Termination w.r.t. Q proof of TRS/currying/D3321.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(p, app(s, x)) → x
app(fact, 0) → app(s, 0)
app(fact, app(s, x)) → app(app(*, app(s, x)), app(fact, app(p, app(s, x))))
app(app(*, 0), y) → 0
app(app(*, app(s, x)), y) → app(app(+, app(app(*, x), y)), y)
app(app(+, x), 0) → x
app(app(+, x), app(s, y)) → app(s, app(app(+, x), y))
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)

Q is empty.

↳ QTRS

  ↳ Overlay + Local Confluence

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

app(p, app(s, x)) → x
app(fact, 0) → app(s, 0)
app(fact, app(s, x)) → app(app(*, app(s, x)), app(fact, app(p, app(s, x))))
app(app(*, 0), y) → 0
app(app(*, app(s, x)), y) → app(app(+, app(app(*, x), y)), y)
app(app(+, x), 0) → x
app(app(+, x), app(s, y)) → app(s, app(app(+, x), y))
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)

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(p, app(s, x)) → x
app(fact, 0) → app(s, 0)
app(fact, app(s, x)) → app(app(*, app(s, x)), app(fact, app(p, app(s, x))))
app(app(*, 0), y) → 0
app(app(*, app(s, x)), y) → app(app(+, app(app(*, x), y)), y)
app(app(+, x), 0) → x
app(app(+, x), app(s, y)) → app(s, app(app(+, x), y))
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)

The set Q consists of the following terms:

app(p, app(s, x0))
app(fact, 0)
app(fact, app(s, x0))
app(app(*, 0), x0)
app(app(*, app(s, x0)), x1)
app(app(+, x0), 0)
app(app(+, x0), app(s, x1))
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), 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(map, f), app(app(cons, x), xs)) → APP(app(cons, app(f, x)), app(app(map, f), xs))
APP(fact, 0) → APP(s, 0)
APP(app(filter, f), app(app(cons, x), xs)) → APP(app(app(filter2, app(f, x)), f), x)
APP(app(filter, f), app(app(cons, x), xs)) → APP(app(app(app(filter2, app(f, x)), f), x), xs)
APP(app(app(app(filter2, true), f), x), xs) → APP(app(cons, x), app(app(filter, f), xs))
APP(app(*, app(s, x)), y) → APP(app(*, x), y)
APP(app(*, app(s, x)), y) → APP(app(+, app(app(*, x), y)), y)
APP(app(map, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(filter, f), app(app(cons, x), xs)) → APP(filter2, app(f, x))
APP(fact, app(s, x)) → APP(p, app(s, x))
APP(app(app(app(filter2, true), f), x), xs) → APP(app(filter, f), xs)
APP(app(*, app(s, x)), y) → APP(+, app(app(*, x), y))
APP(app(+, x), app(s, y)) → APP(app(+, x), y)
APP(fact, app(s, x)) → APP(*, app(s, x))
APP(app(filter, f), app(app(cons, x), xs)) → APP(app(filter2, app(f, x)), f)
APP(app(app(app(filter2, false), f), x), xs) → APP(filter, f)
APP(fact, app(s, x)) → APP(fact, app(p, app(s, x)))
APP(app(map, f), app(app(cons, x), xs)) → APP(app(map, f), xs)
APP(app(filter, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(*, app(s, x)), y) → APP(*, x)
APP(app(app(app(filter2, true), f), x), xs) → APP(filter, f)
APP(fact, app(s, x)) → APP(app(*, app(s, x)), app(fact, app(p, app(s, x))))
APP(app(app(app(filter2, true), f), x), xs) → APP(cons, x)
APP(app(app(app(filter2, false), f), x), xs) → APP(app(filter, f), xs)
APP(app(map, f), app(app(cons, x), xs)) → APP(cons, app(f, x))
APP(app(+, x), app(s, y)) → APP(s, app(app(+, x), y))

