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 [19] 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 [1,15] 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(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)
APP(app(fmap, app(app(fcons, f), t)), x) → APP(fmap, t)
APP(app(fmap, app(app(fcons, f), t)), x) → APP(app(fmap, t), 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
          ↳ DependencyGraphProof

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

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)
APP(app(fmap, app(app(fcons, f), t)), x) → APP(fmap, t)
APP(app(fmap, app(app(fcons, f), t)), x) → APP(app(fmap, t), 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 [15,17,22] contains 1 SCC with 3 less nodes.

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

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

APP(app(fmap, app(app(fcons, f), t)), x) → APP(f, x)
APP(app(fmap, app(app(fcons, f), t)), x) → APP(app(fmap, t), 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.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ UsableRulesProof
                ↳ QDP
                  ↳ UsableRulesReductionPairsProof

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

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

R is empty.
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.
By using the usable rules with reduction pair processor [15] with a polynomial ordering [25], all dependency pairs and the corresponding usable rules [17] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

APP(app(fmap, app(app(fcons, f), t)), x) → APP(f, x)
APP(app(fmap, app(app(fcons, f), t)), x) → APP(app(fmap, t), x)
No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [25]:

POL(APP(x1, x2)) = 2·x1 + x2   
POL(app(x1, x2)) = 2·x1 + 2·x2   
POL(fcons) = 0   
POL(fmap) = 0   



↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ UsableRulesProof
                ↳ QDP
                  ↳ UsableRulesReductionPairsProof
                    ↳ QDP
                      ↳ PisEmptyProof

Q DP problem:
P is empty.
R is empty.
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.