Termination w.r.t. Q of the following Term Rewriting System could be proven:

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

app2(app2(plus, 0), y) -> y
app2(app2(plus, app2(s, x)), y) -> app2(s, app2(app2(plus, x), y))
app2(app2(times, 0), y) -> 0
app2(app2(times, app2(s, x)), y) -> app2(app2(plus, app2(app2(times, x), y)), y)
app2(app2(app2(comp, f), g), x) -> app2(f, app2(g, x))
app2(twice, f) -> app2(app2(comp, f), f)

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

app2(app2(plus, 0), y) -> y
app2(app2(plus, app2(s, x)), y) -> app2(s, app2(app2(plus, x), y))
app2(app2(times, 0), y) -> 0
app2(app2(times, app2(s, x)), y) -> app2(app2(plus, app2(app2(times, x), y)), y)
app2(app2(app2(comp, f), g), x) -> app2(f, app2(g, x))
app2(twice, f) -> app2(app2(comp, f), f)

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(times, app2(s, x)), y) -> APP2(times, x)
APP2(app2(plus, app2(s, x)), y) -> APP2(s, app2(app2(plus, x), y))
APP2(twice, f) -> APP2(comp, f)
APP2(twice, f) -> APP2(app2(comp, f), f)
APP2(app2(plus, app2(s, x)), y) -> APP2(app2(plus, x), y)
APP2(app2(app2(comp, f), g), x) -> APP2(f, app2(g, x))
APP2(app2(plus, app2(s, x)), y) -> APP2(plus, x)
APP2(app2(times, app2(s, x)), y) -> APP2(app2(times, x), y)
APP2(app2(times, app2(s, x)), y) -> APP2(app2(plus, app2(app2(times, x), y)), y)
APP2(app2(times, app2(s, x)), y) -> APP2(plus, app2(app2(times, x), y))
APP2(app2(app2(comp, f), g), x) -> APP2(g, x)

The TRS R consists of the following rules:

app2(app2(plus, 0), y) -> y
app2(app2(plus, app2(s, x)), y) -> app2(s, app2(app2(plus, x), y))
app2(app2(times, 0), y) -> 0
app2(app2(times, app2(s, x)), y) -> app2(app2(plus, app2(app2(times, x), y)), y)
app2(app2(app2(comp, f), g), x) -> app2(f, app2(g, x))
app2(twice, f) -> app2(app2(comp, f), f)

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(times, app2(s, x)), y) -> APP2(times, x)
APP2(app2(plus, app2(s, x)), y) -> APP2(s, app2(app2(plus, x), y))
APP2(twice, f) -> APP2(comp, f)
APP2(twice, f) -> APP2(app2(comp, f), f)
APP2(app2(plus, app2(s, x)), y) -> APP2(app2(plus, x), y)
APP2(app2(app2(comp, f), g), x) -> APP2(f, app2(g, x))
APP2(app2(plus, app2(s, x)), y) -> APP2(plus, x)
APP2(app2(times, app2(s, x)), y) -> APP2(app2(times, x), y)
APP2(app2(times, app2(s, x)), y) -> APP2(app2(plus, app2(app2(times, x), y)), y)
APP2(app2(times, app2(s, x)), y) -> APP2(plus, app2(app2(times, x), y))
APP2(app2(app2(comp, f), g), x) -> APP2(g, x)

The TRS R consists of the following rules:

app2(app2(plus, 0), y) -> y
app2(app2(plus, app2(s, x)), y) -> app2(s, app2(app2(plus, x), y))
app2(app2(times, 0), y) -> 0
app2(app2(times, app2(s, x)), y) -> app2(app2(plus, app2(app2(times, x), y)), y)
app2(app2(app2(comp, f), g), x) -> app2(f, app2(g, x))
app2(twice, f) -> app2(app2(comp, f), f)

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 7 less nodes.

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

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

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

The TRS R consists of the following rules:

app2(app2(plus, 0), y) -> y
app2(app2(plus, app2(s, x)), y) -> app2(s, app2(app2(plus, x), y))
app2(app2(times, 0), y) -> 0
app2(app2(times, app2(s, x)), y) -> app2(app2(plus, app2(app2(times, x), y)), y)
app2(app2(app2(comp, f), g), x) -> app2(f, app2(g, x))
app2(twice, f) -> app2(app2(comp, f), f)

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

POL(APP2(x1, x2)) = 1 + x1   
POL(app2(x1, x2)) = 1 + x2   
POL(plus) = 0   
POL(s) = 0   

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(plus, 0), y) -> y
app2(app2(plus, app2(s, x)), y) -> app2(s, app2(app2(plus, x), y))
app2(app2(times, 0), y) -> 0
app2(app2(times, app2(s, x)), y) -> app2(app2(plus, app2(app2(times, x), y)), y)
app2(app2(app2(comp, f), g), x) -> app2(f, app2(g, x))
app2(twice, f) -> app2(app2(comp, f), f)

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

The TRS R consists of the following rules:

app2(app2(plus, 0), y) -> y
app2(app2(plus, app2(s, x)), y) -> app2(s, app2(app2(plus, x), y))
app2(app2(times, 0), y) -> 0
app2(app2(times, app2(s, x)), y) -> app2(app2(plus, app2(app2(times, x), y)), y)
app2(app2(app2(comp, f), g), x) -> app2(f, app2(g, x))
app2(twice, f) -> app2(app2(comp, f), f)

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

POL(APP2(x1, x2)) = 1 + x1   
POL(app2(x1, x2)) = 1 + x2   
POL(s) = 0   
POL(times) = 0   

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(plus, 0), y) -> y
app2(app2(plus, app2(s, x)), y) -> app2(s, app2(app2(plus, x), y))
app2(app2(times, 0), y) -> 0
app2(app2(times, app2(s, x)), y) -> app2(app2(plus, app2(app2(times, x), y)), y)
app2(app2(app2(comp, f), g), x) -> app2(f, app2(g, x))
app2(twice, f) -> app2(app2(comp, f), f)

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

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

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

APP2(app2(app2(comp, f), g), x) -> APP2(f, app2(g, x))
APP2(app2(app2(comp, f), g), x) -> APP2(g, x)

The TRS R consists of the following rules:

app2(app2(plus, 0), y) -> y
app2(app2(plus, app2(s, x)), y) -> app2(s, app2(app2(plus, x), y))
app2(app2(times, 0), y) -> 0
app2(app2(times, app2(s, x)), y) -> app2(app2(plus, app2(app2(times, x), y)), y)
app2(app2(app2(comp, f), g), x) -> app2(f, app2(g, x))
app2(twice, f) -> app2(app2(comp, f), f)

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(app2(comp, f), g), x) -> APP2(f, app2(g, x))
APP2(app2(app2(comp, f), g), x) -> APP2(g, x)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(0) = 0   
POL(APP2(x1, x2)) = 1 + x1   
POL(app2(x1, x2)) = x1 + x2   
POL(comp) = 1   
POL(plus) = 0   
POL(s) = 0   
POL(times) = 1   
POL(twice) = 1   

The following usable rules [14] were oriented: none



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

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

app2(app2(plus, 0), y) -> y
app2(app2(plus, app2(s, x)), y) -> app2(s, app2(app2(plus, x), y))
app2(app2(times, 0), y) -> 0
app2(app2(times, app2(s, x)), y) -> app2(app2(plus, app2(app2(times, x), y)), y)
app2(app2(app2(comp, f), g), x) -> app2(f, app2(g, x))
app2(twice, f) -> app2(app2(comp, f), f)

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.