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(filter, f), nil) -> nil
app2(app2(filter, f), app2(app2(cons, y), ys)) -> app2(app2(app2(filtersub, app2(f, y)), f), app2(app2(cons, y), ys))
app2(app2(app2(filtersub, true), f), app2(app2(cons, y), ys)) -> app2(app2(cons, y), app2(app2(filter, f), ys))
app2(app2(app2(filtersub, false), f), app2(app2(cons, y), ys)) -> app2(app2(filter, f), ys)

Q is empty.


↳ QTRS
  ↳ Non-Overlap Check

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

app2(app2(filter, f), nil) -> nil
app2(app2(filter, f), app2(app2(cons, y), ys)) -> app2(app2(app2(filtersub, app2(f, y)), f), app2(app2(cons, y), ys))
app2(app2(app2(filtersub, true), f), app2(app2(cons, y), ys)) -> app2(app2(cons, y), app2(app2(filter, f), ys))
app2(app2(app2(filtersub, false), f), app2(app2(cons, y), ys)) -> app2(app2(filter, f), ys)

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(filter, f), nil) -> nil
app2(app2(filter, f), app2(app2(cons, y), ys)) -> app2(app2(app2(filtersub, app2(f, y)), f), app2(app2(cons, y), ys))
app2(app2(app2(filtersub, true), f), app2(app2(cons, y), ys)) -> app2(app2(cons, y), app2(app2(filter, f), ys))
app2(app2(app2(filtersub, false), f), app2(app2(cons, y), ys)) -> app2(app2(filter, f), ys)

The set Q consists of the following terms:

app2(app2(filter, x0), nil)
app2(app2(filter, x0), app2(app2(cons, x1), x2))
app2(app2(app2(filtersub, true), x0), app2(app2(cons, x1), x2))
app2(app2(app2(filtersub, false), x0), app2(app2(cons, 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:

APP2(app2(app2(filtersub, false), f), app2(app2(cons, y), ys)) -> APP2(filter, f)
APP2(app2(app2(filtersub, true), f), app2(app2(cons, y), ys)) -> APP2(app2(cons, y), app2(app2(filter, f), ys))
APP2(app2(app2(filtersub, false), f), app2(app2(cons, y), ys)) -> APP2(app2(filter, f), ys)
APP2(app2(filter, f), app2(app2(cons, y), ys)) -> APP2(filtersub, app2(f, y))
APP2(app2(app2(filtersub, true), f), app2(app2(cons, y), ys)) -> APP2(filter, f)
APP2(app2(filter, f), app2(app2(cons, y), ys)) -> APP2(app2(filtersub, app2(f, y)), f)
APP2(app2(app2(filtersub, true), f), app2(app2(cons, y), ys)) -> APP2(app2(filter, f), ys)
APP2(app2(filter, f), app2(app2(cons, y), ys)) -> APP2(f, y)
APP2(app2(filter, f), app2(app2(cons, y), ys)) -> APP2(app2(app2(filtersub, app2(f, y)), f), app2(app2(cons, y), ys))

The TRS R consists of the following rules:

app2(app2(filter, f), nil) -> nil
app2(app2(filter, f), app2(app2(cons, y), ys)) -> app2(app2(app2(filtersub, app2(f, y)), f), app2(app2(cons, y), ys))
app2(app2(app2(filtersub, true), f), app2(app2(cons, y), ys)) -> app2(app2(cons, y), app2(app2(filter, f), ys))
app2(app2(app2(filtersub, false), f), app2(app2(cons, y), ys)) -> app2(app2(filter, f), ys)

The set Q consists of the following terms:

app2(app2(filter, x0), nil)
app2(app2(filter, x0), app2(app2(cons, x1), x2))
app2(app2(app2(filtersub, true), x0), app2(app2(cons, x1), x2))
app2(app2(app2(filtersub, false), x0), app2(app2(cons, 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:

