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:

app2(app2(minus, 0), y) -> 0
app2(app2(minus, app2(s, x)), 0) -> app2(s, x)
app2(app2(minus, app2(s, x)), app2(s, y)) -> app2(app2(minus, x), y)
app2(app2(le, 0), y) -> true
app2(app2(le, app2(s, x)), 0) -> false
app2(app2(le, app2(s, x)), app2(s, y)) -> app2(app2(le, x), y)
app2(app2(app2(if, true), x), y) -> x
app2(app2(app2(if, false), x), y) -> y
app2(perfectp, 0) -> false
app2(perfectp, app2(s, x)) -> app2(app2(app2(app2(f, x), app2(s, 0)), app2(s, x)), app2(s, x))
app2(app2(app2(app2(f, 0), y), 0), u) -> true
app2(app2(app2(app2(f, 0), y), app2(s, z)), u) -> false
app2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> app2(app2(app2(app2(f, x), u), app2(app2(minus, z), app2(s, x))), u)
app2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> app2(app2(app2(if, app2(app2(le, x), y)), app2(app2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z), u)), app2(app2(app2(app2(f, x), u), z), u))

Q is empty.


↳ QTRS
  ↳ Non-Overlap Check

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

app2(app2(minus, 0), y) -> 0
app2(app2(minus, app2(s, x)), 0) -> app2(s, x)
app2(app2(minus, app2(s, x)), app2(s, y)) -> app2(app2(minus, x), y)
app2(app2(le, 0), y) -> true
app2(app2(le, app2(s, x)), 0) -> false
app2(app2(le, app2(s, x)), app2(s, y)) -> app2(app2(le, x), y)
app2(app2(app2(if, true), x), y) -> x
app2(app2(app2(if, false), x), y) -> y
app2(perfectp, 0) -> false
app2(perfectp, app2(s, x)) -> app2(app2(app2(app2(f, x), app2(s, 0)), app2(s, x)), app2(s, x))
app2(app2(app2(app2(f, 0), y), 0), u) -> true
app2(app2(app2(app2(f, 0), y), app2(s, z)), u) -> false
app2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> app2(app2(app2(app2(f, x), u), app2(app2(minus, z), app2(s, x))), u)
app2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> app2(app2(app2(if, app2(app2(le, x), y)), app2(app2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z), u)), app2(app2(app2(app2(f, x), u), z), u))

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:

app2(app2(minus, 0), y) -> 0
app2(app2(minus, app2(s, x)), 0) -> app2(s, x)
app2(app2(minus, app2(s, x)), app2(s, y)) -> app2(app2(minus, x), y)
app2(app2(le, 0), y) -> true
app2(app2(le, app2(s, x)), 0) -> false
app2(app2(le, app2(s, x)), app2(s, y)) -> app2(app2(le, x), y)
app2(app2(app2(if, true), x), y) -> x
app2(app2(app2(if, false), x), y) -> y
app2(perfectp, 0) -> false
app2(perfectp, app2(s, x)) -> app2(app2(app2(app2(f, x), app2(s, 0)), app2(s, x)), app2(s, x))
app2(app2(app2(app2(f, 0), y), 0), u) -> true
app2(app2(app2(app2(f, 0), y), app2(s, z)), u) -> false
app2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> app2(app2(app2(app2(f, x), u), app2(app2(minus, z), app2(s, x))), u)
app2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> app2(app2(app2(if, app2(app2(le, x), y)), app2(app2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z), u)), app2(app2(app2(app2(f, x), u), z), u))

