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

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(app(compose, f), g), x) → app(g, app(f, x))
app(reverse, l) → app(app(reverse2, l), nil)
app(app(reverse2, nil), l) → l
app(app(reverse2, app(app(cons, x), xs)), l) → app(app(reverse2, xs), app(app(cons, x), l))
app(hd, app(app(cons, x), xs)) → x
app(tl, app(app(cons, x), xs)) → xs
last → app(app(compose, hd), reverse)
init → app(app(compose, reverse), app(app(compose, tl), reverse))

Q is empty.

↳ QTRS

  ↳ Overlay + Local Confluence

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

app(app(app(compose, f), g), x) → app(g, app(f, x))
app(reverse, l) → app(app(reverse2, l), nil)
app(app(reverse2, nil), l) → l
app(app(reverse2, app(app(cons, x), xs)), l) → app(app(reverse2, xs), app(app(cons, x), l))
app(hd, app(app(cons, x), xs)) → x
app(tl, app(app(cons, x), xs)) → xs
last → app(app(compose, hd), reverse)
init → app(app(compose, reverse), app(app(compose, tl), reverse))

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(compose, f), g), x) → app(g, app(f, x))
app(reverse, l) → app(app(reverse2, l), nil)
app(app(reverse2, nil), l) → l
app(app(reverse2, app(app(cons, x), xs)), l) → app(app(reverse2, xs), app(app(cons, x), l))
app(hd, app(app(cons, x), xs)) → x
app(tl, app(app(cons, x), xs)) → xs
last → app(app(compose, hd), reverse)
init → app(app(compose, reverse), app(app(compose, tl), reverse))

The set Q consists of the following terms:

app(app(app(compose, x0), x1), x2)
app(reverse, x0)
app(app(reverse2, nil), x0)
app(app(reverse2, app(app(cons, x0), x1)), x2)
app(hd, app(app(cons, x0), x1))
app(tl, app(app(cons, x0), x1))
last
init

Using Dependency Pairs [1,13] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

LAST → APP(app(compose, hd), reverse)
LAST → APP(compose, hd)
APP(app(reverse2, app(app(cons, x), xs)), l) → APP(app(cons, x), l)
INIT → APP(app(compose, tl), reverse)
INIT → APP(compose, tl)
APP(app(app(compose, f), g), x) → APP(f, x)
APP(app(reverse2, app(app(cons, x), xs)), l) → APP(reverse2, xs)
INIT → APP(compose, reverse)
APP(reverse, l) → APP(reverse2, l)
APP(app(reverse2, app(app(cons, x), xs)), l) → APP(app(reverse2, xs), app(app(cons, x), l))
APP(app(app(compose, f), g), x) → APP(g, app(f, x))
APP(reverse, l) → APP(app(reverse2, l), nil)
INIT → APP(app(compose, reverse), app(app(compose, tl), reverse))

The TRS R consists of the following rules:

app(app(app(compose, f), g), x) → app(g, app(f, x))
app(reverse, l) → app(app(reverse2, l), nil)
app(app(reverse2, nil), l) → l
app(app(reverse2, app(app(cons, x), xs)), l) → app(app(reverse2, xs), app(app(cons, x), l))
app(hd, app(app(cons, x), xs)) → x
app(tl, app(app(cons, x), xs)) → xs
last → app(app(compose, hd), reverse)
init → app(app(compose, reverse), app(app(compose, tl), reverse))

The set Q consists of the following terms:

app(app(app(compose, x0), x1), x2)
app(reverse, x0)
app(app(reverse2, nil), x0)
app(app(reverse2, app(app(cons, x0), x1)), x2)
app(hd, app(app(cons, x0), x1))
app(tl, app(app(cons, x0), x1))
last
init

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:

LAST → APP(app(compose, hd), reverse)
LAST → APP(compose, hd)
APP(app(reverse2, app(app(cons, x), xs)), l) → APP(app(cons, x), l)
INIT → APP(app(compose, tl), reverse)
INIT → APP(compose, tl)
APP(app(app(compose, f), g), x) → APP(f, x)
APP(app(reverse2, app(app(cons, x), xs)), l) → APP(reverse2, xs)
INIT → APP(compose, reverse)
APP(reverse, l) → APP(reverse2, l)
APP(app(reverse2, app(app(cons, x), xs)), l) → APP(app(reverse2, xs), app(app(cons, x), l))
APP(app(app(compose, f), g), x) → APP(g, app(f, x))
APP(reverse, l) → APP(app(reverse2, l), nil)
INIT → APP(app(compose, reverse), app(app(compose, tl), reverse))

