Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS
↳1 DependencyPairsProof (⇔)
↳2 QDP
↳3 DependencyGraphProof (⇔)
↳4 AND
    ↳5 QDP
        ↳6 QDPOrderProof (⇔)
        ↳7 QDP
        ↳8 PisEmptyProof (⇔)
        ↳9 TRUE
    ↳10 QDP
        ↳11 QDPOrderProof (⇔)
        ↳12 QDP
        ↳13 PisEmptyProof (⇔)
        ↳14 TRUE

(0) Obligation:

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.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

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

CONCAT(cons(U, V), Y) → CONCAT(V, Y)
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)

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

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 2 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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))

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

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


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.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
CONCAT(x0, x1, x2)  =  CONCAT(x1)

Tags:
CONCAT has argument tags [1,2,2] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(CONCAT(x1, x2)) = 1   
POL(cons(x1, x2)) = 1 + x2   

The following usable rules [FROCOS05] were oriented: none

(7) Obligation:

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))

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

(8) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(9) TRUE

(10) Obligation:

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))

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

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


LESSLEAVES(cons(U, V), cons(W, Z)) → LESSLEAVES(concat(U, V), concat(W, Z))
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
LESSLEAVES(x0, x1, x2)  =  LESSLEAVES(x0)

Tags:
LESSLEAVES has argument tags [3,0,3] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(LESSLEAVES(x1, x2)) = x2   
POL(concat(x1, x2)) = x1 + x2   
POL(cons(x1, x2)) = 1 + x1 + x2   
POL(leaf) = 0   

The following usable rules [FROCOS05] were oriented:

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

(12) Obligation:

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))

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

(13) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(14) TRUE