Termination w.r.t. Q of the following Term Rewriting System could not be shown:

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

app2(app2(app2(fold, f), nil), x) -> x
app2(app2(app2(fold, f), app2(app2(cons, h), t)), x) -> app2(app2(app2(fold, f), t), app2(app2(f, x), h))
app2(sum, l) -> app2(app2(app2(fold, add), l), 0)
app2(app2(app2(fold, mul), l), 1) -> app2(prod, l)

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

app2(app2(app2(fold, f), nil), x) -> x
app2(app2(app2(fold, f), app2(app2(cons, h), t)), x) -> app2(app2(app2(fold, f), t), app2(app2(f, x), h))
app2(sum, l) -> app2(app2(app2(fold, add), l), 0)
app2(app2(app2(fold, mul), l), 1) -> app2(prod, l)

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(app2(fold, mul), l), 1) -> APP2(prod, l)
APP2(sum, l) -> APP2(fold, add)
APP2(sum, l) -> APP2(app2(app2(fold, add), l), 0)
APP2(sum, l) -> APP2(app2(fold, add), l)
APP2(app2(app2(fold, f), app2(app2(cons, h), t)), x) -> APP2(app2(f, x), h)
APP2(app2(app2(fold, f), app2(app2(cons, h), t)), x) -> APP2(f, x)
APP2(app2(app2(fold, f), app2(app2(cons, h), t)), x) -> APP2(app2(fold, f), t)
APP2(app2(app2(fold, f), app2(app2(cons, h), t)), x) -> APP2(app2(app2(fold, f), t), app2(app2(f, x), h))

The TRS R consists of the following rules:

app2(app2(app2(fold, f), nil), x) -> x
app2(app2(app2(fold, f), app2(app2(cons, h), t)), x) -> app2(app2(app2(fold, f), t), app2(app2(f, x), h))
app2(sum, l) -> app2(app2(app2(fold, add), l), 0)
app2(app2(app2(fold, mul), l), 1) -> app2(prod, l)

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(app2(fold, mul), l), 1) -> APP2(prod, l)
APP2(sum, l) -> APP2(fold, add)
APP2(sum, l) -> APP2(app2(app2(fold, add), l), 0)
APP2(sum, l) -> APP2(app2(fold, add), l)
APP2(app2(app2(fold, f), app2(app2(cons, h), t)), x) -> APP2(app2(f, x), h)
APP2(app2(app2(fold, f), app2(app2(cons, h), t)), x) -> APP2(f, x)
APP2(app2(app2(fold, f), app2(app2(cons, h), t)), x) -> APP2(app2(fold, f), t)
APP2(app2(app2(fold, f), app2(app2(cons, h), t)), x) -> APP2(app2(app2(fold, f), t), app2(app2(f, x), h))

The TRS R consists of the following rules:

app2(app2(app2(fold, f), nil), x) -> x
app2(app2(app2(fold, f), app2(app2(cons, h), t)), x) -> app2(app2(app2(fold, f), t), app2(app2(f, x), h))
app2(sum, l) -> app2(app2(app2(fold, add), l), 0)
app2(app2(app2(fold, mul), l), 1) -> app2(prod, l)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 1 SCC with 4 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP

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

APP2(sum, l) -> APP2(app2(app2(fold, add), l), 0)
APP2(app2(app2(fold, f), app2(app2(cons, h), t)), x) -> APP2(app2(f, x), h)
APP2(app2(app2(fold, f), app2(app2(cons, h), t)), x) -> APP2(f, x)
APP2(app2(app2(fold, f), app2(app2(cons, h), t)), x) -> APP2(app2(app2(fold, f), t), app2(app2(f, x), h))

The TRS R consists of the following rules:

app2(app2(app2(fold, f), nil), x) -> x
app2(app2(app2(fold, f), app2(app2(cons, h), t)), x) -> app2(app2(app2(fold, f), t), app2(app2(f, x), h))
app2(sum, l) -> app2(app2(app2(fold, add), l), 0)
app2(app2(app2(fold, mul), l), 1) -> app2(prod, l)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.