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(h, z), app2(e, x)) -> app2(app2(h, app2(c, z)), app2(app2(d, z), x))
app2(app2(d, z), app2(app2(g, 0), 0)) -> app2(e, 0)
app2(app2(d, z), app2(app2(g, x), y)) -> app2(app2(g, app2(e, x)), app2(app2(d, z), y))
app2(app2(d, app2(c, z)), app2(app2(g, app2(app2(g, x), y)), 0)) -> app2(app2(g, app2(app2(d, app2(c, z)), app2(app2(g, x), y))), app2(app2(d, z), app2(app2(g, x), y)))
app2(app2(g, app2(e, x)), app2(e, y)) -> app2(e, app2(app2(g, x), y))

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

app2(app2(h, z), app2(e, x)) -> app2(app2(h, app2(c, z)), app2(app2(d, z), x))
app2(app2(d, z), app2(app2(g, 0), 0)) -> app2(e, 0)
app2(app2(d, z), app2(app2(g, x), y)) -> app2(app2(g, app2(e, x)), app2(app2(d, z), y))
app2(app2(d, app2(c, z)), app2(app2(g, app2(app2(g, x), y)), 0)) -> app2(app2(g, app2(app2(d, app2(c, z)), app2(app2(g, x), y))), app2(app2(d, z), app2(app2(g, x), y)))
app2(app2(g, app2(e, x)), app2(e, y)) -> app2(e, app2(app2(g, x), y))

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(d, z), app2(app2(g, x), y)) -> APP2(app2(d, z), y)
APP2(app2(g, app2(e, x)), app2(e, y)) -> APP2(g, x)
APP2(app2(h, z), app2(e, x)) -> APP2(c, z)
APP2(app2(d, app2(c, z)), app2(app2(g, app2(app2(g, x), y)), 0)) -> APP2(app2(d, z), app2(app2(g, x), y))
APP2(app2(g, app2(e, x)), app2(e, y)) -> APP2(e, app2(app2(g, x), y))
APP2(app2(d, z), app2(app2(g, 0), 0)) -> APP2(e, 0)
APP2(app2(d, z), app2(app2(g, x), y)) -> APP2(g, app2(e, x))
APP2(app2(d, app2(c, z)), app2(app2(g, app2(app2(g, x), y)), 0)) -> APP2(g, app2(app2(d, app2(c, z)), app2(app2(g, x), y)))
APP2(app2(d, app2(c, z)), app2(app2(g, app2(app2(g, x), y)), 0)) -> APP2(app2(d, app2(c, z)), app2(app2(g, x), y))
APP2(app2(d, app2(c, z)), app2(app2(g, app2(app2(g, x), y)), 0)) -> APP2(app2(g, app2(app2(d, app2(c, z)), app2(app2(g, x), y))), app2(app2(d, z), app2(app2(g, x), y)))
APP2(app2(h, z), app2(e, x)) -> APP2(app2(d, z), x)
APP2(app2(h, z), app2(e, x)) -> APP2(app2(h, app2(c, z)), app2(app2(d, z), x))
APP2(app2(h, z), app2(e, x)) -> APP2(h, app2(c, z))
APP2(app2(h, z), app2(e, x)) -> APP2(d, z)
APP2(app2(d, z), app2(app2(g, x), y)) -> APP2(app2(g, app2(e, x)), app2(app2(d, z), y))
APP2(app2(g, app2(e, x)), app2(e, y)) -> APP2(app2(g, x), y)
APP2(app2(d, z), app2(app2(g, x), y)) -> APP2(e, x)
APP2(app2(d, app2(c, z)), app2(app2(g, app2(app2(g, x), y)), 0)) -> APP2(d, z)

The TRS R consists of the following rules:

app2(app2(h, z), app2(e, x)) -> app2(app2(h, app2(c, z)), app2(app2(d, z), x))
app2(app2(d, z), app2(app2(g, 0), 0)) -> app2(e, 0)
app2(app2(d, z), app2(app2(g, x), y)) -> app2(app2(g, app2(e, x)), app2(app2(d, z), y))
app2(app2(d, app2(c, z)), app2(app2(g, app2(app2(g, x), y)), 0)) -> app2(app2(g, app2(app2(d, app2(c, z)), app2(app2(g, x), y))), app2(app2(d, z), app2(app2(g, x), y)))
app2(app2(g, app2(e, x)), app2(e, y)) -> app2(e, app2(app2(g, x), y))

