Termination w.r.t. Q proof of TRS/D3330.trs

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:

:(:(:(:(C, x), y), z), u) → :(:(x, z), :(:(:(x, y), z), u))

Q is empty.

↳ QTRS

  ↳ Overlay + Local Confluence

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

:(:(:(:(C, x), y), z), u) → :(:(x, z), :(:(:(x, y), z), u))

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:

:(:(:(:(C, x), y), z), u) → :(:(x, z), :(:(:(x, y), z), u))

The set Q consists of the following terms:

:(:(:(:(C, x0), x1), 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:

:¹(:(:(:(C, x), y), z), u) → :¹(x, z)
:¹(:(:(:(C, x), y), z), u) → :¹(x, y)
:¹(:(:(:(C, x), y), z), u) → :¹(:(x, z), :(:(:(x, y), z), u))
:¹(:(:(:(C, x), y), z), u) → :¹(:(x, y), z)
:¹(:(:(:(C, x), y), z), u) → :¹(:(:(x, y), z), u)

The TRS R consists of the following rules:

:(:(:(:(C, x), y), z), u) → :(:(x, z), :(:(:(x, y), z), u))

The set Q consists of the following terms:

:(:(:(:(C, x0), x1), x2), x3)

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

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ QDPOrderProof

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

:¹(:(:(:(C, x), y), z), u) → :¹(x, z)
:¹(:(:(:(C, x), y), z), u) → :¹(x, y)
:¹(:(:(:(C, x), y), z), u) → :¹(:(x, z), :(:(:(x, y), z), u))
:¹(:(:(:(C, x), y), z), u) → :¹(:(x, y), z)
:¹(:(:(:(C, x), y), z), u) → :¹(:(:(x, y), z), u)

The TRS R consists of the following rules:

:(:(:(:(C, x), y), z), u) → :(:(x, z), :(:(:(x, y), z), u))

The set Q consists of the following terms:

:(:(:(:(C, x0), x1), x2), x3)

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.

:¹(:(:(:(C, x), y), z), u) → :¹(x, z)
:¹(:(:(:(C, x), y), z), u) → :¹(x, y)
:¹(:(:(:(C, x), y), z), u) → :¹(:(x, z), :(:(:(x, y), z), u))
:¹(:(:(:(C, x), y), z), u) → :¹(:(x, y), z)
:¹(:(:(:(C, x), y), z), u) → :¹(:(:(x, y), z), u)

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

Recursive path order with status [2].
Precedence:

trivial

Status:

:₂: [1,2]

The following usable rules [14] were oriented:

:(:(:(:(C, x), y), z), u) → :(:(x, z), :(:(:(x, y), z), u))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ PisEmptyProof

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

:(:(:(:(C, x), y), z), u) → :(:(x, z), :(:(:(x, y), z), u))

The set Q consists of the following terms:

:(:(:(:(C, x0), x1), x2), x3)

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