2(app'2(minsort, app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, x0), x1)), x2)

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


The following pairs can be strictly oriented and are deleted.


APP'2(app'2(app'2(if_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 by weakly be oriented.

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: Combined order from the following AFS and order.
APP'2(x1, x2)  =  x2
app'2(x1, x2)  =  app'1(x2)
if_rm  =  if_rm
true  =  true
add  =  add
rm  =  rm
eq  =  eq
false  =  false
0  =  0
s  =  s

Lexicographic Path Order [19].
Precedence:
rm > app'1 > true > ifrm
rm > app'1 > false > ifrm
rm > add > ifrm
eq > true > ifrm
eq > false > ifrm
0 > true > ifrm
0 > false > ifrm
s > false > ifrm

The following usable rules [14] were oriented:

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)



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

The set Q consists of the following terms:

app'2(app'2(eq, 0), 0)
app'2(app'2(eq, 0), app'2(s, x0))
app'2(app'2(eq, app'2(s, x0)), 0)
app'2(app'2(eq, 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(min, app'2(app'2(add, x0), nil))
app'2(min, app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, true), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, false), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(rm, x0), nil)
app'2(app'2(rm, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, false), x0), app'2(app'2(add, x1), x2))
app'2(app'2(minsort, nil), nil)
app'2(app'2(minsort, app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, x0), x1)), x2)

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

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

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))

The set Q consists of the following terms:

app'2(app'2(eq, 0), 0)
app'2(app'2(eq, 0), app'2(s, x0))
app'2(app'2(eq, app'2(s, x0)), 0)
app'2(app'2(eq, 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(min, app'2(app'2(add, x0), nil))
app'2(min, app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, true), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(if_min, false), app'2(app'2(add, x0), app'2(app'2(add, x1), x2)))
app'2(app'2(rm, x0), nil)
app'2(app'2(rm, x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, true), x0), app'2(app'2(add, x1), x2))
app'2(app'2(app'2(if_rm, false), x0), app'2(app'2(add, x1), x2))
app'2(app'2(minsort, nil), nil)
app'2(app'2(minsort, app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, true), app'2(app'2(add, x0), x1)), x2)
app'2(app'2(app'2(if_minsort, false), app'2(app'2(add, x0), x1)), x2)

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