Termination w.r.t. Q proof of TRS/higher-order/LifantsevEx9Maps.trs

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(map_1, f), app(app(cons, h), t)) → app(app(cons, app(f, h)), app(app(map_1, f), t))
app(app(app(map_2, f), c), app(app(cons, h), t)) → app(app(cons, app(app(f, h), c)), app(app(app(map_2, f), c), t))
app(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → app(app(cons, app(app(app(f, g), h), c)), app(app(app(app(map_3, f), g), c), t))

Q is empty.

↳ QTRS

  ↳ Overlay + Local Confluence

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

app(app(map_1, f), app(app(cons, h), t)) → app(app(cons, app(f, h)), app(app(map_1, f), t))
app(app(app(map_2, f), c), app(app(cons, h), t)) → app(app(cons, app(app(f, h), c)), app(app(app(map_2, f), c), t))
app(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → app(app(cons, app(app(app(f, g), h), c)), app(app(app(app(map_3, f), g), c), t))

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(map_1, f), app(app(cons, h), t)) → app(app(cons, app(f, h)), app(app(map_1, f), t))
app(app(app(map_2, f), c), app(app(cons, h), t)) → app(app(cons, app(app(f, h), c)), app(app(app(map_2, f), c), t))
app(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → app(app(cons, app(app(app(f, g), h), c)), app(app(app(app(map_3, f), g), c), t))

The set Q consists of the following terms:

app(app(map_1, x0), app(app(cons, x1), x2))
app(app(app(map_2, x0), x1), app(app(cons, x2), x3))
app(app(app(app(map_3, x0), g), x1), app(app(cons, x2), x3))

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(map_1, f), app(app(cons, h), t)) → APP(app(map_1, f), t)
APP(app(app(map_2, f), c), app(app(cons, h), t)) → APP(app(app(map_2, f), c), t)
APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(app(cons, app(app(app(f, g), h), c)), app(app(app(app(map_3, f), g), c), t))
APP(app(app(map_2, f), c), app(app(cons, h), t)) → APP(f, h)
APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(app(app(f, g), h), c)
APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(app(f, g), h)
APP(app(app(map_2, f), c), app(app(cons, h), t)) → APP(app(f, h), c)
APP(app(map_1, f), app(app(cons, h), t)) → APP(cons, app(f, h))
APP(app(app(map_2, f), c), app(app(cons, h), t)) → APP(cons, app(app(f, h), c))
APP(app(app(map_2, f), c), app(app(cons, h), t)) → APP(app(cons, app(app(f, h), c)), app(app(app(map_2, f), c), t))
APP(app(map_1, f), app(app(cons, h), t)) → APP(app(cons, app(f, h)), app(app(map_1, f), t))
APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(cons, app(app(app(f, g), h), c))
APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(app(app(app(map_3, f), g), c), t)
APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(f, g)
APP(app(map_1, f), app(app(cons, h), t)) → APP(f, h)

The TRS R consists of the following rules:

app(app(map_1, f), app(app(cons, h), t)) → app(app(cons, app(f, h)), app(app(map_1, f), t))
app(app(app(map_2, f), c), app(app(cons, h), t)) → app(app(cons, app(app(f, h), c)), app(app(app(map_2, f), c), t))
app(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → app(app(cons, app(app(app(f, g), h), c)), app(app(app(app(map_3, f), g), c), t))

The set Q consists of the following terms:

app(app(map_1, x0), app(app(cons, x1), x2))
app(app(app(map_2, x0), x1), app(app(cons, x2), x3))
app(app(app(app(map_3, x0), g), x1), app(app(cons, x2), x3))

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(map_1, f), app(app(cons, h), t)) → APP(app(map_1, f), t)
APP(app(app(map_2, f), c), app(app(cons, h), t)) → APP(app(app(map_2, f), c), t)
APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(app(cons, app(app(app(f, g), h), c)), app(app(app(app(map_3, f), g), c), t))
APP(app(app(map_2, f), c), app(app(cons, h), t)) → APP(f, h)
APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(app(app(f, g), h), c)
APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(app(f, g), h)
APP(app(app(map_2, f), c), app(app(cons, h), t)) → APP(app(f, h), c)
APP(app(map_1, f), app(app(cons, h), t)) → APP(cons, app(f, h))
APP(app(app(map_2, f), c), app(app(cons, h), t)) → APP(cons, app(app(f, h), c))
APP(app(app(map_2, f), c), app(app(cons, h), t)) → APP(app(cons, app(app(f, h), c)), app(app(app(map_2, f), c), t))
APP(app(map_1, f), app(app(cons, h), t)) → APP(app(cons, app(f, h)), app(app(map_1, f), t))
APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(cons, app(app(app(f, g), h), c))
APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(app(app(app(map_3, f), g), c), t)
APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(f, g)
APP(app(map_1, f), app(app(cons, h), t)) → APP(f, h)