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(d, z), app2(app2(g, x), y)) -> APP2(app2(d, z), y)
APP2(app2(g, app2(e, x)), app2(e, y)) -> APP2(g, x)
APP2(app2(h, z), app2(e, x)) -> APP2(c, z)
APP2(app2(d, app2(c, z)), app2(app2(g, app2(app2(g, x), y)), 0)) -> APP2(app2(d, z), app2(app2(g, x), y))
APP2(app2(g, app2(e, x)), app2(e, y)) -> APP2(e, app2(app2(g, x), y))
APP2(app2(d, z), app2(app2(g, 0), 0)) -> APP2(e, 0)
APP2(app2(d, z), app2(app2(g, x), y)) -> APP2(g, app2(e, x))
APP2(app2(d, app2(c, z)), app2(app2(g, app2(app2(g, x), y)), 0)) -> APP2(g, app2(app2(d, app2(c, z)), app2(app2(g, x), y)))
APP2(app2(d, app2(c, z)), app2(app2(g, app2(app2(g, x), y)), 0)) -> APP2(app2(d, app2(c, z)), app2(app2(g, x), y))
APP2(app2(d, app2(c, z)), app2(app2(g, app2(app2(g, x), y)), 0)) -> APP2(app2(g, app2(app2(d, app2(c, z)), app2(app2(g, x), y))), app2(app2(d, z), app2(app2(g, x), y)))
APP2(app2(h, z), app2(e, x)) -> APP2(app2(d, z), x)
APP2(app2(h, z), app2(e, x)) -> APP2(app2(h, app2(c, z)), app2(app2(d, z), x))
APP2(app2(h, z), app2(e, x)) -> APP2(h, app2(c, z))
APP2(app2(h, z), app2(e, x)) -> APP2(d, z)
APP2(app2(d, z), app2(app2(g, x), y)) -> APP2(app2(g, app2(e, x)), app2(app2(d, z), y))
APP2(app2(g, app2(e, x)), app2(e, y)) -> APP2(app2(g, x), y)
APP2(app2(d, z), app2(app2(g, x), y)) -> APP2(e, x)
APP2(app2(d, app2(c, z)), app2(app2(g, app2(app2(g, x), y)), 0)) -> APP2(d, z)

The TRS R consists of the following rules:

app2(app2(h, z), app2(e, x)) -> app2(app2(h, app2(c, z)), app2(app2(d, z), x))
app2(app2(d, z), app2(app2(g, 0), 0)) -> app2(e, 0)
app2(app2(d, z), app2(app2(g, x), y)) -> app2(app2(g, app2(e, x)), app2(app2(d, z), y))
app2(app2(d, app2(c, z)), app2(app2(g, app2(app2(g, x), y)), 0)) -> app2(app2(g, app2(app2(d, app2(c, z)), app2(app2(g, x), y))), app2(app2(d, z), app2(app2(g, x), y)))
app2(app2(g, app2(e, x)), app2(e, y)) -> app2(e, app2(app2(g, x), y))

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 13 less nodes.

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

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

APP2(app2(g, app2(e, x)), app2(e, y)) -> APP2(app2(g, x), y)

The TRS R consists of the following rules:

app2(app2(h, z), app2(e, x)) -> app2(app2(h, app2(c, z)), app2(app2(d, z), x))
app2(app2(d, z), app2(app2(g, 0), 0)) -> app2(e, 0)
app2(app2(d, z), app2(app2(g, x), y)) -> app2(app2(g, app2(e, x)), app2(app2(d, z), y))
app2(app2(d, app2(c, z)), app2(app2(g, app2(app2(g, x), y)), 0)) -> app2(app2(g, app2(app2(d, app2(c, z)), app2(app2(g, x), y))), app2(app2(d, z), app2(app2(g, x), y)))
app2(app2(g, app2(e, x)), app2(e, y)) -> app2(e, app2(app2(g, x), y))

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 strictly oriented and are deleted.


APP2(app2(g, app2(e, x)), app2(e, y)) -> APP2(app2(g, x), y)
The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
APP2(x1, x2)  =  APP1(x2)
app2(x1, x2)  =  app1(x2)
g  =  g
e  =  e

Lexicographic Path Order [19].
Precedence:
APP1 > app1
APP1 > g
e > app1
e > g

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(h, z), app2(e, x)) -> app2(app2(h, app2(c, z)), app2(app2(d, z), x))
app2(app2(d, z), app2(app2(g, 0), 0)) -> app2(e, 0)
app2(app2(d, z), app2(app2(g, x), y)) -> app2(app2(g, app2(e, x)), app2(app2(d, z), y))
app2(app2(d, app2(c, z)), app2(app2(g, app2(app2(g, x), y)), 0)) -> app2(app2(g, app2(app2(d, app2(c, z)), app2(app2(g, x), y))), app2(app2(d, z), app2(app2(g, x), y)))
app2(app2(g, app2(e, x)), app2(e, y)) -> app2(e, app2(app2(g, x), y))

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(d, z), app2(app2(g, x), y)) -> APP2(app2(d, z), y)
APP2(app2(d, app2(c, z)), app2(app2(g, app2(app2(g, x), y)), 0)) -> APP2(app2(d, z), app2(app2(g, x), y))
APP2(app2(d, app2(c, z)), app2(app2(g, app2(app2(g, x), y)), 0)) -> APP2(app2(d, app2(c, z)), app2(app2(g, x), y))

The TRS R consists of the following rules:

app2(app2(h, z), app2(e, x)) -> app2(app2(h, app2(c, z)), app2(app2(d, z), x))
app2(app2(d, z), app2(app2(g, 0), 0)) -> app2(e, 0)
app2(app2(d, z), app2(app2(g, x), y)) -> app2(app2(g, app2(e, x)), app2(app2(d, z), y))
app2(app2(d, app2(c, z)), app2(app2(g, app2(app2(g, x), y)), 0)) -> app2(app2(g, app2(app2(d, app2(c, z)), app2(app2(g, x), y))), app2(app2(d, z), app2(app2(g, x), y)))
app2(app2(g, app2(e, x)), app2(e, y)) -> app2(e, app2(app2(g, x), y))

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 strictly oriented and are deleted.


