Termination w.r.t. Q proof of TRS/higher-order/ToyamaRTA04Ex7Sorting.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(app(sort, f), g), nil) → nil
app(app(app(sort, f), g), app(app(cons, x), y)) → app(app(app(app(insert, f), g), app(app(app(sort, f), g), y)), x)
app(app(app(app(insert, f), g), nil), y) → app(app(cons, y), nil)
app(app(app(app(insert, f), g), app(app(cons, x), z)), y) → app(app(cons, app(app(f, x), y)), app(app(app(app(insert, f), g), z), app(app(g, x), y)))
app(app(max, 0), y) → y
app(app(max, x), 0) → x
app(app(max, app(s, x)), app(s, y)) → app(app(max, x), y)
app(app(min, 0), y) → 0
app(app(min, x), 0) → 0
app(app(min, app(s, x)), app(s, y)) → app(app(min, x), y)
app(asort, z) → app(app(app(sort, min), max), z)
app(dsort, z) → app(app(app(sort, max), min), z)

Q is empty.

↳ QTRS

  ↳ Overlay + Local Confluence

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

app(app(app(sort, f), g), nil) → nil
app(app(app(sort, f), g), app(app(cons, x), y)) → app(app(app(app(insert, f), g), app(app(app(sort, f), g), y)), x)
app(app(app(app(insert, f), g), nil), y) → app(app(cons, y), nil)
app(app(app(app(insert, f), g), app(app(cons, x), z)), y) → app(app(cons, app(app(f, x), y)), app(app(app(app(insert, f), g), z), app(app(g, x), y)))
app(app(max, 0), y) → y
app(app(max, x), 0) → x
app(app(max, app(s, x)), app(s, y)) → app(app(max, x), y)
app(app(min, 0), y) → 0
app(app(min, x), 0) → 0
app(app(min, app(s, x)), app(s, y)) → app(app(min, x), y)
app(asort, z) → app(app(app(sort, min), max), z)
app(dsort, z) → app(app(app(sort, max), min), z)

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(app(sort, f), g), nil) → nil
app(app(app(sort, f), g), app(app(cons, x), y)) → app(app(app(app(insert, f), g), app(app(app(sort, f), g), y)), x)
app(app(app(app(insert, f), g), nil), y) → app(app(cons, y), nil)
app(app(app(app(insert, f), g), app(app(cons, x), z)), y) → app(app(cons, app(app(f, x), y)), app(app(app(app(insert, f), g), z), app(app(g, x), y)))
app(app(max, 0), y) → y
app(app(max, x), 0) → x
app(app(max, app(s, x)), app(s, y)) → app(app(max, x), y)
app(app(min, 0), y) → 0
app(app(min, x), 0) → 0
app(app(min, app(s, x)), app(s, y)) → app(app(min, x), y)
app(asort, z) → app(app(app(sort, min), max), z)
app(dsort, z) → app(app(app(sort, max), min), z)

The set Q consists of the following terms:

app(app(app(sort, x0), x1), nil)
app(app(app(sort, x0), x1), app(app(cons, x2), x3))
app(app(app(app(insert, x0), x1), nil), x2)
app(app(app(app(insert, x0), x1), app(app(cons, x2), x3)), x4)
app(app(max, 0), x0)
app(app(max, x0), 0)
app(app(max, app(s, x0)), app(s, x1))
app(app(min, 0), x0)
app(app(min, x0), 0)
app(app(min, app(s, x0)), app(s, x1))
app(asort, x0)
app(dsort, x0)

