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:

app(app(iterate, f), x) → app(app(cons, x), app(app(iterate, f), app(f, x)))

Q is empty.


↳ QTRS
  ↳ Overlay + Local Confluence

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

app(app(iterate, f), x) → app(app(cons, x), app(app(iterate, f), app(f, x)))

Q is empty.

The TRS is overlay and locally confluent. By [15] we can switch to innermost.

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof

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

app(app(iterate, f), x) → app(app(cons, x), app(app(iterate, f), app(f, x)))

The set Q consists of the following terms:

app(app(iterate, x0), x1)


Using Dependency Pairs [1,13] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

APP(app(iterate, f), x) → APP(f, x)
APP(app(iterate, f), x) → APP(app(iterate, f), app(f, x))
APP(app(iterate, f), x) → APP(cons, x)
APP(app(iterate, f), x) → APP(app(cons, x), app(app(iterate, f), app(f, x)))

The TRS R consists of the following rules:

app(app(iterate, f), x) → app(app(cons, x), app(app(iterate, f), app(f, x)))

The set Q consists of the following terms:

app(app(iterate, x0), x1)

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

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ EdgeDeletionProof

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

APP(app(iterate, f), x) → APP(f, x)
APP(app(iterate, f), x) → APP(app(iterate, f), app(f, x))
APP(app(iterate, f), x) → APP(cons, x)
APP(app(iterate, f), x) → APP(app(cons, x), app(app(iterate, f), app(f, x)))

The TRS R consists of the following rules:

app(app(iterate, f), x) → app(app(cons, x), app(app(iterate, f), app(f, x)))

The set Q consists of the following terms:

app(app(iterate, x0), x1)

We have to consider all minimal (P,Q,R)-chains.
We deleted some edges using various graph approximations

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ EdgeDeletionProof
            ↳ QDP
              ↳ DependencyGraphProof

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

APP(app(iterate, f), x) → APP(f, x)
APP(app(iterate, f), x) → APP(app(iterate, f), app(f, x))
APP(app(iterate, f), x) → APP(cons, x)
APP(app(iterate, f), x) → APP(app(cons, x), app(app(iterate, f), app(f, x)))

The TRS R consists of the following rules:

app(app(iterate, f), x) → app(app(cons, x), app(app(iterate, f), app(f, x)))

The set Q consists of the following terms:

app(app(iterate, x0), x1)

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 1 SCC with 2 less nodes.

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ EdgeDeletionProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ QDP
                  ↳ QDPOrderProof

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

APP(app(iterate, f), x) → APP(f, x)
APP(app(iterate, f), x) → APP(app(iterate, f), app(f, x))

The TRS R consists of the following rules:

app(app(iterate, f), x) → app(app(cons, x), app(app(iterate, f), app(f, x)))

The set Q consists of the following terms:

app(app(iterate, x0), x1)

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.


APP(app(iterate, f), x) → APP(f, x)
The remaining pairs can at least be oriented weakly.

APP(app(iterate, f), x) → APP(app(iterate, f), app(f, x))
Used ordering: Combined order from the following AFS and order.
APP(x1, x2)  =  APP(x1)
app(x1, x2)  =  app(x2)
iterate  =  iterate
cons  =  cons

Recursive Path Order [2].
Precedence:
[APP1, iterate] > app1
cons > app1


The following usable rules [14] were oriented: none



↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ EdgeDeletionProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ QDP
                  ↳ QDPOrderProof
                    ↳ QDP

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

APP(app(iterate, f), x) → APP(app(iterate, f), app(f, x))

The TRS R consists of the following rules:

app(app(iterate, f), x) → app(app(cons, x), app(app(iterate, f), app(f, x)))

The set Q consists of the following terms:

app(app(iterate, x0), x1)

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