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
  ↳ Non-Overlap Check

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.

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(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)))

The set Q consists of the following terms:

app2(app2(lt, app2(s, x0)), app2(s, x1))
app2(app2(lt, 0), app2(s, x0))
app2(app2(lt, x0), 0)
app2(app2(eq, x0), x0)
app2(app2(eq, app2(s, x0)), 0)
app2(app2(eq, 0), app2(s, x0))
app2(app2(member, x0), null)
app2(app2(member, x0), app2(app2(app2(fork, x1), x2), x3))


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)))

The set Q consists of the following terms:

app2(app2(lt, app2(s, x0)), app2(s, x1))
app2(app2(lt, 0), app2(s, x0))
app2(app2(lt, x0), 0)
app2(app2(eq, x0), x0)
app2(app2(eq, app2(s, x0)), 0)
app2(app2(eq, 0), app2(s, x0))
app2(app2(member, x0), null)
app2(app2(member, x0), app2(app2(app2(fork, x1), x2), x3))

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(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)))

The set Q consists of the following terms:

app2(app2(lt, app2(s, x0)), app2(s, x1))
app2(app2(lt, 0), app2(s, x0))
app2(app2(lt, x0), 0)
app2(app2(eq, x0), x0)
app2(app2(eq, app2(s, x0)), 0)
app2(app2(eq, 0), app2(s, x0))
app2(app2(member, x0), null)
app2(app2(member, x0), app2(app2(app2(fork, x1), x2), x3))

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
  ↳ Non-Overlap Check
    ↳ 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)))

The set Q consists of the following terms:

app2(app2(lt, app2(s, x0)), app2(s, x1))
app2(app2(lt, 0), app2(s, x0))
app2(app2(lt, x0), 0)
app2(app2(eq, x0), x0)
app2(app2(eq, app2(s, x0)), 0)
app2(app2(eq, 0), app2(s, x0))
app2(app2(member, x0), null)
app2(app2(member, x0), app2(app2(app2(fork, x1), x2), x3))

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(lt, app2(s, x)), app2(s, y)) -> APP2(app2(lt, 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)  =  x1
app2(x1, x2)  =  app2(x1, x2)
lt  =  lt
s  =  s

Lexicographic Path Order [19].
Precedence:
trivial


The following usable rules [14] were oriented: none



↳ QTRS
  ↳ Non-Overlap Check
    ↳ 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)))

The set Q consists of the following terms:

app2(app2(lt, app2(s, x0)), app2(s, x1))
app2(app2(lt, 0), app2(s, x0))
app2(app2(lt, x0), 0)
app2(app2(eq, x0), x0)
app2(app2(eq, app2(s, x0)), 0)
app2(app2(eq, 0), app2(s, x0))
app2(app2(member, x0), null)
app2(app2(member, x0), app2(app2(app2(fork, x1), x2), x3))

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

↳ QTRS
  ↳ Non-Overlap Check
    ↳ 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)))

The set Q consists of the following terms:

app2(app2(lt, app2(s, x0)), app2(s, x1))
app2(app2(lt, 0), app2(s, x0))
app2(app2(lt, x0), 0)
app2(app2(eq, x0), x0)
app2(app2(eq, app2(s, x0)), 0)
app2(app2(eq, 0), app2(s, x0))
app2(app2(member, x0), null)
app2(app2(member, x0), app2(app2(app2(fork, x1), x2), x3))

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(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 by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
APP2(x1, x2)  =  APP1(x2)
app2(x1, x2)  =  app2(x1, x2)
member  =  member
fork  =  fork

Lexicographic Path Order [19].
Precedence:
APP1 > [app2, member]
fork > [app2, member]


The following usable rules [14] were oriented: none



↳ QTRS
  ↳ Non-Overlap Check
    ↳ 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)))

The set Q consists of the following terms:

app2(app2(lt, app2(s, x0)), app2(s, x1))
app2(app2(lt, 0), app2(s, x0))
app2(app2(lt, x0), 0)
app2(app2(eq, x0), x0)
app2(app2(eq, app2(s, x0)), 0)
app2(app2(eq, 0), app2(s, x0))
app2(app2(member, x0), null)
app2(app2(member, x0), app2(app2(app2(fork, x1), x2), x3))

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