The set Q consists of the following terms:

app2(app2(minus, 0), x0)
app2(app2(minus, app2(s, x0)), 0)
app2(app2(minus, app2(s, x0)), app2(s, x1))
app2(app2(le, 0), x0)
app2(app2(le, app2(s, x0)), 0)
app2(app2(le, app2(s, x0)), app2(s, x1))
app2(app2(app2(if, true), x0), x1)
app2(app2(app2(if, false), x0), x1)
app2(perfectp, 0)
app2(perfectp, app2(s, x0))
app2(app2(app2(app2(f, 0), x0), 0), x1)
app2(app2(app2(app2(f, 0), x0), app2(s, x1)), x2)
app2(app2(app2(app2(f, app2(s, x0)), 0), x1), x2)
app2(app2(app2(app2(f, app2(s, x0)), app2(s, x1)), x2), x3)


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

APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(f, app2(s, x)), app2(app2(minus, y), x))
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(minus, y), x)
APP2(app2(minus, app2(s, x)), app2(s, y)) -> APP2(minus, x)
APP2(perfectp, app2(s, x)) -> APP2(app2(app2(f, x), app2(s, 0)), app2(s, x))
APP2(perfectp, app2(s, x)) -> APP2(f, x)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(f, x), u)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(le, x)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z), u)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(if, app2(app2(le, x), y))
APP2(app2(le, app2(s, x)), app2(s, y)) -> APP2(le, x)
APP2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> APP2(minus, z)
APP2(perfectp, app2(s, x)) -> APP2(app2(f, x), app2(s, 0))
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(minus, y)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(app2(f, x), u), z)
APP2(app2(minus, app2(s, x)), app2(s, y)) -> APP2(app2(minus, x), y)
APP2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> APP2(app2(f, x), u)
APP2(perfectp, app2(s, x)) -> APP2(app2(app2(app2(f, x), app2(s, 0)), app2(s, x)), app2(s, x))
APP2(app2(le, app2(s, x)), app2(s, y)) -> APP2(app2(le, x), y)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(app2(app2(f, x), u), z), u)
APP2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> APP2(f, x)
APP2(perfectp, app2(s, x)) -> APP2(s, 0)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(le, x), y)
APP2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> APP2(app2(app2(f, x), u), app2(app2(minus, z), app2(s, x)))
APP2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> APP2(app2(minus, z), app2(s, x))
APP2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> APP2(app2(app2(app2(f, x), u), app2(app2(minus, z), app2(s, x))), u)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(f, x)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(app2(if, app2(app2(le, x), y)), app2(app2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z), u)), app2(app2(app2(app2(f, x), u), z), u))
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(if, app2(app2(le, x), y)), app2(app2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z), u))

The TRS R consists of the following rules:

app2(app2(minus, 0), y) -> 0
app2(app2(minus, app2(s, x)), 0) -> app2(s, x)
app2(app2(minus, app2(s, x)), app2(s, y)) -> app2(app2(minus, x), y)
app2(app2(le, 0), y) -> true
app2(app2(le, app2(s, x)), 0) -> false
app2(app2(le, app2(s, x)), app2(s, y)) -> app2(app2(le, x), y)
app2(app2(app2(if, true), x), y) -> x
app2(app2(app2(if, false), x), y) -> y
app2(perfectp, 0) -> false
app2(perfectp, app2(s, x)) -> app2(app2(app2(app2(f, x), app2(s, 0)), app2(s, x)), app2(s, x))
app2(app2(app2(app2(f, 0), y), 0), u) -> true
app2(app2(app2(app2(f, 0), y), app2(s, z)), u) -> false
app2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> app2(app2(app2(app2(f, x), u), app2(app2(minus, z), app2(s, x))), u)
app2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> app2(app2(app2(if, app2(app2(le, x), y)), app2(app2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z), u)), app2(app2(app2(app2(f, x), u), z), u))

The set Q consists of the following terms:

app2(app2(minus, 0), x0)
app2(app2(minus, app2(s, x0)), 0)
app2(app2(minus, app2(s, x0)), app2(s, x1))
app2(app2(le, 0), x0)
app2(app2(le, app2(s, x0)), 0)
app2(app2(le, app2(s, x0)), app2(s, x1))
app2(app2(app2(if, true), x0), x1)
app2(app2(app2(if, false), x0), x1)
app2(perfectp, 0)
app2(perfectp, app2(s, x0))
app2(app2(app2(app2(f, 0), x0), 0), x1)
app2(app2(app2(app2(f, 0), x0), app2(s, x1)), x2)
app2(app2(app2(app2(f, app2(s, x0)), 0), x1), x2)
app2(app2(app2(app2(f, app2(s, x0)), app2(s, x1)), x2), x3)

