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(add, 0), y) -> y
app2(app2(add, app2(s, x)), y) -> app2(s, app2(app2(add, x), y))
app2(app2(mult, 0), y) -> 0
app2(app2(mult, app2(s, x)), y) -> app2(app2(add, app2(app2(mult, x), y)), y)
app2(app2(app2(rec, f), x), 0) -> x
app2(app2(app2(rec, f), x), app2(s, y)) -> app2(app2(f, app2(s, y)), app2(app2(app2(rec, f), x), y))
fact -> app2(app2(rec, mult), app2(s, 0))

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

app2(app2(add, 0), y) -> y
app2(app2(add, app2(s, x)), y) -> app2(s, app2(app2(add, x), y))
app2(app2(mult, 0), y) -> 0
app2(app2(mult, app2(s, x)), y) -> app2(app2(add, app2(app2(mult, x), y)), y)
app2(app2(app2(rec, f), x), 0) -> x
app2(app2(app2(rec, f), x), app2(s, y)) -> app2(app2(f, app2(s, y)), app2(app2(app2(rec, f), x), y))
fact -> app2(app2(rec, mult), app2(s, 0))

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:

APP2(app2(add, app2(s, x)), y) -> APP2(app2(add, x), y)
APP2(app2(app2(rec, f), x), app2(s, y)) -> APP2(f, app2(s, y))
APP2(app2(mult, app2(s, x)), y) -> APP2(add, app2(app2(mult, x), y))
APP2(app2(mult, app2(s, x)), y) -> APP2(app2(add, app2(app2(mult, x), y)), y)
APP2(app2(add, app2(s, x)), y) -> APP2(s, app2(app2(add, x), y))
APP2(app2(add, app2(s, x)), y) -> APP2(add, x)
FACT -> APP2(s, 0)
FACT -> APP2(rec, mult)
APP2(app2(mult, app2(s, x)), y) -> APP2(app2(mult, x), y)
APP2(app2(app2(rec, f), x), app2(s, y)) -> APP2(app2(f, app2(s, y)), app2(app2(app2(rec, f), x), y))
FACT -> APP2(app2(rec, mult), app2(s, 0))
APP2(app2(mult, app2(s, x)), y) -> APP2(mult, x)
APP2(app2(app2(rec, f), x), app2(s, y)) -> APP2(app2(app2(rec, f), x), y)

The TRS R consists of the following rules:

app2(app2(add, 0), y) -> y
app2(app2(add, app2(s, x)), y) -> app2(s, app2(app2(add, x), y))
app2(app2(mult, 0), y) -> 0
app2(app2(mult, app2(s, x)), y) -> app2(app2(add, app2(app2(mult, x), y)), y)
app2(app2(app2(rec, f), x), 0) -> x
app2(app2(app2(rec, f), x), app2(s, y)) -> app2(app2(f, app2(s, y)), app2(app2(app2(rec, f), x), y))
fact -> app2(app2(rec, mult), app2(s, 0))

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:

APP2(app2(add, app2(s, x)), y) -> APP2(app2(add, x), y)
APP2(app2(app2(rec, f), x), app2(s, y)) -> APP2(f, app2(s, y))
APP2(app2(mult, app2(s, x)), y) -> APP2(add, app2(app2(mult, x), y))
APP2(app2(mult, app2(s, x)), y) -> APP2(app2(add, app2(app2(mult, x), y)), y)
APP2(app2(add, app2(s, x)), y) -> APP2(s, app2(app2(add, x), y))
APP2(app2(add, app2(s, x)), y) -> APP2(add, x)
FACT -> APP2(s, 0)
FACT -> APP2(rec, mult)
APP2(app2(mult, app2(s, x)), y) -> APP2(app2(mult, x), y)
APP2(app2(app2(rec, f), x), app2(s, y)) -> APP2(app2(f, app2(s, y)), app2(app2(app2(rec, f), x), y))
FACT -> APP2(app2(rec, mult), app2(s, 0))
APP2(app2(mult, app2(s, x)), y) -> APP2(mult, x)
APP2(app2(app2(rec, f), x), app2(s, y)) -> APP2(app2(app2(rec, f), x), y)

The TRS R consists of the following rules:

app2(app2(add, 0), y) -> y
app2(app2(add, app2(s, x)), y) -> app2(s, app2(app2(add, x), y))
app2(app2(mult, 0), y) -> 0
app2(app2(mult, app2(s, x)), y) -> app2(app2(add, app2(app2(mult, x), y)), y)
app2(app2(app2(rec, f), x), 0) -> x
app2(app2(app2(rec, f), x), app2(s, y)) -> app2(app2(f, app2(s, y)), app2(app2(app2(rec, f), x), y))
fact -> app2(app2(rec, mult), app2(s, 0))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 3 SCCs with 8 less nodes.