The TRS R consists of the following rules:

app(p, app(s, x)) → x
app(fact, 0) → app(s, 0)
app(fact, app(s, x)) → app(app(*, app(s, x)), app(fact, app(p, app(s, x))))
app(app(*, 0), y) → 0
app(app(*, app(s, x)), y) → app(app(+, app(app(*, x), y)), y)
app(app(+, x), 0) → x
app(app(+, x), app(s, y)) → app(s, app(app(+, x), y))
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)

The set Q consists of the following terms:

app(p, app(s, x0))
app(fact, 0)
app(fact, app(s, x0))
app(app(*, 0), x0)
app(app(*, app(s, x0)), x1)
app(app(+, x0), 0)
app(app(+, x0), app(s, x1))
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), 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(map, f), app(app(cons, x), xs)) → APP(app(cons, app(f, x)), app(app(map, f), xs))
APP(fact, 0) → APP(s, 0)
APP(app(filter, f), app(app(cons, x), xs)) → APP(app(app(filter2, app(f, x)), f), x)
APP(app(filter, f), app(app(cons, x), xs)) → APP(app(app(app(filter2, app(f, x)), f), x), xs)
APP(app(app(app(filter2, true), f), x), xs) → APP(app(cons, x), app(app(filter, f), xs))
APP(app(*, app(s, x)), y) → APP(app(*, x), y)
APP(app(*, app(s, x)), y) → APP(app(+, app(app(*, x), y)), y)
APP(app(map, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(filter, f), app(app(cons, x), xs)) → APP(filter2, app(f, x))
APP(fact, app(s, x)) → APP(p, app(s, x))
APP(app(app(app(filter2, true), f), x), xs) → APP(app(filter, f), xs)
APP(app(*, app(s, x)), y) → APP(+, app(app(*, x), y))
APP(app(+, x), app(s, y)) → APP(app(+, x), y)
APP(fact, app(s, x)) → APP(*, app(s, x))
APP(app(filter, f), app(app(cons, x), xs)) → APP(app(filter2, app(f, x)), f)
APP(app(app(app(filter2, false), f), x), xs) → APP(filter, f)
APP(fact, app(s, x)) → APP(fact, app(p, app(s, x)))
APP(app(map, f), app(app(cons, x), xs)) → APP(app(map, f), xs)
APP(app(filter, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(*, app(s, x)), y) → APP(*, x)
APP(app(app(app(filter2, true), f), x), xs) → APP(filter, f)
APP(fact, app(s, x)) → APP(app(*, app(s, x)), app(fact, app(p, app(s, x))))
APP(app(app(app(filter2, true), f), x), xs) → APP(cons, x)
APP(app(app(app(filter2, false), f), x), xs) → APP(app(filter, f), xs)
APP(app(map, f), app(app(cons, x), xs)) → APP(cons, app(f, x))
APP(app(+, x), app(s, y)) → APP(s, app(app(+, x), y))

The TRS R consists of the following rules:

app(p, app(s, x)) → x
app(fact, 0) → app(s, 0)
app(fact, app(s, x)) → app(app(*, app(s, x)), app(fact, app(p, app(s, x))))
app(app(*, 0), y) → 0
app(app(*, app(s, x)), y) → app(app(+, app(app(*, x), y)), y)
app(app(+, x), 0) → x
app(app(+, x), app(s, y)) → app(s, app(app(+, x), y))
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)