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(asort, z) → APP(sort, min)
APP(app(app(sort, f), g), app(app(cons, x), y)) → APP(app(app(sort, f), g), y)
APP(dsort, z) → APP(sort, max)
APP(app(app(sort, f), g), app(app(cons, x), y)) → APP(app(app(app(insert, f), g), app(app(app(sort, f), g), y)), x)
APP(dsort, z) → APP(app(sort, max), min)
APP(app(app(app(insert, f), g), app(app(cons, x), z)), y) → APP(app(app(insert, f), g), z)
APP(app(app(app(insert, f), g), nil), y) → APP(cons, y)
APP(app(app(app(insert, f), g), app(app(cons, x), z)), y) → APP(app(cons, app(app(f, x), y)), app(app(app(app(insert, f), g), z), app(app(g, x), y)))
APP(app(app(app(insert, f), g), app(app(cons, x), z)), y) → APP(app(g, x), y)
APP(app(min, app(s, x)), app(s, y)) → APP(app(min, x), y)
APP(app(max, app(s, x)), app(s, y)) → APP(app(max, x), y)
APP(asort, z) → APP(app(app(sort, min), max), z)
APP(app(app(app(insert, f), g), nil), y) → APP(app(cons, y), nil)
APP(app(app(sort, f), g), app(app(cons, x), y)) → APP(app(insert, f), g)
APP(app(max, app(s, x)), app(s, y)) → APP(max, x)
APP(asort, z) → APP(app(sort, min), max)
APP(app(app(app(insert, f), g), app(app(cons, x), z)), y) → APP(app(f, x), y)
APP(app(min, app(s, x)), app(s, y)) → APP(min, x)
APP(dsort, z) → APP(app(app(sort, max), min), z)
APP(app(app(app(insert, f), g), app(app(cons, x), z)), y) → APP(f, x)
APP(app(app(app(insert, f), g), app(app(cons, x), z)), y) → APP(app(app(app(insert, f), g), z), app(app(g, x), y))
APP(app(app(sort, f), g), app(app(cons, x), y)) → APP(app(app(insert, f), g), app(app(app(sort, f), g), y))
APP(app(app(app(insert, f), g), app(app(cons, x), z)), y) → APP(g, x)
APP(app(app(sort, f), g), app(app(cons, x), y)) → APP(insert, f)
APP(app(app(app(insert, f), g), app(app(cons, x), z)), y) → APP(cons, app(app(f, x), y))

The TRS R consists of the following rules:

app(app(app(sort, f), g), nil) → nil
app(app(app(sort, f), g), app(app(cons, x), y)) → app(app(app(app(insert, f), g), app(app(app(sort, f), g), y)), x)
app(app(app(app(insert, f), g), nil), y) → app(app(cons, y), nil)
app(app(app(app(insert, f), g), app(app(cons, x), z)), y) → app(app(cons, app(app(f, x), y)), app(app(app(app(insert, f), g), z), app(app(g, x), y)))
app(app(max, 0), y) → y
app(app(max, x), 0) → x
app(app(max, app(s, x)), app(s, y)) → app(app(max, x), y)
app(app(min, 0), y) → 0
app(app(min, x), 0) → 0
app(app(min, app(s, x)), app(s, y)) → app(app(min, x), y)
app(asort, z) → app(app(app(sort, min), max), z)
app(dsort, z) → app(app(app(sort, max), min), z)

The set Q consists of the following terms:

app(app(app(sort, x0), x1), nil)
app(app(app(sort, x0), x1), app(app(cons, x2), x3))
app(app(app(app(insert, x0), x1), nil), x2)
app(app(app(app(insert, x0), x1), app(app(cons, x2), x3)), x4)
app(app(max, 0), x0)
app(app(max, x0), 0)
app(app(max, app(s, x0)), app(s, x1))
app(app(min, 0), x0)
app(app(min, x0), 0)
app(app(min, app(s, x0)), app(s, x1))
app(asort, x0)
app(dsort, x0)

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(asort, z) → APP(sort, min)
APP(app(app(sort, f), g), app(app(cons, x), y)) → APP(app(app(sort, f), g), y)
APP(dsort, z) → APP(sort, max)
APP(app(app(sort, f), g), app(app(cons, x), y)) → APP(app(app(app(insert, f), g), app(app(app(sort, f), g), y)), x)
APP(dsort, z) → APP(app(sort, max), min)
APP(app(app(app(insert, f), g), app(app(cons, x), z)), y) → APP(app(app(insert, f), g), z)
APP(app(app(app(insert, f), g), nil), y) → APP(cons, y)
APP(app(app(app(insert, f), g), app(app(cons, x), z)), y) → APP(app(cons, app(app(f, x), y)), app(app(app(app(insert, f), g), z), app(app(g, x), y)))
APP(app(app(app(insert, f), g), app(app(cons, x), z)), y) → APP(app(g, x), y)
APP(app(min, app(s, x)), app(s, y)) → APP(app(min, x), y)
APP(app(max, app(s, x)), app(s, y)) → APP(app(max, x), y)
APP(asort, z) → APP(app(app(sort, min), max), z)
APP(app(app(app(insert, f), g), nil), y) → APP(app(cons, y), nil)
APP(app(app(sort, f), g), app(app(cons, x), y)) → APP(app(insert, f), g)
APP(app(max, app(s, x)), app(s, y)) → APP(max, x)
APP(asort, z) → APP(app(sort, min), max)
APP(app(app(app(insert, f), g), app(app(cons, x), z)), y) → APP(app(f, x), y)
APP(app(min, app(s, x)), app(s, y)) → APP(min, x)
APP(dsort, z) → APP(app(app(sort, max), min), z)
APP(app(app(app(insert, f), g), app(app(cons, x), z)), y) → APP(f, x)
APP(app(app(app(insert, f), g), app(app(cons, x), z)), y) → APP(app(app(app(insert, f), g), z), app(app(g, x), y))
APP(app(app(sort, f), g), app(app(cons, x), y)) → APP(app(app(insert, f), g), app(app(app(sort, f), g), y))
APP(app(app(app(insert, f), g), app(app(cons, x), z)), y) → APP(g, x)
APP(app(app(sort, f), g), app(app(cons, x), y)) → APP(insert, f)
APP(app(app(app(insert, f), g), app(app(cons, x), z)), y) → APP(cons, app(app(f, x), y))

