Termination w.r.t. Q proof of TRS/Rubiobintrees.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:

concat(leaf, Y) → Y
concat(cons(U, V), Y) → cons(U, concat(V, Y))
lessleaves(X, leaf) → false
lessleaves(leaf, cons(W, Z)) → true
lessleaves(cons(U, V), cons(W, Z)) → lessleaves(concat(U, V), concat(W, Z))

Q is empty.

↳ QTRS

  ↳ AAECC Innermost

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

concat(leaf, Y) → Y
concat(cons(U, V), Y) → cons(U, concat(V, Y))
lessleaves(X, leaf) → false
lessleaves(leaf, cons(W, Z)) → true
lessleaves(cons(U, V), cons(W, Z)) → lessleaves(concat(U, V), concat(W, Z))

Q is empty.

We have applied [15,7] to switch to innermost. The TRS R 1 is

concat(leaf, Y) → Y
concat(cons(U, V), Y) → cons(U, concat(V, Y))

The TRS R 2 is

lessleaves(X, leaf) → false
lessleaves(leaf, cons(W, Z)) → true
lessleaves(cons(U, V), cons(W, Z)) → lessleaves(concat(U, V), concat(W, Z))

The signature Sigma is {true, false, lessleaves}

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

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

concat(leaf, Y) → Y
concat(cons(U, V), Y) → cons(U, concat(V, Y))
lessleaves(X, leaf) → false
lessleaves(leaf, cons(W, Z)) → true
lessleaves(cons(U, V), cons(W, Z)) → lessleaves(concat(U, V), concat(W, Z))

The set Q consists of the following terms:

concat(leaf, x0)
concat(cons(x0, x1), x2)
lessleaves(x0, leaf)
lessleaves(leaf, cons(x0, x1))
lessleaves(cons(x0, x1), cons(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:

LESSLEAVES(cons(U, V), cons(W, Z)) → LESSLEAVES(concat(U, V), concat(W, Z))
LESSLEAVES(cons(U, V), cons(W, Z)) → CONCAT(W, Z)
LESSLEAVES(cons(U, V), cons(W, Z)) → CONCAT(U, V)
CONCAT(cons(U, V), Y) → CONCAT(V, Y)

The TRS R consists of the following rules:

concat(leaf, Y) → Y
concat(cons(U, V), Y) → cons(U, concat(V, Y))
lessleaves(X, leaf) → false
lessleaves(leaf, cons(W, Z)) → true
lessleaves(cons(U, V), cons(W, Z)) → lessleaves(concat(U, V), concat(W, Z))

The set Q consists of the following terms:

concat(leaf, x0)
concat(cons(x0, x1), x2)
lessleaves(x0, leaf)
lessleaves(leaf, cons(x0, x1))
lessleaves(cons(x0, x1), cons(x2, x3))

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ EdgeDeletionProof

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

LESSLEAVES(cons(U, V), cons(W, Z)) → LESSLEAVES(concat(U, V), concat(W, Z))
LESSLEAVES(cons(U, V), cons(W, Z)) → CONCAT(W, Z)
LESSLEAVES(cons(U, V), cons(W, Z)) → CONCAT(U, V)
CONCAT(cons(U, V), Y) → CONCAT(V, Y)

The TRS R consists of the following rules:

concat(leaf, Y) → Y
concat(cons(U, V), Y) → cons(U, concat(V, Y))
lessleaves(X, leaf) → false
lessleaves(leaf, cons(W, Z)) → true
lessleaves(cons(U, V), cons(W, Z)) → lessleaves(concat(U, V), concat(W, Z))

The set Q consists of the following terms:

concat(leaf, x0)
concat(cons(x0, x1), x2)
lessleaves(x0, leaf)
lessleaves(leaf, cons(x0, x1))
lessleaves(cons(x0, x1), cons(x2, x3))

We have to consider all minimal (P,Q,R)-chains.
We deleted some edges using various graph approximations

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ EdgeDeletionProof

            ↳ QDP

              ↳ DependencyGraphProof

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

LESSLEAVES(cons(U, V), cons(W, Z)) → LESSLEAVES(concat(U, V), concat(W, Z))
LESSLEAVES(cons(U, V), cons(W, Z)) → CONCAT(U, V)
LESSLEAVES(cons(U, V), cons(W, Z)) → CONCAT(W, Z)
CONCAT(cons(U, V), Y) → CONCAT(V, Y)

The TRS R consists of the following rules:

concat(leaf, Y) → Y
concat(cons(U, V), Y) → cons(U, concat(V, Y))
lessleaves(X, leaf) → false
lessleaves(leaf, cons(W, Z)) → true
lessleaves(cons(U, V), cons(W, Z)) → lessleaves(concat(U, V), concat(W, Z))

The set Q consists of the following terms:

concat(leaf, x0)
concat(cons(x0, x1), x2)
lessleaves(x0, leaf)
lessleaves(leaf, cons(x0, x1))
lessleaves(cons(x0, x1), cons(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

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ EdgeDeletionProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                  ↳ QDP

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

CONCAT(cons(U, V), Y) → CONCAT(V, Y)

The TRS R consists of the following rules:

concat(leaf, Y) → Y
concat(cons(U, V), Y) → cons(U, concat(V, Y))
lessleaves(X, leaf) → false
lessleaves(leaf, cons(W, Z)) → true
lessleaves(cons(U, V), cons(W, Z)) → lessleaves(concat(U, V), concat(W, Z))

The set Q consists of the following terms:

concat(leaf, x0)
concat(cons(x0, x1), x2)
lessleaves(x0, leaf)
lessleaves(leaf, cons(x0, x1))
lessleaves(cons(x0, x1), cons(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.

CONCAT(cons(U, V), Y) → CONCAT(V, Y)

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

Lexicographic Path Order [19].
Precedence:

cons₂ > CONCAT₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ AAECC Innermost

    ↳ 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:

concat(leaf, Y) → Y
concat(cons(U, V), Y) → cons(U, concat(V, Y))
lessleaves(X, leaf) → false
lessleaves(leaf, cons(W, Z)) → true
lessleaves(cons(U, V), cons(W, Z)) → lessleaves(concat(U, V), concat(W, Z))

The set Q consists of the following terms:

concat(leaf, x0)
concat(cons(x0, x1), x2)
lessleaves(x0, leaf)
lessleaves(leaf, cons(x0, x1))
lessleaves(cons(x0, x1), cons(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

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ EdgeDeletionProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

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

LESSLEAVES(cons(U, V), cons(W, Z)) → LESSLEAVES(concat(U, V), concat(W, Z))

The TRS R consists of the following rules:

concat(leaf, Y) → Y
concat(cons(U, V), Y) → cons(U, concat(V, Y))
lessleaves(X, leaf) → false
lessleaves(leaf, cons(W, Z)) → true
lessleaves(cons(U, V), cons(W, Z)) → lessleaves(concat(U, V), concat(W, Z))

The set Q consists of the following terms:

concat(leaf, x0)
concat(cons(x0, x1), x2)
lessleaves(x0, leaf)
lessleaves(leaf, cons(x0, x1))
lessleaves(cons(x0, x1), cons(x2, x3))

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