Termination w.r.t. Q proof of /home/nowonder/forschung/aprove/satrung/aprove/lpar18/RubioSLASHbintrees.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:

concat₂(leaf, Y) -> Y
concat₂(cons₂(U, V), Y) -> cons₂(U, concat₂(V, Y))
lessleaves₂(X, leaf) -> false
lessleaves₂(leaf, cons₂(W, Z)) -> true
lessleaves₂(cons₂(U, V), cons₂(W, Z)) -> lessleaves₂(concat₂(U, V), concat₂(W, Z))

Q is empty.

↳ QTRS

  ↳ Non-Overlap Check

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))
lessleaves₂(X, leaf) -> false
lessleaves₂(leaf, cons₂(W, Z)) -> true
lessleaves₂(cons₂(U, V), cons₂(W, Z)) -> lessleaves₂(concat₂(U, V), concat₂(W, Z))

Q is empty.

The TRS is non-overlapping. Hence, we can switch to innermost.

↳ QTRS

  ↳ Non-Overlap Check

    ↳ 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))
lessleaves₂(X, leaf) -> false
lessleaves₂(leaf, cons₂(W, Z)) -> true
lessleaves₂(cons₂(U, V), cons₂(W, Z)) -> lessleaves₂(concat₂(U, V), concat₂(W, Z))

The set Q consists of the following terms:

concat₂(leaf, x0)
concat₂(cons₂(x0, x1), x2)
lessleaves₂(x0, leaf)
lessleaves₂(leaf, cons₂(x0, x1))
lessleaves₂(cons₂(x0, x1), cons₂(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:

LESSLEAVES₂(cons₂(U, V), cons₂(W, Z)) -> LESSLEAVES₂(concat₂(U, V), concat₂(W, Z))
LESSLEAVES₂(cons₂(U, V), cons₂(W, Z)) -> CONCAT₂(W, Z)
LESSLEAVES₂(cons₂(U, V), cons₂(W, Z)) -> CONCAT₂(U, V)
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))
lessleaves₂(X, leaf) -> false
lessleaves₂(leaf, cons₂(W, Z)) -> true
lessleaves₂(cons₂(U, V), cons₂(W, Z)) -> lessleaves₂(concat₂(U, V), concat₂(W, Z))

The set Q consists of the following terms:

concat₂(leaf, x0)
concat₂(cons₂(x0, x1), x2)
lessleaves₂(x0, leaf)
lessleaves₂(leaf, cons₂(x0, x1))
lessleaves₂(cons₂(x0, x1), cons₂(x2, x3))

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

↳ QTRS

  ↳ Non-Overlap Check

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

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

LESSLEAVES₂(cons₂(U, V), cons₂(W, Z)) -> LESSLEAVES₂(concat₂(U, V), concat₂(W, Z))
LESSLEAVES₂(cons₂(U, V), cons₂(W, Z)) -> CONCAT₂(W, Z)
LESSLEAVES₂(cons₂(U, V), cons₂(W, Z)) -> CONCAT₂(U, V)
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))
lessleaves₂(X, leaf) -> false
lessleaves₂(leaf, cons₂(W, Z)) -> true
lessleaves₂(cons₂(U, V), cons₂(W, Z)) -> lessleaves₂(concat₂(U, V), concat₂(W, Z))

The set Q consists of the following terms:

concat₂(leaf, x0)
concat₂(cons₂(x0, x1), x2)
lessleaves₂(x0, leaf)
lessleaves₂(leaf, cons₂(x0, x1))
lessleaves₂(cons₂(x0, x1), cons₂(x2, x3))

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

  ↳ Non-Overlap Check

    ↳ 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))
lessleaves₂(X, leaf) -> false
lessleaves₂(leaf, cons₂(W, Z)) -> true
lessleaves₂(cons₂(U, V), cons₂(W, Z)) -> lessleaves₂(concat₂(U, V), concat₂(W, Z))

The set Q consists of the following terms:

concat₂(leaf, x0)
concat₂(cons₂(x0, x1), x2)
lessleaves₂(x0, leaf)
lessleaves₂(leaf, cons₂(x0, x1))
lessleaves₂(cons₂(x0, x1), cons₂(x2, x3))

We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be strictly oriented and are deleted.

CONCAT₂(cons₂(U, V), Y) -> CONCAT₂(V, Y)

The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
CONCAT₂(x1, x2) = CONCAT₁(x1)
cons₂(x1, x2) = cons₁(x2)

Lexicographic Path Order [19].
Precedence:

cons₁ > CONCAT₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ Non-Overlap Check

    ↳ 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))
lessleaves₂(X, leaf) -> false
lessleaves₂(leaf, cons₂(W, Z)) -> true
lessleaves₂(cons₂(U, V), cons₂(W, Z)) -> lessleaves₂(concat₂(U, V), concat₂(W, Z))

The set Q consists of the following terms:

concat₂(leaf, x0)
concat₂(cons₂(x0, x1), x2)
lessleaves₂(x0, leaf)
lessleaves₂(leaf, cons₂(x0, x1))
lessleaves₂(cons₂(x0, x1), cons₂(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.

↳ QTRS

  ↳ Non-Overlap Check

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

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

LESSLEAVES₂(cons₂(U, V), cons₂(W, Z)) -> LESSLEAVES₂(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))
lessleaves₂(X, leaf) -> false
lessleaves₂(leaf, cons₂(W, Z)) -> true
lessleaves₂(cons₂(U, V), cons₂(W, Z)) -> lessleaves₂(concat₂(U, V), concat₂(W, Z))

The set Q consists of the following terms:

concat₂(leaf, x0)
concat₂(cons₂(x0, x1), x2)
lessleaves₂(x0, leaf)
lessleaves₂(leaf, cons₂(x0, x1))
lessleaves₂(cons₂(x0, x1), cons₂(x2, x3))

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