The set Q consists of the following terms:

app(p, app(s, x0))
app(fact, 0)
app(fact, app(s, x0))
app(app(*, 0), x0)
app(app(*, app(s, x0)), x1)
app(app(+, x0), 0)
app(app(+, x0), app(s, x1))
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), 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(map, f), app(app(cons, x), xs)) → APP(app(cons, app(f, x)), app(app(map, f), xs))
APP(fact, 0) → APP(s, 0)
APP(app(filter, f), app(app(cons, x), xs)) → APP(app(app(filter2, app(f, x)), f), x)
APP(app(filter, f), app(app(cons, x), xs)) → APP(app(app(app(filter2, app(f, x)), f), x), xs)
APP(app(app(app(filter2, true), f), x), xs) → APP(app(cons, x), app(app(filter, f), xs))
APP(app(*, app(s, x)), y) → APP(app(*, x), y)
APP(app(filter, f), app(app(cons, x), xs)) → APP(filter2, app(f, x))
APP(app(map, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(*, app(s, x)), y) → APP(app(+, app(app(*, x), y)), y)
APP(fact, app(s, x)) → APP(p, app(s, x))
APP(app(app(app(filter2, true), f), x), xs) → APP(app(filter, f), xs)
APP(app(*, app(s, x)), y) → APP(+, app(app(*, x), y))
APP(app(+, x), app(s, y)) → APP(app(+, x), y)
APP(app(filter, f), app(app(cons, x), xs)) → APP(app(filter2, app(f, x)), f)
APP(fact, app(s, x)) → APP(*, app(s, x))
APP(app(app(app(filter2, false), f), x), xs) → APP(filter, f)
APP(fact, app(s, x)) → APP(fact, app(p, app(s, x)))
APP(app(filter, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(map, f), app(app(cons, x), xs)) → APP(app(map, f), xs)
APP(app(*, app(s, x)), y) → APP(*, x)
APP(app(app(app(filter2, true), f), x), xs) → APP(filter, f)
APP(fact, app(s, x)) → APP(app(*, app(s, x)), app(fact, app(p, app(s, x))))
APP(app(app(app(filter2, true), f), x), xs) → APP(cons, x)
APP(app(app(app(filter2, false), f), x), xs) → APP(app(filter, f), xs)
APP(app(map, f), app(app(cons, x), xs)) → APP(cons, app(f, x))
APP(app(+, x), app(s, y)) → APP(s, app(app(+, x), y))

The TRS R consists of the following rules:

app(p, app(s, x)) → x
app(fact, 0) → app(s, 0)
app(fact, app(s, x)) → app(app(*, app(s, x)), app(fact, app(p, app(s, x))))
app(app(*, 0), y) → 0
app(app(*, app(s, x)), y) → app(app(+, app(app(*, x), y)), y)
app(app(+, x), 0) → x
app(app(+, x), app(s, y)) → app(s, app(app(+, x), y))
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)

The set Q consists of the following terms:

app(p, app(s, x0))
app(fact, 0)
app(fact, app(s, x0))
app(app(*, 0), x0)
app(app(*, app(s, x0)), x1)
app(app(+, x0), 0)
app(app(+, x0), app(s, x1))
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 4 SCCs with 17 less nodes.

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ EdgeDeletionProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

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

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

The TRS R consists of the following rules:

app(p, app(s, x)) → x
app(fact, 0) → app(s, 0)
app(fact, app(s, x)) → app(app(*, app(s, x)), app(fact, app(p, app(s, x))))
app(app(*, 0), y) → 0
app(app(*, app(s, x)), y) → app(app(+, app(app(*, x), y)), y)
app(app(+, x), 0) → x
app(app(+, x), app(s, y)) → app(s, app(app(+, x), y))
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)

The set Q consists of the following terms:

app(p, app(s, x0))
app(fact, 0)
app(fact, app(s, x0))
app(app(*, 0), x0)
app(app(*, app(s, x0)), x1)
app(app(+, x0), 0)
app(app(+, x0), app(s, x1))
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)

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:

+1(x, s(y)) → +1(x, y)

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

P(s(x0))
FACT(0)
FACT(s(x0))
*(0, x0)
*(s(x0), x1)
+(x0, 0)
+(x0, s(x1))
map(x0, nil)
map(x0, cons(x1, x2))
filter(x0, nil)
filter(x0, cons(x1, x2))
filter2(true, x0, x1, x2)
filter2(false, x0, x1, x2)

