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(fmap, fnil), x) → nil
app(app(fmap, app(app(fcons, f), t)), x) → app(app(cons, app(f, x)), app(app(fmap, t), x))

Q is empty.


↳ QTRS
  ↳ Overlay + Local Confluence

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

app(app(fmap, fnil), x) → nil
app(app(fmap, app(app(fcons, f), t)), x) → app(app(cons, app(f, x)), app(app(fmap, t), 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(fmap, fnil), x) → nil
app(app(fmap, app(app(fcons, f), t)), x) → app(app(cons, app(f, x)), app(app(fmap, t), x))

The set Q consists of the following terms:

app(app(fmap, fnil), x0)
app(app(fmap, app(app(fcons, x0), x1)), x2)


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(fmap, app(app(fcons, f), t)), x) → APP(app(fmap, t), x)
APP(app(fmap, app(app(fcons, f), t)), x) → APP(fmap, t)
APP(app(fmap, app(app(fcons, f), t)), x) → APP(cons, app(f, x))
APP(app(fmap, app(app(fcons, f), t)), x) → APP(app(cons, app(f, x)), app(app(fmap, t), x))
APP(app(fmap, app(app(fcons, f), t)), x) → APP(f, x)

The TRS R consists of the following rules:

app(app(fmap, fnil), x) → nil
app(app(fmap, app(app(fcons, f), t)), x) → app(app(cons, app(f, x)), app(app(fmap, t), x))

The set Q consists of the following terms:

app(app(fmap, fnil), x0)
app(app(fmap, app(app(fcons, x0), x1)), x2)

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(fmap, app(app(fcons, f), t)), x) → APP(app(fmap, t), x)
APP(app(fmap, app(app(fcons, f), t)), x) → APP(fmap, t)
APP(app(fmap, app(app(fcons, f), t)), x) → APP(cons, app(f, x))
APP(app(fmap, app(app(fcons, f), t)), x) → APP(app(cons, app(f, x)), app(app(fmap, t), x))
APP(app(fmap, app(app(fcons, f), t)), x) → APP(f, x)

The TRS R consists of the following rules:

app(app(fmap, fnil), x) → nil
app(app(fmap, app(app(fcons, f), t)), x) → app(app(cons, app(f, x)), app(app(fmap, t), x))

The set Q consists of the following terms:

app(app(fmap, fnil), x0)
app(app(fmap, app(app(fcons, x0), x1)), x2)

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(fmap, app(app(fcons, f), t)), x) → APP(app(fmap, t), x)
APP(app(fmap, app(app(fcons, f), t)), x) → APP(fmap, t)
APP(app(fmap, app(app(fcons, f), t)), x) → APP(cons, app(f, x))
APP(app(fmap, app(app(fcons, f), t)), x) → APP(app(cons, app(f, x)), app(app(fmap, t), x))
APP(app(fmap, app(app(fcons, f), t)), x) → APP(f, x)

The TRS R consists of the following rules:

app(app(fmap, fnil), x) → nil
app(app(fmap, app(app(fcons, f), t)), x) → app(app(cons, app(f, x)), app(app(fmap, t), x))

The set Q consists of the following terms:

app(app(fmap, fnil), x0)
app(app(fmap, app(app(fcons, x0), x1)), x2)

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 1 SCC with 3 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(fmap, app(app(fcons, f), t)), x) → APP(app(fmap, t), x)
APP(app(fmap, app(app(fcons, f), t)), x) → APP(f, x)

The TRS R consists of the following rules:

app(app(fmap, fnil), x) → nil
app(app(fmap, app(app(fcons, f), t)), x) → app(app(cons, app(f, x)), app(app(fmap, t), x))

The set Q consists of the following terms:

app(app(fmap, fnil), x0)
app(app(fmap, app(app(fcons, x0), x1)), x2)

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(fmap, app(app(fcons, f), t)), x) → APP(app(fmap, t), x)
APP(app(fmap, app(app(fcons, f), t)), x) → APP(f, x)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
APP(x1, x2)  =  x1
app(x1, x2)  =  app(x1, x2)
fmap  =  fmap
fcons  =  fcons

Lexicographic path order with status [19].
Quasi-Precedence:
trivial

Status:
trivial


The following usable rules [14] were oriented: none



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

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

app(app(fmap, fnil), x) → nil
app(app(fmap, app(app(fcons, f), t)), x) → app(app(cons, app(f, x)), app(app(fmap, t), x))

The set Q consists of the following terms:

app(app(fmap, fnil), x0)
app(app(fmap, app(app(fcons, x0), x1)), x2)

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