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:

APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(f, app2(s, x)), app2(app2(minus, y), x))
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(minus, y), x)
APP2(app2(minus, app2(s, x)), app2(s, y)) -> APP2(minus, x)
APP2(perfectp, app2(s, x)) -> APP2(app2(app2(f, x), app2(s, 0)), app2(s, x))
APP2(perfectp, app2(s, x)) -> APP2(f, x)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(f, x), u)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(le, x)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z), u)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(if, app2(app2(le, x), y))
APP2(app2(le, app2(s, x)), app2(s, y)) -> APP2(le, x)
APP2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> APP2(minus, z)
APP2(perfectp, app2(s, x)) -> APP2(app2(f, x), app2(s, 0))
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(minus, y)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(app2(f, x), u), z)
APP2(app2(minus, app2(s, x)), app2(s, y)) -> APP2(app2(minus, x), y)
APP2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> APP2(app2(f, x), u)
APP2(perfectp, app2(s, x)) -> APP2(app2(app2(app2(f, x), app2(s, 0)), app2(s, x)), app2(s, x))
APP2(app2(le, app2(s, x)), app2(s, y)) -> APP2(app2(le, x), y)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(app2(app2(f, x), u), z), u)
APP2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> APP2(f, x)
APP2(perfectp, app2(s, x)) -> APP2(s, 0)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(le, x), y)
APP2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> APP2(app2(app2(f, x), u), app2(app2(minus, z), app2(s, x)))
APP2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> APP2(app2(minus, z), app2(s, x))
APP2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> APP2(app2(app2(app2(f, x), u), app2(app2(minus, z), app2(s, x))), u)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(f, x)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(app2(if, app2(app2(le, x), y)), app2(app2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z), u)), app2(app2(app2(app2(f, x), u), z), u))
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(if, app2(app2(le, x), y)), app2(app2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z), u))

The TRS R consists of the following rules:

app2(app2(minus, 0), y) -> 0
app2(app2(minus, app2(s, x)), 0) -> app2(s, x)
app2(app2(minus, app2(s, x)), app2(s, y)) -> app2(app2(minus, x), y)
app2(app2(le, 0), y) -> true
app2(app2(le, app2(s, x)), 0) -> false
app2(app2(le, app2(s, x)), app2(s, y)) -> app2(app2(le, x), y)
app2(app2(app2(if, true), x), y) -> x
app2(app2(app2(if, false), x), y) -> y
app2(perfectp, 0) -> false
app2(perfectp, app2(s, x)) -> app2(app2(app2(app2(f, x), app2(s, 0)), app2(s, x)), app2(s, x))
app2(app2(app2(app2(f, 0), y), 0), u) -> true
app2(app2(app2(app2(f, 0), y), app2(s, z)), u) -> false
app2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> app2(app2(app2(app2(f, x), u), app2(app2(minus, z), app2(s, x))), u)
app2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> app2(app2(app2(if, app2(app2(le, x), y)), app2(app2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z), u)), app2(app2(app2(app2(f, x), u), z), u))

The set Q consists of the following terms:

app2(app2(minus, 0), x0)
app2(app2(minus, app2(s, x0)), 0)
app2(app2(minus, app2(s, x0)), app2(s, x1))
app2(app2(le, 0), x0)
app2(app2(le, app2(s, x0)), 0)
app2(app2(le, app2(s, x0)), app2(s, x1))
app2(app2(app2(if, true), x0), x1)
app2(app2(app2(if, false), x0), x1)
app2(perfectp, 0)
app2(perfectp, app2(s, x0))
app2(app2(app2(app2(f, 0), x0), 0), x1)
app2(app2(app2(app2(f, 0), x0), app2(s, x1)), x2)
app2(app2(app2(app2(f, app2(s, x0)), 0), x1), x2)
app2(app2(app2(app2(f, app2(s, x0)), app2(s, x1)), x2), x3)

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph contains 3 SCCs with 24 less nodes.