The TRS R consists of the following rules:

app(app(app(sort, f), g), nil) → nil
app(app(app(sort, f), g), app(app(cons, x), y)) → app(app(app(app(insert, f), g), app(app(app(sort, f), g), y)), x)
app(app(app(app(insert, f), g), nil), y) → app(app(cons, y), nil)
app(app(app(app(insert, f), g), app(app(cons, x), z)), y) → app(app(cons, app(app(f, x), y)), app(app(app(app(insert, f), g), z), app(app(g, x), y)))
app(app(max, 0), y) → y
app(app(max, x), 0) → x
app(app(max, app(s, x)), app(s, y)) → app(app(max, x), y)
app(app(min, 0), y) → 0
app(app(min, x), 0) → 0
app(app(min, app(s, x)), app(s, y)) → app(app(min, x), y)
app(asort, z) → app(app(app(sort, min), max), z)
app(dsort, z) → app(app(app(sort, max), min), z)

The set Q consists of the following terms:

app(app(app(sort, x0), x1), nil)
app(app(app(sort, x0), x1), app(app(cons, x2), x3))
app(app(app(app(insert, x0), x1), nil), x2)
app(app(app(app(insert, x0), x1), app(app(cons, x2), x3)), x4)
app(app(max, 0), x0)
app(app(max, x0), 0)
app(app(max, app(s, x0)), app(s, x1))
app(app(min, 0), x0)
app(app(min, x0), 0)
app(app(min, app(s, x0)), app(s, x1))
app(asort, x0)
app(dsort, x0)

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(asort, z) → APP(sort, min)
APP(dsort, z) → APP(sort, max)
APP(app(app(sort, f), g), app(app(cons, x), y)) → APP(app(app(sort, f), g), y)
APP(app(app(sort, f), g), app(app(cons, x), y)) → APP(app(app(app(insert, f), g), app(app(app(sort, f), g), y)), x)
APP(dsort, z) → APP(app(sort, max), min)
APP(app(app(app(insert, f), g), nil), y) → APP(cons, y)
APP(app(app(app(insert, f), g), app(app(cons, x), z)), y) → APP(app(app(insert, f), g), z)
APP(app(app(app(insert, f), g), app(app(cons, x), z)), y) → APP(app(cons, app(app(f, x), y)), app(app(app(app(insert, f), g), z), app(app(g, x), y)))
APP(app(app(app(insert, f), g), app(app(cons, x), z)), y) → APP(app(g, x), y)
APP(app(min, app(s, x)), app(s, y)) → APP(app(min, x), y)
APP(app(max, app(s, x)), app(s, y)) → APP(app(max, x), y)
APP(asort, z) → APP(app(app(sort, min), max), z)
APP(app(app(sort, f), g), app(app(cons, x), y)) → APP(app(insert, f), g)
APP(app(app(app(insert, f), g), nil), y) → APP(app(cons, y), nil)
APP(asort, z) → APP(app(sort, min), max)
APP(app(max, app(s, x)), app(s, y)) → APP(max, x)
APP(dsort, z) → APP(app(app(sort, max), min), z)
APP(app(min, app(s, x)), app(s, y)) → APP(min, x)
APP(app(app(app(insert, f), g), app(app(cons, x), z)), y) → APP(app(f, x), y)
APP(app(app(app(insert, f), g), app(app(cons, x), z)), y) → APP(f, x)
APP(app(app(app(insert, f), g), app(app(cons, x), z)), y) → APP(app(app(app(insert, f), g), z), app(app(g, x), y))
APP(app(app(sort, f), g), app(app(cons, x), y)) → APP(app(app(insert, f), g), app(app(app(sort, f), g), y))
APP(app(app(app(insert, f), g), app(app(cons, x), z)), y) → APP(g, x)
APP(app(app(sort, f), g), app(app(cons, x), y)) → APP(insert, f)
APP(app(app(app(insert, f), g), app(app(cons, x), z)), y) → APP(cons, app(app(f, x), y))

