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(f, app2(app2(cons, nil), y)) -> y
app2(f, app2(app2(cons, app2(f, app2(app2(cons, nil), y))), z)) -> app2(app2(app2(copy, n), y), z)
app2(app2(app2(copy, 0), y), z) -> app2(f, z)
app2(app2(app2(copy, app2(s, x)), y), z) -> app2(app2(app2(copy, x), y), app2(app2(cons, app2(f, y)), z))

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

app2(f, app2(app2(cons, nil), y)) -> y
app2(f, app2(app2(cons, app2(f, app2(app2(cons, nil), y))), z)) -> app2(app2(app2(copy, n), y), z)
app2(app2(app2(copy, 0), y), z) -> app2(f, z)
app2(app2(app2(copy, app2(s, x)), y), z) -> app2(app2(app2(copy, x), y), app2(app2(cons, app2(f, y)), z))

Q is empty.

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

APP2(f, app2(app2(cons, app2(f, app2(app2(cons, nil), y))), z)) -> APP2(copy, n)
APP2(app2(app2(copy, app2(s, x)), y), z) -> APP2(app2(cons, app2(f, y)), z)
APP2(app2(app2(copy, 0), y), z) -> APP2(f, z)
APP2(f, app2(app2(cons, app2(f, app2(app2(cons, nil), y))), z)) -> APP2(app2(app2(copy, n), y), z)
APP2(app2(app2(copy, app2(s, x)), y), z) -> APP2(app2(copy, x), y)
APP2(app2(app2(copy, app2(s, x)), y), z) -> APP2(cons, app2(f, y))
APP2(app2(app2(copy, app2(s, x)), y), z) -> APP2(f, y)
APP2(app2(app2(copy, app2(s, x)), y), z) -> APP2(copy, x)
APP2(f, app2(app2(cons, app2(f, app2(app2(cons, nil), y))), z)) -> APP2(app2(copy, n), y)
APP2(app2(app2(copy, app2(s, x)), y), z) -> APP2(app2(app2(copy, x), y), app2(app2(cons, app2(f, y)), z))

The TRS R consists of the following rules:

app2(f, app2(app2(cons, nil), y)) -> y
app2(f, app2(app2(cons, app2(f, app2(app2(cons, nil), y))), z)) -> app2(app2(app2(copy, n), y), z)
app2(app2(app2(copy, 0), y), z) -> app2(f, z)
app2(app2(app2(copy, app2(s, x)), y), z) -> app2(app2(app2(copy, x), y), app2(app2(cons, app2(f, y)), 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(f, app2(app2(cons, app2(f, app2(app2(cons, nil), y))), z)) -> APP2(copy, n)
APP2(app2(app2(copy, app2(s, x)), y), z) -> APP2(app2(cons, app2(f, y)), z)
APP2(app2(app2(copy, 0), y), z) -> APP2(f, z)
APP2(f, app2(app2(cons, app2(f, app2(app2(cons, nil), y))), z)) -> APP2(app2(app2(copy, n), y), z)
APP2(app2(app2(copy, app2(s, x)), y), z) -> APP2(app2(copy, x), y)
APP2(app2(app2(copy, app2(s, x)), y), z) -> APP2(cons, app2(f, y))
APP2(app2(app2(copy, app2(s, x)), y), z) -> APP2(f, y)
APP2(app2(app2(copy, app2(s, x)), y), z) -> APP2(copy, x)
APP2(f, app2(app2(cons, app2(f, app2(app2(cons, nil), y))), z)) -> APP2(app2(copy, n), y)
APP2(app2(app2(copy, app2(s, x)), y), z) -> APP2(app2(app2(copy, x), y), app2(app2(cons, app2(f, y)), z))

The TRS R consists of the following rules:

app2(f, app2(app2(cons, nil), y)) -> y
app2(f, app2(app2(cons, app2(f, app2(app2(cons, nil), y))), z)) -> app2(app2(app2(copy, n), y), z)
app2(app2(app2(copy, 0), y), z) -> app2(f, z)
app2(app2(app2(copy, app2(s, x)), y), z) -> app2(app2(app2(copy, x), y), app2(app2(cons, app2(f, y)), z))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph contains 1 SCC with 9 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPAfsSolverProof

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

APP2(app2(app2(copy, app2(s, x)), y), z) -> APP2(app2(app2(copy, x), y), app2(app2(cons, app2(f, y)), z))

The TRS R consists of the following rules:

app2(f, app2(app2(cons, nil), y)) -> y
app2(f, app2(app2(cons, app2(f, app2(app2(cons, nil), y))), z)) -> app2(app2(app2(copy, n), y), z)
app2(app2(app2(copy, 0), y), z) -> app2(f, z)
app2(app2(app2(copy, app2(s, x)), y), z) -> app2(app2(app2(copy, x), y), app2(app2(cons, app2(f, y)), z))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using an argument filtering and a montonic ordering, at least one Dependency Pair of this SCC can be strictly oriented.

APP2(app2(app2(copy, app2(s, x)), y), z) -> APP2(app2(app2(copy, x), y), app2(app2(cons, app2(f, y)), z))
Used argument filtering: APP2(x1, x2)  =  x1
app2(x1, x2)  =  app2(x1, x2)
copy  =  copy
s  =  s
f  =  f
cons  =  cons
nil  =  nil
n  =  n
Used ordering: Quasi Precedence: [app_2, n] cons > copy 


↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPAfsSolverProof
            ↳ QDP
              ↳ PisEmptyProof

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

app2(f, app2(app2(cons, nil), y)) -> y
app2(f, app2(app2(cons, app2(f, app2(app2(cons, nil), y))), z)) -> app2(app2(app2(copy, n), y), z)
app2(app2(app2(copy, 0), y), z) -> app2(f, z)
app2(app2(app2(copy, app2(s, x)), y), z) -> app2(app2(app2(copy, x), y), app2(app2(cons, app2(f, y)), 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.