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))


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 contains 2 SCCs with 2 less nodes.

↳ QTRS
  ↳ Non-Overlap Check
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
QDP
                ↳ QDPAfsSolverProof
              ↳ 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.
By using an argument filtering and a montonic ordering, at least one Dependency Pair of this SCC can be strictly oriented.

CONCAT2(cons2(U, V), Y) -> CONCAT2(V, Y)
Used argument filtering: CONCAT2(x1, x2)  =  x1
cons2(x1, x2)  =  cons1(x2)
Used ordering: Quasi Precedence: trivial


↳ QTRS
  ↳ Non-Overlap Check
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
                ↳ QDPAfsSolverProof
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.