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

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

APP2(app2(le, app2(s, x)), app2(s, y)) -> APP2(app2(le, x), y)

The TRS R consists of the following rules:

app2(app2(minus, 0), y) -> 0
app2(app2(minus, app2(s, x)), 0) -> app2(s, x)
app2(app2(minus, app2(s, x)), app2(s, y)) -> app2(app2(minus, x), y)
app2(app2(le, 0), y) -> true
app2(app2(le, app2(s, x)), 0) -> false
app2(app2(le, app2(s, x)), app2(s, y)) -> app2(app2(le, x), y)
app2(app2(app2(if, true), x), y) -> x
app2(app2(app2(if, false), x), y) -> y
app2(perfectp, 0) -> false
app2(perfectp, app2(s, x)) -> app2(app2(app2(app2(f, x), app2(s, 0)), app2(s, x)), app2(s, x))
app2(app2(app2(app2(f, 0), y), 0), u) -> true
app2(app2(app2(app2(f, 0), y), app2(s, z)), u) -> false
app2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> app2(app2(app2(app2(f, x), u), app2(app2(minus, z), app2(s, x))), u)
app2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> app2(app2(app2(if, app2(app2(le, x), y)), app2(app2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z), u)), app2(app2(app2(app2(f, x), u), z), u))

The set Q consists of the following terms:

app2(app2(minus, 0), x0)
app2(app2(minus, app2(s, x0)), 0)
app2(app2(minus, app2(s, x0)), app2(s, x1))
app2(app2(le, 0), x0)
app2(app2(le, app2(s, x0)), 0)
app2(app2(le, app2(s, x0)), app2(s, x1))
app2(app2(app2(if, true), x0), x1)
app2(app2(app2(if, false), x0), x1)
app2(perfectp, 0)
app2(perfectp, app2(s, x0))
app2(app2(app2(app2(f, 0), x0), 0), x1)
app2(app2(app2(app2(f, 0), x0), app2(s, x1)), x2)
app2(app2(app2(app2(f, app2(s, x0)), 0), x1), x2)
app2(app2(app2(app2(f, app2(s, x0)), app2(s, x1)), x2), x3)

We have to consider all minimal (P,Q,R)-chains.
By using an argument filtering and a montonic ordering, at least one Dependency Pair of this SCC can be strictly oriented.

APP2(app2(le, app2(s, x)), app2(s, y)) -> APP2(app2(le, x), y)
Used argument filtering: APP2(x1, x2)  =  x2
app2(x1, x2)  =  app1(x2)
s  =  s
Used ordering: Quasi Precedence: trivial 


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

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

app2(app2(minus, 0), y) -> 0
app2(app2(minus, app2(s, x)), 0) -> app2(s, x)
app2(app2(minus, app2(s, x)), app2(s, y)) -> app2(app2(minus, x), y)
app2(app2(le, 0), y) -> true
app2(app2(le, app2(s, x)), 0) -> false
app2(app2(le, app2(s, x)), app2(s, y)) -> app2(app2(le, x), y)
app2(app2(app2(if, true), x), y) -> x
app2(app2(app2(if, false), x), y) -> y
app2(perfectp, 0) -> false
app2(perfectp, app2(s, x)) -> app2(app2(app2(app2(f, x), app2(s, 0)), app2(s, x)), app2(s, x))
app2(app2(app2(app2(f, 0), y), 0), u) -> true
app2(app2(app2(app2(f, 0), y), app2(s, z)), u) -> false
app2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> app2(app2(app2(app2(f, x), u), app2(app2(minus, z), app2(s, x))), u)
app2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> app2(app2(app2(if, app2(app2(le, x), y)), app2(app2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z), u)), app2(app2(app2(app2(f, x), u), z), u))