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

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
+1(x1, x2) = +1(x1, x2)
s(x1) = s(x1)

Lexicographic path order with status [19].
Precedence:

s₁ > +1₂

Status:

s₁: multiset
+1₂: [1,2]

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

                  ↳ QDP

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

app(p, app(s, x)) → x
app(fact, 0) → app(s, 0)
app(fact, app(s, x)) → app(app(*, app(s, x)), app(fact, app(p, app(s, x))))
app(app(*, 0), y) → 0
app(app(*, app(s, x)), y) → app(app(+, app(app(*, x), y)), y)
app(app(+, x), 0) → x
app(app(+, x), app(s, y)) → app(s, app(app(+, x), y))
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)

The set Q consists of the following terms:

app(p, app(s, x0))
app(fact, 0)
app(fact, app(s, x0))
app(app(*, 0), x0)
app(app(*, app(s, x0)), x1)
app(app(+, x0), 0)
app(app(+, x0), app(s, x1))
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), 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.

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ EdgeDeletionProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPOrderProof

                  ↳ QDP

                  ↳ QDP

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

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

The TRS R consists of the following rules:

app(p, app(s, x)) → x
app(fact, 0) → app(s, 0)
app(fact, app(s, x)) → app(app(*, app(s, x)), app(fact, app(p, app(s, x))))
app(app(*, 0), y) → 0
app(app(*, app(s, x)), y) → app(app(+, app(app(*, x), y)), y)
app(app(+, x), 0) → x
app(app(+, x), app(s, y)) → app(s, app(app(+, x), y))
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)

The set Q consists of the following terms:

app(p, app(s, x0))
app(fact, 0)
app(fact, app(s, x0))
app(app(*, 0), x0)
app(app(*, app(s, x0)), x1)
app(app(+, x0), 0)
app(app(+, x0), app(s, x1))
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)

*1(s(x), y) → *1(x, y)

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

P(s(x0))
FACT(0)
FACT(s(x0))
*(0, x0)
*(s(x0), x1)
+(x0, 0)
+(x0, s(x1))
map(x0, nil)
map(x0, cons(x1, x2))
filter(x0, nil)
filter(x0, cons(x1, x2))
filter2(true, x0, x1, x2)
filter2(false, x0, x1, x2)

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

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
*1(x1, x2) = *1(x1)
s(x1) = s(x1)

Lexicographic path order with status [19].
Precedence:

s₁ > *1₁

Status:

s₁: multiset
*1₁: [1]

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

                  ↳ QDP

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

app(p, app(s, x)) → x
app(fact, 0) → app(s, 0)
app(fact, app(s, x)) → app(app(*, app(s, x)), app(fact, app(p, app(s, x))))
app(app(*, 0), y) → 0
app(app(*, app(s, x)), y) → app(app(+, app(app(*, x), y)), y)
app(app(+, x), 0) → x
app(app(+, x), app(s, y)) → app(s, app(app(+, x), y))
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)

The set Q consists of the following terms:

app(p, app(s, x0))
app(fact, 0)
app(fact, app(s, x0))
app(app(*, 0), x0)
app(app(*, app(s, x0)), x1)
app(app(+, x0), 0)
app(app(+, x0), app(s, x1))
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), 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.

