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))
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
  ↳ 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))

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:

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))

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:

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))

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))
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.
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 Order [17,21] with Interpretation:

POL( CONCAT2(x1, x2) ) = x1 + 3


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



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))
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.
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:

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))

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.


LESSLEAVES2(cons2(U, V), cons2(W, Z)) -> LESSLEAVES2(concat2(U, V), concat2(W, Z))
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( concat2(x1, x2) ) = max{0, 2x1 + x2 - 3}


POL( leaf ) = 3


POL( cons2(x1, x2) ) = 2x1 + x2 + 3


POL( LESSLEAVES2(x1, x2) ) = max{0, x1 - 2}



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