The TRS R consists of the following rules:

app(app(map_1, f), app(app(cons, h), t)) → app(app(cons, app(f, h)), app(app(map_1, f), t))
app(app(app(map_2, f), c), app(app(cons, h), t)) → app(app(cons, app(app(f, h), c)), app(app(app(map_2, f), c), t))
app(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → app(app(cons, app(app(app(f, g), h), c)), app(app(app(app(map_3, f), g), c), t))

The set Q consists of the following terms:

app(app(map_1, x0), app(app(cons, x1), x2))
app(app(app(map_2, x0), x1), app(app(cons, x2), x3))
app(app(app(app(map_3, x0), g), x1), app(app(cons, x2), x3))

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(map_1, f), app(app(cons, h), t)) → APP(app(map_1, f), t)
APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(app(cons, app(app(app(f, g), h), c)), app(app(app(app(map_3, f), g), c), t))
APP(app(app(map_2, f), c), app(app(cons, h), t)) → APP(app(app(map_2, f), c), t)
APP(app(app(map_2, f), c), app(app(cons, h), t)) → APP(f, h)
APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(app(app(f, g), h), c)
APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(app(f, g), h)
APP(app(app(map_2, f), c), app(app(cons, h), t)) → APP(app(f, h), c)
APP(app(map_1, f), app(app(cons, h), t)) → APP(cons, app(f, h))
APP(app(app(map_2, f), c), app(app(cons, h), t)) → APP(cons, app(app(f, h), c))
APP(app(app(map_2, f), c), app(app(cons, h), t)) → APP(app(cons, app(app(f, h), c)), app(app(app(map_2, f), c), t))
APP(app(map_1, f), app(app(cons, h), t)) → APP(app(cons, app(f, h)), app(app(map_1, f), t))
APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(cons, app(app(app(f, g), h), c))
APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(f, g)
APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(app(app(app(map_3, f), g), c), t)
APP(app(map_1, f), app(app(cons, h), t)) → APP(f, h)

The TRS R consists of the following rules:

app(app(map_1, f), app(app(cons, h), t)) → app(app(cons, app(f, h)), app(app(map_1, f), t))
app(app(app(map_2, f), c), app(app(cons, h), t)) → app(app(cons, app(app(f, h), c)), app(app(app(map_2, f), c), t))
app(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → app(app(cons, app(app(app(f, g), h), c)), app(app(app(app(map_3, f), g), c), t))

The set Q consists of the following terms:

app(app(map_1, x0), app(app(cons, x1), x2))
app(app(app(map_2, x0), x1), app(app(cons, x2), x3))
app(app(app(app(map_3, x0), g), x1), app(app(cons, x2), x3))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 1 SCC with 7 less nodes.

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ EdgeDeletionProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

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

APP(app(map_1, f), app(app(cons, h), t)) → APP(app(map_1, f), t)
APP(app(app(map_2, f), c), app(app(cons, h), t)) → APP(app(app(map_2, f), c), t)
APP(app(app(map_2, f), c), app(app(cons, h), t)) → APP(f, h)
APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(app(app(f, g), h), c)
APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(app(f, g), h)
APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(app(app(app(map_3, f), g), c), t)
APP(app(app(map_2, f), c), app(app(cons, h), t)) → APP(app(f, h), c)
APP(app(map_1, f), app(app(cons, h), t)) → APP(f, h)

The TRS R consists of the following rules:

app(app(map_1, f), app(app(cons, h), t)) → app(app(cons, app(f, h)), app(app(map_1, f), t))
app(app(app(map_2, f), c), app(app(cons, h), t)) → app(app(cons, app(app(f, h), c)), app(app(app(map_2, f), c), t))
app(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → app(app(cons, app(app(app(f, g), h), c)), app(app(app(app(map_3, f), g), c), t))

The set Q consists of the following terms:

app(app(map_1, x0), app(app(cons, x1), x2))
app(app(app(map_2, x0), x1), app(app(cons, x2), x3))
app(app(app(app(map_3, x0), g), x1), app(app(cons, x2), x3))

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