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(lt, app2(s, x)), app2(s, y)) -> app2(app2(lt, x), y)
app2(app2(lt, 0), app2(s, y)) -> true
app2(app2(lt, y), 0) -> false
app2(app2(eq, x), x) -> true
app2(app2(eq, app2(s, x)), 0) -> false
app2(app2(eq, 0), app2(s, x)) -> false
app2(app2(member, w), null) -> false
app2(app2(member, w), app2(app2(app2(fork, x), y), z)) -> app2(app2(app2(if, app2(app2(lt, w), y)), app2(app2(member, w), x)), app2(app2(app2(if, app2(app2(eq, w), y)), true), app2(app2(member, w), z)))

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

app2(app2(lt, app2(s, x)), app2(s, y)) -> app2(app2(lt, x), y)
app2(app2(lt, 0), app2(s, y)) -> true
app2(app2(lt, y), 0) -> false
app2(app2(eq, x), x) -> true
app2(app2(eq, app2(s, x)), 0) -> false
app2(app2(eq, 0), app2(s, x)) -> false
app2(app2(member, w), null) -> false
app2(app2(member, w), app2(app2(app2(fork, x), y), z)) -> app2(app2(app2(if, app2(app2(lt, w), y)), app2(app2(member, w), x)), app2(app2(app2(if, app2(app2(eq, w), y)), true), app2(app2(member, w), z)))

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(member, w), app2(app2(app2(fork, x), y), z)) -> APP2(app2(app2(if, app2(app2(lt, w), y)), app2(app2(member, w), x)), app2(app2(app2(if, app2(app2(eq, w), y)), true), app2(app2(member, w), z)))
APP2(app2(member, w), app2(app2(app2(fork, x), y), z)) -> APP2(app2(member, w), x)
APP2(app2(member, w), app2(app2(app2(fork, x), y), z)) -> APP2(if, app2(app2(eq, w), y))
APP2(app2(member, w), app2(app2(app2(fork, x), y), z)) -> APP2(eq, w)
APP2(app2(member, w), app2(app2(app2(fork, x), y), z)) -> APP2(app2(member, w), z)
APP2(app2(member, w), app2(app2(app2(fork, x), y), z)) -> APP2(app2(if, app2(app2(lt, w), y)), app2(app2(member, w), x))
APP2(app2(member, w), app2(app2(app2(fork, x), y), z)) -> APP2(lt, w)
APP2(app2(lt, app2(s, x)), app2(s, y)) -> APP2(lt, x)
APP2(app2(member, w), app2(app2(app2(fork, x), y), z)) -> APP2(app2(lt, w), y)
APP2(app2(member, w), app2(app2(app2(fork, x), y), z)) -> APP2(app2(app2(if, app2(app2(eq, w), y)), true), app2(app2(member, w), z))
APP2(app2(member, w), app2(app2(app2(fork, x), y), z)) -> APP2(if, app2(app2(lt, w), y))
APP2(app2(lt, app2(s, x)), app2(s, y)) -> APP2(app2(lt, x), y)
APP2(app2(member, w), app2(app2(app2(fork, x), y), z)) -> APP2(app2(if, app2(app2(eq, w), y)), true)
APP2(app2(member, w), app2(app2(app2(fork, x), y), z)) -> APP2(app2(eq, w), y)

The TRS R consists of the following rules:

app2(app2(lt, app2(s, x)), app2(s, y)) -> app2(app2(lt, x), y)
app2(app2(lt, 0), app2(s, y)) -> true
app2(app2(lt, y), 0) -> false
app2(app2(eq, x), x) -> true
app2(app2(eq, app2(s, x)), 0) -> false
app2(app2(eq, 0), app2(s, x)) -> false
app2(app2(member, w), null) -> false
app2(app2(member, w), app2(app2(app2(fork, x), y), z)) -> app2(app2(app2(if, app2(app2(lt, w), y)), app2(app2(member, w), x)), app2(app2(app2(if, app2(app2(eq, w), y)), true), app2(app2(member, w), z)))

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(member, w), app2(app2(app2(fork, x), y), z)) -> APP2(app2(app2(if, app2(app2(lt, w), y)), app2(app2(member, w), x)), app2(app2(app2(if, app2(app2(eq, w), y)), true), app2(app2(member, w), z)))
APP2(app2(member, w), app2(app2(app2(fork, x), y), z)) -> APP2(app2(member, w), x)
APP2(app2(member, w), app2(app2(app2(fork, x), y), z)) -> APP2(if, app2(app2(eq, w), y))
APP2(app2(member, w), app2(app2(app2(fork, x), y), z)) -> APP2(eq, w)
APP2(app2(member, w), app2(app2(app2(fork, x), y), z)) -> APP2(app2(member, w), z)
APP2(app2(member, w), app2(app2(app2(fork, x), y), z)) -> APP2(app2(if, app2(app2(lt, w), y)), app2(app2(member, w), x))
APP2(app2(member, w), app2(app2(app2(fork, x), y), z)) -> APP2(lt, w)
APP2(app2(lt, app2(s, x)), app2(s, y)) -> APP2(lt, x)
APP2(app2(member, w), app2(app2(app2(fork, x), y), z)) -> APP2(app2(lt, w), y)
APP2(app2(member, w), app2(app2(app2(fork, x), y), z)) -> APP2(app2(app2(if, app2(app2(eq, w), y)), true), app2(app2(member, w), z))
APP2(app2(member, w), app2(app2(app2(fork, x), y), z)) -> APP2(if, app2(app2(lt, w), y))
APP2(app2(lt, app2(s, x)), app2(s, y)) -> APP2(app2(lt, x), y)
APP2(app2(member, w), app2(app2(app2(fork, x), y), z)) -> APP2(app2(if, app2(app2(eq, w), y)), true)
APP2(app2(member, w), app2(app2(app2(fork, x), y), z)) -> APP2(app2(eq, w), y)