APP2(app2(app2(filtersub, false), f), app2(app2(cons, y), ys)) -> APP2(filter, f)
APP2(app2(app2(filtersub, true), f), app2(app2(cons, y), ys)) -> APP2(app2(cons, y), app2(app2(filter, f), ys))
APP2(app2(app2(filtersub, false), f), app2(app2(cons, y), ys)) -> APP2(app2(filter, f), ys)
APP2(app2(filter, f), app2(app2(cons, y), ys)) -> APP2(filtersub, app2(f, y))
APP2(app2(app2(filtersub, true), f), app2(app2(cons, y), ys)) -> APP2(filter, f)
APP2(app2(filter, f), app2(app2(cons, y), ys)) -> APP2(app2(filtersub, app2(f, y)), f)
APP2(app2(app2(filtersub, true), f), app2(app2(cons, y), ys)) -> APP2(app2(filter, f), ys)
APP2(app2(filter, f), app2(app2(cons, y), ys)) -> APP2(f, y)
APP2(app2(filter, f), app2(app2(cons, y), ys)) -> APP2(app2(app2(filtersub, app2(f, y)), f), app2(app2(cons, y), ys))

The TRS R consists of the following rules:

app2(app2(filter, f), nil) -> nil
app2(app2(filter, f), app2(app2(cons, y), ys)) -> app2(app2(app2(filtersub, app2(f, y)), f), app2(app2(cons, y), ys))
app2(app2(app2(filtersub, true), f), app2(app2(cons, y), ys)) -> app2(app2(cons, y), app2(app2(filter, f), ys))
app2(app2(app2(filtersub, false), f), app2(app2(cons, y), ys)) -> app2(app2(filter, f), ys)

The set Q consists of the following terms:

app2(app2(filter, x0), nil)
app2(app2(filter, x0), app2(app2(cons, x1), x2))
app2(app2(app2(filtersub, true), x0), app2(app2(cons, x1), x2))
app2(app2(app2(filtersub, false), x0), app2(app2(cons, x1), x2))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 1 SCC with 6 less nodes.

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

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

APP2(app2(app2(filtersub, false), f), app2(app2(cons, y), ys)) -> APP2(app2(filter, f), ys)
APP2(app2(app2(filtersub, true), f), app2(app2(cons, y), ys)) -> APP2(app2(filter, f), ys)
APP2(app2(filter, f), app2(app2(cons, y), ys)) -> APP2(f, y)

The TRS R consists of the following rules:

app2(app2(filter, f), nil) -> nil
app2(app2(filter, f), app2(app2(cons, y), ys)) -> app2(app2(app2(filtersub, app2(f, y)), f), app2(app2(cons, y), ys))
app2(app2(app2(filtersub, true), f), app2(app2(cons, y), ys)) -> app2(app2(cons, y), app2(app2(filter, f), ys))
app2(app2(app2(filtersub, false), f), app2(app2(cons, y), ys)) -> app2(app2(filter, f), ys)

The set Q consists of the following terms:

app2(app2(filter, x0), nil)
app2(app2(filter, x0), app2(app2(cons, x1), x2))
app2(app2(app2(filtersub, true), x0), app2(app2(cons, x1), x2))
app2(app2(app2(filtersub, false), x0), app2(app2(cons, 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.


APP2(app2(app2(filtersub, false), f), app2(app2(cons, y), ys)) -> APP2(app2(filter, f), ys)
APP2(app2(app2(filtersub, true), f), app2(app2(cons, y), ys)) -> APP2(app2(filter, f), ys)
APP2(app2(filter, f), app2(app2(cons, y), ys)) -> APP2(f, 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)
filtersub  =  filtersub
false  =  false
cons  =  cons
filter  =  filter
true  =  true

Lexicographic Path Order [19].
Precedence:
filtersub > app2

The following usable rules [14] were oriented: none



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

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

app2(app2(filter, f), nil) -> nil
app2(app2(filter, f), app2(app2(cons, y), ys)) -> app2(app2(app2(filtersub, app2(f, y)), f), app2(app2(cons, y), ys))
app2(app2(app2(filtersub, true), f), app2(app2(cons, y), ys)) -> app2(app2(cons, y), app2(app2(filter, f), ys))
app2(app2(app2(filtersub, false), f), app2(app2(cons, y), ys)) -> app2(app2(filter, f), ys)

The set Q consists of the following terms:

app2(app2(filter, x0), nil)
app2(app2(filter, x0), app2(app2(cons, x1), x2))
app2(app2(app2(filtersub, true), x0), app2(app2(cons, x1), x2))
app2(app2(app2(filtersub, false), x0), app2(app2(cons, 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.