The TRS R consists of the following rules:

app(app(app(sort, f), g), nil) → nil
app(app(app(sort, f), g), app(app(cons, x), y)) → app(app(app(app(insert, f), g), app(app(app(sort, f), g), y)), x)
app(app(app(app(insert, f), g), nil), y) → app(app(cons, y), nil)
app(app(app(app(insert, f), g), app(app(cons, x), z)), y) → app(app(cons, app(app(f, x), y)), app(app(app(app(insert, f), g), z), app(app(g, x), y)))
app(app(max, 0), y) → y
app(app(max, x), 0) → x
app(app(max, app(s, x)), app(s, y)) → app(app(max, x), y)
app(app(min, 0), y) → 0
app(app(min, x), 0) → 0
app(app(min, app(s, x)), app(s, y)) → app(app(min, x), y)
app(asort, z) → app(app(app(sort, min), max), z)
app(dsort, z) → app(app(app(sort, max), min), z)

The set Q consists of the following terms:

app(app(app(sort, x0), x1), nil)
app(app(app(sort, x0), x1), app(app(cons, x2), x3))
app(app(app(app(insert, x0), x1), nil), x2)
app(app(app(app(insert, x0), x1), app(app(cons, x2), x3)), x4)
app(app(max, 0), x0)
app(app(max, x0), 0)
app(app(max, app(s, x0)), app(s, x1))
app(app(min, 0), x0)
app(app(min, x0), 0)
app(app(min, app(s, x0)), app(s, x1))
app(asort, x0)
app(dsort, x0)

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 3 SCCs with 14 less nodes.

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ EdgeDeletionProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                  ↳ QDP

                  ↳ QDP

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

APP(app(min, app(s, x)), app(s, y)) → APP(app(min, x), y)

The TRS R consists of the following rules:

app(app(app(sort, f), g), nil) → nil
app(app(app(sort, f), g), app(app(cons, x), y)) → app(app(app(app(insert, f), g), app(app(app(sort, f), g), y)), x)
app(app(app(app(insert, f), g), nil), y) → app(app(cons, y), nil)
app(app(app(app(insert, f), g), app(app(cons, x), z)), y) → app(app(cons, app(app(f, x), y)), app(app(app(app(insert, f), g), z), app(app(g, x), y)))
app(app(max, 0), y) → y
app(app(max, x), 0) → x
app(app(max, app(s, x)), app(s, y)) → app(app(max, x), y)
app(app(min, 0), y) → 0
app(app(min, x), 0) → 0
app(app(min, app(s, x)), app(s, y)) → app(app(min, x), y)
app(asort, z) → app(app(app(sort, min), max), z)
app(dsort, z) → app(app(app(sort, max), min), z)

The set Q consists of the following terms:

app(app(app(sort, x0), x1), nil)
app(app(app(sort, x0), x1), app(app(cons, x2), x3))
app(app(app(app(insert, x0), x1), nil), x2)
app(app(app(app(insert, x0), x1), app(app(cons, x2), x3)), x4)
app(app(max, 0), x0)
app(app(max, x0), 0)
app(app(max, app(s, x0)), app(s, x1))
app(app(min, 0), x0)
app(app(min, x0), 0)
app(app(min, app(s, x0)), app(s, x1))
app(asort, x0)
app(dsort, x0)

We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13]. Here, we combined the reduction pair processor with the A-transformation [14] which results in the following intermediate Q-DP Problem.
Q DP problem:
The TRS P consists of the following rules:

MIN(s(x), s(y)) → MIN(x, y)

R is empty.
The set Q consists of the following terms:

sort(x0, x1, nil)
sort(x0, x1, cons(x2, x3))
insert(x0, x1, nil, x2)
insert(x0, x1, cons(x2, x3), x4)
max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
min(0, x0)
min(x0, 0)
min(s(x0), s(x1))
asort(x0)
dsort(x0)

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

The following pairs can be oriented strictly and are deleted.