The TRS R consists of the following rules:

app2(app2(lt, app2(s, x)), app2(s, y)) -> app2(app2(lt, x), y)
app2(app2(lt, 0), app2(s, y)) -> true
app2(app2(lt, y), 0) -> false
app2(app2(eq, x), x) -> true
app2(app2(eq, app2(s, x)), 0) -> false
app2(app2(eq, 0), app2(s, x)) -> false
app2(app2(member, w), null) -> false
app2(app2(member, w), app2(app2(app2(fork, x), y), z)) -> app2(app2(app2(if, app2(app2(lt, w), y)), app2(app2(member, w), x)), app2(app2(app2(if, app2(app2(eq, w), y)), true), app2(app2(member, w), z)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 2 SCCs with 11 less nodes.

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

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

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

The TRS R consists of the following rules:

app2(app2(lt, app2(s, x)), app2(s, y)) -> app2(app2(lt, x), y)
app2(app2(lt, 0), app2(s, y)) -> true
app2(app2(lt, y), 0) -> false
app2(app2(eq, x), x) -> true
app2(app2(eq, app2(s, x)), 0) -> false
app2(app2(eq, 0), app2(s, x)) -> false
app2(app2(member, w), null) -> false
app2(app2(member, w), app2(app2(app2(fork, x), y), z)) -> app2(app2(app2(if, app2(app2(lt, w), y)), app2(app2(member, w), x)), app2(app2(app2(if, app2(app2(eq, w), y)), true), app2(app2(member, w), z)))

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

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

The following usable rules [14] were oriented: none



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

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

app2(app2(lt, app2(s, x)), app2(s, y)) -> app2(app2(lt, x), y)
app2(app2(lt, 0), app2(s, y)) -> true
app2(app2(lt, y), 0) -> false
app2(app2(eq, x), x) -> true
app2(app2(eq, app2(s, x)), 0) -> false
app2(app2(eq, 0), app2(s, x)) -> false
app2(app2(member, w), null) -> false
app2(app2(member, w), app2(app2(app2(fork, x), y), z)) -> app2(app2(app2(if, app2(app2(lt, w), y)), app2(app2(member, w), x)), app2(app2(app2(if, app2(app2(eq, w), y)), true), app2(app2(member, w), z)))

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

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

APP2(app2(member, w), app2(app2(app2(fork, x), y), z)) -> APP2(app2(member, w), x)
APP2(app2(member, w), app2(app2(app2(fork, x), y), z)) -> APP2(app2(member, w), z)

The TRS R consists of the following rules:

app2(app2(lt, app2(s, x)), app2(s, y)) -> app2(app2(lt, x), y)
app2(app2(lt, 0), app2(s, y)) -> true
app2(app2(lt, y), 0) -> false
app2(app2(eq, x), x) -> true
app2(app2(eq, app2(s, x)), 0) -> false
app2(app2(eq, 0), app2(s, x)) -> false
app2(app2(member, w), null) -> false
app2(app2(member, w), app2(app2(app2(fork, x), y), z)) -> app2(app2(app2(if, app2(app2(lt, w), y)), app2(app2(member, w), x)), app2(app2(app2(if, app2(app2(eq, w), y)), true), app2(app2(member, w), z)))

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

POL(APP2(x1, x2)) = 1 + x1 + x1·x2   
POL(app2(x1, x2)) = x1 + x1·x2   
POL(fork) = 1   
POL(member) = 1   

The following usable rules [14] were oriented: none



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

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

app2(app2(lt, app2(s, x)), app2(s, y)) -> app2(app2(lt, x), y)
app2(app2(lt, 0), app2(s, y)) -> true
app2(app2(lt, y), 0) -> false
app2(app2(eq, x), x) -> true
app2(app2(eq, app2(s, x)), 0) -> false
app2(app2(eq, 0), app2(s, x)) -> false
app2(app2(member, w), null) -> false
app2(app2(member, w), app2(app2(app2(fork, x), y), z)) -> app2(app2(app2(if, app2(app2(lt, w), y)), app2(app2(member, w), x)), app2(app2(app2(if, app2(app2(eq, w), y)), true), app2(app2(member, w), z)))

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.