The set Q consists of the following terms:

app2(app2(minus, 0), x0)
app2(app2(minus, app2(s, x0)), 0)
app2(app2(minus, app2(s, x0)), app2(s, x1))
app2(app2(le, 0), x0)
app2(app2(le, app2(s, x0)), 0)
app2(app2(le, app2(s, x0)), app2(s, x1))
app2(app2(app2(if, true), x0), x1)
app2(app2(app2(if, false), x0), x1)
app2(perfectp, 0)
app2(perfectp, app2(s, x0))
app2(app2(app2(app2(f, 0), x0), 0), x1)
app2(app2(app2(app2(f, 0), x0), app2(s, x1)), x2)
app2(app2(app2(app2(f, app2(s, x0)), 0), x1), x2)
app2(app2(app2(app2(f, app2(s, x0)), app2(s, x1)), x2), x3)

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

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

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

APP2(app2(minus, app2(s, x)), app2(s, y)) -> APP2(app2(minus, x), y)

The TRS R consists of the following rules:

app2(app2(minus, 0), y) -> 0
app2(app2(minus, app2(s, x)), 0) -> app2(s, x)
app2(app2(minus, app2(s, x)), app2(s, y)) -> app2(app2(minus, x), y)
app2(app2(le, 0), y) -> true
app2(app2(le, app2(s, x)), 0) -> false
app2(app2(le, app2(s, x)), app2(s, y)) -> app2(app2(le, x), y)
app2(app2(app2(if, true), x), y) -> x
app2(app2(app2(if, false), x), y) -> y
app2(perfectp, 0) -> false
app2(perfectp, app2(s, x)) -> app2(app2(app2(app2(f, x), app2(s, 0)), app2(s, x)), app2(s, x))
app2(app2(app2(app2(f, 0), y), 0), u) -> true
app2(app2(app2(app2(f, 0), y), app2(s, z)), u) -> false
app2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> app2(app2(app2(app2(f, x), u), app2(app2(minus, z), app2(s, x))), u)
app2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> app2(app2(app2(if, app2(app2(le, x), y)), app2(app2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z), u)), app2(app2(app2(app2(f, x), u), z), u))

The set Q consists of the following terms:

app2(app2(minus, 0), x0)
app2(app2(minus, app2(s, x0)), 0)
app2(app2(minus, app2(s, x0)), app2(s, x1))
app2(app2(le, 0), x0)
app2(app2(le, app2(s, x0)), 0)
app2(app2(le, app2(s, x0)), app2(s, x1))
app2(app2(app2(if, true), x0), x1)
app2(app2(app2(if, false), x0), x1)
app2(perfectp, 0)
app2(perfectp, app2(s, x0))
app2(app2(app2(app2(f, 0), x0), 0), x1)
app2(app2(app2(app2(f, 0), x0), app2(s, x1)), x2)
app2(app2(app2(app2(f, app2(s, x0)), 0), x1), x2)
app2(app2(app2(app2(f, app2(s, x0)), app2(s, x1)), x2), x3)

We have to consider all minimal (P,Q,R)-chains.
By using an argument filtering and a montonic ordering, at least one Dependency Pair of this SCC can be strictly oriented.

