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:
[CONCAT1, cons1]


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.