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ EdgeDeletionProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

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

APP(fact, app(s, x)) → APP(fact, app(p, app(s, x)))

The TRS R consists of the following rules:

app(p, app(s, x)) → x
app(fact, 0) → app(s, 0)
app(fact, app(s, x)) → app(app(*, app(s, x)), app(fact, app(p, app(s, x))))
app(app(*, 0), y) → 0
app(app(*, app(s, x)), y) → app(app(+, app(app(*, x), y)), y)
app(app(+, x), 0) → x
app(app(+, x), app(s, y)) → app(s, app(app(+, x), y))
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)

The set Q consists of the following terms:

app(p, app(s, x0))
app(fact, 0)
app(fact, app(s, x0))
app(app(*, 0), x0)
app(app(*, app(s, x0)), x1)
app(app(+, x0), 0)
app(app(+, x0), app(s, x1))
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)

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

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ EdgeDeletionProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPOrderProof

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

APP(app(app(app(filter2, true), f), x), xs) → APP(app(filter, f), xs)
APP(app(map, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(filter, f), app(app(cons, x), xs)) → APP(app(app(app(filter2, app(f, x)), f), x), xs)
APP(app(app(app(filter2, false), f), x), xs) → APP(app(filter, f), xs)
APP(app(filter, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(map, f), app(app(cons, x), xs)) → APP(app(map, f), xs)

The TRS R consists of the following rules:

app(p, app(s, x)) → x
app(fact, 0) → app(s, 0)
app(fact, app(s, x)) → app(app(*, app(s, x)), app(fact, app(p, app(s, x))))
app(app(*, 0), y) → 0
app(app(*, app(s, x)), y) → app(app(+, app(app(*, x), y)), y)
app(app(+, x), 0) → x
app(app(+, x), app(s, y)) → app(s, app(app(+, x), y))
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)

The set Q consists of the following terms:

app(p, app(s, x0))
app(fact, 0)
app(fact, app(s, x0))
app(app(*, 0), x0)
app(app(*, app(s, x0)), x1)
app(app(+, x0), 0)
app(app(+, x0), app(s, x1))
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), 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(map, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(filter, f), app(app(cons, x), xs)) → APP(app(app(app(filter2, app(f, x)), f), x), xs)
APP(app(filter, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(map, f), app(app(cons, x), xs)) → APP(app(map, f), xs)

The remaining pairs can at least be oriented weakly.

APP(app(app(app(filter2, true), f), x), xs) → APP(app(filter, f), xs)
APP(app(app(app(filter2, false), f), x), xs) → APP(app(filter, f), xs)

Used ordering: Combined order from the following AFS and order.
APP(x1, x2) = APP(x2)
app(x1, x2) = app(x1, x2)
filter2 = filter2
true = true
filter = filter
map = map
cons = cons
false = false
fact = fact
0 = 0
s = s
nil = nil
* = *
+ = +
p = p

Lexicographic path order with status [19].
Precedence:

APP₁ > app₂
filter2 > app₂
true > app₂
filter > app₂
map > app₂
cons > app₂
false > app₂
fact > app₂
0 > app₂
s > app₂
* > app₂
+ > app₂

Status:

APP₁: [1]
map: multiset
fact: multiset
0: multiset
s: multiset
nil: multiset
p: multiset
*: multiset
cons: multiset
filter: multiset
true: multiset
false: multiset
app₂: [1,2]
+: multiset
filter2: multiset

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ EdgeDeletionProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

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

APP(app(app(app(filter2, true), f), x), xs) → APP(app(filter, f), xs)
APP(app(app(app(filter2, false), f), x), xs) → APP(app(filter, f), xs)

The TRS R consists of the following rules:

app(p, app(s, x)) → x
app(fact, 0) → app(s, 0)
app(fact, app(s, x)) → app(app(*, app(s, x)), app(fact, app(p, app(s, x))))
app(app(*, 0), y) → 0
app(app(*, app(s, x)), y) → app(app(+, app(app(*, x), y)), y)
app(app(+, x), 0) → x
app(app(+, x), app(s, y)) → app(s, app(app(+, x), y))
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)

The set Q consists of the following terms:

app(p, app(s, x0))
app(fact, 0)
app(fact, app(s, x0))
app(app(*, 0), x0)
app(app(*, app(s, x0)), x1)
app(app(+, x0), 0)
app(app(+, x0), app(s, x1))
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 0 SCCs with 2 less nodes.