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

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

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

The TRS R consists of the following rules:

app2(app2(add, 0), y) -> y
app2(app2(add, app2(s, x)), y) -> app2(s, app2(app2(add, x), y))
app2(app2(mult, 0), y) -> 0
app2(app2(mult, app2(s, x)), y) -> app2(app2(add, app2(app2(mult, x), y)), y)
app2(app2(app2(rec, f), x), 0) -> x
app2(app2(app2(rec, f), x), app2(s, y)) -> app2(app2(f, app2(s, y)), app2(app2(app2(rec, f), x), y))
fact -> app2(app2(rec, mult), app2(s, 0))

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.


APP2(app2(add, app2(s, x)), y) -> APP2(app2(add, x), y)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( s ) = 2


POL( APP2(x1, x2) ) = x1 + x2


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


POL( app2(x1, x2) ) = max{0, x1 + 2x2 - 1}



The following usable rules [14] were oriented: none



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

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

app2(app2(add, 0), y) -> y
app2(app2(add, app2(s, x)), y) -> app2(s, app2(app2(add, x), y))
app2(app2(mult, 0), y) -> 0
app2(app2(mult, app2(s, x)), y) -> app2(app2(add, app2(app2(mult, x), y)), y)
app2(app2(app2(rec, f), x), 0) -> x
app2(app2(app2(rec, f), x), app2(s, y)) -> app2(app2(f, app2(s, y)), app2(app2(app2(rec, f), x), y))
fact -> app2(app2(rec, mult), app2(s, 0))

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

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

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

The TRS R consists of the following rules:

app2(app2(add, 0), y) -> y
app2(app2(add, app2(s, x)), y) -> app2(s, app2(app2(add, x), y))
app2(app2(mult, 0), y) -> 0
app2(app2(mult, app2(s, x)), y) -> app2(app2(add, app2(app2(mult, x), y)), y)
app2(app2(app2(rec, f), x), 0) -> x
app2(app2(app2(rec, f), x), app2(s, y)) -> app2(app2(f, app2(s, y)), app2(app2(app2(rec, f), x), y))
fact -> app2(app2(rec, mult), app2(s, 0))

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.


APP2(app2(mult, app2(s, x)), y) -> APP2(app2(mult, x), y)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( s ) = 2


POL( APP2(x1, x2) ) = x1 + x2


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


POL( app2(x1, x2) ) = max{0, x1 + 2x2 - 1}



The following usable rules [14] were oriented: none



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

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

app2(app2(add, 0), y) -> y
app2(app2(add, app2(s, x)), y) -> app2(s, app2(app2(add, x), y))
app2(app2(mult, 0), y) -> 0
app2(app2(mult, app2(s, x)), y) -> app2(app2(add, app2(app2(mult, x), y)), y)
app2(app2(app2(rec, f), x), 0) -> x
app2(app2(app2(rec, f), x), app2(s, y)) -> app2(app2(f, app2(s, y)), app2(app2(app2(rec, f), x), y))
fact -> app2(app2(rec, mult), app2(s, 0))

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

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

APP2(app2(app2(rec, f), x), app2(s, y)) -> APP2(f, app2(s, y))
APP2(app2(app2(rec, f), x), app2(s, y)) -> APP2(app2(f, app2(s, y)), app2(app2(app2(rec, f), x), y))
APP2(app2(app2(rec, f), x), app2(s, y)) -> APP2(app2(app2(rec, f), x), y)

The TRS R consists of the following rules:

app2(app2(add, 0), y) -> y
app2(app2(add, app2(s, x)), y) -> app2(s, app2(app2(add, x), y))
app2(app2(mult, 0), y) -> 0
app2(app2(mult, app2(s, x)), y) -> app2(app2(add, app2(app2(mult, x), y)), y)
app2(app2(app2(rec, f), x), 0) -> x
app2(app2(app2(rec, f), x), app2(s, y)) -> app2(app2(f, app2(s, y)), app2(app2(app2(rec, f), x), y))
fact -> app2(app2(rec, mult), app2(s, 0))

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