The TRS R consists of the following rules:

app(app(app(compose, f), g), x) → app(g, app(f, x))
app(reverse, l) → app(app(reverse2, l), nil)
app(app(reverse2, nil), l) → l
app(app(reverse2, app(app(cons, x), xs)), l) → app(app(reverse2, xs), app(app(cons, x), l))
app(hd, app(app(cons, x), xs)) → x
app(tl, app(app(cons, x), xs)) → xs
last → app(app(compose, hd), reverse)
init → app(app(compose, reverse), app(app(compose, tl), reverse))

The set Q consists of the following terms:

app(app(app(compose, x0), x1), x2)
app(reverse, x0)
app(app(reverse2, nil), x0)
app(app(reverse2, app(app(cons, x0), x1)), x2)
app(hd, app(app(cons, x0), x1))
app(tl, app(app(cons, x0), x1))
last
init

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:

LAST → APP(app(compose, hd), reverse)
LAST → APP(compose, hd)
INIT → APP(app(compose, tl), reverse)
APP(app(reverse2, app(app(cons, x), xs)), l) → APP(app(cons, x), l)
INIT → APP(compose, tl)
APP(app(app(compose, f), g), x) → APP(f, x)
INIT → APP(compose, reverse)
APP(app(reverse2, app(app(cons, x), xs)), l) → APP(reverse2, xs)
APP(reverse, l) → APP(reverse2, l)
APP(app(reverse2, app(app(cons, x), xs)), l) → APP(app(reverse2, xs), app(app(cons, x), l))
APP(app(app(compose, f), g), x) → APP(g, app(f, x))
APP(reverse, l) → APP(app(reverse2, l), nil)
INIT → APP(app(compose, reverse), app(app(compose, tl), reverse))

The TRS R consists of the following rules:

app(app(app(compose, f), g), x) → app(g, app(f, x))
app(reverse, l) → app(app(reverse2, l), nil)
app(app(reverse2, nil), l) → l
app(app(reverse2, app(app(cons, x), xs)), l) → app(app(reverse2, xs), app(app(cons, x), l))
app(hd, app(app(cons, x), xs)) → x
app(tl, app(app(cons, x), xs)) → xs
last → app(app(compose, hd), reverse)
init → app(app(compose, reverse), app(app(compose, tl), reverse))

The set Q consists of the following terms:

app(app(app(compose, x0), x1), x2)
app(reverse, x0)
app(app(reverse2, nil), x0)
app(app(reverse2, app(app(cons, x0), x1)), x2)
app(hd, app(app(cons, x0), x1))
app(tl, app(app(cons, x0), x1))
last
init

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 2 SCCs with 10 less nodes.

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ EdgeDeletionProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                  ↳ QDP

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

APP(app(reverse2, app(app(cons, x), xs)), l) → APP(app(reverse2, xs), app(app(cons, x), l))

The TRS R consists of the following rules:

app(app(app(compose, f), g), x) → app(g, app(f, x))
app(reverse, l) → app(app(reverse2, l), nil)
app(app(reverse2, nil), l) → l
app(app(reverse2, app(app(cons, x), xs)), l) → app(app(reverse2, xs), app(app(cons, x), l))
app(hd, app(app(cons, x), xs)) → x
app(tl, app(app(cons, x), xs)) → xs
last → app(app(compose, hd), reverse)
init → app(app(compose, reverse), app(app(compose, tl), reverse))

The set Q consists of the following terms:

app(app(app(compose, x0), x1), x2)
app(reverse, x0)
app(app(reverse2, nil), x0)
app(app(reverse2, app(app(cons, x0), x1)), x2)
app(hd, app(app(cons, x0), x1))
app(tl, app(app(cons, x0), x1))
last
init

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:

REVERSE2(cons(x, xs), l) → REVERSE2(xs, cons(x, l))

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

compose(x0, x1, x2)
reverse(x0)
reverse2(nil, x0)
reverse2(cons(x0, x1), x2)
hd(cons(x0, x1))
tl(cons(x0, x1))
last
init

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

The following pairs can be oriented strictly and are deleted.