APP(app(min, app(s, x)), app(s, y)) → APP(app(min, x), y)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
MIN(x1, x2) = x2
s(x1) = s(x1)

Lexicographic path order with status [19].
Precedence:

trivial

Status:

trivial

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ EdgeDeletionProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ PisEmptyProof

                  ↳ QDP

                  ↳ QDP

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

app(app(app(sort, f), g), nil) → nil
app(app(app(sort, f), g), app(app(cons, x), y)) → app(app(app(app(insert, f), g), app(app(app(sort, f), g), y)), x)
app(app(app(app(insert, f), g), nil), y) → app(app(cons, y), nil)
app(app(app(app(insert, f), g), app(app(cons, x), z)), y) → app(app(cons, app(app(f, x), y)), app(app(app(app(insert, f), g), z), app(app(g, x), y)))
app(app(max, 0), y) → y
app(app(max, x), 0) → x
app(app(max, app(s, x)), app(s, y)) → app(app(max, x), y)
app(app(min, 0), y) → 0
app(app(min, x), 0) → 0
app(app(min, app(s, x)), app(s, y)) → app(app(min, x), y)
app(asort, z) → app(app(app(sort, min), max), z)
app(dsort, z) → app(app(app(sort, max), min), z)

The set Q consists of the following terms:

app(app(app(sort, x0), x1), nil)
app(app(app(sort, x0), x1), app(app(cons, x2), x3))
app(app(app(app(insert, x0), x1), nil), x2)
app(app(app(app(insert, x0), x1), app(app(cons, x2), x3)), x4)
app(app(max, 0), x0)
app(app(max, x0), 0)
app(app(max, app(s, x0)), app(s, x1))
app(app(min, 0), x0)
app(app(min, x0), 0)
app(app(min, app(s, x0)), app(s, x1))
app(asort, x0)
app(dsort, x0)

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

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ EdgeDeletionProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPOrderProof

                  ↳ QDP

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

APP(app(max, app(s, x)), app(s, y)) → APP(app(max, x), y)

The TRS R consists of the following rules:

app(app(app(sort, f), g), nil) → nil
app(app(app(sort, f), g), app(app(cons, x), y)) → app(app(app(app(insert, f), g), app(app(app(sort, f), g), y)), x)
app(app(app(app(insert, f), g), nil), y) → app(app(cons, y), nil)
app(app(app(app(insert, f), g), app(app(cons, x), z)), y) → app(app(cons, app(app(f, x), y)), app(app(app(app(insert, f), g), z), app(app(g, x), y)))
app(app(max, 0), y) → y
app(app(max, x), 0) → x
app(app(max, app(s, x)), app(s, y)) → app(app(max, x), y)
app(app(min, 0), y) → 0
app(app(min, x), 0) → 0
app(app(min, app(s, x)), app(s, y)) → app(app(min, x), y)
app(asort, z) → app(app(app(sort, min), max), z)
app(dsort, z) → app(app(app(sort, max), min), z)

The set Q consists of the following terms:

app(app(app(sort, x0), x1), nil)
app(app(app(sort, x0), x1), app(app(cons, x2), x3))
app(app(app(app(insert, x0), x1), nil), x2)
app(app(app(app(insert, x0), x1), app(app(cons, x2), x3)), x4)
app(app(max, 0), x0)
app(app(max, x0), 0)
app(app(max, app(s, x0)), app(s, x1))
app(app(min, 0), x0)
app(app(min, x0), 0)
app(app(min, app(s, x0)), app(s, x1))
app(asort, x0)
app(dsort, x0)

MAX(s(x), s(y)) → MAX(x, y)

R is empty.
The set Q consists of the following terms:

sort(x0, x1, nil)
sort(x0, x1, cons(x2, x3))
insert(x0, x1, nil, x2)
insert(x0, x1, cons(x2, x3), x4)
max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
min(0, x0)
min(x0, 0)
min(s(x0), s(x1))
asort(x0)
dsort(x0)

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

The following pairs can be oriented strictly and are deleted.

