Termination w.r.t. Q proof of TRS/AG01#3.14.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:

concat₂(leaf, y) -> y
concat₂(cons₂(u, v), y) -> cons₂(u, concat₂(v, y))
less_leaves₂(x, leaf) -> false
less_leaves₂(leaf, cons₂(w, z)) -> true
less_leaves₂(cons₂(u, v), cons₂(w, z)) -> less_leaves₂(concat₂(u, v), concat₂(w, z))

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

concat₂(leaf, y) -> y
concat₂(cons₂(u, v), y) -> cons₂(u, concat₂(v, y))
less_leaves₂(x, leaf) -> false
less_leaves₂(leaf, cons₂(w, z)) -> true
less_leaves₂(cons₂(u, v), cons₂(w, z)) -> less_leaves₂(concat₂(u, v), concat₂(w, z))

Q is empty.

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

LESS_LEAVES₂(cons₂(u, v), cons₂(w, z)) -> CONCAT₂(u, v)
LESS_LEAVES₂(cons₂(u, v), cons₂(w, z)) -> LESS_LEAVES₂(concat₂(u, v), concat₂(w, z))
LESS_LEAVES₂(cons₂(u, v), cons₂(w, z)) -> CONCAT₂(w, z)
CONCAT₂(cons₂(u, v), y) -> CONCAT₂(v, y)

The TRS R consists of the following rules:

concat₂(leaf, y) -> y
concat₂(cons₂(u, v), y) -> cons₂(u, concat₂(v, y))
less_leaves₂(x, leaf) -> false
less_leaves₂(leaf, cons₂(w, z)) -> true
less_leaves₂(cons₂(u, v), cons₂(w, z)) -> less_leaves₂(concat₂(u, v), concat₂(w, z))

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

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

LESS_LEAVES₂(cons₂(u, v), cons₂(w, z)) -> CONCAT₂(u, v)
LESS_LEAVES₂(cons₂(u, v), cons₂(w, z)) -> LESS_LEAVES₂(concat₂(u, v), concat₂(w, z))
LESS_LEAVES₂(cons₂(u, v), cons₂(w, z)) -> CONCAT₂(w, z)
CONCAT₂(cons₂(u, v), y) -> CONCAT₂(v, y)

The TRS R consists of the following rules:

concat₂(leaf, y) -> y
concat₂(cons₂(u, v), y) -> cons₂(u, concat₂(v, y))
less_leaves₂(x, leaf) -> false
less_leaves₂(leaf, cons₂(w, z)) -> true
less_leaves₂(cons₂(u, v), cons₂(w, z)) -> less_leaves₂(concat₂(u, v), concat₂(w, z))

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

          ↳ QDP

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

CONCAT₂(cons₂(u, v), y) -> CONCAT₂(v, y)

The TRS R consists of the following rules:

concat₂(leaf, y) -> y
concat₂(cons₂(u, v), y) -> cons₂(u, concat₂(v, y))
less_leaves₂(x, leaf) -> false
less_leaves₂(leaf, cons₂(w, z)) -> true
less_leaves₂(cons₂(u, v), cons₂(w, z)) -> less_leaves₂(concat₂(u, v), concat₂(w, z))

Q is empty.
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.

CONCAT₂(cons₂(u, v), y) -> CONCAT₂(v, y)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(CONCAT₂(x₁, x₂)) = 2·x₁
POL(cons₂(x₁, x₂)) = 1 + 2·x₂

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ PisEmptyProof

          ↳ QDP

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

concat₂(leaf, y) -> y
concat₂(cons₂(u, v), y) -> cons₂(u, concat₂(v, y))
less_leaves₂(x, leaf) -> false
less_leaves₂(leaf, cons₂(w, z)) -> true
less_leaves₂(cons₂(u, v), cons₂(w, z)) -> less_leaves₂(concat₂(u, v), concat₂(w, z))

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

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

LESS_LEAVES₂(cons₂(u, v), cons₂(w, z)) -> LESS_LEAVES₂(concat₂(u, v), concat₂(w, z))

The TRS R consists of the following rules:

concat₂(leaf, y) -> y
concat₂(cons₂(u, v), y) -> cons₂(u, concat₂(v, y))
less_leaves₂(x, leaf) -> false
less_leaves₂(leaf, cons₂(w, z)) -> true
less_leaves₂(cons₂(u, v), cons₂(w, z)) -> less_leaves₂(concat₂(u, v), concat₂(w, z))

Q is empty.
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.

LESS_LEAVES₂(cons₂(u, v), cons₂(w, z)) -> LESS_LEAVES₂(concat₂(u, v), concat₂(w, z))

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(LESS_LEAVES₂(x₁, x₂)) = x₁
POL(concat₂(x₁, x₂)) = x₁ + x₂
POL(cons₂(x₁, x₂)) = 1 + x₁ + x₂
POL(leaf) = 0

The following usable rules [14] were oriented:

concat₂(leaf, y) -> y
concat₂(cons₂(u, v), y) -> cons₂(u, concat₂(v, y))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ PisEmptyProof

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

concat₂(leaf, y) -> y
concat₂(cons₂(u, v), y) -> cons₂(u, concat₂(v, y))
less_leaves₂(x, leaf) -> false
less_leaves₂(leaf, cons₂(w, z)) -> true
less_leaves₂(cons₂(u, v), cons₂(w, z)) -> less_leaves₂(concat₂(u, v), concat₂(w, z))

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