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:

app(app(lt, app(s, x)), app(s, y)) → app(app(lt, x), y)
app(app(lt, 0), app(s, y)) → true
app(app(lt, y), 0) → false
app(app(eq, x), x) → true
app(app(eq, app(s, x)), 0) → false
app(app(eq, 0), app(s, x)) → false
app(app(member, w), null) → false
app(app(member, w), app(app(app(fork, x), y), z)) → app(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z)))

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

app(app(lt, app(s, x)), app(s, y)) → app(app(lt, x), y)
app(app(lt, 0), app(s, y)) → true
app(app(lt, y), 0) → false
app(app(eq, x), x) → true
app(app(eq, app(s, x)), 0) → false
app(app(eq, 0), app(s, x)) → false
app(app(member, w), null) → false
app(app(member, w), app(app(app(fork, x), y), z)) → app(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z)))

Q is empty.

Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

APP(app(member, w), app(app(app(fork, x), y), z)) → APP(if, app(app(lt, w), y))
APP(app(lt, app(s, x)), app(s, y)) → APP(lt, x)
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(app(lt, w), y)
APP(app(lt, app(s, x)), app(s, y)) → APP(app(lt, x), y)
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(if, app(app(eq, w), y))
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(eq, w)
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(app(if, app(app(eq, w), y)), true)
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(app(member, w), x)
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z))
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(lt, w)
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(app(eq, w), y)
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z)))
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(app(if, app(app(lt, w), y)), app(app(member, w), x))
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(app(member, w), z)

The TRS R consists of the following rules:

app(app(lt, app(s, x)), app(s, y)) → app(app(lt, x), y)
app(app(lt, 0), app(s, y)) → true
app(app(lt, y), 0) → false
app(app(eq, x), x) → true
app(app(eq, app(s, x)), 0) → false
app(app(eq, 0), app(s, x)) → false
app(app(member, w), null) → false
app(app(member, w), app(app(app(fork, x), y), z)) → app(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(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:

APP(app(member, w), app(app(app(fork, x), y), z)) → APP(if, app(app(lt, w), y))
APP(app(lt, app(s, x)), app(s, y)) → APP(lt, x)
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(app(lt, w), y)
APP(app(lt, app(s, x)), app(s, y)) → APP(app(lt, x), y)
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(if, app(app(eq, w), y))
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(eq, w)
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(app(if, app(app(eq, w), y)), true)
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(app(member, w), x)
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z))
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(lt, w)
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(app(eq, w), y)
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z)))
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(app(if, app(app(lt, w), y)), app(app(member, w), x))
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(app(member, w), z)

The TRS R consists of the following rules:

app(app(lt, app(s, x)), app(s, y)) → app(app(lt, x), y)
app(app(lt, 0), app(s, y)) → true
app(app(lt, y), 0) → false
app(app(eq, x), x) → true
app(app(eq, app(s, x)), 0) → false
app(app(eq, 0), app(s, x)) → false
app(app(member, w), null) → false
app(app(member, w), app(app(app(fork, x), y), z)) → app(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] 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:

APP(app(lt, app(s, x)), app(s, y)) → APP(app(lt, x), y)

The TRS R consists of the following rules:

app(app(lt, app(s, x)), app(s, y)) → app(app(lt, x), y)
app(app(lt, 0), app(s, y)) → true
app(app(lt, y), 0) → false
app(app(eq, x), x) → true
app(app(eq, app(s, x)), 0) → false
app(app(eq, 0), app(s, x)) → false
app(app(member, w), null) → false
app(app(member, w), app(app(app(fork, x), y), z)) → app(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


APP(app(lt, app(s, x)), app(s, y)) → APP(app(lt, x), y)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(APP(x1, x2)) = (4)x_1 + (3)x_2   
POL(app(x1, x2)) = 4 + (2)x_2   
POL(lt) = 0   
POL(s) = 0   
The value of delta used in the strict ordering is 44.
The following usable rules [17] 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:

app(app(lt, app(s, x)), app(s, y)) → app(app(lt, x), y)
app(app(lt, 0), app(s, y)) → true
app(app(lt, y), 0) → false
app(app(eq, x), x) → true
app(app(eq, app(s, x)), 0) → false
app(app(eq, 0), app(s, x)) → false
app(app(member, w), null) → false
app(app(member, w), app(app(app(fork, x), y), z)) → app(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(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:

APP(app(member, w), app(app(app(fork, x), y), z)) → APP(app(member, w), x)
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(app(member, w), z)

The TRS R consists of the following rules:

app(app(lt, app(s, x)), app(s, y)) → app(app(lt, x), y)
app(app(lt, 0), app(s, y)) → true
app(app(lt, y), 0) → false
app(app(eq, x), x) → true
app(app(eq, app(s, x)), 0) → false
app(app(eq, 0), app(s, x)) → false
app(app(member, w), null) → false
app(app(member, w), app(app(app(fork, x), y), z)) → app(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


APP(app(member, w), app(app(app(fork, x), y), z)) → APP(app(member, w), x)
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(app(member, w), z)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(APP(x1, x2)) = (4)x_2   
POL(fork) = 0   
POL(app(x1, x2)) = 1 + (4)x_1 + (4)x_2   
POL(member) = 0   
The value of delta used in the strict ordering is 84.
The following usable rules [17] 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:

app(app(lt, app(s, x)), app(s, y)) → app(app(lt, x), y)
app(app(lt, 0), app(s, y)) → true
app(app(lt, y), 0) → false
app(app(eq, x), x) → true
app(app(eq, app(s, x)), 0) → false
app(app(eq, 0), app(s, x)) → false
app(app(member, w), null) → false
app(app(member, w), app(app(app(fork, x), y), z)) → app(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(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.