APP(app(reverse2, app(app(cons, x), xs)), l) → APP(app(reverse2, xs), app(app(cons, x), l))

The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
REVERSE2(x1, x2) = REVERSE2(x1)
cons(x1, x2) = cons(x1, x2)

Recursive path order with status [2].
Quasi-Precedence:

[REVERSE2₁, cons_2]

Status:

cons₂: [2,1]
REVERSE2₁: multiset

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ EdgeDeletionProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ PisEmptyProof

                  ↳ QDP

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

app(app(app(compose, f), g), x) → app(g, app(f, x))
app(reverse, l) → app(app(reverse2, l), nil)
app(app(reverse2, nil), l) → l
app(app(reverse2, app(app(cons, x), xs)), l) → app(app(reverse2, xs), app(app(cons, x), l))
app(hd, app(app(cons, x), xs)) → x
app(tl, app(app(cons, x), xs)) → xs
last → app(app(compose, hd), reverse)
init → app(app(compose, reverse), app(app(compose, tl), reverse))

The set Q consists of the following terms:

app(app(app(compose, x0), x1), x2)
app(reverse, x0)
app(app(reverse2, nil), x0)
app(app(reverse2, app(app(cons, x0), x1)), x2)
app(hd, app(app(cons, x0), x1))
app(tl, app(app(cons, x0), x1))
last
init

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

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

APP(app(app(compose, f), g), x) → APP(g, app(f, x))
APP(app(app(compose, f), g), x) → APP(f, x)

The TRS R consists of the following rules:

app(app(app(compose, f), g), x) → app(g, app(f, x))
app(reverse, l) → app(app(reverse2, l), nil)
app(app(reverse2, nil), l) → l
app(app(reverse2, app(app(cons, x), xs)), l) → app(app(reverse2, xs), app(app(cons, x), l))
app(hd, app(app(cons, x), xs)) → x
app(tl, app(app(cons, x), xs)) → xs
last → app(app(compose, hd), reverse)
init → app(app(compose, reverse), app(app(compose, tl), reverse))

The set Q consists of the following terms:

app(app(app(compose, x0), x1), x2)
app(reverse, x0)
app(app(reverse2, nil), x0)
app(app(reverse2, app(app(cons, x0), x1)), x2)
app(hd, app(app(cons, x0), x1))
app(tl, app(app(cons, x0), x1))
last
init

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(app(compose, f), g), x) → APP(g, app(f, x))
APP(app(app(compose, f), g), 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) = APP(x1)
app(x1, x2) = app(x1, x2)
compose = compose
reverse2 = reverse2
cons = cons
reverse = reverse
nil = nil
tl = tl
hd = hd

Recursive path order with status [2].
Quasi-Precedence:

[app₂, reverse2] > [APP₁, compose]
[app₂, reverse2] > cons
[app₂, reverse2] > [reverse, nil]

Status:

APP₁: multiset
hd: multiset
app₂: multiset
compose: multiset
tl: multiset
reverse2: multiset
nil: multiset
cons: multiset
reverse: multiset

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ EdgeDeletionProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ PisEmptyProof

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

app(app(app(compose, f), g), x) → app(g, app(f, x))
app(reverse, l) → app(app(reverse2, l), nil)
app(app(reverse2, nil), l) → l
app(app(reverse2, app(app(cons, x), xs)), l) → app(app(reverse2, xs), app(app(cons, x), l))
app(hd, app(app(cons, x), xs)) → x
app(tl, app(app(cons, x), xs)) → xs
last → app(app(compose, hd), reverse)
init → app(app(compose, reverse), app(app(compose, tl), reverse))

The set Q consists of the following terms:

app(app(app(compose, x0), x1), x2)
app(reverse, x0)
app(app(reverse2, nil), x0)
app(app(reverse2, app(app(cons, x0), x1)), x2)
app(hd, app(app(cons, x0), x1))
app(tl, app(app(cons, x0), x1))
last
init

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