APP(app(max, app(s, x)), app(s, y)) → APP(app(max, x), y)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
MAX(x1, x2) = x2
s(x1) = s(x1)

Lexicographic path order with status [19].
Precedence:

trivial

Status:

trivial

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ EdgeDeletionProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ PisEmptyProof

                  ↳ QDP

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

app(app(app(sort, f), g), nil) → nil
app(app(app(sort, f), g), app(app(cons, x), y)) → app(app(app(app(insert, f), g), app(app(app(sort, f), g), y)), x)
app(app(app(app(insert, f), g), nil), y) → app(app(cons, y), nil)
app(app(app(app(insert, f), g), app(app(cons, x), z)), y) → app(app(cons, app(app(f, x), y)), app(app(app(app(insert, f), g), z), app(app(g, x), y)))
app(app(max, 0), y) → y
app(app(max, x), 0) → x
app(app(max, app(s, x)), app(s, y)) → app(app(max, x), y)
app(app(min, 0), y) → 0
app(app(min, x), 0) → 0
app(app(min, app(s, x)), app(s, y)) → app(app(min, x), y)
app(asort, z) → app(app(app(sort, min), max), z)
app(dsort, z) → app(app(app(sort, max), min), z)

The set Q consists of the following terms:

app(app(app(sort, x0), x1), nil)
app(app(app(sort, x0), x1), app(app(cons, x2), x3))
app(app(app(app(insert, x0), x1), nil), x2)
app(app(app(app(insert, x0), x1), app(app(cons, x2), x3)), x4)
app(app(max, 0), x0)
app(app(max, x0), 0)
app(app(max, app(s, x0)), app(s, x1))
app(app(min, 0), x0)
app(app(min, x0), 0)
app(app(min, app(s, x0)), app(s, x1))
app(asort, x0)
app(dsort, x0)

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

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ EdgeDeletionProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

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

APP(app(app(app(insert, f), g), app(app(cons, x), z)), y) → APP(app(g, x), y)
APP(app(app(app(insert, f), g), app(app(cons, x), z)), y) → APP(app(app(app(insert, f), g), z), app(app(g, x), y))
APP(app(app(sort, f), g), app(app(cons, x), y)) → APP(app(app(sort, f), g), y)
APP(asort, z) → APP(app(app(sort, min), max), z)
APP(app(app(sort, f), g), app(app(cons, x), y)) → APP(app(app(app(insert, f), g), app(app(app(sort, f), g), y)), x)
APP(dsort, z) → APP(app(app(sort, max), min), z)
APP(app(app(app(insert, f), g), app(app(cons, x), z)), y) → APP(app(f, x), y)
APP(app(app(app(insert, f), g), app(app(cons, x), z)), y) → APP(g, x)
APP(app(app(app(insert, f), g), app(app(cons, x), z)), y) → APP(f, x)

The TRS R consists of the following rules:

app(app(app(sort, f), g), nil) → nil
app(app(app(sort, f), g), app(app(cons, x), y)) → app(app(app(app(insert, f), g), app(app(app(sort, f), g), y)), x)
app(app(app(app(insert, f), g), nil), y) → app(app(cons, y), nil)
app(app(app(app(insert, f), g), app(app(cons, x), z)), y) → app(app(cons, app(app(f, x), y)), app(app(app(app(insert, f), g), z), app(app(g, x), y)))
app(app(max, 0), y) → y
app(app(max, x), 0) → x
app(app(max, app(s, x)), app(s, y)) → app(app(max, x), y)
app(app(min, 0), y) → 0
app(app(min, x), 0) → 0
app(app(min, app(s, x)), app(s, y)) → app(app(min, x), y)
app(asort, z) → app(app(app(sort, min), max), z)
app(dsort, z) → app(app(app(sort, max), min), z)

The set Q consists of the following terms:

app(app(app(sort, x0), x1), nil)
app(app(app(sort, x0), x1), app(app(cons, x2), x3))
app(app(app(app(insert, x0), x1), nil), x2)
app(app(app(app(insert, x0), x1), app(app(cons, x2), x3)), x4)
app(app(max, 0), x0)
app(app(max, x0), 0)
app(app(max, app(s, x0)), app(s, x1))
app(app(min, 0), x0)
app(app(min, x0), 0)
app(app(min, app(s, x0)), app(s, x1))
app(asort, x0)
app(dsort, x0)

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