APP2(app2(minus, app2(s, x)), app2(s, y)) -> APP2(app2(minus, x), y)
Used argument filtering: APP2(x1, x2)  =  x2
app2(x1, x2)  =  app1(x2)
s  =  s
Used ordering: Quasi Precedence: trivial 


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

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

app2(app2(minus, 0), y) -> 0
app2(app2(minus, app2(s, x)), 0) -> app2(s, x)
app2(app2(minus, app2(s, x)), app2(s, y)) -> app2(app2(minus, x), y)
app2(app2(le, 0), y) -> true
app2(app2(le, app2(s, x)), 0) -> false
app2(app2(le, app2(s, x)), app2(s, y)) -> app2(app2(le, x), y)
app2(app2(app2(if, true), x), y) -> x
app2(app2(app2(if, false), x), y) -> y
app2(perfectp, 0) -> false
app2(perfectp, app2(s, x)) -> app2(app2(app2(app2(f, x), app2(s, 0)), app2(s, x)), app2(s, x))
app2(app2(app2(app2(f, 0), y), 0), u) -> true
app2(app2(app2(app2(f, 0), y), app2(s, z)), u) -> false
app2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> app2(app2(app2(app2(f, x), u), app2(app2(minus, z), app2(s, x))), u)
app2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> app2(app2(app2(if, app2(app2(le, x), y)), app2(app2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z), u)), app2(app2(app2(app2(f, x), u), z), u))

The set Q consists of the following terms:

app2(app2(minus, 0), x0)
app2(app2(minus, app2(s, x0)), 0)
app2(app2(minus, app2(s, x0)), app2(s, x1))
app2(app2(le, 0), x0)
app2(app2(le, app2(s, x0)), 0)
app2(app2(le, app2(s, x0)), app2(s, x1))
app2(app2(app2(if, true), x0), x1)
app2(app2(app2(if, false), x0), x1)
app2(perfectp, 0)
app2(perfectp, app2(s, x0))
app2(app2(app2(app2(f, 0), x0), 0), x1)
app2(app2(app2(app2(f, 0), x0), app2(s, x1)), x2)
app2(app2(app2(app2(f, app2(s, x0)), 0), x1), x2)
app2(app2(app2(app2(f, app2(s, x0)), app2(s, x1)), x2), x3)

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

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

APP2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> APP2(app2(app2(app2(f, x), u), app2(app2(minus, z), app2(s, x))), u)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z), u)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(app2(app2(f, x), u), z), u)

The TRS R consists of the following rules:

app2(app2(minus, 0), y) -> 0
app2(app2(minus, app2(s, x)), 0) -> app2(s, x)
app2(app2(minus, app2(s, x)), app2(s, y)) -> app2(app2(minus, x), y)
app2(app2(le, 0), y) -> true
app2(app2(le, app2(s, x)), 0) -> false
app2(app2(le, app2(s, x)), app2(s, y)) -> app2(app2(le, x), y)
app2(app2(app2(if, true), x), y) -> x
app2(app2(app2(if, false), x), y) -> y
app2(perfectp, 0) -> false
app2(perfectp, app2(s, x)) -> app2(app2(app2(app2(f, x), app2(s, 0)), app2(s, x)), app2(s, x))
app2(app2(app2(app2(f, 0), y), 0), u) -> true
app2(app2(app2(app2(f, 0), y), app2(s, z)), u) -> false
app2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> app2(app2(app2(app2(f, x), u), app2(app2(minus, z), app2(s, x))), u)
app2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> app2(app2(app2(if, app2(app2(le, x), y)), app2(app2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z), u)), app2(app2(app2(app2(f, x), u), z), u))

The set Q consists of the following terms:

app2(app2(minus, 0), x0)
app2(app2(minus, app2(s, x0)), 0)
app2(app2(minus, app2(s, x0)), app2(s, x1))
app2(app2(le, 0), x0)
app2(app2(le, app2(s, x0)), 0)
app2(app2(le, app2(s, x0)), app2(s, x1))
app2(app2(app2(if, true), x0), x1)
app2(app2(app2(if, false), x0), x1)
app2(perfectp, 0)
app2(perfectp, app2(s, x0))
app2(app2(app2(app2(f, 0), x0), 0), x1)
app2(app2(app2(app2(f, 0), x0), app2(s, x1)), x2)
app2(app2(app2(app2(f, app2(s, x0)), 0), x1), x2)
app2(app2(app2(app2(f, app2(s, x0)), app2(s, x1)), x2), x3)

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