APP2(app2(d, app2(c, z)), app2(app2(g, app2(app2(g, x), y)), 0)) -> APP2(app2(d, z), app2(app2(g, x), y))
The remaining pairs can at least by weakly be oriented.

APP2(app2(d, z), app2(app2(g, x), y)) -> APP2(app2(d, z), y)
APP2(app2(d, app2(c, z)), app2(app2(g, app2(app2(g, x), y)), 0)) -> APP2(app2(d, app2(c, z)), app2(app2(g, x), y))
Used ordering: Combined order from the following AFS and order.
APP2(x1, x2)  =  x1
app2(x1, x2)  =  app1(x2)
d  =  d
g  =  g
c  =  c
0  =  0
e  =  e

Lexicographic Path Order [19].
Precedence:
c > d > g > app1
c > d > g > e

The following usable rules [14] were oriented:

app2(app2(g, app2(e, x)), app2(e, y)) -> app2(e, app2(app2(g, x), y))



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

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

APP2(app2(d, z), app2(app2(g, x), y)) -> APP2(app2(d, z), y)
APP2(app2(d, app2(c, z)), app2(app2(g, app2(app2(g, x), y)), 0)) -> APP2(app2(d, app2(c, z)), app2(app2(g, x), y))

The TRS R consists of the following rules:

app2(app2(h, z), app2(e, x)) -> app2(app2(h, app2(c, z)), app2(app2(d, z), x))
app2(app2(d, z), app2(app2(g, 0), 0)) -> app2(e, 0)
app2(app2(d, z), app2(app2(g, x), y)) -> app2(app2(g, app2(e, x)), app2(app2(d, z), y))
app2(app2(d, app2(c, z)), app2(app2(g, app2(app2(g, x), y)), 0)) -> app2(app2(g, app2(app2(d, app2(c, z)), app2(app2(g, x), y))), app2(app2(d, z), app2(app2(g, x), y)))
app2(app2(g, app2(e, x)), app2(e, y)) -> app2(e, app2(app2(g, x), y))

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 strictly oriented and are deleted.


APP2(app2(d, z), app2(app2(g, x), y)) -> APP2(app2(d, z), y)
APP2(app2(d, app2(c, z)), app2(app2(g, app2(app2(g, x), y)), 0)) -> APP2(app2(d, app2(c, z)), app2(app2(g, x), y))
The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
APP2(x1, x2)  =  x2
app2(x1, x2)  =  app2(x1, x2)
d  =  d
g  =  g
c  =  c
0  =  0
e  =  e

Lexicographic Path Order [19].
Precedence:
app2 > e > g
d > g
c > g
0 > g

The following usable rules [14] were oriented:

app2(app2(g, app2(e, x)), app2(e, y)) -> app2(e, app2(app2(g, x), y))



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

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

app2(app2(h, z), app2(e, x)) -> app2(app2(h, app2(c, z)), app2(app2(d, z), x))
app2(app2(d, z), app2(app2(g, 0), 0)) -> app2(e, 0)
app2(app2(d, z), app2(app2(g, x), y)) -> app2(app2(g, app2(e, x)), app2(app2(d, z), y))
app2(app2(d, app2(c, z)), app2(app2(g, app2(app2(g, x), y)), 0)) -> app2(app2(g, app2(app2(d, app2(c, z)), app2(app2(g, x), y))), app2(app2(d, z), app2(app2(g, x), y)))
app2(app2(g, app2(e, x)), app2(e, y)) -> app2(e, app2(app2(g, x), y))

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(h, z), app2(e, x)) -> APP2(app2(h, app2(c, z)), app2(app2(d, z), x))

The TRS R consists of the following rules:

app2(app2(h, z), app2(e, x)) -> app2(app2(h, app2(c, z)), app2(app2(d, z), x))
app2(app2(d, z), app2(app2(g, 0), 0)) -> app2(e, 0)
app2(app2(d, z), app2(app2(g, x), y)) -> app2(app2(g, app2(e, x)), app2(app2(d, z), y))
app2(app2(d, app2(c, z)), app2(app2(g, app2(app2(g, x), y)), 0)) -> app2(app2(g, app2(app2(d, app2(c, z)), app2(app2(g, x), y))), app2(app2(d, z), app2(app2(g, x), y)))
app2(app2(g, app2(e, x)), app2(e, y)) -> app2(e, app2(app2(g, x), y))

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